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PAGE 1 1 ENDMEMBER AND PROPORTION ESTIMATION USING PHYSICS BASED MACROSCOPIC AND MICROSCOPIC MIXTURE MODELS By RYAN CLOSE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE R EQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011 PAGE 2 2 2011 Ryan Close PAGE 3 3 A CKNOWLEDGMENTS I w ould like to thank my family, friends, and mentors for providing encouragement and support throughout my academic career. Specifi cally, I want to thank my parents, Don and Marian Close, my sister, Katherine Close, and my grandparents, Hal and Ann Ingman and Don and Peggy Close. A special thanks to my grandfathers, whose influence first led me to engineering and science. I would l ike to thank my advisor s Dr. Paul Gader and Dr. Joe Wilson for all of their guidance and support. I would also like to thank my committee members, D r. Daniel Lee Dr. Douglas Dankel and Dr. Arunava Banerjee for all of their help and valuable suggestio ns. Additionally, I would like to thank all of my current and former labmates for the many times in which they have assisted in my research and served as a sounding board for ideas. A bove all, I thank God, for providing me the opportunity to pursue this r esearch and granting me the capability to proceed successfully. PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 3 LIST OF TABLES ................................ ................................ ................................ ............ 6 LIST OF FIGURES ................................ ................................ ................................ .......... 7 LIST OF ABBREVIATIONS ................................ ................................ ........................... 11 ABSTRACT ................................ ................................ ................................ ................... 13 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 15 Hyperspectral Data ................................ ................................ ................................ 15 Mixture Based Models ................................ ................................ ............................ 17 Problem Statement ................................ ................................ ................................ 18 2 LITERATURE REVIEW ................................ ................................ .......................... 19 Macroscopic Mixtures ................................ ................................ ............................. 20 Introduction to Macroscopic Mixtures ................................ ............................... 20 Statistical Approaches to the LMM Inversion Problem ................................ ..... 22 I terated constrained endmembers algorithm ................................ .............. 22 Sparsity promoting iterated constrained endmember ................................ 24 Geometric Approaches to the LMM Inversion Problem ................................ .... 24 Pixel purity index ................................ ................................ ........................ 25 N FINDR ................................ ................................ ................................ .... 26 Vertex co mponent analysis ................................ ................................ ........ 26 Microscopic Mixtures ................................ ................................ .............................. 27 Introduction to Microscopic Mixing ................................ ................................ ... 27 Reflectance functions ................................ ................................ ................. 28 Single scattering albedo ................................ ................................ ............ 31 Existing Endmember and Proportion Estimation Methods ................................ 32 Endmember estimation ................................ ................................ .............. 32 Proportion estimation ................................ ................................ ................. 33 Hyperspectral No nlinear Mixture Models ................................ ................................ 39 Virtual Endmembers ................................ ................................ ......................... 39 Kernel Methods for Endmember and Proportion Estimation ............................. 40 Neural Network Based Proportion Estimation ................................ .................. 41 Manifold Learning in Hyperspectral Data ................................ .......................... 42 ISOMAP ................................ ................................ ................................ ..... 44 Sub manifolds ................................ ................................ ............................ 44 Manifold based endmember estimation ................................ ..................... 45 PAGE 5 5 3 TECHNICAL APPROACH ................................ ................................ ...................... 48 Review of Macroscopic and Microscopic Mixtures ................................ .................. 48 Macroscopic Mixtures ................................ ................................ ....................... 49 Microscopic Mixtures ................................ ................................ ........................ 49 Discrete Mixture Estimation ................................ ................................ .................... 53 Proportion Estimation ................................ ................................ ....................... 54 Endmember Estimation ................................ ................................ .................... 56 Multi Mixture Pixel Estimation ................................ ................................ ................. 58 Multi Mixture Pix el Model ................................ ................................ ................. 58 Proportion Estimation ................................ ................................ ....................... 59 Endmember Estimation ................................ ................................ .................... 61 4 EXP ERIMENTAL RESULTS ................................ ................................ ................... 67 Synthetic Data sets ................................ ................................ ................................ 68 Description of Data sets ................................ ................................ ................... 68 Endmember Estimation ................................ ................................ .................... 69 Initialization and parameter robustness ................................ ..................... 71 Microscopic mixture effects ................................ ................................ ........ 73 Endmember estimation accuracy ................................ ............................... 78 Proportion Estimation ................................ ................................ ....................... 79 Microscopic mixture effects ................................ ................................ ........ 81 Estimating microscopic mixture concentration ................................ ........... 81 Proportion estimation accuracy ................................ ................................ .. 83 RELAB Data sets ................................ ................................ ................................ .... 84 Description of Data Sets ................................ ................................ ................... 84 Endmember Estimation ................................ ................................ .................... 85 Proportion Estimation ................................ ................................ ....................... 85 Aerial Hyperspectral Data Sets ................................ ................................ ............... 87 Baseball Diamond Scene ................................ ................................ ................. 88 Beach Scene ................................ ................................ ................................ .... 89 5 CONCLUSION ................................ ................................ ................................ ...... 124 LIST OF REFERENCES ................................ ................................ ............................. 128 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 140 PAGE 6 6 LIST OF TABLES Table page 4 1 Algorithm parameter settings. ................................ ................................ .......... 91 4 2 RELAB data set true proportions ................................ ................................ ........ 91 4 3 Comparison of proportion estimates of RELAB mixtures using mass fraction and F parameter values. A) Oliv ine & Enstatite mixture. B ) Olivine & Magnetite mixture ................................ ................................ ............................... 92 PAGE 7 7 LIST OF FIGURES Figure page 2 1 Two linear simplexes formed from macroscopic mixtures of five endm embers with uniform proportions ................................ ................................ ..................... 47 2 2 Two nonlinear simplexes formed from microscopic mixtures of five endmembers with uniform proportions ................................ ............................... 47 3 1 Derivative of reflectance function over range of albedo values .......................... 63 3 2 Macroscopic mixture points formed from uniform proportions of two endmembers ................................ ................................ ................................ ....... 63 3 3 Reflectance curve with microscopic mixture points formed from uniform proportions of two endme mbers ................................ ................................ ......... 64 3 4 Three dimensional plot of mi croscopic mixture of two endmembe rs .................. 64 3 5 Diagram of multi mixture pixel ................................ ................................ ............ 65 3 6 Diagram of multi mixture pixel ................................ ................................ ............ 65 3 7 Diagram of multi mixture pixel ................................ ................................ ............ 66 4 1 Synthetic endmember ref lectance spectra ................................ ......................... 93 4 2 1 st and 2 nd principal compo nents of synthetic CMM data set ............................ 93 4 3 1 st and 3 rd principal compo nents of synthetic CMM data set ............................ 94 4 4 1 st and 2 nd principal comp onent of synthetic MMP data set .............................. 94 4 5 1 st and 3 rd principal comp onent of synthetic MMP data set .............................. 95 4 6 Mean MAB E over range of annealing rates ................................ ........................ 95 4 7 Mean MABE of synthetic CMM data sets over range of settings ..................... 96 4 8 Histogram of ICE MA BE using random starting points ................................ ....... 96 4 9 Histogram of RLMM GD MABE using random starts ................................ .......... 97 4 10 Histogram of AD R LMM GD MABE using random starts ................................ ... 97 4 11 Histogram of DME MABE using r andom starts ................................ ................... 98 4 12 Histogram of MPE MABE using random starts ................................ ................... 98 PAGE 8 8 4 13 RLMM GD estimated endmembers for CMM data set (1 st and 2 nd principal components) ................................ ................................ ................................ ....... 99 4 14 RLMM GD estimated endmembers for CMM data set (1 st and 3 rd principal components) ................................ ................................ ................................ ....... 99 4 15 RLMM GD estimated endmem bers for CMM data set ( spectra) ....................... 100 4 16 RLMM GD estimated endmembers for CMM data set ................................ ...... 100 4 17 RLMM GD estimated endmembers for CMM data set ................................ ...... 101 4 18 RLMM GD estimated endmembers for CMM data set ................................ ...... 101 4 19 RLMM GD estimated endmember spectra with multiple parameters ............ 102 4 20 Mean MABE of VCA, DME, and MPE ................................ .............................. 102 4 21 Mean MAB E of ICE, RLMM GD, DME, and MPE ................................ ............. 103 4 22 Mean M ABE of AD RLMM GD, DME, and MPE ................................ ............... 103 4 23 Histogram of V CA MABE values for CMM data set ................................ .......... 104 4 24 H istogram of I CE MABE values for CMM data set ................................ ........... 104 4 25 Histogram of RLMM GD MABE values for CMM data set ................................ 105 4 26 Histogram of AD RLMM GD MABE values for CMM data set .......................... 105 4 27 Histogram of DME MABE values for CMM dat a set ................................ ......... 106 4 28 Histogram of M PE MABE valu es for CMM data set ................................ .......... 106 4 29 Mean MABE difference between VCA and D ME and MPE .............................. 107 4 30 Mean MABE differe nce between ICE and DME and MPE ................................ 107 4 31 Mean MABE difference between R LMM GD and DME and MPE ..................... 108 4 32 Mean MABE difference bet ween AD RLMM GD and DME a nd MPE .............. 108 4 33 Histogram of MABE difference between VCA a nd DME and MPE for CMM data set ................................ ................................ ................................ ............. 109 4 34 Histogram of MABE difference between ICE a nd DME and MPE for CMM data set ................................ ................................ ................................ ............. 109 PAGE 9 9 4 35 Histogram of MABE difference between RLMM GD a nd DME and MPE for CMM data set ................................ ................................ ................................ ... 110 4 36 Histogram of MABE difference between AD RLMM GD a nd DME and MPE for CMM data set ................................ ................................ .............................. 110 4 37 Histogram of MABE difference between VCA, ICE, RLMM GD, AD RLMM GD, and D ME and MPE for M MP HMP data set ................................ .............. 111 4 38 Comparison of nonlinearity in albedo domain and reflectance domain mixtures ................................ ................................ ................................ ............ 111 4 39 Microscopic propor tions resulting from LMM, AD LMM, DME, and M PE over all synthetic data sets ................................ ................................ ....................... 112 4 40 Proportion RMSE resulting from LMM, AD LMM, DME, and M PE over all synthetic data sets ................................ ................................ ............................ 112 4 41 Spectra o f Enstatite and Olivine mixture ................................ ........................... 113 4 42 Spectra o f Magnetite and Olivine mixture ................................ ......................... 113 4 43 Wavelength scatter plot o f Magnetite and Olivine mixture ................................ 114 4 44 Endmember estimation of RELA B Magnetite and Olivine mixture .................... 114 4 45 Endmember estimation of RELA B Enstatite and Olivine mixture ...................... 115 4 46 RELAB data sets e ndmember estimation error chart ................................ ....... 115 4 47 Baseball diamond scene ................................ ................................ .................. 116 4 48 Endmembers estima ted from baseball diamond scene ................................ .... 116 4 49 Points l abeled by DME as microscopically mixed ................................ ............. 117 4 50 MPE microscopic mixture proportions ................................ .............................. 117 4 51 MPE grass macroscopic proportions ................................ ................................ 118 4 52 MPE grass microscopic proportions ................................ ................................ 118 4 53 MPE dirt macroscopic proportions ................................ ................................ .... 119 4 54 MPE dirt microscopic proportions ................................ ................................ ..... 119 4 55 MPE clay macroscopic proportions ................................ ................................ .. 120 4 56 MPE clay micros copic proportions ................................ ................................ .... 120 PAGE 10 10 4 57 B each scene ................................ ................................ ................................ ..... 121 4 58 Microscopic proportions estimated using MPE ................................ ................. 121 4 59 Macroscopic proportion map for sand endmembe r ................................ .......... 122 4 60 Microscopic proportion map for sand endmember ................................ ............ 122 4 61 Macroscopic proportion map for sediment endmember ................................ .... 123 4 62 Microscopic proportion map for sediment endmember ................................ ..... 