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071c9b3ce359052d817401bebc3d7b5a02fc9574 95217 F20101222_AAAUOF kim_i_Page_23.jpg ce27dc17c5ab0dc30ba246cc55b56b28 9aaf230cfdaaf09df06d4c4fbf96ecb07a2f2d52 F20101222_AAAUJJ kim_i_Page_39.tif 2b569d8b06f69f85f3cb57a306b03c7b 518a745bd463f348cce7808a1bf31e1cab87ded8 57927 F20101222_AAAUOG kim_i_Page_25.jpg 4745d9bb89f46e0232155bdda876c7cb 5b0edbfcbfd6539bac2e487f5ce8f868c32f0457 IMAGE DENOISING USING HISTOGRAMBASED NOISE ESTIMATION By IGIL KIM A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF ENGINEER UNIVERSITY OF FLORIDA 2008 2008 IGil Kim To my family, especially my wife Jiyoung and little Julia ACKNOWLEDGMENTS I am deeply grateful to my mentor Dr. Paul W. Chun from the bottom of my heart. He always encouraged me and gave me valuable advice. I also offer my gratitude to Dr. Fred J. Taylor, my supervisory committee chair, for his direction and guidance. I would like to thank my committee members. Finally, I express my gratitude to my father, Dr. Koojin Kim. He instilled in me in the great love I have for the natural world and taught me to be curious about science. Special thanks go to my family and friends for their love, support, and sacrifices that allowed me to be me throughout my years of study. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ..............................................................................................................4 LIST OF TABLES ......... ..... .... ....................................................6 LIST OF A BBREV IA TION S ......... ............. .............................................................9 A B S T R A C T ........................................... ................................................................. 1 1 CHAPTER 1 INTRODUCTION ............... .......................................................... 13 Conventional Noise Level Estimation Methods ............... .............................................13 Conventional N oise R education M ethods.................................................................... ...... 14 2 BACKGROUNG AND RERATED REASEARCH.................................................15 Bosco's N oise Estim ation (2005, IEEE) ........................................ .......................... 15 Amer's Noise Level Estimation (2005, IEEE) .............. ..........................................16 Shin's Noise Level Estimation (2005, IEEE)...................................................................... 17 3 NOISE LEVEL ESTIMATION USING HISTOGRAM COMPRESSION ........................20 Drawbacks of Conventional Noise Level Estimation Methods...........................................20 Proposed Noise Level Estimation Method based on Histogram Compression...................22 4 NOISE REMOVAL FILTERING.......................................................... ...............34 B ilateral N oise R education Filtering .......................................................................... ....... 34 Im prove ent of B lateral Filtering ........................................................................... .... ... 36 Im pulse N oise R em oval .................. ...... ...... ............. ................. ............ 37 Performance Test of the Proposed Noise Reduction Method by PSNR .............. ..............38 5 C O N C L U S IO N .............................................................................. .......................... .. 4 0 APPENDIX: TEST IMAGES FOR PERFORMANCE COMPARISON.............. ................ 41 L IST O F R E F E R E N C E S .................................................................................... .....................43 B IO G R A PH IC A L SK E T C H .............................................................................. .....................45 LIST OF TABLES Table 61 PSNR for test im ages ......... ... .......... ...... .......... .. ........... ......... 39 page LIST OF FIGURES Figure page 21 Differences computation in homogeneous areas .................................... ............... 15 22 Absolute noise histogram .............................................................................. 16 23 Fine structure and texture im ages. ............................................. ............................ 17 2 4 S p ecial m ask s ........................................................................ 17 25 Shin's noise estim ation algorithm ......................................................................... ... ... 19 31 F our types of im age .................... ... ........ ................................................ .......... ..... .... 2 1 32 H om ogeneou s regions............................................................................. .....................22 33 C olor digital im age acquisition ............................................................... .....................23 34 Histogram compression on luminance of noisy image................................ ..............24 35 Relationship between ox and aw................ .....................................27 36 Graphical analysis of the histogram compression for four different types of noisy im ag es .......................................................... .................................. 2 8 37 Graphical analysis of the effect of histogram compression............... ......................29 38 H om ogeneous im age perform ance test.................................... ........................... ......... 30 39 Lesshomogeneous image performance test ......................................... ...............31 310 Complex Structure image performance test............................................. ...............31 311 Nonhomogeneous image performance test............................................. ...............32 312 Homogeneous and less homogeneous image performance test......................................33 313 Complex structure, and nonhomogeneous image performance test..............................33 41 Pixel selection for filtering based on a ........................................ ......................... 36 42 C central pixel outlier detection ......................................... .............................................37 43 Impulse noise reduction by using detecting central pixel outlier ....................................38 Ai The Kodak homogeneous and nonhomogeneous photo images....................................41 A2 Complex structure nonhomogeneous photo images............. .............. ............... 42 LIST OF ABBREVIATIONS b,, Nonoverlapping image block B Blue component of RGB color space B* Homogeneous block Ek Absolute difference error between true and estimated noise level f[k] Original signal A f[k] Estimated original signal by filtering g[k] Contaminated signal G Green component of RGB color space h[k, ] Local filter MSE Mean squared error n[k] Noise signal Ps Weight of the proposed method PSNR Peak signaltonoise ratio (dB) R Red component of RGB color space RGB RGB red, green, and blue color space W White Gaussian noise W[k, ] Weight of bilateral filter W [k, ] Space weight of bilateral filter WR [k, ;] Range weight of bilateral filter X Original noisefree image Y Noisy image Y, Luminance of YUV color space YUV Luminance and chrominance color space z mmin Z  , Histogram modified noisy image Standard deviation of intensity for each block bi Minimum standard deviation Standard deviation of signal X, Y, and W Neighboring point Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Engineer IMAGE DENOISING USING HISTOGRAMBASED NOISE ESTIMATION By IGil Kim August 2008 Chair: Fred J. Taylor Major: Electrical and Computer Engineering Exact noise level estimation is very useful in digital image processing. For example, some noise removal algorithms use a noise level estimation to adjust the aggressiveness of noise removal. If an estimated noise level is too low, too much noise will remain in the denoised image. If an estimated noise level is too high, the features of the original images will be removed from the denoised image. Accurate noise level estimation will produce better results in the restored image. Most conventional noise estimation methods are only focused on estimating the noise level from a noisy image including a lot of homogeneous (flat) regions. So, the typical methods induce excessive overestimation and underestimation if the given noisy image doesn't have any homogeneous regions. I propose a new noise level estimation method that uses histogram modification to find more exact noise level from the given noisy image. This new noise estimation method uses a proposed noise reduction filter based on a bilateral filter. It makes use of blockbased noise estimation, in which an input image is assumed to be contaminated by the additive white Gaussian noise and impulse noise. A noise reduction filtering process is performed by the filter bilateralbased. Coefficients of the noise filter are selected as functions of the standard deviation of the Gaussian noise that is estimated from a given noisy image by the proposed method. To accurately estimate the amount of noise in various types of noisy images, images are categorized into four different types based on their homogeneity: homogeneous, less homogeneous, complex structure, and nonhomogeneous images. In the proposed method, by using histogram modification on a noisy image, fluctuation in a nonhomogeneous region can be effectively suppressed without influencing its noise level. This process increases homogeneous regions in a given noisy image and improves the performance of noise estimation for most noisy images regardless of their homogeneity. To show the effectiveness of the proposed method, it is tested on various kinds of noisy images ranging from homogeneous to nonhomogeneous and its performance is compared with that of three conventional noise estimation methods. The proposed noise estimation and reduction method can be efficiently used in various image and videobased applications such as digital cameras and digital television and is superior because of its performance and simplicity. CHAPTER 1 INTRODUCTION Nowadays many people use camera phones and lowend digital cameras in their everyday life. These devices are particularly subject to noise reduction, when images are acquired in low lighting conditions. However these devices are often used in poor lightingconditions due to their portability. In order to obtain an acceptable picture, sometimes it is necessary to amplify the image signal taken under low light environments such as indoor scenes. However, when boosting an already degraded image signal, noise in the image also can be amplified. An effective noise reduction filter should change its strength according to the noise level in the noisy image. Here, we need some measure to estimate the noise level in the given noisy image. In general, we can get noise level information from the noise standard deviation a. Hence many noise filters depend on a to adaptively change their smoothing effects [1][4]. Conventional Noise Level Estimation Methods Conventional noise level estimation techniques can be categorized into three main classes [8]: Blockbased, filterbased, and adaptivebased methods. Before 1990, filterbased methods were commonly used. Such methods perform a prefiltering operation in which the noisy image is blurred to suppress image structure. A difference image is computed by subtracting the filtered image from the original one and then noise level is estimated using the difference image, which is assumed to contain only the noise signal. After 1990, blockbased methods started to play a role in noise level estimation. Briefly, the blockbased method partitions the image into a sequence of blocks. The estimation of standard deviation of a noisy image is carried out by the properly calculating the weighted noise level obtained by averaging the noise levels of the most homogeneous blocks. An adaptive method is a hybrid method which uses elements of the filter based and the blockbased methods. Its performance depends on the first noise level estimation. Efficient techniques based on the block based approach are described in [5] and [6]. A comparison of various methods for determining an approximation of the noise level in an image is given in [7]. Conventional Noise Reduction Methods A large number of different noise reduction methods have been proposed so far. Traditional denoising methods can be generalized into two main groups: spatial domain filtering and transform domain filtering. Spatial domain filtering methods have long been the mainstay of signal denoising and manipulate the noisy signal in a direct fashion. Conventional linear spatial filters like Gaussian filters try to suppress noise by smoothing the signal. While this works well in the situations where signal variation is low, such spatial filters result in undesirable blurring of the signal in situations where signal variation is high. To overcome these drawbacks, a number of new spatial filtering methods such as total variation techniques [9], anisotropic filtering techniques [10], and bilateral filtering techniques [11,12] have been introduced to suppress noise while preserving signal characteristics in the regions of high signal variation. Bilateral filtering is a nonlinear and noniterative filtering technique which utilizes both spatial and amplitudinal distances to preserve signal detail. On the other hand, transform domain filtering methods transform the noisy signal into the frequency domain and manipulate the frequency coefficients to suppress signal noise before the signal is transformed back into the spatial domain. These techniques had been introduced by Wiener filtering [13] and waveletbased methods [14,15] I propose novel approaches to noise level estimation and noise reduction that improve bilateral filtering. The proposed methods are robust in situations where the given noisy image is characterized by nonhomogeneity and provide improved noise suppression. CHAPTER 2 BACKGROUND AND RERATED RESEARCH In this section, I introduce three recent noise estimation algorithms, showing good performance in IEEE journal. Bosco's and Amer's method are blockbased approaches, and Shin's method is adaptive approach. These three conventional methods will be compared with my proposed method in chapters 3 and 4. Bosco's Noise Estimation (2005, IEEE) In most of blockbased noise estimation methods, noise level is obtained by averaging the noise levels of the most homogeneous blocks. In order to overcome inaccuracy of averaging, Bosco proposed new method using difference histogram approximation. The detail algorithm is as follows: 1. Tessellate an input noisy image into a number of 3x3(5x5) nonoverlapping image blocks 2. Compute absolute differences between the central pixel and its neighborhood for each blocks (Figure 21) 3. Select a block as a homogeneous block if the 8 differences are very small 4. Compute noise histogram for selected homogeneous blocks from accumulated differences (Figure 22) 5. Obtain noise level(a) by considering 68% of the noise samples. The index on the xaxis of the noise histogram allowing reaching the 68% of the noise samples Figure 21. Differences computation in homogeneous areas In Bosco's method, histogram approximation method is exploited instead of averaging local standard deviations usually used in conventional method for noise level estimation. In some of case, it provides better result than other conventional methods. The main weakness of this method is that it takes more time and storage to accumulate differences. Example: Noise Level Estimation for the Green Sample Color Channel i llliei 24 251 7 Figure 22. Absolute noise histogram. The index on the xaxis allowed to reach the 68% of the total samples represents the current noise level estimation. Amer's Noise Level Estimation (2005, IEEE) Selection of homogeneous areas in the given noisy image is very important in most of conventional noise estimation methods. Accuracy of blockbased noise level estimation highly relies on selecting homogeneous blocks. However, many conventional methods were found to have difficulties finding homogeneous region in noisy image containing fine structures or textures (Figure 23). Amer introduced a new approach to overcome this drawback using special masks to find homogeneous regions. The following steps describe the proposed method: 1. Tessellate an input noisy image into a number of 3 x3(5 x5) nonoverlapping image blocks 2. Use eight highpass operators and special masks for covers to stabilize the homogeneity estimation for each block. (Figure 24) 3. Compute summation of the absolute values of all eight quantities from step 2) 4. Select the block as a homogeneous block if the summation is less than threshold 5. Average local standard deviation of the selected homogeneous blocks Amer's method gives better noise estimation performance for the images which have texture and fine structure by using special masking if the noise level of noisy image is not too high. The number of detected homogeneous regions is rapidly decreased when given noise level is high. So it is hard to find homogeneous areas in this case. ,.~;ieu;; gw~aiaAS'F Figure 23. Fine structure and texture images. current pixel Figure 24. Special masks Shin's Noise Level Estimation (2005, IEEE) Shin's noise estimation algorithm is based on both blockbased and filteringbased approaches. It selects homogeneous blocks by a blockbased approach and then filters the mask 1 mask 2 mask 3 mask 4 mask 5 mask 6 mask 7 mask 8 H~ffl91 selected homogeneous blocks using a filteringbased approach. Blockbased methods require a low computational load whereas filteringbased approaches yield a stable estimate. Fig. 2 shows the flowchart of Shin's estimation noise estimation method using adaptive Gaussian filtering. Each step of Shin's method is described as follows. 1. Tessellate an input noisy image into a number of 3 x 16 nonoverlapping image blocks (b,). 2. Compute the standard deviation (a,) of intensity for each block bi and find the minimum standard deviation (amin). 3. Select homogeneous blocks (B*) whose standard deviations of intensity are close to Umin 4. Compute Gaussian filter coefficients (olst is set to omin) 5. Gaussian filtering on the selected blocks (B*). 6. Compute the standard deviation(U2nd) of the difference image between noisy and filtered images within selected blocks (B*), which gives the estimated noise standard deviation (< =U2nd) In general, blockbased approaches tend to overestimate the noise level in good quality images and underestimate it in highly noisy images. This underestimation can be compensated by adaptive Gaussian filtering of Shin's method. However, its performance is highly depend on the first estimated noise level (ua,t) and execution time is usually longer than other blockbased approaches because of the second filteringbased estimation. Noisy image I(i.j) Figure 25. Block diagram of Shin's noise estimation algorithm CHAPTER 3 NOISE LEVEL ESTIMATION USING HISTOGRAM COMPRESSION In this chapter I will address the drawbacks of conventional methods for noise level estimation in digital images degraded by noise and compare performances of each method with the proposed method for four types of noisy image. Drawbacks of Conventional Noise Level Estimation Methods Most conventional methods for estimating the noise standard deviation are generally based on the following steps: 1. Detect homogeneous regions in the noisy image (because fluctuations in flat areas, pixels are supposed to be due exclusively to noise). 2. Compute the local standard deviation in the detected homogeneous regions. 3. Repeat steps 1) and 2) until the whole image has been processed. 4. Finally estimate the noise level (or standard deviation of noise) by averaging the computed local standard deviations. This method has two major drawbacks. First, if there are no homogeneous regions or too small areas of flat region in a given noisy image, it is hard to detect the homogeneous regions. An insufficient number of detected homogeneous regions may result in overestimation or underestimation. This will induce wrong estimation for noise level. Second, the process of detecting homogeneous regions in a noisy image requires a high number of computations. The detection process for a nonhomogeneous image may be just time wasted work and the noise estimation based on such detection has a high probability of being wrong. Theses drawbacks originate from the fact that most conventional noise estimation methods are only focused on noisy images including a lot of homogeneous regions. In order to make this problem clear, I have categorized noisy images into four types as following: 1. Homogeneous image: image contains a lot of homogeneous regions. 2. Less homogeneous image: image contains very few homogeneous regions in a limited area. 3. Complex structure image: original noisefree image has very complex structure, so it is not easy to find homogeneous regions due to local fluctuations. 