123 PAGE 11 11 LIST OF ABBREVIATION S AD LMM Albedo domain linear mixture model AD MMM Albedo domain microscopic mixture model AD RLMM GD Albedo domain regularized linear mixture model gradient descent BRDF Bidirectional reflectance distribution function CMM Combined macroscopic and microscopic D ME Discrete mixture estimation HNN Hopfield neural network HNN/MLP Hopfield neural network / multi layer perceptron HSI Hyperspectral image ICE Iterated constrained endmembers IR Infrared KFCLS Kernel fully constrained least squares LIDAR Light detection a nd ranging LMM Linear mixture model LSE Least squared error MABE Mean absolute band error MLP Multi layer perceptron MMP Multi mixture pixel MMP HMP Multi mixture pixel with high microscopic proportion MNF Minimum noise fraction MPE Multi mixture pixel est imation MSE Mean squared error PCA Principal component analysis PPI Pixel purity index PAGE 12 12 SPICE Sparsity promoting iterated constrained endmembers SPT Sparsity promoting term SSA Single scattering albedos SSD Sum of squared distances SVDD Support vector data description RBF Radial basis function RBF L Radial basis function linear RBF NL Radial basis function nonlinear RBFNN Radial basis function neural network RD MMM Reflectance domain microscopic mixture model RELAB Reflectance experiment laboratory RLMM GD R egularized linear mixture model gradient descent RMSE Root mean square error RSS Residual sum of squares VCA Vertex component analysis PAGE 13 13 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfil lment of the Requirements for the Degree of Doctor of Philosophy ENDMEMBER AND PROPORTION ESTIMATION USING PHYSICS BASED MACROSCOPIC AND MICROSCOPIC MIXTURE MODELS By Ryan Close December 2011 Chair: Paul Gader Cochair: Joseph Wilson Major: Computer Engi neering Methods of incorporating macroscopic and microscopic mixture models into hyperspectral image (HSI) endmember and proportion estimation is presented and discussed. These methods utilize the linear mixture model (LMM) and the bidirectional reflecta nce distribution function to model macroscopic and microscopic mixtures respectively. This allows for estimation of endmembers and their proportions from HSI s composed of both macroscopic and microscopic mixtures. A vast majority of HSI endmember and p roportion estimation methods rely on the LMM to describe pixel spectra resulting from mixtures of endmembers. However, nonlinear mixtures have been shown to exist in HSIs. Specifically, the microscopic mixture is known to cause nonlinear mixing relations hips between endmembers. Methods using microscopic mixture models exist but rely on severely limiting assumptions or estimations of the nonlinearity present in microscopic mixtures. The methods presented in this research explicitly account for the nonlin earity caused by microscopic mixing. The mixtures are therefore, estimated directly from the data PAGE 14 14 without the need for a priori knowledge of the mixture types. In addition, the explicit modeling of both mixture types allows for direct estimation of endm embers. Results are presented using synthetic data sets which consist of macroscopic and microscopic mixtures, to demonstrate the increased accuracy in endmember and proportion estimation using th e s e new physics based methods over existing linear method s. In addition, results are presented using a well known laborator y data set Using these results and other published results from this data set increased accuracy in proportion estimation over other microscopic mixture methods is shown. Finally, a qua litative analysis of the new methods is performed using a n HSI collected from an airborne platform. PAGE 15 15 CHAPTER 1 INTRODUCTION Hyperspectral Data Hyperspectral image (HSI) data is characterized by a set of spatially organized pixels [ 1 2 ] E ach pixel has an associated reflectance spectrum, estimated from radiance measurements collected by the sensor [ 1 ] Each pixel spectrum often with a spatial resolution of 1 30 square meters [ 1 2 ] Often the goal of HSI analysis is to identify the substances and their abundances from a given scene [ 1 ] This task is complicated by the mixtures and interactions presen t in hyperspectral data. This fact necessitates the ill posed inve rsion problem of estimating substance spectra and abundances of each substance present in a scene [ 1 ] Hyperspectral sensors are co mmonly used from air and space borne platforms [ 1 2 ] This causes limited spatial and spectral resolution s As a result it is not possible to fully resolve the spectral signature of every substance or compound imaged by the sensor [ 1 ] T herefore, the endmember conce pt is introduced. An endmember is defined (for the purposes of this research) a s a substance within an HSI that can be described by a single observable spectral signature. Examples of the endmember concept would be a single s ubstance or compound, such as clay, or a static mixture, such as a uniform mixture of sand and silt ( i.e., no variations in mixture proportions between the mixed endmembers are present in the scene) The ill posed inversion problem illustrated in Equatio n 1 1 is restated as the estimation of an HSI endmembers and their proportions present in each pixel subject to a set of PAGE 16 16 underlying functions [ 1 ] This p roblem is often known as endmember extraction and pixel unmixing. ( 1 1 ) A vast majority of research to date has attempted to solve this inversion problem by modeling the function as a linear mixture of endmembers shown in Equation 1 2 and Equation 1 3 [ 1 3 6 ] This type of mixtu re is known as a macroscopic mixture [ 7 ] While the linear approach is amen a ble to mathematical analysis, it has been repeatedly shown that nonlinear mixing is prevalent in hyperspectral data [ 1 8 22 ] Specifically, multi scattering between endmember particles or mi croscopic mixing, is known to be pervasive in many scenes and to form nonlinear mixtures [ 1 7 20 34 ] ( 1 2 ) ( 1 3 ) Attempts to incorporate the nonlinearity of microscopic mixing into the unmixing problem have largely been black box approach es Trad itional methods of reproducing an u nknown nonlinear function such as neural networks [ 35 38 ] and kernel methods [ 23 25 28 39 41 ] have been used to learn the function in Equation 1 1 Recent research has introduced man ifolds to model the nonlinearity in hyperspectral data. Typically used in hyperspectral classification applications, endmember and proportion estimation has also been pursued with manifold based models [ 42 44 ] These methods all attempt to account for the nonlinearity with a generic machine learning model to PAGE 17 17 approximate the mixture however they do not explicitly model the underlying physics of the mixture Mixtu re Based Models Often, many plausible endmembers and proportions exist that can explain the data [ 3 4 ] This is a common problem when addressing ill posed inversion problems. This fact necessitates the inclusion of constraints into the unmixing problem to ensure a physically realizable and probable solution. Basing the unmixing model on physics based models derive d from the theories of reflectance spectroscopy achieve this goal. [ 1 23 25 32 ] Using reflectance spectroscopy to analyze an HSI is known as image spectroscopy. Algorithms based on these models constrain the inversion to be governed by the physical properties of the environment in which the data were col lected, thereby finding solutions relevant to a real world explanation of the data. [ 1 ] The linear mixture model (LMM) mentioned abov e effectively models macroscopic mixtures in hyperspectral data. [ 1 ] Macroscopic mixtures also referred to as areal or checkerboard mixtures are formed when the spectral signatures of two or more endmembers are measured and represented by a single pixel. [ 1 ] Thi s causes an averaging of spectral signatures as described by the LMM This type of mixing is caused by the coarse resolution of the sensor and is not a physical mixing of endmembers. In contrast, microscopic mixtures also known as intimate mixtures, do represent a particles of two or more endmembers are homogenously mixed together causing multi scattering between the particles of the endmembers. [ 7 30 32 ] Multi scattering PAGE 18 18 scattering albedos (SSA) [ 7 30 ] The reflectance spectrum resulting from the average SSA is manifeste d as a nonlinear mixing of the endmembers. Utilizing various remote sensing assumptions the reflectance spectra can be estimated using bidirectional reflectance distribution function. [ 7 30 32 ] In the past two decades of HSI analysis research the macroscopic mixture has largely been regarded as the domi nant mixture present. [ 1 4 45 58 ] This is a direct consequence of the coarse pixel resolution of commonly used hyperspectral sensors. As sensor resolution continues to improve, the macroscopic mixture is becoming less prominent and the effects of physically occurring mixtures, such as the microscopic mixture, are becoming dominant. [ 8 13 15 18 23 25 27 28 42 44 59 70 ] This necessitates the ability to estimate endmembers an d proportions while fully accounting for the physical mixtures specifically microscopic mixtures Problem Statement Currently, model s and associated algorithm s for estimating endmembers in the presence of both macroscopic and microscopic mixtures do not exi s t. Additionally, existing methods for estimating proportions in the presence of both macroscopic and microscopic mixtures rely on approximate models that do not explicitly represent the mixtures present in hyperspectral data. Using the reflectance m odels describing macroscopic and microscopic mixing, this research will investigate computational model s to estimate endmember s and proportions in the presence of both mixtures. PAGE 19 19 CHAPTER 2 LITERATURE REVIEW Hyperspectral image analysis can take many forms dependi ng on the desired application such as anomaly and target detection [ 2 ] or monitoring variations in vegetation cover [ 15 ] Similarly, the breadth of research in HSI analysis is extensive, including feature extraction [ 1 ] detecting the number of endmembers [ 71 72 ] band selection [ 5 ] dimensionality analysis [ 73 ] spectral libraries for unmixing [ 74 ] anomaly and target detection [ 2 ] etc. However, a common goal of HSI applications is to characte rize some aspect of a scene via air or space borne collections. [ 1 2 75 ] An HSI collection can be generalized into two procedures. First, a given area is scanned b y the hyperspectral sensor and its radiance, or raw energy, is measured. [ 1 ] Secondly, a post processing step is used to convert the radiance values to reflectance; this process takes into account collection specific atmospheric and illumination parameters. [ 1 2 ] Some applications use measured radiance as an input data source, however converted reflectance data is assumed when utilizing reflectance spectroscopy models. [ 1 ] To fully characterize an HSI the endmembers, proportions, and mixtures present in a scene must be determined [ 1 ] An endmember is defined, for the purposes of this research, as a substance within an HSI that can be described by a single observable spectral signature. Examples of the endmember concept are single substances or com pounds, such as clay, or a static mixture, such as a uniform mixture of sand and silt ( i.e., no variations in mixture proportions between the endmembers are present in the PAGE 20 20 HSI ). The concepts of proportions and mixtures will be conveyed through the discuss ion of the mixture models. Assuming no prior knowledge of the area scanned this problem is an ill posed inversion probl em. [ 1 3 ] As a result there exist infinitely many plausible solutions to the problem. The principles and theories of reflectance spectroscopy are used to create models constrained by the laws of physics [ 1 7 ] These models result in ph ysically plausible en dmember, proportion and mixture estimations [ 7 ] This approach to analyzin g HSIs can be referred to as image spectroscopy Macroscopic Mixtures Introduction to M acroscopic M ix tures A common mixture type in HSI s is the macroscopic mixture, also known as an areal or checkerboa rd mixture. [ 1 ] This type of mixture is caused by the presence of two or more discrete endmembers instantaneous field of view [ 1 ] In a macroscopic mixture t he endmembers are not physically mixed ; h owever the spectral signatures of each endmember are mixed within the pixel [ 1 2 ] M ixing of this type is caused by the finite spatial resolution of optical detectors. Thus, a hyperspectral s mixing present in an HSI [ 1 ] Macroscopically mixed re flectance spectra ( ) are modeled by a linear combination of spectra from each of the constituent endmembers ( ) as s hown in Equation 2 1 [ 1 2 7 20 22 ] The geometric cross section ( ) of the k th endmember is the physical realization of the proportion of a mixed spectra composed of the PAGE 21 21 ( 2 1 ) T he macroscopic mixture model k nown as the LMM, serves as the foundation for many approaches to HSI endmember and proportion estimation. [ 1 2 76 78 ] The LMM ( Equation 2 2 ) represents the i th pixel of an HSI ( ) a s the linear combination of endmembers ( ) and their respective proportions ( ) with additive zero mean Gaussian noise ( ) [ 3 4 ] The LMM has the constraints and This linear relationship makes the model amenable to mathematical analysis, which largely contributes to its popularity. ( 2 2 ) The formulation of the LMM (Equation 2 2 ), with its associated constraints, depicts an HSI pixel as a convex combination of its e ndmembers. The LMM, therefore, implies a convex hull assumption over the set of pixels composing an HSI [ 1 2 76 78 ] Consequentially, the LMM forms an M 1 linear simplex within a D dimensional space where D is the number of wavelengths (or bands) measured by the hyperspectral sensor. [ 52 ] The HSI endmembers are the vertices of this simplex An example of two such simplex es created by the LMM can be seen Figure 2 1 Numerous HSI endmember and proportion estimation algorithms, based on the LMM, have been published [ 1 4 6 45 47 49 58 76 82 ] This includes r ecent algorithms that incorporat e spatial informat ion into the LMM [ 82 91 ] The algorithms formed from the LMM can be generalized into two categories of autonomous approaches: sta tistical and geome tric [ 55 77 78 ] PAGE 22 22 Stati stical A pproach es to the LMM I nversion P roblem S tatistical based approach es minimize or maximize a given objective function. [ 3 4 45 47 49 50 ] The objective function to minimize is often the square d error b etwee n the spectral signature of each pixel in the HSI and the reconstructed spectra [ 3 ] The reconstructe d spectra are formed from the mixture of the estimated endmembers and their respective proportions. The statistical methods often incorporate simultaneous estimation of endmembers and proportions A popular method using this approach is t he Iterated Const rained Endmembers (ICE) algorithm [ 3 ] The ICE algorithm has also spawned many variants that address shortfalls in the model. For example, the Sparsity Promoting Iterated Constrained Endmember (SPICE) algorithm uses a Bayesian approach to automatically determine the number of endmembers while simultaneously estimating the endmembers and proportions [ 4 ] Both the ICE and SPICE algorithms are described in t he following sections. Iterated constrained endmembers algorithm The ICE algorithm minimize s a residual sum of squares (RSS) term, shown in Equation 2 3 This is accomplished by estimating a set of endmembers ( ) and the proportions ( ) that best explain the spectral data for each pixel. [ 3 ] ( 2 3 ) The model, in an attempt to minimize the RSS, can theoretically create an infinitely large convex h ull that completely contains the given data points While these endmembers create a zero error solution, they are not unique and have no guarantee of being probable or even plausible. The model therefore requires additional PAGE 23 23 considerations in the objectiv e to constrain the set of possible solutions. T he sum of squared distances (SSD) term is used for this constraint shown in Equation 2 4 ( 2 4 ) The SSD term is proportional to the volume of the area bounded by the endmembers and thus, when minimized, provides endmember s that are tightly fitted around the data. It can be shown that the SSD formulation ( Equation 2 4 ) is equivalent to the formulation shown in Equation 2 5 where is the sum of variances. ( 2 5 ) Usi ng this new formulation for SSD ( ignoring the constant) and the RSS a new objective function is formed This new objective is referred to as the regularized RSS and is shown in E quation 2 6 with being a regularization parameter. ( 2 6 ) Using this new regularized formulation ICE estimates endmembers by minimiz ing the objective function, with respect to the fixed. Minimization of this objective function is accomplished using a closed form solution. Next, the proportion values are estimated with quadratic programming, while holding the endmembers fixed. This alternating optimization algorithm is continued until a convergence criterion is met Given a set of data points, this model may estimate a singular set of endmembers that represent s a plausible, an d even probable, explanation of the data. However, no guarantee exists that these are actual endmembers. This model also suffers from the PAGE 24 24 assumption that only linear relationships are present in the given HSI data, i.e., only macroscopic mixtures exist. S parsity promoting iterated constrained endmember The SPICE algorithm is a popular variant of the ICE approach to HSI endmember and proportion estimation [ 4 ] SPICE uses a Bayesian approach t o autonomously determine the number of endmembers In contrast, other techniques require the number of endmembers as i nput parameter s to the algorithm This capability is introduced through the sparsity promoting term (SPT) shown in Equation 2 7 and Equation 2 8 The parameter is an input associated with the degree t o which the proportions are driven to zero. ( 2 7 ) ( 2 8 ) The SPICE objective function, given by Equation 2 9 is updated in a manner similar to the ICE algorithm. As with ICE, the SPICE algorithm will output the endmembers and proportions estimated from the HSI d ata. ( 2 9 ) Geometric A pproaches to the LMM Inversion P roblem Geometric approach es exploit the convex hull geometry formed by the endmembers and their linear mixtures. [ 51 58 ] Th e endmembers, which appear as vertices, form the linear simplex described by this geometry This simplex reveals an implicit a ssumption of the geometric algorithms the pure pixel assumption. Th e pure pixel assumption requires that a given HSI have at least one pure pixel of each PAGE 25 25 endmember present in the scene ( i.e., ) This fact illustrates the prominent disadvantage of the geometric algorithms over the statistical approaches. A notable advantage of a geometric approach is an increased robustness to small data sets Unlike statistical methods, geom etric approaches do not require the optimization of an objective function through learning spectra and proportions. The geometric category of algorithms exploits the intrinsic geometry created by the LMM or similar properties of the simplex to find endm ember spectra existing in the given data Once the pure pixel assumption is satisfied, geometric approaches are able to find endmembers with in a smaller data set than would be required by a statistical approach Using these endmembers the proportions can then be estimated. As with the statistically based models many variations appear in literature. Three popular algorithms are described : the pixel purity index (PPI) [ 53 ] N FINDR [ 57 ] and vertex component analysis (VCA) [ 54 ] Pixel purity index The P PI algo rithm attempts to