4. Nonhomogeneous image: there are no homogeneous regions in the given noisy image. A B C . D Figure 31. Four types of image. A) homogeneous image, B) less homogeneous image, C) complex structure image, D) nonhomogeneous image Figures 31 shows examples of four different types of image based on homogeneity. There are homogeneous regions in Figure 31 A) and B), but it is not easy to find homogeneous regions in Figure 31 C) and D) by using conventional noise estimation methods. Figure 32 shows homogeneous regions detected by Bosco[l] method and violet blocks in each image are the detected areas. A B C D Figure 32. Homogeneous regions. A) homogeneous image, B) less homogeneous image, C) complex structure image, D) nonhomogeneous image Proposed Noise Level Estimation Method based on Histogram Compression In this section the proposed noise estimation method is shown to overcome the drawbacks of conventional noise estimation methods by means of histogram compression on intensity of image that tries to exploit correlation among R, G, and B components of color image regardless of the number of homogeneous regions in the given noisy image. The best way to estimate an exact noise level from a given noisy image is to make the image homogeneous without influencing its noise deviation. In other words, if we can suppress deviation in the original noisefree image without changing deviation of noise in a noisy image, it is easy to find more exact noise levels from the suppressed noisy image. Nowadays most digital images are in color and each pixel of the image has red (R), green (G), and blue (B) components. Each RGB component in a noisy color image has a noise and the noise tends to have Gaussian distribution in a digital image acquisition system. Optical Axis Noise X 0l Y Color Filter X = 0.3Rx R channel: Rx + WR Y = 0.3(Rx + WR) + 0.5Gx G channel : Gx + WG + 0.5(Gx + WG) + 0.3Bx B channel : Bx + WB + 0.3(Bx + WB) where OWR = OWG = OWB Figure 33. Color Digital image Acquisition Figure 33 shows the structure of a current digital image acquisition system. Generally most common noise in digital image acquisition can be modeled by a white Gaussian noise with the same standard deviation ow. In order to reduce the correlation between the RGB components, a conversion from RGB to YUV color space as follows: Y, (Luminance) = 0.3R, + 0.6G, + 0.1B, U, (Chrominance) = B, Y, V, (Chrominance2) = R, Y, However, if component noise is uncorrelated in RGB space, it is correlated in YUV space. Also the component noise and RGB component are independent to each other, so they are uncorrelated in RGB space. A linear operation such as histogram modification in YUV space may affect uncorrected RGB components and correlated component noise. Histogram compression, one of several histogram modification methods, can be useful to suppress deviation of RGB component without affecting deviation of component noise. (a=0.01, b 128) Y aX+ b X 0.2Rx + 0.5Gx +0.3Bx 0.0, b 128) Y 0.2Ry + 0.5Gy +0.3By Figure 34. Histogram Compression on Luminance of Noisy Image. The histogram plots the number of pixels in the image (vertical axis) with a particular brightness value (horizontal axis) The noisy image and original noisefree image have luminance correlation among RGB components in YUV color space. X 0.2Rx + 0.5Gx + 0.3Bx Y=X+W (1) = 0.2Ry+ 0.5Gy+ 0.3By (2) = 0.2(Rx+ W) + 0.5(Gx+ WG) + 0.3(Bx+ WB) (3) where Xis original noisefree image and Yis a noisy image. Wis an additive white Gaussian noise. Because X and Ware independent of each other, variation of Y has the following relation to variation of X. 2 2 2 aY x + aW (4) If we do histogram compression on the luminance of noisy image Y Z = aYb, (5) a= Y Z (a)Y+ b Z=Ya = 0.2*Rz + 0.5*Gz + 0.3*Bz (6) = 0.2(Ry a) + 0.5(G a) + 0.3(By a) (7) where Z is compressed noisy image in YUV. From equation (6) and (7), the R component of Z, R, becomes Rz Ry a =Rx+ WR(1a)Y b = Rx+ WR(a)(X+ W) b = [Rx(a)X+b] + [WR(1a)(O.2WR+0.5WG+0.3WB)] If the histogram compression equation (5) has parameters a=0.01 and b=128, the term (1a) can be ignored as in the following equation: Rz z [RxX+128] + [0.8WRO.5WGO.3WB)] = P+ W (8) where P = RxX+128 and Wp 0.8WR0.5WG0.3WB. Rz can be divided into the RGB component term P and noise term Wp. From equation (5) and a=0.01, the variance relation between Z and Y becomes az2 = O.Olaz2 0 a2 << 2 (9) In other words, equation (9) means that the variation of Yis much higher than that of Z and also the deviations of R components of Y and Z have a similar relation. OR2 ORY2 (10) a2 + UP2 << RX2 + UW from equation (2),(3), and (8) ap2 + 0.64 U TR2+0.25 OUWG2+0.09 UaB2 << U2 + Ow White Gaussian noise in Figure 33 has the same standard deviation (aw = UwR = UwG = UWB), so equation (10) becomes ap2 + 0.98ow2 << ORX2 + ow2 The noise term aw2 can be eliminated by approximation op2 o o2 (11) The deviation relation of equation (8) is OR2 = 2 2+ O2 (12) Equations (11) and (12) show that the standard deviation of the R component is suppressed by histogram compression on luminance of a noisy image but the standard deviation is not suppressed. The numbers of homogeneous regions in a histogramcompressed noisy image are increased with remaining component noise. In order to see the effect of histogram compression more clearly, graphical analysis by using relations among ox, ay, and cw in equation (4) is helpful to understand the proposed method. Figure 34 shows the relation and the values of oy on each pixel of noisy image are distance from the origin by using the Pythagoras relationship between ax and oy. Each blue point shows the relative locations of local standard deviation ax, yo, and Cw of 3x3 blocks for R components. Even if ax and ow are unknown in a given noisy image, we can get the values of local standard deviation ax before the noise level cw is added to original noise free image. If blue points are close to the vertical axis, the local blocks of the points are homogeneous including homogeneous regions in the noisy image. There are a lot of homogeneous blocks in Figure 34 because the given noisy image is homogeneous. Hence, many blue points can be found near the vertical axis. image id=3, window size=5given std =15 .X S 1:1i 3l X1:1 X X :.i .X 1 3x3 block. .... Figure 35. Relationship between uxand aw. Each blue pixel shows relative locations of local standard deviation of 3 x 3 block. However, the result of this graphical analysis may be different for the four types of image based on homogeneity as categorized in the previous section. In the case of a nonhomogeneous image, most of the blue points masking be apart from the vertical axis. Figure 35 shows the graphical analysis for the for different types of image and the effect of histogram compression on R components. The blue points in a red box indicate that the local blocks of the points are homogeneous. In the case of complex structures and nonhomogeneous images, there are very few homogeneous blocks. So it is not easy to find homogeneous regions using conventional 0 1 :I ' l:l 40 50 i ..:0 ... .. noise estimation methods. However, after histogram compression, ox can be suppressed without influencing distribution of ow in all four types of images. The total number of points which are included in the red box in a complex structure and nonhomogeneous images is increased. In other words, homogeneous regions are increased in the given noisy images by histogram compression. Homogeneous tu ,.',... , Less homogeneous Complex Structure Nonhomogeneous A(% Histogram Histogram Histogram Histogram CompFigure 36. Graphical analysis of the histogram compression for four different types of noisy images. deviations before and after histogram compression for original noisefree image X, noisy image Y, and additive white Gaussian noise W separately. Figure 36 shows the results. It is apparent that can be increased y histogram compression even if there are very few such homogeneous regions Figure 36. Graphical analysis of the histogram compression for four different types of noisy images. in the given notherisy image. With the increased number of histogram compression is regions, it becomes easier deviato more exactly estimate the overam oppression for original noisefree image X, noisy image Y,ise level. and additive white Gaussian noise W separately. Figure 36 shows the results. It is apparent that the local standard deviation of the original noisefree image oa can be suppressed by histogram compression. Histogram compression doesn't affect the local standard deviation of white Gaussian noise. From two graphical analyses in Figure 35 and 36, we can see that homogeneous regions can be increased by histogram compression even if there are very few such homogeneous regions in the given noisy image. With the increased number of homogenous regions, it becomes easier to more exactly estimate the overall noise level. x x Figure 37. Effect of histogram Hstogram compression. Ir Chiompression n lmin e of noisy im e. W W' r 'ar i I . Ir Y Y Figure 37. Effect of histogram compression. The proposed noise level estimation method is based on the following steps: 1. Perform histogram compression on luminance of noisy image. 2. Detect homogeneous regions in the image gotten from the step (1): homogeneous blocks can be selected by the local standard deviation which is less than a threshold 3. Estimate noise level from the detected homogeneous regions: estimated noise level is calculated by averaging the local standard deviation of detected homogeneous blocks or finding the most frequent local standard deviation by using histogram approximation. Performance of the proposed noise level estimation method was tested by comparing the results with those obtained using other three noise estimation methods: Bosco[l], Amer[2], and Shin[3], introduced in chapter 2. 15  T10  S10 15 A 50 15 B Given standard deviation Grven standard deviation image id33. lndow sze5 .blk number=15606 15 n 0 15 15 C Given standard deviation Figure 38. Performance Test for Homogeneous Image A) Bosco's method, B) Amer's method, C) Shin's method, D) Proposed method Figure 37 illustrates the performance comparison for the homogeneous image shown in Figure 31. Values on the horizontal axis represent actual given noise levels (standard deviations) and those on the vertical axis represent estimated noise levels (standard deviations). Red line is an ideal case of noise estimation in which the estimated amount of noise is equal to the amount of noise actually added. Blue points represent actual estimated noise levels obtained by using each NLE method. Conventional noise level estimation methods show good performance because there are a lot of homogeneous regions. However, underestimation appears at high noise levels. The proposed method shows better performance at both low and high noise levels. For the lesshomogeneous image test shown in Figure 38, overestimation at low noise levels and underestimation at high noise levels using the conventional methods are induced. However, the proposed method still shows good performance. image id=13, window size=5, threshold =20 ,blk number =15606 10 10 5 10 5 A 12 15 Given standard deviation Given standard deviation 0  j:   image id=I3, window size=5,blk number =15606 n m= . m, : 20~~ i''*   S5 10 1 5 10 15 Given standard deviation A Given standard deviation B image id=13, window size5 blk number =1706 20 i .................  .... ..... ..... 16      ." 10 2    S .~".     I   S5 10 15 CL 1 . Given standard deviation C D : 1 Figure 39. Performance Test for Lesshomogeneous Image. A) Bosco's method, B) Amer's method, C) Shin's method, D) Proposed method image id=60, window s oe=5 sig max =20 blk number =9975 image id=6O, window size=5, threshold =25 ,blk number 9975 2 0     .    .. .. .. .. .. . 10 I7  0 5 10 15 5 10 1 Given standard deviation A Given standard deviation B image id=0, window size5 ,blk number =9975 ,=,0 ,, :.= 251   S..... .  ...  0 5 10 15 0 1, D Gven standard deviation 0v standard d i at 1 i Figure 310. Performance Test for Complex Structure Image. A) Bosco's method, B) Amer's method, C) Shin's method, D) Proposed method method, C) Shin s method, D) Proposed method image id=30, window size=5, threshold =25 blk number =15741 S 1      i    t"           o4 ...       SA B Given standard deviation Given standard deviation image td=30, window size=5 ,blk number =15741 image id=30, window size=5 ................ co ........  25 j  u252 .. .C D S5 10 1 Given standard deviation : 5 ie s Ia I d ,, ai n Figure 311. Performance Test for Nonhomogeneous Image. A) Bosco's method, B) Amer' method, C) Shin's method, D) Proposed method The proposed method proves particularly strong when applied to a complex structure or nonhomogeneous image as seen in Figure 39 and 310. In these cases, some of the conventional methods show less acceptable performance due to excessive overestimation and underestimation. Neither is a problem in using the proposed method. To evaluate the performance of the algorithm, the estimation Ek= OOe error is first calculated. Ek is the difference between the true and the estimated noise level. The average and 0 5 10 the standard deviation of the estimation error are then computed from all the measures. In Figure 311, the performance comparison is the actual test result for 24 Kodak homogeneous and less homogeneous photo images in Append Figure 3A) which are used as test images in image processing. The left plot A) in Figure 311 reveals that the estimation error using the proposed processing. The left plot A) in Figure 311 reveals that the estimation error using the proposed method is lower than that of other NLE methods for all noise levels. Interestingly, in the right plot B), the standard deviation of the error using the proposed method is significantly less. 45 1 l 8 arner amer / 4 ........... ..... bosco  16 ......  ...... bosco  proposed / proposed 35  4    sogang  S12  . ...  4   4  "           D 5 4     2 05 10 1 5 10 15 Given standard devation A Gien standard devation B Figure 312. Performance Test for Homogeneous and Less homogeneous Image. A) Mean of absolute error of difference between actual noise and estimated noise level, B) Standard deviation of error bosco 20 i ^=^ 2 proposed 0sogang 15     15 10    10( SG5 10 1 Given standard deviation Figure 313. Performance Test for Complex structure, and Nonhomogeneous Images. A) Mean of absolute error of difference between actual noise and estimated noise level, B) Standard deviation of error Figure 312 shows the performance comparison for complex structure and non homogeneous images in Appendix A (Figure A2) which can not be successfully tested in conventional noise estimation methods. The proposed method shows significantly better noise level estimation for less homogeneous, complex structure, and nonhomogeneous images, all of which remain problematic in the area of current image processing. CHAPTER 4 NOISE REMOVAL FILTERING The proposed noise level estimation method described in chapter 4 has been inserted in a noise reduction system for white Gaussian noise. Basically its performance depends on the estimation of standard deviation a for a noisy image. In order to remove noise based on the estimated noise level using the proposed method, a bilateral noise reduction filter is used because it is a nonlinear filtering technique which utilizes both spatial and amplitudinal distances to better preserve signal detail. Bilateral Noise Reduction Filtering Consider a 2D signal fthat has been degraded by a white Gaussian noise n. The contaminated signal g can be expressed as follows: g[k] f[k] + n[k] where k=(x,y) The goal of noise reduction from a noisy image is to suppress noise n and extract original noise free imageffrom noisy image g. In spatial filtering techniques, an estimate of f is obtained by applying a local filter h to g. f[k] = h[k, ] x g[k] In a conventional linear spatial filtering method, the local filter is defined based on spatial distances between the particular point in the signal at center pixel location (x,y) and its neighboring points. In the case of Gaussian filtering, the local filter is defined as in the following equation: h[k, ] = exp 4s where c represents a neighboring point. Such filters operate under the assumption that an amplitudinal variation within the neighborhood is small and that the noise signal has large amplitudinal variations. The noise signal can be suppressed by smoothing the signal over the local neighborhood. The problem of this assumption is that important signal detail is also characterized by large amplitudinal variation. Therefore, such filters may induce an undesirable blurring of signal detail. A simple and effective solution for this problem is to use bilateral filtering, which is firstly introduced by Tomasi et al. [11] and developed from the Bayesian approach by Elad [12]. In bilateral filtering, a local filter is defined based on a combination of the spatial distances and the amplitudinal distances between a center point at (x, y) and its neighboring points. This can be formulated as a product of two local filters. One is an enforcing spatial locality and the other is an enforcing amplitudinal locality. In the Gaussian case, the bilateral filter can be defined by the following equations: A7 N f[k]= N W[k,] g[k ] NW[k, ;]/ N ;=N W[k, = W, [k, ] WR[k, ] 2 W [k, ]= exp 2 2" W [Y[k]Y[k ;]2] WR[k,] = exp 2 The main advantage of defining the filter in this manner is that it allows for nonlinear filtering to enforce both spatial and amplitudinal locality at the same time. The estimated amplitude at a particular point is influenced by neighboring points if the neighboring points have similar amplitudes which are more than those with distant different amplitudes. This can reduce smoothing across signal regions which have large but consistent amplitudinal variations, thus it is better to preserve such signal detail. Furthermore, the normalization term of the above formulation helps the bilateral filter smooth away small amplitudinal differences associated with the noise in smooth regions. Improvement of Bilateral Filtering It was advantageous to improve the bilateral noise removal filtering to obtain a more robust detection of the outliers. The proposed method uses the estimated local standard deviation a. The bilateral filter averages only pixels that are similar to the central pixel in the filter mask. For example, assume that the mask is a 5x5 window including 9 valid pixels as seen in Figure 4 1. In order to improve the performance of bilateral filtering, two slightly different central pixels can be considered instead of P. Those are the obiased center Pa and P+a. An interval whose width is directly proportional to a is used to select the pixels that can be averaged safely. Only similar pixels are selected for filtering by the interval centered on P because it maximizes the number of selected pixels. ,. ,. ,. Pu P P+a Figure 41. Pixel Selection for Filtering Based on a Pixels are successively averaged using different weights depending on their spatial locality and locality. The final filtered pixel is obtained by the following equation: ZW [k, ;]x Px h[k,;]= N =N where W[k, represents the weight associated to thk,e th pixel. N N where W[k, gf] represents the weight associated to the th pixel. JL JL JL JL X J Impulse Noise Removal If the central pixel P is an outlier (defected by impulse noise), unfortunately there are no pixels similar to it. In this case the obiased centers of P in the previous section are not useful to improve bilteral filtering because there will be no pixels which are included by the corresponding intervals. outlier Min Max Po P P+a Figure 42. Central Pixel Outlier Detection Unfortunately, an outlier can be located in the center pixel of the filter mask in Figure 42. To overcome this problem, we need to consider the minimum (Min) and the maximum (Max) value which is contained in the neighbor pixels of the central pixel. These values are useful to determine whether the central element in the mask is correct or affected by impulse noise. We can determine that a center pixel may be an outlier if Po is greater than the maximum value in the neighbor pixels. After the center pixel is classified as defective, it should be replaced by a weighted average of the remaining neighbor pixels. Figure 43 shows the result of proposed impulse noise reduction and bilateral noise reduction in a noisy image including impulse noise and white Gaussian noise. Impulse noise is eliminated by using the proposed method with reduction of white Gaussian noise in the cropped and magnified part of a noisy image. Impulse noise reduction Figure 43. Impulse Noise Reduction by using Detecting Central Pixel Outlier Performance Test of the Proposed Noise Reduction Method by PSNR PSNR is a measurement of the similarity between two images. In PSNR higher numbers are better. If PSNR is higher than 30dB, it is hard to distinguish between the two images with the human eye. General PSNR formula is the following: I m1 n1 2 MSE = 1 Y I(i, j) K(i, j mn 0o j= MAX MAX PSNR = 10 *log,0(A 2 = 20 *logo( ) MSE VMSE where I and K are images compared to each other to show their similarity. The proposed method was applied to four different types of noisy image with different characteristics. Each test image is contaminated by white Gaussian noise with standard deviation of 5 and impulse noise. The PSNR of the restored image was measured for the proposed method as well as conventional noise estimation methods and general bilateral filtering method. A summary of the results is shown in Table 61. The proposed method achieves good PSNR gains over other methods for all of the test images. Furthermore, the proposed method achieves better PSNR gains over less homogeneous, complex structure, and nonhomogeneous noisy images. Table 61. PSNR for test images Method PSNR (DB) BOSCO AMER SHIN PROPOSED Image type Homogeneous 34.5325 34.5667 34.5790 34.5790 Less homogeneous 28.9874 28.9983 28.9283 29.2625 Complexstructure 29.2839 29.2047 29.1903 30.2863 Nonhomogeneous 27.5907 27.5440 27.5327 28.0638 CHAPTER 5 CONCLUSION The proposed noise level estimation and noise reduction method is valid for estimating the noise level in images affected by additive white Gaussian noise and impulse noise. Conventional noise estimation methods perform well only when applied to a homogeneous image. However, the proposed method can estimate more exact noise level from homogeneous to non homogeneous images and also shows good performance in noise reduction. It is particularly strong in estimating the noise level in complex structure and nonhomogeneous image. The proposed noise estimation method requires fewer computational resources than conventional methods because there is no special process required to detect homogeneous regions in a nonhomogeneous noisy image. Its parallelism structure would speed up performance on H/W implementation. APPENDIX A TEST IMAGES FOR PERFORMANCE COMPARISON mmm nmm r kodir1OT kodim02 kodim03 kodim04 kodim05 kodim07 kodimOB kodim09 kodiml0 kodim11 kodiml3 kodiml4 kodiml5 kodiml6 kodimi? kodim06 kodiml2 kodiml8 I .EmmE kodiml9 kodim20 kodim21 kodim22 kodim23 kodim24 Figure A1. 