find the pixels in a hyperspectral scene that have the purest representation of an endmember. [ 53 ] The algorithm first uses minimum n oise f raction (MNF) to reduce the dimensionality of the data and perform noise whitening. Next, the data is projected onto numerous random ly created vectors With every vector the pixels creating extrema l values are recorded. After this process has run a predetermined number of times the results are analyzed. The pixels shown to have had frequent extrema l values are determined to represent the purest pixels. This extremity score is related to t convex geometry analysis. [ 52 ] The PPI algorithm was originall y designed for target detection, however the PPI has PAGE 26 26 been used, often with the addition of other algorithms or manual input to extract endmembers and model the background subspace of hyperspectral scene s N FINDR The N FINDR algorithm is based on the assum ption that the largest volume simplex in the spectral feature space is formed by the endmembers of the scene [ 57 ] The algorithm starts by initially selecting a gr oup of pixels from the HSI these are the is calculated. A new pixel is then selected to replace one of the current endmembers T he volume of the simplex create d with the replaced endmember is calculated. If the new volume is l arger than the previous volume the replacement stays otherwise the previous endmember set is restored This process is repeated for all of pixels in the scene T he set of endmembers (pixels) remaining at the end of the algorithm create s a simplex e nclosing the largest possible volume T hus they are the endmembers for t he given HSI scene. Vertex component analysis As with PPI and N FINDR, VCA exploits the simplex geometry of a hyperspectral mixture and attempts to find the vertices of the simplex Thus again requiring the pure pixel assumption. [ 54 ] After initialization, the VCA algorithm projects data onto a direction orthogonal to the simplex spanned by the existing set of endmembers. The pixels cor responding to the extrem a l values in the projection s are labeled endmembers. This process continues until a predetermined number of endmembers in the scene is found. PAGE 27 27 Micro scopic M ixtures R esult ing f rom the prevalence of macroscopic mixing in hyperspectral scenes and the mathematical amenability of the LMM a majority of endmember and proportion estimation research has been focused on macroscopic mixtures [ 1 ] However, research has shown that nonlinear mixtures are present in hyperspectral scenes and have a substantial effect on measured spectral reflectance signatures over a given scene. [ 1 16 17 ] Keshava and Mustard describe the effects of nonlinear mixing by stating [ 1 ] : In tests of linear versus nonlinear mixing on laboratory data, the fractions calculated may be in error by as much as 30% absolute. In addition, th e linear model can cause considerable ambiguity and false fractions when used on nonlinear mixtures. Absorption bands and continua in nonlinear mixtures cannot be adequately fit with a linear model. Microscopic mixtures, also called intimate mixtures, are known to be present in many HSI scenes. [ 1 7 20 24 30 34 ] Unlike the macroscopic mixture microscopic mixtures form non linear mixtures of endmember spectra [ 7 21 22 30 31 34 ] Consequently the LMM has been shown inadequate for modeling this type of mixing in hyperspectral data [ 1 7 20 24 30 34 ] While microscopic mixing does represent physical mixtures of endmembers, it is important to note that it is not a chem ical reaction ( i.e., the constituent endmember substances still exist). Introduction to M icroscopic M ixing Microscopic mixing occur s when particles of different endmember s are homogenously distributed within a instantaneous field of view Light i ncident on this mixed surface ( assumed a semi infinite medium ) is scattered. The scattered radiance (power per unit area per unit solid angle of the detector) measured by a sensor ( ) is the sum of the singly scattered radiance ( ) and the multiply scatter radiance ( ) (Equation 2 10 ). [ 7 30 ] The close proximity of endmember s particles occurring in a PAGE 28 28 microscopic mixture cause s multi scatter between particles with different scattering properti es [ 1 22 ] Th is multi scattering between particles causes a nonlinear mix ture in reflectance spectra To accurately estima te endmembers and proportions within the context of microscopic mixture reflectance spectra, a robust model for reflectance is required. However, many types of reflectance exist. ( 2 10 ) Reflectance functions The type of reflectance is characterized by the collimation of the light source and the detector ( or sensor ) The naming convention used to distinguish th e various reflectance types is by preceding two adjectives These adjectives (listed in decreasing order of collimation) are typically : directional, conical, or hemispherical The first adjective describes the degree of culmination of the source, and t he second the detector If both adjectives are the same a bi is used in place of the first adjective Because perfect collimation and perfect diffuseness do not exist in reality all reflectances are biconical. [ 7 30 ] However, on a clear, sunny day, the sun can be reasonably approximated as a collimated source ( i.e., d irectional) and on an overcast day the light scattered by the clouds can be viewed as a diffuse source ( i.e., hemispherical). [ 7 ] The human eye looking at the ground can be perceived as a directional detector; likewise an air or space borne sensor looking at the ground approximates a directional detector [ 7 ] Therefore, under these basic remote sensing assumptions (including assuming minimal cloud cover ), the bidirectional reflectance distribution function (B R D F) is the appropriate model for HSI applications. BRDF has been va lidated through experimentation and is widely accepted as an accurate model [ 7 20 35 37 ] BRDF is given by Equation PAGE 29 29 2 11 where and are the cosine of the angles of incidence and emergence, respectively, is the phase angle is the phase function for particles is the backscatter function given by Equation 2 12 is C isotropic multi scattering given by Equation 2 13 and is the average SSA given by Equation 2 14 with and being the power of th e incident irradiance scattered and absorbed by the particle, respectively [ 7 30 31 ] The p hase angle and the angles of incidence and emergence describe the geometries of the light source and detect or Scattering properties of the surface are described by the function. The backscatter function models the opposition effect as given by Equation 2 12 where describes the magnitude of the opposition effect and depends on the inter particle spacing ( 2 11 ) ( 2 12 ) ( 2 13 ) ( 2 14 ) The BRDF shown in Equation 2 11 is given as ance spectroscopy paper [ 30 ] To unclutter the notation, the input parameters to the reflectance function will be dropped. Additionally, to avoid conflict with future uses of and will be used in place of and to represent the cosine of the angles of incidence an d emergence. The BRDF with altered notations is shown in Equation 2 15 PAGE 30 30 ( 2 15 ) A perfectly Lambert ian surface reflects all light incident on it and appears equally bright when viewed fro m any given angle. [ 7 30 ] Therefore, the BRDF of a Lambert reflectance is given by Equation 2 16 Using the BRDF (Equation 2 15 ) and the Lambert reflectance (Equation 2 16 ) the reflectance coefficient can be f ormulated as shown in Equation 2 17 [ 7 30 ] The reflectance coefficient is the brightness relative to an identically illuminated Lambert surface. This normalized form of the BRDF is convenient for remote sensing applications and often used in literature for HSI analysis. [ 22 25 32 35 ] ( 2 16 ) ( 2 17 ) Under remote sensing assumptions, the angle of incidence is commonly 15 0 to 40 0 and the angle of emergence 0 0 under these conditions the phase ang l e is between 15 0 and 40 0 [ 32 ] Additionally, under remote sensing assumptions, these angles are assumed constant over a given collection Mustard and Pieters demonstrated that a t phase angle s greater than 15 0 the was negligible. [ 32 ] Additionally, under the assumpt ion of isotropic scattering ( all light is scattered equally in all directions ) the phase function is defined to be [ 7 30 ] Using these remote sensing assumptions and the con sequential results for and a simplified reflectance coefficient function can be derived as shown in Equation 2 18 where the subscript is used to signify the wavelength dependence of the reflectance function. Also, was added as to represent the single scattering PAGE 31 31 albedo as an input parameter to the function. As the reflectance coefficient function, under the described assumptions, is the only model for reflectance (Equation 2 18 ) used in this research it will simply be referred to as the reflectance function. ( 2 18 ) Single scattering a lbedo The SSA of a particle (Equation 2 14 ) is defined by Hapke as [ 7 ] : The ratio of the total amount of power scattered to the total power removed from t he wave is the particle single scattering albedo scattering albedo are functions of wavelength. The reflectance of a microscopic mixture is described by the reflectance function (Equation 2 18 ) and the average SSA of the mixture shown in Equation 2 19 [ 7 22 32 ] The average SSA is a linear combination of endmember SSAs where and are the mass fraction, single particle density, and average effective particle size for a given mixture component ( i.e. endmember). [ 7 ] ( 2 19 ) The average SSA can also be represented as a relative geometric cross section ( ) or endmember proportion ( ), shown in Equation 2 20 for the k th component. [ 7 ] This representation is important because often in remote sensing the particle density and particle size of endmembers is not known. Therefore, the particle density and particle size are assumed roughly equal for all endmembers. Under this assumption, the proportion of an endmember composing a mixed pixel ( i.e., the relative geometric cross section of the pixel composed of the endmembe r) is equal to its mass fraction. PAGE 32 32 ( 2 20 ) Using the relative geometric cross section representation for each endmember, the equation for the average effective SSA can be written as Equation 2 21 This formulation leads to constraints similar to those of macroscopic mixing in the reflectance domain: and ( 2 21 ) Existing E ndmember and Proportion Estimation Methods Until recently, microscopic mixtures research was largely confined to the fields of geophysics and spectrography. The princi pal data source in these fields of research has been either laboratory data o f known microscopic mixtures or telescopic data collected from lunar surfaces known to be dominated by microscopic mixing [ 20 22 30 32 34 ] Consequently, most published works for estimating endmembers and proportions of microscopic mixtures require severely limiting assumption s These assumptions prevent most existing methods of endmember and proportion estimation for microscopic mixtures f rom being feasible for real world HSI sc enes. The following sections describe the existing methods for endmember and proportion estimation of microscopic mixtures. Endmember estimation A large majority of endmember and proportion estimation methods rely on the endmembers being provided a priori [ 23 25 32 35 37 ] Only two published methods were found to estimate endmembers from microscopic mixtures. [ 26 27 29 ] Both PAGE 33 33 methods however, assume that all pixels are formed from one mixture type (macroscopic or microscopic) and require the pure pixel assumption. Each of these methods is a two step pr ocess. First, the given hyperspectral data is converted into SSA using a reflectance function appropriate for the given data. [ 26 27 29 ] In most applications the BRDF is used for th is conversion, however Smith, Johnson and Adams used data collected from a hemispherical detector and therefore used a direc tional hemispherical model of reflectance. In either case, the given reflectance data was converted to average SSA. Secondly, both methods use a common LMM endmember estimation technique on the converted SSA data. Smith, Johnson, and Adams use principa l component analysis ( PCA ) to find the endmembers, assuming a pure pixel exists [ 26 29 ] Using an eigenvalue approach on the converted SSA data and the original reflectance data, the alg orithm determines if the entire data set is best modeled using SSA or reflectance. Thus, the algorithm determines the endmembers and the mixture type of the data set as a whole. Jose, et al use the simplex identification via split augmented Lagrangian ( SISAL) algorithm to estimate endmembers in the converted SSA data. [ 27 ] As with the PCA method, the SISAL algorithm requires a pure pixel assumption. These algorithms represent a first attempt to estimate endmembers from microscopically mixed data. However, they both require that the data contains only macroscopic or microscopic mixtures. This assumption makes these methods invalid for data sets collected over a scene of more than one mixture type. Propo rtion estimation In 2010 Broadwater and Banerjee introduced the first algorithm capable of unmixing hyperspectral data composed of macroscopic and microscopic mixtures. [ 24 PAGE 34 34 25 ] Until then all methods of microscopic mixture proportion estimation require d the same ass umption as the microscopic mixture endmember estimation methods ; a ll data sets must contain only macroscopically mixed data or microscopically mixed data. [ 22 26 27 29 32 33 35 ] Of these approaches, two are commonly referenced for comparison with new techniques of unmixing, Mustard and Pieters (LSE) unmixing of SSA data [ 32 ] and Guilfoyle, et al. use of a radial basis function neural network (RBFNN) [ 35 ] In the following sections the se two approaches will be discussed. Next, the Hopfield neural network/ multi layer perceptron (HNN/MLP) nonlinear mixture model of Plaza, et al. [ 37 ] will be introduced and discussed ; the results of this paper are also use d for comparison with new microscopic unmixing algorithms. Finally, the physic s based kernel approaches of Broadwater and Banerjee will be presented. [ 23 25 ] LSE Mustard and P ieters collected highly accurate laboratory bidirectiona l reflectance measurements of various crushed mineral species, known to mix microscopically. [ 32 ] Two of these bina ry mineral mixtures have become a de facto standard for assessing the accuracy of microscopic mixture proportion estimation. [ 23 24 32 35 37 ] To unmix these mineral mixtures, Mustard and Pieters used the BRDF of Equation 2 18 to convert the hyperspectral reflectance data to SSA. Once transformed to SSA a LMM is assumed with a given set of endmember spectra, likewise converted to SSA. Using this SSA data and endmembers a LSE approach was used to solve the inversion problem for estimating proportions. [ 32 ] Mustard and Pieters noted that in a subset of mineral mixtures a sum to one constraint was not needed in the PAGE 35 35 inversion, however when Magnetite was used in a mixture the sum to one constraint was required. RBFNN. Gu ilfoyle, et al. used a n RBFNN to interpolate the microscopic propo rtions based off a given training set. [ 35 ] The RBFNN approach takes as input the (Equation 2 18 ) a set of tr aining reflectance spectra are created from the microscopic mixing of the given endmembers. The proportions used to create the training microscopic mixture data are randomly selected. The constructed microscopic mixture spectra are used as centers for th e radial basis functions used in the RBFNN. The weights of the RBFNN are then updated using the known mixing proportions of the created mixture spectra. This RBFNN, constructed with microscopic mixture data, is labeled the radial basis function nonlinear (RBF NL) network. For comparison of results, a radial basis function linear (RBF L) network was created using the same technique as the RBF NL network. The RBF L was created with training data constructed using the LMM instead of the BRDF. To estimate m ixture proportions, a hyperspectral sample is tak en as input to the trained RBFNNs. The RBF NL and RBF L each output a set of estimated proportion values for the input sample. [ 35 ] In the paper, Guilfoyle, et al. did not discuss a method to determine which RBFNN output should be used. The mean squared error (MSE) of the reflectance experiment laboratory ( RELAB ) data used with the RBFNNs (and the known mixture proportions) compares the networks to each other and to Mustard and Pieters LSE approach. It was concluded that on the RELAB data, the RBF NL network outperformed the RBF L network and the RBF NL performed similarly to Mustard and PAGE 36 36 Pieters. [ 35 ] These results are not s urprising due to the fact that they both use the same BRDF to estimate microscopic mixtures, take endmembers as input to th e unmixing, and assume that a data set contains only one mixture type (macroscopic or microscopic). [ 32 35 ] HNN/MLP Plaza, et al. were the first to publish an approach to proportion estimation assuming that macroscopic and microscopic m ixtures could be present in a given scene, or data set [ 37 ] This approach requires the endmembers be supplied externally to the algorithm. In the case of the RELAB data the endmembers were given as input, for HSI scenes the automated morphological endmember extraction algorithm was used to find endmembers as a preprocessing step. Onc e a set of endmembers has been established, the HNN/MLP first uses a Hopfield neural network (HNN) to unmix the macroscopic proportions (referred to in the [ 37 ] The HNN uses gradient descent learning based on error minimization. [ 37 ] The HNN uses an iterative process. Once a convergence criterion is met the HNN outputs a set of estimated linear mixing proportio ns The output proportions from the HNN are used as inputs to the multi layer perceptron (ML P). The MLP is used to refine the linearly estimated proportions of the HNN to account for the nonlinearity in the data. The MLP is constructed using the known p roportions of the training data with back propagation and an adjustable learning rate. This fact reveals a fatal flaw in this approach. Instead of using a model to estimate the proportions from a known endmember set, this approach must use actual microsc opic mixture data to interpolate proportions of new mixture data. This level of PAGE 37 37 highly accurate ground truth is seldom available for large data sets or aerial HSI