24 Kodak Homogeneous and Nonhomogeneous Photo Images Figure A2. Complex Structure Nonhomogeneous Photo Images ;r J 'F. r L LIST OF REFERENCES [1] J.Brailean, R.Kleihorst, S.Efstratiadis, A. Katsaggelos, R. Lagendijk, "Noise Reduction Filter for Dynamic Image Sequences: A Review", Proceedings of the IEEE, Vol.83, No.9, Sept. 1995 [2] S. Battiato, A.Bosco, M. Mancuso, G. Spampinato, "Adaptive Temporal Filtering for CFA Video Sequences", In Proceedings of IEEE ACIVS'02 Advanced Concepts for Intelligent Vision Systems 2002, pp. 1924, Ghent University, Belgium, September 2002 [3] A. Bosco, K. Findlater, S. Battiato, A. Castorina A Noise Reduction Filter for FullFrame Imaging Devices" IEEE Transactions on Consumer Electronics Vol. 49, Issue 3, August 2003 [4] A. Bosco, K. Findlater, S. Battiato, A. Castorina, "A Temporal Noise Reduction Filter Based on FullFrame Data Image Sensors" in Proceedings of IEEE ICCE 2003 [5] A. Amer, A. Mitiche, and E. Dubois, "Reliable and Fast StructureOriented Video Noise Estimation," in Proc. IEEE Int. Conf. Image Processing, Montreal, Quebec, Canada, Sep. 2002 [6] G. Messina, A. Bosco, A. Bruna, G. Spampinato, Fast Method for Noise Level Estimation and Integrated Noise Reduction, IEEE Transactions on Consumer Electronics, vol. 51, No. 3, pp. 1028 1033, August, 2005. [7] D.H. Shin, R.H Park, S. Yang, J.H. Jung, BlockBased Noise Estimation Using Adaptive Gaussian Filtering, in IEEE Transactions on Consumer Electronics, Vol. 51, No. 1, February, 2005 [8] S. I. Olsen, "Estimation of Noise in Images: An Evaluation," Graphical Models and Image Process., vol. 55, pp. 319323, July 1993 [9] L. Rudin, S. Osher, Total variation based image restoration with free local constraints, in: Proceedings of the IEEE ICIP, vol. 1, 1994, pp. 3135. [10] S. Greenberg, D. Kogan, Improved structureadaptive anisotropic filter, Pattern Recognition Lett. 27 (1) (2006) 5965. [11] C. Tomasi, R.Manduchi, Bilateral filtering for gray and color images, in: Proceedings of the ICCV, 1998, pp. 836846. [12] M. Elad, On the origin of the bilateral filter and ways to improve it, IEEE Trans. Image Process. 11 (10) (2002) 11411151. [13] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, Wiley, New York, 1949. [14] J. Portilla, V. Strela, M. Wainwright, E. Simoncelli, Image denoising using scale mixtures of Gaussians in the wavelet domain, IEEE Trans. Image Process. 12 (11) (2003) 13381351. [15] Q. Li, C. He, Application of wavelet threshold to image denoising, in: Proceedings of the ICICIC, vol. 2, 2006, pp. 693696. BIOGRAPHICAL SKETCH IGil Kim obtained his B.S. degree in the Department of Electronics Engineering from Hong Ik University, Korea, in 1999. After graduation from the university, he moved to the United States to pursue his graduate studies. He received his M.S degree in Electrical Engineering from University of Southern California in 2002. He served as a software engineer at the digital media R&D center in Samsung, Korea, from 2003 to 2005. He was admitted to the University of Florida in the fall of 2005 and has worked on numerous projects in the fields of image processing and pattern recognition in the department of electrical and computer engineering while completing his engineering degree. PAGE 1 1 IMAGE DENOISING USING HISTOGRAMBAS ED NOISE ESTIMATION By IGIL KIM A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORI DA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF ENGINEER UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 IGil Kim PAGE 3 3 To my family, especially my wife Jiyoung and little Julia PAGE 4 4 ACKNOWLEDGMENTS I am deeply grateful to my mentor Dr. Paul W. Chun from the bottom of my heart. He always encouraged me and gave me valuable advi ce. I also offer my gratitude to Dr. Fred J. Taylor, my supervisory committee chair, for hi s direction and guidance. I would like to thank my committee members. Finally, I express my gr atitude to my father, Dr. Koojin Kim. He instilled in me in the great love I have for th e natural world and taught me to be curious about science. Special thanks go to my family and fr iends for their love, support, and sacrifices that allowed me to be me throughout my years of study. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........6 LIST OF ABBREVIATIONS..........................................................................................................9 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION..................................................................................................................13 Conventional Noise Level Estimation Methods.....................................................................13 Conventional Noise Reduction Methods................................................................................14 2 BACKGROUNG AND RERATED REASEARCH..............................................................15 Boscos Noise Estimation (2005, IEEE)................................................................................15 Amers Noise Level Estimation (2005, IEEE).......................................................................16 Shins Noise Level Estimation (2005, IEEE).........................................................................17 3 NOISE LEVEL ESTIMATION USI NG HISTOGRAM COMPRESSION..........................20 Drawbacks of Conventional Nois e Level Estimation Methods..............................................20 Proposed Noise Level Estimation Met hod based on Histogram Compression......................22 4 NOISE REMOVAL FILTERING..........................................................................................34 Bilateral Noise Reduction Filtering........................................................................................34 Improvement of Bilateral Filtering.........................................................................................36 Impulse Noise Removal..........................................................................................................37 Performance Test of the Propos ed Noise Reduction Method by PSNR .................................38 5 CONCLUSION..................................................................................................................... ..40 APPENDIX: TEST IMAGES FOR PERFORMANCE COMPARISON.....................................41 LIST OF REFERENCES............................................................................................................. ..43 BIOGRAPHICAL SKETCH.........................................................................................................45 PAGE 6 6 LIST OF TABLES Table page 61 PSNR for test images....................................................................................................... ..39 PAGE 7 7 LIST OF FIGURES Figure page 21 Differences computation in homogeneous areas...............................................................15 22 Absolute noise histogram...................................................................................................16 23 Fine structure and texture images......................................................................................17 24 Special masks.............................................................................................................. .......17 25 Shins noise estimation algorithm......................................................................................19 31 Four types of image........................................................................................................ ...21 32 Homogene ous regions........................................................................................................22 33 Color digital image acquisition..........................................................................................23 34 Histogram compression on lu minance of noisy image.....................................................24 35 Relationship between X and W.........................................................................................27 36 Graphical analysis of th e histogram compression for four different types of noisy images......................................................................................................................... .......28 37 Graphical analysis of the e ffect of histogram compression...............................................29 38 Homogeneous image performance test..............................................................................30 39 Lesshomogeneous im age performance test......................................................................31 310 Complex Structure im age performance test.......................................................................31 311 Nonhomogeneous im age performance test.......................................................................32 312 Homogeneous and less homoge neous image performance test.........................................33 313 Complex structure, and nonhom ogeneous image performance test.................................33 41 Pixel selection for filtering based on ..............................................................................36 42 Central pixel outlier detection............................................................................................37 43 Impulse noise reduction by using detecting central pixel outlier......................................38 PAGE 8 8 A1 The Kodak homogeneous and nonhomogeneous photo images.......................................41 A2 Complex structure nonhomogeneous photo images.........................................................42 PAGE 9 9 LIST OF ABBREVIATIONS bij Nonoverlapping image block B Blue component of RGB color space B Homogeneous block Ek Absolute difference error between true and estimated noise level f[k] Original signal ] k [ f Estimated original signal by filtering g[k] Contaminated signal G Green component of RGB color space ] k [ h Local filter MSE Mean squared error n[k] Noise signal Ps Weight of the proposed method PSNR Peak signaltonoise ratio (dB) R Red component of RGB color space RGB RGB red, green, and blue color space W White Gaussian noise ] k [ W Weight of bilateral filter ] k [ Ws Space weight of bilateral filter ] k [ WR Range weight of bilateral filter X Original noisefree image Y Noisy image Yi Luminance of YUV color space YUV Luminance and chrominance color space PAGE 10 10 Z Histogram modified noisy image ij Standard deviation of intensity for each block bij min Minimum standard deviation W Y X, Standard deviation of signal X, Y, and W Neighboring point PAGE 11 11 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Engineer IMAGE DENOISING USING HISTOGR AMBASED NOISE ESTIMATION By IGil Kim August 2008 Chair: Fred J. Taylor Major: Electrical and Computer Engineering Exact noise level estimation is very useful in digital image processing. For example, some noise removal algorithms use a noise level esti mation to adjust the aggressiveness of noise removal. If an estimated noise level is too low, too much noise will remain in the denoised image. If an estimated noise level is too high, the featur es of the original images will be removed from the denoised image. Accurate noise level estimati on will produce better results in the restored image. Most conventional noise estimation methods ar e only focused on estimating the noise level from a noisy image including a lot of homogeneous (flat) regions. So, th e typical methods induce excessive overestimation and underestimation if the given noisy image doesnt have any homogeneous regions. I propose a new noise level estimation method that uses histogram modification to find more exact noise level from the given noisy im age. This new noise estimation method uses a proposed noise reduction filter ba sed on a bilateral filter. It ma kes use of blockbased noise estimation, in which an input image is assumed to be contaminated by the additive white Gaussian noise and impulse noise. A noise reduct ion filtering process is performed by the filter bilateralbased. Coefficients of the noise filter ar e selected as functions of the standard deviation PAGE 12 12 of the Gaussian noise that is estimated from a given noisy image by the proposed method. To accurately estimate the amount of noise in various types of noisy images, images are categorized into four different types based on their hom ogeneity: homogeneous, less homogeneous, complex structure, and nonhomogeneous images. In the proposed method, by using histogram modification on a noisy image, fluctuation in a nonhomogeneous region can be effectively suppressed without influencing its noise level. Th is process increases ho mogeneous regions in a given noisy image and improves the performance of noise estimation for most noisy images regardless of their homogeneity. To show the effectiveness of the proposed method, it is tested on va rious kinds of noisy images ranging from homogeneous to nonhomogeneous and its performance is compared with that of three conventional noise estimation methods. The proposed noise estimation and reduction method can be efficiently used in variou s image and videobased applications such as digital cameras and digital television and is supe rior because of its performance and simplicity. PAGE 13 13 CHAPTER 1 INTRODUCTION Nowadays many people use camera phones and lo wend digital cameras in their everyday life. These devices are particularly subject to noise reduction, when images are acquired in lowlighting conditions. However these devices are often used in poor lightingconditions due to their portability. In order to obtain an acceptable picture, sometimes it is necessary to amplify the image signal taken under low light environments su ch as indoor scenes. However, when boosting an already degraded image signal, noise in the image also can be amplified. An effective noise reduction filter should change its strength accordi ng to the noise level in the noisy image. Here, we need some measure to estimate the noise leve l in the given noisy image. In general, we can get noise level information from the noise standard deviation Hence many noise filters depend on to adaptively change their smoothing effects [1][4]. Conventional Noise Level Estimation Methods Conventional noise level estima tion techniques can be categoriz ed into three main classes [8]: Blockbased, filterbased, and adaptivebased methods. Before 1990, filterbased methods were commonly used. Such methods perform a prefiltering operation in which the noisy image is blurred to suppress image struct ure. A difference image is comput ed by subtractin g the filtered image from the original one and then noise leve l is estimated using the difference image, which is assumed to contain only the noise signal. After 1990, blockbased met hods started to play a role in noise level estimation. Briefly, the bl ockbased method partitions the image into a sequence of blocks. The estimation of standard deviation of a noisy image is carried out by the properly calculating the weighted no ise level obtained by averaging the noise levels of the most homogeneous blocks. An adaptive method is a hybr id method which uses elements of the filterbased and the blockbased methods Its performance depends on the first noise level estimation. PAGE 14 14 Efficient techniques based on the block based approach are describe d in [5] and [6]. A comparison of various methods for determining an approximation of the noise level in an image is given in [7]. Conventional Noise Reduction Methods A large number of different noise reducti on methods have been proposed so far. Traditional denoising methods can be generalized into two main groups: spatial domain filtering and transform domain filtering. Spatial domain filte ring methods have long been the mainstay of signal denoising and manipulate the noisy signal in a direct fashi on. Conventional linear spatial filters like Gaussian filters try to suppress noi se by smoothing the signal. While this works well in the situations where signal variation is low, su ch spatial filters result in undesirable blurring of the signal in situations where signal variation is high. To overc ome these drawbacks, a number of new spatial filtering methods such as tota l variation techniques [9], anisotropic filtering techniques [10], and bilateral f iltering techniques [11,12] have been introduced to suppress noise while preserving signal characterist ics in the regions of high signal variation. Bilate ral filtering is a nonlinear and noniterative filtering techniqu e which utilizes both spatial and amplitudinal distances to preserve signal detail. On th e other hand, transform domain filtering methods transform the noisy signal into the frequency do main and manipulate the frequency coefficients to suppress signal noise before the signal is transformed back into the spatial domain. These techniques had been introduced by Wiener f iltering [13] and waveletbased methods [14,15] I propose novel approaches to noise level estimation and noise reduction that improve bilateral filtering. The proposed methods are robust in situations where the given noisy image is characterized by nonhomogeneity and pr ovide improved noise suppression. PAGE 15 15 CHAPTER 2 BACKGROUNG AND RERATED REASEARCH In this section, I introduce three recent noise estima tion algorithms, showing good performance in IEEE journal. Boscos and Am ers method are blockbased approaches, and Shins method is adaptive approach. These thr ee conventional methods will be compared with my proposed method in chapters 3 and 4. Boscos Noise Estimation (2005, IEEE) In most of blockbased noise estimation met hods, noise level is obtai ned by averaging the noise levels of the most homogeneous blocks. In order to overcome inaccuracy of averaging, Bosco proposed new method using difference hist ogram approximation. The detail algorithm is as follows: 1. Tessellate an input noisy image into a numbe r of 3(5) nonoverlapping image blocks 2. Compute absolute differences between the centr al pixel and its neighborhood for each blocks (Figure 21) 3. Select a block as a homogeneous block if the 8 differences are very small 4. Compute noise histogram for selected homoge neous blocks from accumulated differences (Figure 22) 5. Obtain noise level( ) by considering 68% of the noise sa mples. The index on the xaxis of the noise histogram allowing reaching the 68% of the noise samples Figure 21. Differences com putation in homogeneous areas PAGE 16 16 In Boscos method, histogram approximation me thod is exploited instead of averaging local standard deviations usually used in conventional method for noise level estimation. In some of case, it provides bette r result than other conv entional methods. The main weakness of this method is that it takes more time an d storage to accumulate differences. Figure 22. Absolute noise histogram. The index on the xaxis allowed to reach the 68% of the total samples represents the current noise level estimation. Amers Noise Level Estimation (2005, IEEE) Selection of homogeneous areas in the given noisy image is very important in most of conventional noise estimation methods. Accuracy of blockbased noise level estimation highly relies on selecting homogeneous blocks. Howeve r, many conventional methods were found to have difficulties finding homoge neous region in noisy image c ontaining fine structures or textures (Figure 23). Amer introduced a new appr oach to overcome this drawback using special masks to find homogeneous regions. The follo wing steps describe the proposed method: 1. Tessellate an input noisy image into a number of 3 3(5 5) nonoverlapping image blocks 2. Use eight highpass operators and special mask s for corners to stabilize the homogeneity estimation for each block. (Figure 24) PAGE 17 17 3. Compute summation of the absolute values of all eight quantities from step 2) 4. Select the block as a homogeneous block if the summation is less than threshold 5. Average local standard deviation of the selected homogeneous blocks Amers method gives better noise estimation perfor mance for the images which have texture and fine structure by using special masking if the noi se level of noisy image is not too high. The number of detected homogeneous re gions is rapidly decreased when given noise level is high. So it is hard to find homogeneous areas in this case. Figure 23. Fine struct ure and texture images. Figure 24. Special masks Shins Noise Level Estimation (2005, IEEE) Shins noise estimation algorithm is based on both blockbased and filteringbased approaches. It selects homogeneous blocks by a blockbased approach and then filters the PAGE 18 18 selected homogeneous blocks using a filteringbased approach. Blockbased methods require a low computational load whereas filteringbased approaches yield a stable estimate. Fig. 2 shows the flowchart of Shins estima tion noise estimation method using adaptive Gaussian filtering. Each step of Shins method is described as follows. 1. Tessellate an input noisy image into a number of 3 nonoverlapping image blocks (bij). 2. Compute the standard deviation (ij) of intensity for each block bij and find the minimum standard deviation (min). 3. Select homogeneous blocks (B*) whose standard deviations of intensity are close to min 4. Compute Gaussian filt er coefficients (1st is set to min) 5. Gaussian filtering on th e selected blocks (B*). 6. Compute the standard deviation(2nd) of the difference image between noisy and filtered images within selected blocks (B*), which gives the estimated noise standard deviation (n =2nd) In general, blockbased approaches tend to overestimate the nois e level in good quality images and underestimate it in highly noisy im ages. This underestimation can be compensated by adaptive Gaussian filtering of Shins method. However, its performance is highly depend on the first estimated noise level (1st) and execution time is usually longer than other blockbased approaches because of the sec ond filteringbased estimation. PAGE 19 19 Figure 25. Block diagram of Sh ins noise estimation algorithm PAGE 20 20 CHAPTER 3 NOISE LEVEL ESTIMATION USING HISTOGRAM COMPRESSION In this chapter I will address the drawb acks of conventional methods for noise level estimation in digital images degraded by noise and compare performances of each method with the proposed method for four types of noisy image. Drawbacks of Conventional Noise Level Estimation Methods Most conventional methods for estimating the noi se standard deviation are generally based on the following steps: 1. Detect homogeneous regions in the noisy image (becau se fluctuations in flat areas, pixels are supposed to be due exclusively to noise). 2. Compute the local standard deviation in the detected homogeneous regions. 3. Repeat steps 1) and 2) until the whole image has been processed. 