collections (the author is unaware of the existence of any suc h collection). Plaza, et al. ra n experiments to determine the amount of training data required by their algorithm to successfully unmix the RELAB data. [ 37 ] Their experiments showed that to make accurate proportion estimations three to five samples were required from the RELAB data sets ; the binary mixture RELAB data sets that were used contained only five samples. Physics based kernels. In machine learning literature kernel techniques are often used to model nonlinear systems. [ 92 93 ] This ability to effectively model linear and nonlinear data has made kernels popular in hyperspectral research as well. K ernels have primarily been used in HSI research for classification applications. [ 94 ] However, kernels have also been use d to estimate endmembers and proportions in hyperspectral data [ 23 25 39 41 ] The kernels use d in these approaches were typically developed in the machine learning field for use in nonlinear problems in which the underlying relationships between data points was not known. Therefore, these kernels were generic and not specifically suited to unmixi ng when the underlying mixture model is known. Broadwater and Banerjee developed a physics based kernel, shown in Equation 2 22 in which the function is a mapping from reflectance to SSA based on the BRDF of Equation 2 18 [ 23 ] The physics based kernel was compared to three common generic kernel functions: the line ar kernel, the polynomial kernel, and the radial basis function (RBF). They demonstrated, using the RELAB binary mixtures, that the physics ba sed kernel was better suited to proportion estimation of microscopic mixtures. [ 23 ] PAGE 38 38 Additionally, they compared the proportions estimated with the physics based kernel to those estimated using the RBFNN and the HNN/MLP. When comparing across both RELAB data sets used, the physics based kernel was able to m ore accurately estimate the proportions. [ 23 ] ( 2 22 ) The physics based kernel however, has similar limitations to the previously discussed methods. Namely, the endmembers must be externally supplied and the input data is assumed to be entirely microscopically mixed. Subsequent papers, by Broadwater and Banerjee, acknowledged the need for an approach that accounted for macroscopic and microscopic mixtures within a single algorithm, which lead to the generalized ke rnel for areal (macroscopic) and intimate (microscopic) mixtures (areal/intimate kernel) [ 24 25 ] This kernel, shown in Equation 2 23 will approximate the LMM when is very small and the BRDF (Equation 2 18 ) when is large [ 24 ] To estimate the p roportions of hyperspectral data, kernel fully constrained least squares (KFCLS) is used with the areal/intimate kernel, shown in E quation 2 24 The parameter is optimized by minimizing the objective function shown in Equation 2 25 ( 2 23 ) ( 2 24 ) ( 2 25 ) PAGE 39 39 Hyperspectral Nonlinear Mixture Models In recent years, the existence of nonlinear mixtures within HSIs has become generally accepted as fact. [ 1 7 16 22 25 30 32 39 42 44 60 65 70 95 105 ] Microscopic mixtures are one cause of nonlinear mixing. Therefore, the BRDF is used to m odel nonlinear mixtures, of this type. [ 1 7 23 25 30 32 37 ] However, nonlinear mixtures are also caused by multiple reflections and transmissions from scattering surfaces. [ 14 16 18 19 ] This results in a new mo del for nonlinear mixtures of this type. [ 13 14 ] Additionally, general machine learning nonlinear models have been used to account for nonlinearity present in HSIs. [ 23 25 35 38 40 44 59 62 65 93 99 102 106 119 ] This section presents an overview of nonlinear models, excluding those based on the BRDF, used in hyperspectral mixture analysis. Virtual Endmember s N onlinear mixtures in hyperspectral data can be caused by multiple reflections and transmissions between scattering surfaces. [ 8 18 20 ] This nonlinear mixing in hyperspectral data is often re ferred to as multi scattering, forward scattering or back scattering This form of scattering occur s when light is reflected off one surface and strikes a second and eve n third surface. [ 11 14 15 18 ] In turn the light striking the second or third surface is reflected in such a way that it is then detected by the sensor. In this scenario the second or third surface can also be illuminated directly, by the light source, causing a normal reflection in addition to the multi scattering reflection. [ 8 15 18 19 ] This form of nonlinear mixing causes additional photon interactions on the second and third surfaces. [ 8 15 18 19 ] The light incident on a given area of the second surface is now greater than that of the area of surface not impacted by multi scattering. PAGE 40 40 Secondly, the multiplicatively sca ttered light impacting the second surface does not have the same signature as the light being emitted over the scene. Therefore, a multiplicative filtering effect is created on the second and third surfaces. [ 8 15 18 19 ] In hyperspectral literature this nonlinear mixture is addressed using a variety of techniqu es [ 8 15 18 19 ] These methods all attempt to model sing le scattering components with trad itional endmembers and multiple scattering components with what has been termed a virtual endmember [ 8 15 20 ] Simply put, a virtual endmember is the second and third order polynomial expansion of the endmembers of a given scene. As shown by Equation 2 26 a virtual endmember attempts to model the mult iple scattering components by the element wise product of two endmembers. ( 2 26 ) Through the use of the virtual endmember, this nonlinear mixing become s linear in the proportion values assigned to the endmembers and virtual endmembers. Using this concept endmember extraction and unmixing efforts are again framed as a LMM problem. [ 8 15 18 19 ] The various papers that address this model use an assortment of techniques to determine which endmembers to specifica lly model as virtual endmembers or methods to better account for the physical basis of the model. A common technique is to relax the constraints on the proportion values to allow the model to account for the extra photon interactions. [ 8 15 18 19 ] Kernel Methods for Endmember and Proportion Estimation Kernel me thods are naturally suited to solving nonlin ear problems. By projecting data points into a nonlinear kernel space relationships in the original feature space may become linear. [ 92 93 ] The methods rely on an appropriate kernel function to represent the similarity or dissimilarity between data points. [ 120 121 ] Typically the RBF is used PAGE 41 41 because of its direct utilization of squared distance between points. [ 23 ] However kernel functions have been specifically designe d to model hyperspectral data, such as the angular kernel [ 94 ] or recent work by Broadwater and Banerjee in which a kernel was proposed to model intimate mixtures [ 23 24 ] Within hyperspectral image analysis, kernel methods are typically used for ground cover classification [ 102 105 113 122 123 ] however recent attempts have been made to estimate endmembers and proportions within the kernel space. [ 23 25 39 41 ] Extract ion of endmembers in the kernel space was don e by using the support vector data description (SVDD) algorithm to find the data points that surround the remaining points. [ 39 ] Proportions of these candidate endmember points could be used to fully explain th e data. While this approach did allow for nonlinear extraction of possible endmembers that describe the data, there is no basis in the model to determine that these points are in fact endmembers. This model fits a common theme in which the data is explai ned via a given model but little to no connection is made with the underlying physics that created the data. Estimation of proportions in the kernel space is similar to estimation in the feature space. [ 40 ] While using kernels does allow for nonlinear relationships these estimation methods are not founded on principles of the underlying system that created the data. As hyperspectral endmember and pro portion estimation is an underdetermined inversion problem many possible solutions can be found that fit the data. [ 1 3 ] Neural Network Based Proportion Estimation As with kernel based methods, neural networks provide a technique that is capable of modeling linear and nonlinear relationships in the given data. [ 93 ] Typically, neural networks are used for ground cover classification in hyperspectral literature but PAGE 42 42 also have a known cap ability to learn unknown functions. [ 35 38 92 93 ] This takes the form of passing inputs to the function into the neural ne t and training it with desired outputs of the function. Once the neural network is trained, new inputs are given The expectation is that if the neural network is properly trained, for the new inputs, then the proper function output will result ( i.e., th e neural network will approximate a nonlinear mixing function in hyperspectral data) Neural network proportion estimation makes the assumption that an unknown nonlinear function ( ) defines the interaction between endmembers ( ) and proportions ( ) sh own in Equation 2 27 [ 36 38 ] To learn this function training data is used consisting of a set of candidate endmembers and given pro portions. This technique is also used as a refinement step after a LMM endmember extraction and unmixing method is used. [ 37 38 ] ( 2 27 ) The nonlinear function being approximated by the neural network is assumed unknown and therefore not explicitly modeled. As mentioned before, this mea ns that this approach is not constrained by the physical theories of reflectance and therefore is not likely to accurately model this system when interpolating hyperspectral data. In addition, this technique also requires a given set of endmembers and pro portions in order to train the network, either hand chosen or output from a previously run model. Manifold Learning in Hyperspectral Data In recent years manifold learning has become a popular method to deal with nonlinearities in hyperspectral data. [ 100 124 125 ] This is primarily due to the manifold s ability to represent an underlying lower dimensional linear or nonlinear PAGE 43 43 structure containing the data in a higher dimensional feature space Exploiting this intrinsic structure allows the hyperspectral data to be expressed in a lower dimensional space wh ile preserving the underlying structure of the data. [ 42 44 95 97 99 101 105 112 113 115 119 122 ] The manifold learning trend in hyperspectral literature began with three articles [ 100 124 125 ] These articles discussed applications of manifold learning to nonlinear dimensionality reduction of high dimension data sets In addition to showing the utility of these methods, the isometric feature mapping, or ISOMAP, algorithm was introduced. [ 125 ] The ISOMAP algorithm became the basis of the first gen eration of hyperspectral manifold based techniques. [ 95 98 101 115 119 ] Similar to many of the other nonlinear approaches to modeling hyperspectral data, manifold learning is not based on a physical model of the system being sensed. This fact limits endmember extracting and unmixing abilities with models b ased solely upon learning the manifold structure embedding the hyperspectral data. However, the manifold s ability to represent an underlying nonlinear structure in a high dimensional space does make manifold learning ideal for supervised classification t asks [ 99 101 105 112 113 116 119 122 ] semi supervised classification [ 106 126 ] bathymetry mapping [ 95 97 98 115 ] intrinsic dimensionality determination [ 59 60 63 ] dimensionality reduction and feature extraction [ 61 62 66 ] anomaly detection [ 64 65 ] and other hyperspectral image processing applications [ 67 70 ] PAGE 44 44 ISOMAP The ISOMAP algorithm has been the most prevalent me thod for learning a manifold in hyperspectral feature space. [ 59 60 95 98 101 115 119 125 ] The ISOMAP algorithm is composed of three steps [ 125 ] : 1. Construct a neighborhood graph from the given data. The graph can be a k nearest neighbors graph (connect two data points, nodes, if one is a k nearest neighbor) or an graph (connect all nodes within distance of each other) 2. Compute the shortest path on the graph between all points (nodes). If two nodes are connected by an edge then the path distance between them is the direct distance between the nodes. If two nodes are not directly connected then the path distance is the sum of edges composing the shortest path between the two nodes. 3. Construct the d dimensional embedding by using multi dimensional scaling (MDS) on the paths created in step 2. Once the ISOMAP a lgorithm has been completed, the data will be represented by a lower dimensional embedding (the intrinsic dimensionality) via preserving the geodesic distances of the data. Hyperspectral literature research involving this technique has focused on applying it to new problems [ 61 68 69 ] and making it more scalable to the large data set problems common in hyperspectral data analysis [ 59 60 96 118 119 ] Sub m anifolds As research has continued in applications of the ISOMAP algorithm and techniques t o make it scalable for large data sets it was discovered that the global manifold structure from a hyperspectral scene had natural partitions in the data. [ 59 60 96 ] Recent research has discovered that the global manifold has curv ature in its structure; as a result of these curves the local intrinsic d imensionality of the data is PAGE 45 45 higher than that of the global manifold. Using this information the manifold can be partitioned into sub manifolds. [ 59 60 96 ] It was also discovered that these sub manifolds had different properties and thus used a different optimal parameter for constructing the ISOMAP n earest neighbor graph Use of different parameters for the sub manifolds resulted in increased effectiveness of the manifold techniques. [ 96 ] Investigation was not done into what causes the curvature or the sub manifolds that exhibit different properties in the feature space but it could be hypothesized that these represent different m ixtures in the hyperspectral image scene. Manifold b ased endmember estimation In the past year research has been published in which a manifold structure was assumed for the hyperspectral data and endmember extraction was att empted on the manifold. [ 42 44 ] The nonlinear mixing was assumed to be an unknown function that warps the LMM as shown in Equation 2 28 ( 2 28 ) To extract endmembers using this assumption a simplex volume maximization approach was used. In this algorithm, hyperspectral data points are randomly chosen and the volume calculated with those points as the vertices. At each iteration a new point is substituted for an existing vertex. If the new vertex creates a larger volume it i s kept as a candidate endmember, otherwise the old vertex is used in th e next iteration. This technique is altered to work with the manifold structure by using the geodesic distances between points instead of their feature space distances. [ 43 ] This allows the PAGE 46 46 endmember extraction algorithm to model nonlinearity in the distances between data points. T his approach represent s a first concept in attempting to exploi t the manifold structure of the data for endmember extraction. However, i t fails to model the nonlinearities in a system constrained by the mixtures creating the data, thus there is no guarantee that the inversion process will create a physically probable set of endmembers to explain the hyperspectral scene. Further, by using the simplex volume maximization method it inherently relies on the pure pixel assumption, which can be problematic. PAGE 47 47 Figure 2 1 Two linear simplexes formed from macroscopic mixtures of five endmembers with uniform proportions Figure 2 2 Two non lin ear simplexes formed from microscopic mixtures of five endmembers with uniform propo rtions. PAGE 48 48 CHAPTER 3 TECHNICAL APPROACH New methods for hyperspectral endmember and proportion estimation are introduced in this chapter Founded on the models derived from reflectance spectroscopy, t hese new algorithms incorporate nonlinear microscopic mixtures into existing linear model s for macroscopic mixtures. The microscopic mixture model describes the physics of how light incident on a surface will mix between particles of different endmembers. As a result these models represent a wider range of physical mix tures present in an HSI Experiments in Chapter 4 demonstrate that these new algorithms allow for increased accuracy in estimating endmember s and proportions. The f irst algorithm introduced is the discrete mixture estimation (DM E ) method The DME label s a pixel as being best described by either a macroscopic or microscopic mixture. Using this label the endmembers and their respective proportions are estimated with the appropriate mixture model Next a new HSI pixel model is introduced, the multi mixtu re pixel (MMP) The MMP model represents each of the spectra in HSI pixel s as being produced by a mixture of mixtures. Thus, a pixel is composed of a macroscopic mixture of endmembers and a microscopic mixture The MMP is discussed within the contexts o f its assoc iated algorithm, the multi mixture pixel estimation (MPE) method Re view of Macroscopic and Microscopic Mixtures A brief review of macroscopic and microscopic mixtures is presented in this section. A more comprehensive description of these mi xtures and their associated models is given in Chapter 2 PAGE 49 49 Macroscopic M ixtures M acroscopic mixing occurs when two or more discrete endmembers reside with in This caus es a linear mix ing of the spectral reflectance signatures from each of the endmembers The macroscopic mixture is generally modeled with the LMM shown in Equation 3 1 where is the proportion of endmember for the given pixel with additive zero mean Gaussian noise [ 1 3 ] The LMM is constrained as given by Equation 3 2 This type of mixing is a result of the finite spatial resolution of the sensor and does not indicate a physical mixing or interaction between endmembers. [ 1 ] ( 3 1 ) ( 3 2 ) Geometrically, ignoring the noise term, the LMM forms a linear simplex in the feature space, where each of the dimensions in the hyperspect ral feature vector corresponds to a wavelength (or band) measured by the sensor. T wo such s implexes are illustrated in Figure 2 1 Each simplex is defined by its vertices, which in the case of HSI analysis, are the endmembers. Microscopic M ixtures Microscopic mixtures (or intimate