4. Finally estimate the noise level (or standard deviation of noise) by averaging the computed local standard deviations. This method has two major drawbacks. First, if there are no homogeneous regions or too small areas of flat region in a given noisy imag e, it is hard to detect the homogeneous regions. An insufficient number of detected homogeneous regions may result in overestimation or underestimation. This will induce wrong estimation for noise level. Seco nd, the process of detecting homogeneous regions in a noisy image requires a high number of computations. The detection process for a nonhomogeneous image ma y be just time wasted work and the noise estimation based on such detection has a high pr obability of being wrong. Theses drawbacks originate from the fact that most conventi onal noise estimation methods are only focused on noisy images including a lot of homogeneous regions. In order to make this problem clear, I have categorized noisy images into four types as following: 1. Homogeneous image: image contains a lot of homogeneous regions. PAGE 21 21 2. Less homogeneous image: image contains very few homogeneous regions in a limited area. 3. Complex structure image: original noisefree im age has very complex st ructure, so it is not easy to find homogeneous regions due to local fluctuations. 4. Nonhomogeneous image: there are no homoge neous regions in the given noisy image. A B C D Figure 31. Four types of imag e. A) homogeneous image, B) less homogeneous image, C) complex structure image, D) nonhomogeneous image Figures 31 shows examples of four diffe rent types of image based on homogeneity. There are homogeneous regions in Figure 31 A) and B), but it is not easy to find homogeneous regions in Figure 31 C) and D) by using conventional noise estimation methods. Figure 32 shows homogeneous regions detected by Bosco[1] method and violet blocks in each image are the detected areas. PAGE 22 22 A B C D Figure 32. Homogeneous regions A) homogeneous image, B) less homogeneous image, C) complex structure image, D) nonhomogeneous image Proposed Noise Level Estimation Met hod based on Histogram Compression In this section the propos ed noise estimation method is shown to overcome the drawbacks of conventional noise estimation methods by means of histogram compression on intensity of image that tries to exploit correlation among R, G, and B components of color image regardless of the number of homogeneous regions in the given noisy image. The best way to estimate an exact noise leve l from a given noisy image is to make the image homogeneous without influencing its noise deviation. In other words, if we can suppress deviation in the original noisef ree image without changing deviati on of noise in a noisy image, it is easy to find more exact noise leve ls from the suppressed noisy image. PAGE 23 23 Nowadays most digital images are in co lor and each pixel of the image has red (R), green (G), and blue (B) components. Each RGB component in a noisy color image has a noise and the noise tends to have Gaussian distribution in a digital image acquisition system. Figure 33. Color Digital image Acquisition Figure 33 shows the structure of a current digital image acquisition system. Generally most common noise in digital im age acquisition can be modeled by a white Gaussian noise with the same standard deviation W. In order to reduce th e correlation between the RGB components, a conversion from RGB to YUV color space as follows: Yi (Luminance) = 0.3Ri + 0.6Gi + 0.1Bi Ui (Chrominance1) = Bi Yi Vi (Chrominance2) = Ri Yi However, if component noise is uncorrelated in RGB space, it is correlated in YUV space. Also the component noise and RGB component are independent to each other, so they are Noise R channel : RX + WR G channel : GX + WG B channel : BX + WB where WR = WG = WB X Y Y = 0.3(RX + WR) + 0.5(GX + WG) + 0.3(BX + WB) X = 0.3RX + 0.5GX + 0.3BX PAGE 24 24 uncorrelated in RGB space. A linear operation such as histogram modification in YUV space may affect uncorrected RGB components and correlated com ponent noise. Histogram compression, one of several histogram modification methods can be useful to suppress deviation of RGB component without affecting de viation of component noise. Figure 34. Histogram Compressi on on Luminance of Noisy Imag e. The histogram plots the number of pixels in the image (vertical axis) with a particular brightness value (horizontal axis) The noisy image and original noisefree image have luminance correlation among RGB components in YUV color space. X = 0.2RX + 0.5GX + 0.3BX Y = X + W (1) = 0.2RY+0.5GY+0.3BY (2) = 0.2(RX+WR) + 0.5(GX+WG) + 0.3(BX+WB) (3) where X is original noisefree image and Y is a noisy image. W is an additive white Gaussian noise. Because X and W are independent of each other, variation of Y has the following relation to variation of X. Y 2 = X 2 + W 2 (4) If we do histogram compression on the luminance of noisy image Y Z = aYb, (5) Y = 0.2RY + 0.5GY +0.3BYX = 0.2R X + 0.5G X +0.3B X Y = aX + b (a=0.01, b = 128) PAGE 25 25 = Y Z = (1a)Y + b Z = Y = 0.2*RZ + 0.5*GZ + 0.3*BZ (6) = 0.2(RY) + 0.5(GY) + 0.3(BY) (7) where Z is compressed noisy image in YUV. From equation (6) and (7), the R component of Z, Rz becomes RZ = RY= RX+WR(1a)Y b = RX+WR(1a)(X+W) b = [RX(1a)X+b] + [WR(1a)(0.2WR+0.5WG+0.3WB)] If the histogram compression equation (5) has parameters a=0.01 and b=128, the term (1a) can be ignored as in the following equation: RZ [RXX+128] + [0.8WR0.5WG0.3WB)] = P + Wp (8) where P = RXX+128 and Wp = 0.8WR0.5WG0.3WB. Rz can be divided into the RGB component term P and noise term Wp. From equation (5) and a=0.01, the variance relation between Z and Y becomes z 2 = 0.01 zY 2 z 2 << Y 2 (9) In other words, equation (9) means that the variation of Y is much higher than that of Z and also the deviations of R components of Y and Z have a similar relation. RZ 2 << RY 2 (10) P 2 + WP 2 << RX 2 + W 2 from equation (2),(3), and (8) P 2 + 0.64 WR 2+0.25 WG 2+0.09 WB 2 << RX 2 + W 2 PAGE 26 26 White Gaussian noise in Figure 33 has the same standard deviation (W = WR = WG = WB), so equation (10) becomes P 2 + 0.98 W 2 << RX 2 + W 2 The noise term W 2 can be eliminated by approximation P 2 << RX 2 (11) The deviation relation of equation (8) is RZ 2 = P 2 + W 2 (12) Equations (11) and (12) show that the standard deviation of the R component is suppressed by histogram compression on luminance of a noisy image but the standard deviation is not suppressed. The numbers of homogeneous region s in a histogramcompressed noisy image are increased with remaining component noise. In order to see the effect of histogram compression more cl early, graphical analysis by using relations among X, Y, and W in equation (4) is helpful to understand the proposed method. Figure 34 shows the relation and the values of Y on each pixel of noisy image are distance from the origin by using th e Pythagoras relationship between X and Y Each blue point shows the relative locations of local standard deviation X, Y, and W of 3x3 blocks for R components. Even if X and W are unknown in a given noisy imag e, we can get the values of local standard deviation X before the noise level W is added to original noise free image. If blue points are close to the vertical axis, the loca l blocks of the points are homogeneous including homogeneous regions in the noisy image. There are a lot of ho mogeneous blocks in Figure 34 because the given noisy image is homogeneous. He nce, many blue points can be found near the vertical axis. PAGE 27 27 Figure 35. Relationship between X and W Each blue pixel shows relative locations of local standard deviation of 3 x 3 block. However, the result of this graphical analysis may be differe nt for the four types of image based on homogeneity as categorized in the previ ous section. In the case of a nonhomogeneous image, most of the blue points may be apart from the vertical axis. Figure 35 shows the graphical analysis for the four different types of image and the effect of histogram compression on R components. The blue points in a red box indi cate that the local blocks of the points are homogeneous. In the case of complex structures and nonhomogeneous images, there are very few homogeneous blocks. So it is not easy to find homogeneous regi ons using conventional noise estimation methods. However, after histogram compression, X can be suppressed without influencing distribution of W in all four types of images. Th e total number of points which are included in the red box in a complex structure and nonhomogeneous images is increased. In other words, homogeneous regions are increas ed in the given noisy images by histogram compression. x w Y 3x3 block masking for getting local Y PAGE 28 28 Figure 36. Graphical analysis of the histogram compression for four different types of noisy images. Another way to show the effect of histogr am compression is to compare local standard deviations before and after histogram compression for original noisefree image X, noisy image Y, and additive white Gaussian noise W separately. Figure 36 shows the results. It is apparent that the local standard deviation of the original noisefree image X can be suppressed by histogram compression. Histogram compression doesnt affect the local standard deviation of white Gaussian noise. From two graphical analyses in Figure 35 and 36, we can see that homogeneous regions can be increased by histogram compression even if there are very few such homogeneous regions in the given noisy image. With the increased number of homoge nous regions, it becomes easier to more exactly estimate the overall noise level. Histogram Compression Suppressed Suppressed Suppressed Suppressed Histogram Compression Histogram Compression Histogram CompressionHomogeneous Less homogeneous Complex Structure Nonhomogeneous PAGE 29 29 Figure 37. Effect of histogram compression. The proposed noise level estimation me thod is based on the following steps: 1. Perform histogram compression on luminance of noisy image. 