mixtures) occur when the particles of two or more endmembers are homogenously intermixed together such that particles of the component endmembers are in close proximity to each other [ 1 ] The light incident on This phenomenon, called multi scatter, creates a nonlinear mixing in reflectance. [ 7 ] PAGE 50 50 M icroscopic mixtures represent a physical mixing of endmembers within the scattering medium. [ 1 ] Reflectance of the microscopic mixture can be modeled by the nonlinear reflectance function shown in Equation 3 3 and Equation 3 4 where is the average single scattering albedo (SSA or simply albedo ) between endmembers the angle of incidence, and is the angle of emergence [ 7 30 ] As light is multi scattered between s particles their SSAs linearly mix as shown in Equation 3 5 and Equation 3 6 where is the proportion of the k th endmember [ 7 ] In these equations t he SSA is shown with a subscript to signify the wavelength dependent properties of SSA s. This reflectance equation was developed by Hapke and Mustard using various remote sensing assumptions that have been validated through e xperimentation (see Chapter 2 for details) [ 1 7 22 30 32 ] ( 3 3 ) ( 3 4 ) ( 3 5 ) ( 3 6 ) The reflectance function (Equation 3 3 ) use s an approximation to function for isotropic multi scattering shown in Equation 3 7 ( 3 7 ) PAGE 51 51 T he deri vative of Equation 3 3 with respect to the SSA ( ), is shown in Equation 3 8 Inspection of Equation 3 8 leads to the conclusion that the derivative is always greater tha n 0, s ince and all other quantities are nonnegative [ 30 ] Figure 3 1 demonstrates this over the range of SSA values Thus, the reflectance function (Equation 3 3 ) is monotonically increasing. ( 3 8 ) A microscopic mixture can be seen in Figure 2 2 usi ng the same endmembers and proportions as the macroscopic mixture in Figure 2 1 Geometrically the microscopic mixture form s a perturbed si mplex ( i.e., warping of the macroscopic mixture linear simplex ) However, since the same endmembers are used for both mixtures, the vertices of the microscopic mixture warped simplex are the same as the macroscopic mixture linear simplex Additionally, i nspection of Figure 2 1 and Figure 2 2 reveals the non uniform distribution of sample points, created from uniform proportions caused by the microscopic mixture. Specifically the sample points are dens est near endmembers with low albedo. In reflectance spectroscopy this is known as the spectral dominance of low albedo endmembers. This relationship can be seen in Figure 3 1 by observing that the slope of the derivative is alw ays increasing and that the derivate is always less than one for any albedo below .8 The graph shown in Figure 3 1 was determined using angles of incidence and emergence. Different angles of incidence and emergence, howeve r, will decrease this spectral dominance of low albedo endmembers effect. PAGE 52 52 To further characterize the nonlinearity present in microscopic mixtures the following example is given. Assume two synthetic endmembers of three bands each. The first band s of ea ch endmember, and nine macroscopic mixture points created from uniform proportions, are shown in Figure 3 2 Notice that the reflectance values for each mixture point are linearly related. Figure 3 3 sh ows the first band of the same two endmembers microscopically mixed with the proportions used in Figure 3 2 Notice that the mixture points are uniformly spaced along the x axis, the albedo values. These same mixture points, how ever, are not uniformly spaced along the y axis, the reflectance values. The black line in this figure illustrates the reflectance curve created from the reflectance function (Equation 3 3 ). Notice, that for any two albedos the mixture points between them will always have increasing reflectance values, from the lowest albedo to the highest. The reflectance values for all three bands of the microscopic mixture shown in Figure 3 3 are plotted in Figure 3 4 The mixture points between the two endmembers reside on a curved line in the 3 dimensional space The 1 dimensional line between the two endmembers is understood as representing the M 1 dimensional manifold between M endmember s (where M is 2 in this example) [ 52 ] The line is curved as a result of the microscopic m ixing. A reflec tance domain microscopic mixture model (RD MMM) is shown in Equation 3 9 where is the reflectance function given in Equation 3 3 and the constraints of Equation 3 6 hold for all The SSA of the k th endmember is shown in Equation 3 10 ( 3 9 ) ( 3 10 ) PAGE 53 53 An albedo domain microscopic mixture model (AD MMM) derived from Equation 3 9 is shown in Equation 3 11 The additive noise term s in the RD MMM ( ) and AD MMM ( ) are assumed a zero mean Gaussian. [ 1 3 22 25 32 ] This commonly used assumption, however cannot be accurate Assuming is a zero mean Gaussian ran dom variable B ecause the reflectance function (Equation 3 3 ) describing the relationship between the reflectance domain and albedo domain is nonlinear and monotonically increasing the distribution describing the random variable cannot be Gaussian. The models and associated algorithms described in this section use the zero mean Gaussian assumption for the error in both albedo and reflectance domains. However, f uture research could investigate the albedo domain error di stribution and its effects on microscopic mixture model estimation algorithms. S ince the reflectance function (Equation 3 3 ) is strictly monotonically inc reasing, minimizing the RSS of the RD MMM ( Equation 3 9 ) result s in the same proportions ( ) as minimizing the RSS of the AD MMM ( Equation 3 11 ) ( 3 11 ) Discrete Mixtur e Estimation The first method developed in this research is the DM E algorithm This algorithm constitutes a step forward in HSI spectroscopy methods that explicitly include macroscopic and microscopic mixtures. DM E uses a pixel level approach to label ea ch input HSI pixel or data sample as either macroscopically or microscopically m ixed. Previous ly published methods to include b oth mixture models required an image level PAGE 54 54 estimation of mixture type, i.e., an HSI was entirely macroscopically or microscopica lly mixed. [ 26 29 32 35 ] R ecent ly Broadwater and Banerjee created a kernel function that allowed for a pixel level approach to estimating mixture types. [ 24 25 ] This physics based kernel approach relies on an approximation of the reflectance function and externall y provided endmembers [ 23 25 ] A comprehensive survey of microscopic endmember and proportion estimation models and algorithms is presented in Chapter 2 Using the RD MMM constrain s the endmember and proportion estimation inversion problem to a strict physics based model. DME eliminates the assumption of a single mixture type within an HSI without the need to approximate the re flectance function In addition to proportion estimation, DM E allows for direct estimation of endmembers in the presence of b oth mixture types. Use of the ICE framework allows DME to estimate endmember s without the pure pixel assumption as is required w ith existing microscopic mixture endmember estimation methods [ 3 26 29 ] Proportion Estimation To estimate proportions using DME, a set of endmembers must be provided (later sections will describe how the endmember set is estimated using DME) Using these endmembers the proportions of each pixel will be estima ted A pixel level indicator variable is used to label the mixture type of each pixel. The RSS objective function used by DME is given by Equation 3 12 In this model the RSS for the macroscopic mixture ( ), derived from the LMM, is given by Equation 3 13 The RSS for the microscopic mixture ( ), derived from the RD MMM, is shown in Equation 3 14 The variable ( ) determines the mixture type that represents the i th pixel. PAGE 55 55 ( 3 12 ) ( 3 13 ) ( 3 14 ) DME relies on the assumption that a pixel is largely dominated by a single mixture type. In its current configuration DME labels a pixel as either a macroscopic or microscopic mixture; however it is amenable to adding new mixture terms. This allows future modifications to include additional spectral phenomena or mixtures Th e indicator variables ( ) are determined by assigning the minimum RSS mixture term and the remaining terms 0 for all pixels The macroscopic proportions ( ) are estimated using quadratic programming to minimize (Equation 3 13 ). However, the microscopic proportions ( ) appear in the argument to the nonlinear reflectance function Consequentially, quadratic programming cannot be used with Equation 3 14 Therefore, the microscopic proportions are estimated from the albedo domain RSS ( ) derived from the AD MMM ( Equation 3 11 ) The microscopic proportions are estimated using quadratic programming to minimize Equ ation 3 15 ( 3 15 ) The proportion estimation pie ce of DME is as follows: 1. Take as input a set of pixels from an HSI scene and a set of endmembers PAGE 56 56 2. Use quadratic programming to estimate the macroscopic proportions ( ) by minimizing Equation 3 13 3. Use quadr atic programming to estimate the microscopic proportions ( ) by minimizing Equation 3 15 4. A ssign the indicator variables ( ) by setting the indicator variable to 1 corresponding to the minimum RSS term from Equation 3 12 and the rest 0 Endmember Estimation DME is able to use a set of endmembers supplied externally, such as from another endmember estim ation algorithm or library of spectral signatures. H owever, in many applications a full set of spectral signatures from a given image may not be available. M any endmember estimation algorithms also rely on the pure pixel assumption. T his assumption, however, is often not valid for all endmembers present i n an HSI The ICE framework is used in DME to estimate the endmembers from the HSI, versus selecting pixels as endmembers. The inclusion of the RD MMM, along with the LMM, allows DME to utilize implicit endmember information from macroscopic and microsc opic mixtures DME uses the RSS objective function (Equation 3 12 ) within the regularized ICE framework, discussed in Chapter 2 to estimate the endmembers as shown in Equation 3 16 [ 3 ] The SSD between the endmembers shown in Equation 3 17 is used to regu larize endmember est imation The parameter controls the tradeoff between the RSS and SSD terms (Equation 3 16 ). ( 3 16 ) ( 3 17 ) To estimat as shown in Equation 3 18 for the j th endmember where is a learning rate. [ 93 ] The derivative of with respect PAGE 57 57 to endmember is given by Equation 3 19 where and In this equation is the derivative of the reflectance function (Equation 3 8 ) evaluated at the albedo domain microscopic mixture as given by Equation 3 5 ( i.e., ) and is the slope of the inverse reflectance function evaluated at the j th endmember The second derivative of with respect to endmembe r is given by Equation 3 20 To estimate endmembers, assuming proportions are not known a p riori an alternating optimization technique is used between estimating endmembers and their proportions ( 3 18 ) ( 3 19 ) ( 3 20 ) The complete DME algorithm is as follows: 1. Initialize by selecting a starting set of pixels as endmembers 2. Use the DME proportion estimation to estimate the proportions of each pixel with the current endmembers 3. Use a Newton step to update the e ndmembers 4. Repeat steps 1 3 until a stopping criteria is met PAGE 58 58 Multi Mixture Pixel Estimation Multi Mixture Pixel Model In the previous algorithm DME each pixel is assumed to be macro scopic ally or microscopically mixed This assumption is valid for some HSIs. H owever the assumption is violated in HSIs that contain overlapping heterogeneous regions of different mixture types In these scenes a pixel may be composed of a macroscopic mixture, a microscopic mixture, or a combination of the two mixture types. A second violation to this assumption occurs when one or more endmembers of a microscopic mixture clumps together. These particles cause macroscopic mixing with the surrounding mi croscopic mixture. Th e s e observation s led to the MMP model shown in Equation 3 21 T his new model characterizes a pixel as a macroscopic mixture of endmemb ers and a microscopic mixture i.e., a mixture of mixtures The MMP m odel (Equation 3 21 ) has not been previously considered in HSI analysis. E ndmembers and albedos have the same relationship as described previously in Equation 3 10 ( 3 21 ) T he macroscopic proportions ( ) in the MMP model are dependent on the reflectance of the component microscopic mixture Therefore, the macro scopic proportions are dependent on the microscopic proportions ( ). E stimation of the microscopic proportions is an ill posed inversion problem Ignoring noise, c onsider the following case, as illustrated in Figure 3 5 A MMP is composed of two endmembers, and The MMP is constrained by the set of possible macroscopic mixture s between the endmembers illustrated as a dashed blue line and the set of possible PAGE 59 59 microscopic mixture s between the endmember s illustrated as a solid red line Recall that the MMP model treats the microscopic mixture as an endmember. Therefor e, the MMP model can represent with 0 residual error using and for any point o n the arc More precisely, ( ) increases the MMP mixture point moves closer to the microscopic mixture line as shown in Figure 3 6 Therefore the plausible proportions of the microscopic mixture become increasingly constrained and even unique in the case of a 100% microscopic mixture i.e., the arc of plausible points shrinks As a n MMP mixt microscopic mixture proportion lessens the MMP mixture point moves closer to the macroscopic mixture as shown in Figure 3 7 This causes the plausible microscopic proportions to be less constrained i.e., the arc of plausible points increases However, as the microscopic mixture proportion decreases ( i.e., the MMP is dominated by a macroscopic mixture) the microscopic Therefore, the error caused by the ill posed inversion of estimating the microscopic proportions is a small factor in accurately estimating the mixture proportions. Proportion Estimation As with DME, proportion estimation with MPE requires a current set of endmembers to be provided. Estimation of proportions using MPE is composed of two steps. First the reflectance signature of the microscopic mixture component of the MMP must be estimated To determine this reflectance signature, the microscopic mixture proportions a re estimated using the AD MMM Secondly, the macroscopic PAGE 60 60 prop ortions are estimated by minimizing the RSS objective function shown in Equation 3 22 derived from the MMP model (Equation 3 21 ) ( 3 22 ) HSI pixels, described by the MMP model reside within the region bounded by the macroscopic and microscopic mixtures of a given set of endmembers. This relationship is illustrated in Figure 3 5 and Figure 3 6 A unique estimation of the microscopic mixture proportions is not possible with the information contained within an HSI pixel. T herefore it is likely that some amount of estimation error will be introduced into the microscopic proportion estimation This is due to the ill posed inversion. The m icroscopic mixture proportions are estimated by using quadratic programing to minim iz e the RSS in Equation 3 15 This can be considered in Figure 3 5 and Figure 3 6 as the point residing on the microscopic mixture curve that is closest to the MMP mixture point The macroscopic mixture proportions of the endmembers and the microscopic mixture are estimated by using quadratic programming to minimize the RS S objective in Equation 3 22 The proportions estimation component of MPE is as follows : 1. Take as input a set of pixels from an HSI and a set of endmembers 2. Use quadratic programming to estimate the microscopic mixture proportions ( ) by minimizing Equation 3 15 PAGE 61 61 3. Use quadratic programming to estimate the macroscopic mixture proportions ( ) by minimizing Equation 3 22 The resulting proportions are in terpreted as follows. The macroscopic proportions 1 through M ( ) represent the proportions of endmembers macroscopically mixed. The M+1 macroscopic proportion ( ) is the estimated proportion of the pixel composed of a microsco pic mixture Therefore, is the proportion of the pixel described by a microscopic mixture of the k th endmember. Endmember Estimation Similar to the DME, MPE can use externally provided endmembers from either a spectral library or an i ndependent endmember estimation technique. In many situations spectral libraries are unavailable and external methods fail due to pure pixel or linear model assumptions. MPE uses the MMP model to estimate endmembers from the macroscopic and micr oscopic m ixture information inherently contained in an HSI pixel Estimation of endme mbers using MPE utilizes the ICE regularized framework as shown in Equation 3 16 and Equation 3 17 [ 3 ] The endmembers are estimated using (Equation 3 18 ). The derivative of for the j th endmember is shown in Equation 3 23 where and In this formulation is the derivative of the reflectance function (Equati on 3 8 ) at the albedo domain microscopic mixture ( i.e., ) and is the slope of the inverse reflectance function of the j th endmember. The second derivative of is given by Equation 3 24 To estimate endmembers, assuming proportions are not known a p riori an alternating optimization technique is used between estimating endmembers and their proportions. PAGE 62 62 ( 3 23 ) ( 3 24 ) The complete MPE algorithm is as follows: 1. Initialize by selecting a starting set of HSI pixels as the endmembers 2. U se the MPE proportion estimation algorithm to estimate the proportions of each pixel with the current endmembers 3. Use to update the endmembers 4. Repeat steps 1 3 until a stopping criteria is met PAGE 63 63 Figure 3 1 Derivative of reflectance function over range of albedo values. Figure 3 2 Macroscopic mixture points formed from uniform proportions of two endmembers, band one is plotted PAGE 64 64 Figure 3 3 Reflectance curve with mi croscopic mixture points formed from uniform proportions of two endmembers, band 1 is plotted Figure 3 4 Three dimensional pl ot of microscopic mixture of two endmembers, all three bands are plotted. PAGE 65 65 Figure 3 5 Diagram of multi mixture pixel (black star ) contained within the region bounded by the possible macroscopic mixtures (d ashed blue line) and the possible micros copic mixtures (solid red line) with plausible constituent microscopic mixture (purple square). Figure 3 6 Diagram of multi mixture pixel x contained within the reg ion bounded by the possible macroscopic mixtures (dashed blue line) and the possible microscopic mixtures (solid red line) with plausible constituent microscopic mixture s ( green arc ). PAGE 66 66 Figure 3 7 Diagram o f multi mixture pixel x contained within the region bounded by the possible macroscopic mixtures (dashed blue line) and the possible microscopic mixtures (solid red line) with plausible constituent microscopic mixtures (green arc). PAGE 67 67 CHAPTER 4 EXPERIMENTAL RESULTS E xperiments were conducted to assess the ability of the introduced algorithms to estimate p roportions and endmembers in hyperspectral data The algorithm accuracies are function s of the ir underlying model s and the means of estimating their parameters. Quantitative and qualitative analysis was used in evaluating the algorithms accuracy. The quantitative investigation of the proposed methods used a set of synthetically created data sets and the RELAB da ta of known mineral mixtures The qualitative analy sis was performed on two scenes from the MUUFL Gulfport HSI collection. Quantitative experiments allow for e xact measurements of errors in estimating the proportions and endmembers with a given data set These exact measurements are possible because each data set has known proportions and endmembers. A bility of the algorithms to estimate endmembers and proportions from airborne HSIs is assessed using qualitative analysis. Qualitative analysis is necessary due to the lack of exact measurements of endmembe rs and microscopic mixture proportions from airborne data collections The synthetic data sets were used to compare DME and MPE against LMM based algorithms. I t was found that LMM based algorithms caused increased error in estimating endmembers and propor tions in data containing macroscopic and microscopic mixtures. DME demonstrated an ability to estimate endmembers, mixture types, and mixture proportions accurately in mixture data points dominated by discrete combinations of macroscopic and microscopic m ixtures. MPE was found to have similar accuracy to DME but with improved accuracy when estimating endmembers and PAGE 68 68 proportions in multi mixture data points. Using the RELAB data sets, DME and MPE were compared against existing models for estimating proport ions in microscopically mixed data. These experiments indicate that DME and MPE both were able to estimate proportions of microscopic mixtures with more accuracy than existing algorithms. Synthetic Data sets The use of laboratory data sets is a very effec tive method for analyzing algorithms accuracies However, due to the high cost in creating these data sets they are composed of a small number of samples Therefore a set of synthetic data sets was created to supplement the analysis of the laboratory ex periments. The synthetic data sets allow for creation of a variety of conditions in which to test the algorithms ability to estimate proportions and endmembers. Description of Data sets The synthetic data sets are broken into five categories: macroscopic only, microscopic only, combined macroscopic and microscopic (CMM), multi mixture pixel (MMP), and MMP with high microscopic proportion (MMP HMP). Each data set contains 1,000 data points created using three 38 dimensional endmembers obtained from the AS TER spectral library, shown in Figure 4 1 [ 127 ] These endmembers represent the reflectance signatures of red smooth faced brick, aspha lt roofing shingle and white marble; however, they are referred to in this document simply as endmember 1, 2, and 3, respectively. Because these endmembers are reflectance signatures each band of th e endmembers has a range from zero to one. The macrosc opic and microscopic mixture data sets were created using the LMM of Equation 3 1 and the RD MMM of Equation 3 9 respectively. Detailed descriptions of the LMM and the RD MMM are given in Chapter 3. In dependent zero mean Gaussian noise, with standard PAGE 69 69 deviation of .001, was added to each mixture data point. The proportions for each model were drawn from a uniform distribution. The CMM data set was formed from the union of 500 independently created mix ture data points of each of the macroscopic and microscopic mixtures. The CMM data set can be viewed from the perspective of its first three principal components in Figure 4 2 and Figure 4 3 The blue x each of these figur es represent the macroscopic mix ture data points and therefore can also be used to illustrate the macroscopic mixture only data set the figure s are microscopic mixture data points and can be used to visu alize the microscopic only data set The MMP and MMP HMP were created using the multi mixture pixel model of Equation 3 21 The macroscopic and microscopic mixture proportions for the MMP data set were sampled from a uniform dist ribution T herefore the MMP data set proportions between the three endmembers and the microscopic mixture are ~25%. The proportions for the MMP HMP data set were sampled from a Dirichlet distribution in which the microscopic proportion was more likely. Therefore, the MMP HMP data set is representative of a multi mixture pixel in an environment dominated by microscopic mixtures, i.e., the microscopic proportion is ~92%. These two data sets will allow for analysis of the algorithms under contrasting micr oscopic mixing proportions of multi mixed pixels The MMP data set can be seen in Figure 4 4 and Figure 4 5 Endmember Estimation The synthetic data set s were used to assess the accuracy of DME and MPE i n comparison to endmember estimation methods based on the LMM. These experiments were conducted using all five synthetic data set categories. The DME and MPE algorithms results were contrasted with the ICE and VCA algorithms. ICE and VCA PAGE 70 70 constitute repr esentative endmember estimation methods from both the statistical and geometrical families of HSI endmember estimation methods. Additionally, results are compared to the regularized linear mixture model gradient descent (R LMM GD) algorithm and the albedo domain RLMM GD (AD RLMM GD) algorithm R LMM GD is similar to ICE with the exception that it uses a gradient descent approach to estimate endmembers with in the ICE objective function. Details of the ICE algorithm can be found in Chapter 2. The AD RLMM G D algorithm converts all input reflectance domain data into albedo domain using the invers e of the r eflectance function from Equation 3 3 Using the albedo domain data, the RLMM GD algorithm is run to estimate endmembers and prop ortions Using RLMM GD and AD RLMM GD allowed for comparison with LMM based algorithms that used gradient descent to estimate endmembers in both reflectance and albedo domains. Comparing DME and MPE against RLMM GD and AD RLMM GD will negate any advantag es that may be gained from using a gradient descent optimization. To quantify the error between estimated and true endmember s the mean absolute band error (MABE) is introduced. The MABE, shown in Equation 4 1 quantifies the err or in each of the dimensions between the true endmember ( ) and the estimated endmember ( ). Similar to an L 1 norm, the MABE provides for a robust evaluation of the error over a set of M estimated endmembers. ( 4 1 ) This section use s the synthetic data sets to investigate : 1. Initialization of parameters for DME and MPE PAGE 71 71 2. The failures, caused by microscopic mixing, of exist ing endmember estimation methods based on the LMM 3. The accuracy of DME and MPE in comparison to endmember estimation methods based solely on the LMM Initialization and p arameter r obustness As with many endmember estimation algorithms, ICE, RLMM GD, AD RLMM GD, DME and MPE require initialization and parameter settings. [ 3 ] Following the ICE framework, RLMM GD, AD RLMM GD, DME, and MPE use an alternating optimization scheme to estimate the endmembers and proportions [ 3 ] A convergence criterion is used in the algorithms such t hat when the regularized RSS (Equation 3 16 ) error is reduced by less than the algorithms terminate. Gradient descent optimization is used to estimate the endmembers in the RLMM GD, AD RLMM GD, DME, and MPE algorithms. Specifically, in these experiments, all in batch mode for minimizing the regularize d RSS objective function. An annealing learning rate was used to control the step size. The learning rate started at 10 and was divided by two until either a step was found that reduced the objective function or the updated learning rate w as less than If no step was made that reduced the objective function then the algorithm was terminated. Figure 4 6 shows the demonstrate d robustness to the annealing rate used. The results in Figure 4 6 were obtained using five CMM data sets Regularization parameter settings. In the ICE, RLMM GD, AD RLMM GD, DME, and MPE algorithms the parameter acts as a regularization parameter offsetting the RSS error term an d the SSD term, as shown in Equation 3 16 and repeated here in Equation 4 2 for convenience The SSD penalizes the objective function as the distances between endmembers gets larger. This constrains the endmembers to fit PAGE 72 72 tightly around the d ata. The larger the the more the objective function minimize s the distances between endmembers versus the RSS error. ( 4 2 ) To evaluate the algorithms robustness to the para meter the following experiments were conducted Each algorithm was run on f ive CMM synthetic data sets Using the CMM synthetic data sets for these experiments provided a mix of macroscopic and microscopic mixture data points to optimize over. The algor ithms were initialized using PPI. [ 53 ] These experiments were repeated for nine different se ttings of the parameter, the results are shown in Figure 4 7 The setting creating the smallest mean MABE for each algorithm was used Based on the results of this experiment, the values given in Table 4 1 are used in all experiments presented in this document. Initialization of starting endmembers. ICE, RLMM GD, AD RLMM GD, DME, and MPE all use an alternating optimization scheme to iteratively estimate proportions and endmembers. Further, RLMM GD, AD RLMM G D, DME, and MPE use a gradient descent optimization to estimate endmembers. One inherent problem with alternating optimization and gradient descent techniques is local minima. Consequently, initialization of starting endmembers has a large impact on the accuracy of the final estimated endmembers. Choosing random input data points as starting endmembers is often the simplest solution. However, this approach, in all five algorithms, leads to large inconsistencies in estimating endmembers from a single data set, as demonstrated by Figure 4 8 Figure 4 12 The histograms are the result of running each algorithm on a single CMM data PAGE 73 73 set 50 times. Each algorithm run used thre e randomly chosen data points as starting endmembers. Comparison of these histograms to the results obtained using PPI ( Figure 4 7 ) reveals that using random data points as starting endmembers decreases the accura cy in endmember estimation for all considered algorithms. As shown in Figure 4 7 PPI works well for initializing these algorithms even when including microscopic mixtures PPI finds points near the vertices and r eturns spect ra from actual data points. The s e spectra, when microscopically mixed, more accurat ely reflect the microscopic mixture created by the actual endmembers ( i.e., no ambiguous errors are introduced to the spectra through trying to linearly estimat e the endmembers). PPI is used to initialize all experiments presented in this document. A common problem in endmember estimation is determining the number of endmembers in a mixture. [ 3 4 49 71 72 ] Using a n incorrect number of endmembers will result in anomalous or missing endmembers from a mixture which in turn can cause incorrect proportion estimates for the estimated endmembers. In this research the correct number of endmembers will be provided as an in put parameter, as has been done in other microscopic mixture research. [ 23 24 32 35 38 ] Future work, however, can use the SPICE framework to allow DME and MPE to automatically determine the numb er of endmembers. [ 4 ] Microscopic mixture effects To c larify analysis of experimental results, it is advantageous to categorize the types of effects that microscopic mix tures create These effects are discussed with respect to the errors they cause by assuming a LMM. A macroscopic mixture of M endmembers is known to form a linear M 1 simplex embedded in D dimensions. [ 52 ] Assuming a three endme mber mixture, t he first and second principal components PAGE 74 74 therefore fully describe the linear simplex (ignoring noise). A microscopic mixture of the same endmembers warps the linear simplex as discussed in Chapter 3 Using the synthetic data sets, three different effects can be observed due to this warping. These effects, and their magnitudes, are directly related to the single scattering albedos and multi scattering properties of the mixing endmembers and collection parameters such as the angles of in cidence and emergence. Convex combination error. The first error will be referred to as a convex combination error (the se names are not a recognized convention but used in this document for convenience) This error is defined by mixture data points that form a convex set in the linear span of the M 1 principal components but are not constrained to be convex combinations of the mixture endmembers. This violates the convex combination assumption s made in the LMM. [ 3 ] The convex combination error can be observed in Figure 4 2 Th is figure shows the first and second principal compo nents of a representative CMM mixture with three endmembers The one and three form a line of randomly distributed points resulting from the convex combination of endmembers The microscopic m the same two endmembers, curve outside of the linear simplex formed by the endmembers. These mixture data points form a convex set but violate the convex combination constraints. Because this error occurs in the line ar span of the M 1 principal components, geometrical methods will be directly impacted as well. In these synthetic data sets t his type of error cause d distortions in the estimated endmember spectra, i.e., increased or decreased reflectance in a subset of the PAGE 75 75 Convex combination error is shown demonstrated on a CMM data set in Figure 4 13 and Figure 4 15 The image in Figure 4 13 shows the microscopic mixture data points curving outside of the simplex boundary between endmembers three and one Optimization of the objective function based on the LMM caused the en dmember estimates to spread out, attempting to create a linear simplex that enclose d all of the mixture data points. This error, observed in the M 1 principal axis of variation, causes the errors in estimated endmember spectra shown in Figure 4 15 In this example all three endmembers show a skewing of spectra in a subset of their bands. Convex set error. The second type of error will be referred to as convex set error. This error is defined by mixture data points, created by microscopic mixing, that form a non convex set. This violates the LMM assumption that mixture data points form a convex set. Additionally, t his type of effect increases the intrinsic dimensionality of the mixture from M 1 dimensions to at least M dimensions. Convex set error can be observe d in the first and third principal components of the CMM data set shown in Figure 4 3 It is observed that the microscopic mixture data points form a non convex set. Comparatively this error represent ed a small variation in the overall mixture. In the synthetic data sets discussed in this document, the additional principal component accounts for less than 1% of the total variance in a given data set Convex set error can be seen in the endmember estimation results shown in Figure 4 14 This figure shows the first and third principal components of the data, true endmembers, and estimated endmembers resulting from the same experiment shown in PAGE 76 76 Figure 4 13 The estimated endmemb ers minimize linear error between all mixture data points by averaging over the 3 rd principal component. Linear proportions error. The first two types of error s describe d perturbations, caused by microscopic mixing, of the linear simplex. The third typ e of error, linear proportions error, is defined as mixture data points in which the endmember proportion s are not proportional to the relative distances between endmembers. In reflectance spectroscopy literature, this is termed the spectral dominance of low albedo materials in microscopic mixing. [ 32 ] This error can be observed in Figure 4 2 and Figure 4 3 by noting that the macroscopic and microscopic proportions were drawn from the same uniform distribution. The macroscopic mixture points in the first, second, and third principal components appear uniformly distrib uted. However, t he microscopic mixture points have the highest density near the endmember with the lowest albedo, and consequently lowest reflectance, endmember one The microscopic mixture points are very sparse closer to endmember three which has the highest albedo /reflectance. Linear proportions error wa s characterized in these data sets by a uniform shift in the amplitude of the estimated endmember spectra. Th is error can be seen in the first three principal components, shown in Figure 4 16 and Figure 4 17 These figures show the results of the same experiment as shown in Figure 4 13 and Figure 4 14 except that in this experiment a of was used. The images in Figure 4 16 and Figure 4 17 show that the endmember three estimation was pulled in closer to the largest density of mixture data points, when compared to Figure 4 13 and Figure 4 14 Figure 4 18 PAGE 77 77 shows the eff ect of this error on the estimated endmember spectra When compared to Figure 4 15 a uniform reduction in magnitude of endmember three can be observed. The chart in Figure 4 7 shows increased MABE for the larger parameter s, this increased error is largely caused by linear proportions error Figure 4 19 shows the estimated endmembers from the same data set as the experiment s in Figure 4 13 Figure 4 18 using values through These spectra demonstrate the increased error in estimation of endmember three caused by linear p roportions error for larger values and the contrasting increases and decreases in convex combination error Through an understanding of the causes of microscopic mixing it is clear that microscopic mixtures are present in most HSI s [ 1 7 30 31 ] However, t hroughout published literature it is rare to find instances whe re these specific errors are addressed. This can be explained in some HSI s due to low proportion of microscopic mixing or small contrast between microscopically mixed endmembers. Both of thes e reduce the magnitude of error created by the microscopic mixture Additionally, i n many cases adequate ground truth is not available to assess the accuracy of estimated endmembers fr o m HSIs. However, recently the impact of nonlinear mixtures in HSIs ha s garnered much attention [ 1 8 19 28 33 35 38 41 42 44 59 60 69 95 98 101 104 105 116 117 119 122 ] This therefore necessitates that these nonlinear microscopic mixing effects be used to increase endmember estimation accuracy and not simply dismissed. PAGE 78 78 Endmember e stimation accuracy In this section results are presented to assess the accuracy of DME and MPE in estimating endmembers. For comparison, the results of DME and MPE will be contr asted to the results of VCA, ICE, RLMM GD, and AD RLMM GD To compare the algorithms 50 synthetic data set s of each category were generated. Each of the six algorithms was run on the data set s and the resulting MABE value s calculated. The algorithms we re initialized using the same starting initial endmembers, determined by using PPI and the parameter values given in Table 4 1 The mean MABE value s for each data set are given in Figure 4 20 Figure 4 22 For ease of readability, the results were broken into three separate charts. The first chart ( Figure 4 20 ), illustrates the results for VCA, a representative geometric method based on the LMM. The second chart ( Figure 4 21 ), shows the results for ICE and RLMM GD, two representative statistical methods based on the LMM. Finally, the third chart ( Figure 4 22 ) gives the results for AD RLMM GD, an albedo domain method based on the LMM. The error bars in each chart represent one standard deviation from the mean MABE over a given data set category. The distribution of MABE values for a given algorithm and data set however, are not Gaussian distri butions. The histograms for each algorithm run on the CMM data set s is given in Figure 4 23 Figure 4 28 Further analysis treats the MABE values resulting from any two algorithms on the same data set as a paired test. The charts in Figure 4 29 Figure 4 32 illustrate the difference in MABE values between the LMM based algorithms and DME and MPE. From these results it can be observed that when macr oscopic mixtures dominated the data set ( i.e., macroscopic only and MMP), the reflectance domain LMM algorithms performed similar to DME and MPE. These observations are contrasted with the poor PAGE 79 79 performance of the reflectance domain LMM algorithms (VCA, IC E, RLLM GD) on data set s dominated by microscopic mixtures ( i.e., microscopic only and MMP HMP). Likewise, the albedo domain LMM algorithm (AD RLMM GD) performed similar to DME and MPE on those data set s dominated by microscopic mixtures and poorly on tho se dominated by macroscopic mixtures. The CMM data set s combine equal numbers of macroscopic and microscopic mixture data samples, therefore, they provide a single data set to evaluate the results of these algorithms on both mixture types. The paired MAB E difference histograms of the LMM algorithms and DME and MPE are given Figure 4 33 Figure 4 36 The charts in Figure 4 29 Figure 4 32 also indicate that DME did not consistently perform better than the LMM algorithms on the MMP HMP data set s This results from the multi mixture pixel model that formed the s e data set s The histogram in Figure 4 37 illustrates the paired MAB E difference between MPE and all other algorithms, including DME. These results demonstrate that MPE consistently created lower MABE than the other algorithms when estimating endmembers of a multi mixture pixel Proportion Estimation In this section the accuracy of DME and MPE in estimating endmember proportions is assessed To compare with experiments performed in other published research, each algorithm being evaluated took as input the correct endmember spectra for each data set [ 23 24 32 35 37 ] Each algorithm estimated proportions of the mixture data, using their respective approaches, and report ed error in proportion estimation using the root mean square error (RMSE). The RMSE, shown in Equation 4 3 is commonly used to measure proportion estimation accuracy, where is the number of data set samples, is the number of endmembers, is the true proportion PAGE 80 80 of the sample, and is the estima ted proportion. [ 23 24 32 35 37 ] The sample proportion estimates are calculated for DME using Equation 4 4 and for MPE with Equation 4 5 where and are the estimated macroscopic and microscopic proportions for the models, respectively. ( 4 3 ) ( 4 4 ) ( 4 5 ) In these experiments 50 data sets we re generated from each synthetic data set category. DME and MPE were run on all data sets and their mean RMSE of proportions estimates calculated. When using the given true endmembers the standard deviation of proportion estimates was less than .01 in all cases. Therefore, the error bars were not used in the results charts. T o evaluate DME and MPE the LMM and the albedo domain (AD LMM) were included for comparisons The LMM and AD LMM use quadratic programming to estimate the endmember proportions of t he given data points. This method of linear unmixing is commonly used in published literature, including the ICE algorithm. [ 3 ] The AD LMM, however, first converts the endmembers and mixture data points into the albedo domain using the invers e of the reflectance function from Equation 3 3 In the following sections the effects of microscopic mixtures on proportion estimation is discussed, as well as an analysis of the accuracy of DME and MPE for estimating microscopic mixture concentrations and estimating mixture proportions. PAGE 81 81 Microscopic mixture effects Microscopic mixtures a ffe ct linear proportion estimation in two ways. The first is caused by the previously mentioned convex combination and convex set errors These errors were visualized in Figure 4 2 and Figure 4 3 As both figures show, microscopic mixtures create a warped simplex. When unmixing hyperspectral data, using a linear model, it is assumed (by definition of the linear model) that all data points are contained within the linear simplex describe d by the endmembers [ 1 ] Therefore, when estimating the proportions of a microscopic mixture data point using a linear model the data point is actuall y the projection of the microscopic mixture point onto the linear simplex. This projection does not preserve the true relationship between endmembers represented by the data point and thus error is introduced. [ 1 ] Secondly, the previously discussed linear proportions error causes inaccuracies in estimating proportions when using a linear model. The spectral dominance of low albedo sub stances causes a non uniform distribution of mixture points. [ 32 ] The linear model makes the assumption that a data point is a linear combination of its constituent endmembers, i.e., the p roportions of each endmember are equivalent to the relative distance between the mixture point and the vertices of the linear simplex. As shown in Figure 4 38 a microscopic mixture of 50% of each of two endmembers in the albedo domain (shown as the cyan dashed line) is a linear mixture. However, when converted to reflectance domain the signature of the mixture point (cyan dotted line) does not represent a linear 50% mixture of each endmember (red dotted line) Estimating m icroscopic m ixture c oncentration In estimating endmember proportions of a mixed data set, some knowledge about the mixture types present in the data must be either assumed or estimated. I n the case PAGE 82 82 of the LMM the mixture type is assumed to be 100% macroscopically mixed. Likewise, in the AD LMM the mixture type is assumed to be 100% microscopically mixed. Broadwater was the first to present a n algorithm that was ab le to identify a mixture data point or pixel, as macroscopic ally or microscopic ally mixed [ 2 4 25 ] In his paper the mixture identified for each pixel was used to characterize the mixing patterns in the given HSI Th i s information provide s a unique analysis and understanding of the HSI. These labels can also be used as a diagnostic tool to better understand the mixtures present in a mixed data point and evaluate the accuracy of the estimation algorithm bui lds upon this capability by using the MMP model. MPE is able to estimate the proportion of microscopic mixing occurring in a single mixed data point. This allows new observations and analysis of the disbursement patterns of various substances throughout a scene. DME discriminates each sample as who lly ma croscopic ally or mi croscopic ally mixed T herefore, DME estimates microscopic mixture proportions for a given data set a s the percentage of mixture data points labeled as microscopic ally mixed MPE estim ates samples as partially microscopically mixed ; therefore the microscopic proportion estimate of a data set is the mean microscopic proportion over every sample in the data set The results of the mean estimated microscopic proportions are shown in Figure 4 39 LMM and AD LMM are unable to estimate mixture types. Therefore, as indicated by the graph, produce large errors in estimating mixture types when compared to the known mixture type proportions DME and MPE were able to accurately estimate the proportion of mixture data points representing microscopic mixtures in the macroscopic, PAGE 83 83 microscopic, and CMM data sets MMP HMP data set, which is dominated by microscopic mixtures, i s accurate. However, DME discrete mixture assumption fails as is indicated by the large error in estimating the microscopic proportion in the MMP data sets. MPE proportion estimation shows an error of approximately 6. 0 % 8 .0 % estimated microscopic pro portion between the MMP and MMP HMP data sets T his error is due to the ill posed inversion problem of estimating the microscopic proportions. Proportion estimation accuracy Using the results of the same experiments run for the mixture type concentratio n the accuracy in proportion estimation is assessed. The results of the proportion estimation experiments are shown in Figure 4 40 Th is chart shows the RMSE for each of the LMM, AD LMM DME and MPE proportion estimation algor ithms for each randomly generated data set These results indicate large RMSE in proportion estimations for LMM and AD LMM when the data set dominate mixture type is not in line with the mixture type assumption made by the algorithm. Both DME and MPE dem onstrate an ability to accurately estimate endmember proportions in the macroscopic, CMM, and microscopic data sets. DME and MPE indicate increased errors in estimation in the MMP and MMP HMP data sets. The increased error in estimating proportions of th ese data sets is understood in DME as a failed assumption of discrete mixtures. MPE, however, indicates more accuracy in estimating proportions in the MMP and MMP HMP data sets with some error in estimation, as previously discussed. Based on these results the following conclusions can be made. Microscopic PAGE 84 84 violations cause increased error in estimating endmembers and their proportions when microscopic mixing is present. DME a nd MPE are able to accurately estimate endmembers and their proportions when macroscopic and/or microscopic mixtures are present. Additionally, when a mixture data point is composed of a multi mixture pixel the DME can show increased errors in estimating endmembers and proportions. However, MPE is able to estimate these endmembers and proportions with decreased error. RELAB Data sets Description of Data S ets Quantitative analysis of DME and MPE was performed on mixtures of Olivine with Enstatite and Oli vine with Magnetite. The mixtures where prepared and measured by Professors John Mustard and Carle Pieters of Brown University, as specified in their paper [ 32 ] The spectra were collected with RELAB, a high resolution bidirectional reflectance spectrometer with a specified angle of emergence of 0 0 and angle of incidence of 30 0 [ 32 ] Each mixture data set is composed of spectra from five mixed samples (proportions of which are given in Table 4 2 ) and two endmembers, as shown in Figure 4 41 and Figure 4 42 The spectra have 211 dimensions with wavelengths ranging from 400 nm to 2,500 nm. The precise measurements of the mineral mixture proportions and the ir spectra make these data sets ideal for analysis of hyperspectral unmixing algorithms. These mineral sets are known to mix microscopically and each sample was crushed prior to mixing, to ensure a homogenous mixture. A scatter plot of two wavelengths of the Olivine with Magnetite data set ( Figure 4 43 ) demonstrates the nonlinearity in the mixtures. Additionally, the endmembers have a variety of absorption features (including PAGE 85 85 the contamination feature in the Olivine spectra) and both small and large contrasts between endmember albedos. Consequently these samples have become a standard data set for evaluating microscopic unmixing algorithms. [ 23 24 32 35 37 ] Endmember Estimation Experiments were conducted to compare endmember estimation of DME and MPE against ICE using the RELAB data. All three algorithms used the parameters given in Table 4 1 (as previously determined) and sample points on e and five as initial endmembers The results of the experiments on the Magnetite and Olivine data set are shown in Figure 4 44 and the Enstatite and Olivine data set in Figure 4 45 The MABE for both e xperiments is given in Figure 4 46 These results indicate that in both experiments DME and MPE were able to estimate the endmembers with more accuracy than ICE. In the Magnetite and Olivine experiment, DME resulted in a lower MA BE. In the Enstatite and Olivine experiment, MPE resulted in a lower MABE. As will be shown in the next section, this phenomenon is mimicked in the proportion estimation experiments. Proportion Estimation The RELAB Enstatite with Olivine and Magnetite wi th Olivine binary mixtures have become the de facto standard of comparison between microscopic proportion estimation algorithms. [ 23 24 32 35 37 ] Experiments with these data sets allow a precise comparison between DME and MPE and other attempts to estimate proportions with microscopic mixing. In addition results from the LMM algorithm are included for comparisons. The proportion estimation results are shown in Table 4 3 part A shows results from the Enstatite & Olivine mixture and part B the Magnetite & Olivine mixture. PAGE 86 86 Proportion RMSE results were included from or calculated from published results using the LSE [ 32 ] kernel methods [ 23 25 ] RBFNN [ 35 ] and HNN/MLP [ 37 ] algorithms. See Chapter 2 for details on these algorithms. In the published works using kernel methods and neural networks to unmix microscopically mixed hyperspectral data, RMSE comparisons were made using the known mass fractions of the RELAB samples. [ 23 24 35 37 ] However, since these unmixing algorithm s estimate microscopic mixture proportions, often referred to as F parameters, an assumption of similar particle densities and diameters between endmembers is required (details discussed in Chapter 2). [ 7 30 32 ] The RELAB data sets were prepared such that particle diameters are similar but not particle densities [ 32 ] Therefore, a more appropriate measure of accuracy in estimating RELAB proportions is to calculate RMSE betwe en estimated proportions and published F parameters for the data sets. Table 4 3 lists proportion RMSE s between published LSE, RBFNN, and HNN/MLP estimated proportions and F parameter truth for comparison with the experimentally estimated DME and MPE proportions. [ 32 35 37 ] Howev er, the estimated proportions using the various kernel methods were not available and the reference papers did not describe the experiments in enough detail to allow reproduction of the experiments [ 23 24 ] Therefore, t he proportion RMSE using mass fractions is given, as published, for the kernel and neural network approaches [ 23 24 35 37 ] Addit ionally, Table 4 3 lists the percentage of the data set estimated by the algorithm as a microscopic mixture. These results indicate that in the Magnetite and Olivine experiments DME estimated the proportions with the most accura cy, followed by LSE and MPE. Further PAGE 87 87 insight into this result can be drawn from the microscopic mixture estimate for both DME and MPE. DME estimated a 100% microscopic mixture This indicates that the algo rithm would unmix the data set similarly to the LSE algorithm, which leads to the similar results between those algorithms shown in Table 4 3 The microscopic mixture for MPE is 98.3%. Based on this estimation, it can be concluded that previously discussed error in estimating microscopic mixing proportions caused increased proportion estimation error. Therefore, MPE displayed more RMSE than DME and LSE. The results for the Enstatite and Olivine data set show a contrasting example. In these results ( Table 4 3 ), MPE produced the lowest RMSE followed by DME. The microscopic mixture estimation results indicate that both DME and MPE estimated this data set as having a large macroscopic mixing component. This is likely due to the small contrast between m ixing endmembers. The ability of DME and MPE to include macroscopic mixing in their estimations allowed for large improvements in RMSE over the LSE algorithm. Of the algorithms evaluated, DME and MPE were the only ones able to estimate the mixture propor tions accurately over both data sets. Aerial Hyperspectral Data S ets To assess the ability of DME and MPE to operate on aerial hyperspectral data the MUUFL Gulfport collection was used. The MUUFL Gulfport data was collected with a CASI hyperspectral sens or over Gulfport, MS. The data provides 72 bands ranging from 368 nm to 1 043 nm with a spatial resolution of 1 m x 1 m. Qualitative analysis of the data using DME and MPE estimated proportions and endmembers indicated a plausible explanation of the HSI. PAGE 88 88 Baseball Diamond Scene The first scene discussed is the baseball diamond scene, shown in Figure 4 47 The pixels used in this experiment were specifically selected so that just the baseball diamond was included, i.e., the basepl ate and trees were excluded. The endmembers estimated by DME MPE and ICE are shown in Figure 4 48 All three methods estimated endmembers similar in spectral shape, which is a consequence of a large amount of macroscopic mixin g over microscopic. The spectra indicate grass, dirt and clay endmembers. The grass endmember can be identified by the bump observed at approximately 550 nm and the increased reflectance in the near infrared (IR). The clay endmember is characterized by relatively high reflectance from approximately 550 nm through the IR. The negative values present in the spectra are a result of errors in the conversion process of estimating pixel reflectance values from measured radiances. In this data set the conver sion was performed using the empirical line method for atmospheric correction. The relative smoothness of the spectra also indicates that plausible endmember spectra were extracted. DME detected microscopic mixture pixels (shown in red) can be observed in Figure 4 49 DME was able to find a few microscopically mixed pixels but a majority of the pixels in this scene where more than 50% macroscopically mixed. MPE s microscopic mixture mapping is shown in Figure 4 50 To further understand these results the MPE macroscopic proportions map for the grass endmember is shown in Figure 4 51 and the microscopic proportion map in Figure 4 52 These maps indica te extensive macroscopic and microscopic mixing in the outfield and infield. These results make sense, especially when considering the macroscopic and PAGE 89 89 microscopic proportions of the dirt endmember, Figure 4 53 and Figure 4 54 respectively. The dirt endmember shows slight macroscopic and microscopic mixing across the entire scene except for an area between bases in the middle of the base path The dirt endmember also shows increased macroscopic mixing in the area on the outside of the base path that is darker in the color image of the scene ( Figure 4 47 ). A most likely explanation would be that dirt from the outfield has over time spread into the clay base path. The proport ion maps for the clay endmember are shown in Figure 4 55 for the macroscopic proportions and Figure 4 56 for the microscopic proportions. Both proportion maps illustrate a dominance of the clay endmember in the base path, mostly observed between first, second, and third base. However, a faint base path between home plate and first and home plate and third can be observed. Also, pixels containing clay endmember proportions can be seen in the middle of th e infield, the mound. Beach Scene The second scene looked at in these experiments was the beach scene, shown in Figure 4 57 In this experiment only pixels covering the beach were included. To estimate the endmembers V CA was used. [ 54 ] Two endmembers were found : a sand endmember and a sediment endmember. MPE was used to unmix the pixels given the VC A estimated endmembers. The microscopic mixture proportions map is shown in Figure 4 58 From this image it can be seen that the sediment is originating from the construction area to the left of the b each. The sediment is dispe rsed down the beach by wind a nd water. PAGE 90 90 The dispersion pattern of the sediment can be observed by referencing the sand macroscopic and microscopic proportion maps, Figure 4 59 and Figure 4 60 respective ly, and the sediment macroscopic and microscopic proportion maps, Figure 4 61 and Figure 4 62 respectively. It can be observed from the microscopic proportion maps of each endmember that while sand is t he dominant endmember, concentrations of the sediment are found near the origin point and in concentration points along the beach. Also, the microscopic proportion maps reveal that along the waterline traces of the sediment can be seen mixing microscopica lly with the sand. This leads to the conclusion that the sediment ble d into the water an d was deposited along the water line of the beach. Using the macroscopic proportion maps areas of sediment can be observed to be macroscopically mixing along the fence line. This result can be explained by wind that blows the sediment and sand that is then trapped by the fence. Th erefore, more concentrated areas of sediment are created thus causing macroscopic mixtures. Due to a lack of precise mixture ground truth a n exact analysis of DME and MPE on the aerial HSI data is not possible. However, a plausible explanation was presented for each scene. Further, analysis of sediment distribution made possible by estimating microscopic mixture was presented. PAGE 91 91 Table 4 1 Algorithm parameter settings. Table 4 2 RELAB data set true proportions. PAGE 92 92 Table 4 3 Comparison of propo rtion estimates of RELAB mixtures using mass fraction and F parameter values A) Olivine & Enstatite mixture. B) Olivine & Magnetite mixture. [ Results adapted from J. F. Mustard and C. M. Pieters, "Quantitative Abundance Estimates From Bidirectional Re flectance Measurements," Journal of Geophysical Research, vol. 92, pp. E617 E626, March 30, 1987 1987. Results adapted from J. Broadwater and A. Banerjee, "A comparison of kernel functions for intimate mixture models," in Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, 2009. WHISPERS '09. First Workshop on, 2009, pp. 1 4. *** Results adapted from J. Broadwater and A. Banerjee, "A generalized kernel for areal and intimate mixtures," in Hyperspectral Image and Signal Processing: Evo lution in Remote Sensing (WHISPERS), 2010 2nd Workshop on, 2010, pp. 1 4. **** K. J. Guilfoyle, M. L. Althouse, and I. C. Chein, "A quantitative and comparative analysis of linear and nonlinear spectral mixture models using radial basis function neural ne tworks," Geoscience and Remote Sensing, IEEE Transactions on, vol. 39, pp. 2314 2318, 2001. ***** J. Plaza, P. Martinez, R. Perez, and A. Plaza, "Nonlinear Neural Network Mixture Models for Fractional Abundance Estimation in AVIRIS Hyperspectral Images," presented at the NASA Jet Propulsion Laboratory AVIRIS Airborne Earth Science Workshop, Pasadena, California, 2004. ] A B PAGE 93 93 Figure 4 1 Synthetic endmember reflectance spectra. Figure 4 2 1 st and 2 nd principal components of synthetic CMM data set PAGE 94 94 Figure 4 3 1 st and 3 rd principal components of synthetic CMM data set Figure 4 4 1 st and 2 nd principal component of synthetic MMP data set PAGE 95 95 Figure 4 5 1 st and 3 rd principal component of synthetic MMP data set Figure 4 6 Mean MABE over range of annealing rates. PAGE 96 96 Figure 4 7 Mean MABE of synthetic CMM data sets over range of settings. Figure 4 8 H istogram of ICE MABE using random starting points. PAGE 97 97 Figure 4 9 Histogram of RLMM GD MABE using random starts. Figure 4 10 Histogram of AD RLMM GD MABE us ing random starts. PAGE 98 98 Figure 4 11 Histogram of DME MABE using random starts. Figure 4 12 Histogram of M P E MABE using random starts. PAGE 99 99 Figure 4 13 RLMM GD estimated endmembers for CMM data set (1 st and 2 nd principal components). Figure 4 14 RLMM GD estimated endmembers for CMM data set (1 st and 3 rd principal components). PAGE 100 100 Figure 4 15 RLMM GD estimated endmembers for CMM data set (spectra). Figure 4 16 RLMM GD estimated endmembers for CMM data set with (1 st and 2 nd principal components). PAGE 101 101 Figure 4 17 RLMM GD estimated endmembers for CMM data set with (1 st and 3 rd principal components). Figure 4 18 RLMM GD estimated endmembers for CMM data set with (spectra). PAGE 102 102 Figure 4 19 RLMM GD estimated endmember spectra with multiple parameters. Figure 4 20 Mean MABE of VCA, DME, and MPE. PAGE 103 103 Figure 4 21 Mean MABE of ICE, RLMM GD, DME, and MPE. Figure 4 22 Mean MABE of AD RLM M GD, DME, and MPE. PAGE 104 104 Figure 4 23 Histogram of VCA MABE values for CMM data set Figure 4 24 Histogram of ICE MABE values for CMM data set. PAGE 105 105 Figure 4 25 Histogram of RLMM GD MABE values for CMM data set. Figure 4 26 Histogram of AD RLMM GD MABE values for CMM data set. PAGE 106 106 Figure 4 27 Histogram of DM E MABE values for CMM data set. Figure 4 28 Histogram of MPE MABE values for CMM data set. PAGE 107 107 Figure 4 29 Mean MA BE difference between VCA and DME and MPE. Figure 4 30 Mean MABE difference between ICE and DME and MPE. PAGE 108 108 Figure 4 31 Mean MABE difference between RLMM G D and DME and MPE. Figure 4 32 Mean MABE difference between AD RLMM GD and DME and MPE. 0.005 0 0.005 0.01 0.015 0.02 0.025 Macro Only CMM Micro Only MMP MMPHMP Mean MABE Difference ADRLMMGD DME ADRLMMGD MPE PAGE 109 109 Figure 4 33 Histogram of MABE difference between VCA and DME and MPE for CMM data set Figure 4 34 Histogra m of MABE difference between ICE and DME and MPE for CMM data set. PAGE 110 110 Figure 4 35 Histogram of MABE difference b etween RLMM GD and DME and MPE for CMM data set. Figure 4 36 Histogram of MABE difference between AD RLMM GD and DME and MPE for CMM data set. PAGE 111 111 Figure 4 37 Histogram of MABE difference between VCA, ICE RLMM GD, AD RLMM GD, and DME and MPE for MMP HMP data set. Figure 4 38 Comparison of nonlinearity in albedo domain and reflectance domain mixtures Each mixture was created with a 50% proportion of each endmember. Angles of incidence and emergence were both PAGE 112 112 Figure 4 39 Microscopic proportion s resulting from LMM AD LMM, DME and MPE over all synt hetic data sets Figure 4 40 Proportion RMSE resulting from LMM, AD LMM, DME, and MPE over all synthetic data sets. PAGE 113 113 Figure 4 41 Spectra of Enstatite and Olivine mixture. Figure 4 42 Spectra of Magnetite and Olivine mixture. PAGE 114 114 Figure 4 43 Wavelength scatter plot of Magnetite and Olivine mixture. Figure 4 44 Endmember estimation of RELAB Magnetite and Olivine mixture. PAGE 115 115 Figure 4 45 Endmember estimation of RELAB Enstatite and Olivine mixture. Figure 4 46 RELAB data sets endmember estimation error chart. PAGE 116 116 Figure 4 47 Baseball diamond scene. Figure 4 48 Endmem bers estimated from baseball diamond scene. PAGE 117 117 Figure 4 49 Points labeled by DME (red) as microscopically mixed. Figure 4 50 MPE mi croscopic mixture propor tions Darker red pixels indicate higher estimated proportion s PAGE 118 118 Figure 4 51 MPE grass macroscopic proportions Darker red pixels indicate higher estimated proportion s Figure 4 52 MPE grass microscopic proportions Darker red pixels indicate higher estimated proportions. PAGE 119 119 Figure 4 53 MPE dirt macroscopic proportions. Darker red pixels indicate higher estimated proportion s Figure 4 54 MPE dirt microscopic proportions. Darker red pixels indicate higher estimated proportions. PAGE 120 120 Figure 4 55 MPE clay macr oscopic proportions. Darker red pixels indicate higher estimated proportions. Figure 4 56 MPE clay microscopic proportions. Darker red pixels indicate higher estimated proportions. PAGE 121 121 Figure 4 57 Beach scene. Figure 4 58 Microscopic proportions estimated using MPE Darker red pixels indicate higher estimated proportions. PAGE 122 122 Figure 4 59 Macroscopic proportion map for sand endmember. Darker red pixels indicate higher estimated proportions. Figure 4 60 Microscopic proportion map for sand endmember. Darke r red pixels indicate higher estimated proportions. PAGE 123 123 Figure 4 61 Macroscopic proportion map for sediment endmember. Darker red pixels indicate higher estimated proportions. Figure 4 62 Microscopic proportion map for sediment endmember. Darker red pixels indicate higher estimated proportions. PAGE 124 124 CHAPTER 5 CONCLUSION Two method s of incorporating macroscopic and microscopic mixture models into hyperspectral pixel unm ixing is presented and discussed. A vast majority of hyperspectral unmixing methods rely on the linear mixture model to describe pixel spectra resulting from mixtures of endmembers. [ 1 4 ] Methods exist to unmix hyperspectral pixels using nonlinear models, but rely on severely limiting assumptions or estimations of the nonlinearity. [ 1 23 24 32 35 37 ] The unmixing methods presented in this document utilize the bidirectional reflectance distribution function to model microscopic mixtures. Using this model, along with the linear mixture model to incorpo rate macroscopic mixtures, the s e method s are able to accurately unmix hyperspectral images composed of both macroscopic and microscopic mixtures. The mixtures are estimated directly from the hyperspectral data without the need for a priori knowledge of th e mixture types. Quantitative r esults are presented using synthetic data set s, of macroscopic and microscopic mixtures, to demonstrate the increase d accuracy in unmixing using the s e new physics based method s over linear methods. In addition, results are presented using a well known laboratory data set Using these results, and other published results from this data set increased accuracy in unmixing over other nonlinear methods is shown. Qualitative results are presented using two different hypersp ectr al image scenes collected f r o m an airborne platform. The first method introduced is the DME algorithm T his algorithm explicitly uses the macroscopic and microscopic mixture models for estimating endmembers and their proportions Inclusion of these mixtu re models allows DME to unmix hyperspectral data in the presence of both macroscopic and microscopic mixtures Quantitative PAGE 125 125 experiments conducted on both synthetic and laboratory data indicate that DME is able to estimate endmembers and proportions with i ncreased accuracy over linear models, when both mixture types are present. Additionally, experiments on laboratory data indicate that DME estimates proportions of microscopically mixed data with more accuracy than existing methods for unmixing microscopic mixture data. A new model for hyperspectral image pixel s is introduced This model, the MMP, models a pixel as a mixture of mixtures. Specifically the M M P describes a hyperspectral pixel as a macroscopic mixture of endmembers and a microsco pic mixture. A new algorithm was designed around the MMP model the MPE algorithm Experiments utilizing the synthetic and laboratory data sets led to two conclusions The first i s that MPE is able to accurately estimate endmembers and proportions of MMP data sets a s well as data sets formed of macroscopic ally and microscopic ally mixed only pixels Secondly, in some cases MPE is able to estimate proportions of laboratory 100% microscopically mixed minerals more accurately than traditional microscopic unmixing method s. During the course of this research, many areas of future work became apparent. The first area of possible future work is to more accurately estimate the angles of incidence and emergence in an aerial hyperspectral collection. An approach to be investi gated is to use light detection and ranging ( LIDAR ) fused with the hyperspectral data to establish the geometry of the scattering surface to the light source and detector. The LIDAR provides 3 dimenstional spatial data that can be used to determine the at titude of the scattering surface. This extracted information, combined with metadata PAGE 126 126 respect to the sensor platform. Increased accuracy in estimating the angles of incid ence and emergence could result in increased accuracies in estimating endmembers and proportions of the collected data. A nother area of further research could be in endmember estimation. Endmember e stimation using DME and MPE is effective but slow in comp arison to other geometric methods which only account for macroscopic mixtures. Further research could look into using a parametric manifold based on the bidirectional reflectance distribution function to estimate endmembers. This type of approach to end member estimation could allow for the continued accuracy of endmember estimation shown with DME and MPE but with increased estimation speed. Also, the current models rely on the number of endmembers to be given as an input parameter. Adapting DME and MPE to use a Bayesian sparsity promoting prior, to mimic a SPICE framework, could allow the algorithms to estimate the number of endmembers autonomously. [ 4 ] Additionally, experiments could be run to further test the robustness of these methods to increased number of endmembers in the hyperspectral mixtures. A third area of follow on research c ould be to include multiple reflections and transmissions into the endmember and proportion estimation process. Thus, allowing the algorithms to account for another common mixture type. [ 14 16 19 ] The virtual endmember concept could be included into DME and MPE to allow microscopic mixtures between endmembers and virtual endmembers caused by multiple reflections and transmission between surfaces. [ 14 16 18 19 ] Experiments on MPE indicate increased accuracy over existing methods when estimating hyperspectral data composed of MMPs. However, th is document discusses PAGE 127 127 the inherent introduction of error by the ill posed inversion problem of estimating the microscopic proportions. Future research could look into methods of decreasing the error in estimating these proportions. One such method could be the use of sampling to allow for simultaneous estimation of the macroscopic and microscopic proportions. This simultaneous estimation could result in greater accuracy in estimating proportions. Another method could be to use spatial data inherent in h yperspectral images. By using spatial information and the assumption that mixtures proportions are similar in spatially close pixels, the microscopic proportions may be estimated with increased accuracy This research utilize s two laboratory data sets a nd two airborne hyperspectral data collection s However, the lack of real world collection data necessitate d the use of synthetic data sets. Future work should collect additional real world data sets of a variety of endmembers and collecting conditions. These additional data sets will allow for more experiments on the existing models but also further understanding of how the bidirectional reflectance distribution function impacts data unbeknownst to researchers unfamiliar with reflectance spectroscopy. Also, collection of data under differing conditions will allow for the inclusion of different reflectance models, e.g., a direction hemispherical reflectance distribution function. 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ASTER Spectral Library Available: speclib.jpl.nasa.gov/ PAGE 140 140 BIOGRAPHICAL SKETCH Ryan Close graduated from the Univer sity of Florida with a Bachelor of Science in c omputer s cience in December 2002 and a Master of Science degree in computer e ng ineering in December 2005. From 2005 to 2008 he worked at ManTech Real time Systems Laboratory as a staff software engineer. Ryan obtained the Doctor of Philosophy degree from the University of Florida in 2011. His research interest s include : machine l earning, automatic target recognition, remote sensing, and hyperspectral image analysis. 