2. Detect homogeneous regions in the image gotten from the step (1): homogeneous blocks can be selected by the local standard devi ation which is less than a threshold 3. Estimate noise level from the detected homoge neous regions: estimated noise level is calculated by averaging the local standard de viation of detected homogeneous blocks or finding the most frequent local standard deviation by using histogram approximation. Performance of the proposed noise level estim ation method was tested by comparing the results with those obtained using other three noise estimation methods: Bosco[1], Amer[2], and Shin[3], introduced in chapter 2. W W X X Y Y Histogram CompressionX W Y X W Y PAGE 30 30 A B C D Figure 38. Performance Test for Homogeneous Image A) Boscos method, B) Amers method, C) Shins method, D) Proposed method Figure 37 illustrates the performance co mparison for the homogeneous image shown in Figure 31. Values on the horizontal axis represent actual given noise levels (standard deviations) and those on the vertical axis represent estimate d noise levels (standard deviations). Red line is an ideal case of noise estimation in which the es timated amount of noise is equal to the amount of noise actually added. Blue poi nts represent actual estimated noise levels obtained by using each NLE method. Conventional noise level estimation methods show good performance because there are a lot of homogeneous regions. However, underestimation appears at high noise levels. The proposed method shows better perfor mance at both low and high noise levels. For the lesshomogeneous image test shown in Figure 38, overestimation at low noise levels and underestimation at high noise leve ls using the conventional methods are induced. However, the proposed method still shows good performance. PAGE 31 31 A B C D Figure 39. Performance Test for Lesshomoge neous Image. A) Boscos method, B) Amers method, C) Shins method, D) Proposed method A B C D Figure 310. Performance Test for Complex Structure Image A) Boscos method, B) Amers method, C) Shins method, D) Proposed method PAGE 32 32 A B C D Figure 311. Performance Test for Nonhomogeneous Image A) Boscos method, B) Amers method, C) Shins method, D) Proposed method The proposed method proves particularly strong when applied to a complex structure or nonhomogeneous image as seen in Figure 39 and 310. In these cases, some of the conventional methods show less acceptable perf ormance due to excessive overestimation and underestimation. Neither is a proble m in using the proposed method. To evaluate the performance of the algorithm, the estimation Ek=ne error is first calculated. Ek is the difference between the true and th e estimated noise level. The average and the standard deviation of the es timation error are then computed from all the measures. In Figure 311, the performance comparison is the actual test result for 24 Kodak homogeneous and less homogeneous photo images in Appendix A ( Figure A1) which are used as test images in image processing. The left plot A) in Figure 311 reveals that the es timation error using the proposed PAGE 33 33 method is lower than that of other NLE methods for all noise levels. Interestingly, in the right plot B), the standard deviati on of the error using the proposed method is significantly less. A B Figure 312. Performance Test for Homogeneous and Less hom ogeneous Image. A) Mean of absolute error of difference between actua l noise and estimated noise level, B) Standard deviation of error A B Figure 313. Performance Test for Complex structure, and Nonhomogeneous Images. A) Mean of absolute error of difference between act ual noise and estimated noise level, B) Standard deviation of error Figure 312 shows the performance comp arison for complex structure and nonhomogeneous images in Appendix A ( Figure A2) which can not be successfully tested in conventional noise estimation methods. The prop osed method shows significantly better noise level estimation for less homogeneous, complex st ructure, and nonhomogeneous images, all of which remain problematic in the area of current image processing. PAGE 34 34 CHAPTER 4 NOISE REMOVAL FILTERING The proposed noise level estimation method descri bed in chapter 4 has been inserted in a noise reduction system for white Gaussian noi se. Basically its performance depends on the estimation of standard deviation for a noisy image. In order to remove noise based on the estimated noise level using the proposed method, a b ilateral noise reduction filter is used because it is a nonlinear filtering technique which uti lizes both spatial and amplitudinal distances to better preserve signal detail. Bilateral Noise Reduction Filtering Consider a 2D signal f that has been degraded by a white Gaussian noise n. The contaminated signal g can be expressed as follows: g[k] = f[k] + n[k] where k=(x,y) The goal of noise reduction from a noisy image is to suppress noise n and extract original noise free image f from noisy image g. In spatial filtering techni ques, an estimate of f is obtained by applying a lo cal filter h to g. f[k] = h[k, ] g[k] In a conventional linear spatial filtering method, the local filter is defined based on spatial distances between the particular point in the signal at center pixel location (x,y) and its neighboring points. In the case of Gaussian filtering, the local filte r is defined as in the following equation: where represents a neighboring point. Such filter s operate under the assumption that an amplitudinal variation within the neighborhood is small and that the noise signal has large amplitudinal variations. The noise signal can be suppressed by smoothi ng the signal over the 2 s 22 exp ] k [ h PAGE 35 35 local neighborhood. The problem of this assumpti on is that important si gnal detail is also characterized by large amplitudina l variation. Therefore, such f ilters may induce an undesirable blurring of signal detail. A simple and effective solution for this problem is to use bilateral filtering, which is firstly introduced by Tomasi et al. [11] and devel oped from the Bayesian approach by Elad [12]. In bilateral filtering, a local fi lter is defined based on a combin ation of the spatial distances and the amplitudinal distances between a center point at (x, y) and its neighboring points. This can be formulated as a product of two local filters One is an enforcing sp atial locality and the other is an enforcing amplitudinal locality. In th e Gaussian case, the bilateral filter can be defined by the following equations: ] k [ W ] k [ W ] k [ WR S The main advantage of defining the filter in this manner is that it allows for nonlinear filtering to enforce both spatial and amplitudinal locality at the same time. The estimated amplitude at a particular point is influenced by neighboring points if the neighboring points have similar amplitudes which are more than t hose with distant different amplitudes. This can reduce smoothing across signal regi ons which have large but consistent amplitudinal variations, thus it is better to pr eserve such signal detail. Furthermore, the N N N N] k [ g ] k [ W ] k [ W 1 ] k [ f 2 R 2 R2 ] ] k [ Y ] k [ Y [ exp ] k [ W 2 s 2 s2 exp ] k [ W PAGE 36 36 normalization term of the above formulation he lps the bilateral filter smooth away small amplitudinal differences associated with the noise in smooth regions. Improvement of Bilateral Filtering It was advantageous to improve the bilatera l noise removal filtering to obtain a more robust detection of the outliers. The proposed method uses the esti mated local standard deviation The bilateral filter averages onl y pixels that are similar to the central pixel in the filter mask. For example, assume that the mask is a 5x5 window including 9 valid pixels as seen in Figure 41. In order to improve the performance of bilatera l filtering, two slightly different central pixels can be considered instead of P. Those are the biased center Pand P+ An interval whose width is directly proportional to is used to select the pixels th at can be averaged safely. Only similar pixels are selected for filtering by the interval centered on P because it maximizes the number of selected pixels. Figure 41. Pixel Selection for Filtering Based on Pixels are successively averaged using differe nt weights depending on th eir spatial locality and locality. The final filtered pixel is obtained by the following equation: N N N N] k [ W P ] k [ W ] k [ h where W[k,] represents the weight associated to the th pixel. P P P+ PAGE 37 37 Impulse Noise Removal If the central pixel P is an outlier (defected by impulse noise), unfortunately there are no pixels similar to it. In this case the biased centers of P in the previous secti on are not useful to improve bilteral filtering because there will be no pixels which are included by the corresponding intervals. Figure 42. Central Pixel Outlier Detection Unfortunately, an outlier can be located in th e center pixel of the filter mask in Figure 42. To overcome this problem, we need to consider the minimum (Min) and the maximum (Max) value which is contained in the ne ighbor pixels of the central pixel. These va lues are useful to determine whether the central elem ent in the mask is correct or affected by impulse noise. We can determine that a center pixel may be an outlier if Pis greater than the maximum value in the neighbor pixels. After the center pixel is classified as defec tive, it should be replaced by a weighted average of the remaining neighbor pixels. Figure 43 shows the result of proposed impul se noise reduction and bilateral noise reduction in a noisy image including impulse noise and white Gaussian noise. Impulse noise is eliminated by using the proposed method with re duction of white Gaussian noise in the cropped and magnified part of a noisy image. outlier Min Max PP P+ PAGE 38 38 Figure 43. Impulse Noise Reduction by using Detecting Central Pixel Outlier Performance Test of the Proposed Noise Reduction Method by PSNR PSNR is a measurement of the similarity between two images. In PSNR higher numbers are better. If PSNR is higher than 30dB, it is hard to distinguish between the two images with the human eye. General PSNR formula is the following: where I and K are images compared to each other to show their similarity. The proposed method was applied to four diffe rent types of noisy image with different characteristics. Each test image is contaminated by white Gaussian noise with standard deviation of 5 and impulse noise. The PSNR of the restored image was m easured for the proposed method as well as conventional noise estimation methods and genera l bilateral filtering method. A summary of the results is shown in Table 61. The proposed method achieves good PSNR gains over other methods for all of the test images. Furthermore, the proposed method achieves better PSNR gains over less homogeneous, complex st ructure, and nonhomogeneous noisy images. Impulse noise reduction 2 1 m 0 i 1 n 0 jj) K(i, j) I(i, mn 1 MSE ) MSE MAX ( log 20 ) MSE MAX ( log 10 PSNRI 10 2 I 10 PAGE 39 39 Table 61. PSNR for test images PSNR (DB) Method Image type BOSCO AMER SHIN PROPOSED Homogeneous 34.5325 34.5667 34.5790 34.5790 Less homogeneous 28.9874 28.9983 28.9283 29.2625 Complexstructure 29.2839 29.2047 29.1903 30.2863 Nonhomogeneous 27.5907 27.5440 27.5327 28.0638 PAGE 40 40 CHAPTER 5 CONCLUSION The proposed noise level estimation and noise reduction method is valid for estimating the noise level in images affected by additive white Gaussian noise and impulse noise. Conventional noise estimation methods perform well only when applied to a homogeneous image. However, the proposed method can estimate more exac t noise level from homogeneous to nonhomogeneous images and also shows good performa nce in noise reduction. It is particularly strong in estimating the noise level in co mplex structure and nonhomogeneous image. The proposed noise estimation method require s fewer computational resources than conventional methods because there is nospecial process required to detect homogeneous regions in a nonhomogeneous noisy image. Its parallelism structure would speed up performance on H/W implementation. PAGE 41 41 APPENDIX A TEST IMAGES FOR PERFORMANCE COMPARISON Figure A1. 24 Kodak Homogeneous and Nonhomogeneous Photo Images PAGE 42 42 Figure A2. Complex Structure Nonhomogeneous Photo Images PAGE 43 43 LIST OF REFERENCES [1] J.Brailean, R.Kleihorst, S.Efst ratiadis, A. Katsaggelos, R. Lagendijk, Noise Reduction Filter for Dynamic Image Sequences: A Review, Proceedings of the IEEE, Vol.83, No.9, Sept. 1995 [2] S. Battiato, A.Bosco, M. Mancuso, G. Spampi nato, Adaptive Temporal Filtering for CFA Video Sequences, In Proceedings of IEEE ACIVS 02 Advanced Concepts for Intelligent Vision Systems 2002, pp. 1924, Ghent Univer sity, Belgium, September 2002 [3] A. Bosco, K. Findlater, S. Battiato, A. 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He, Application of wavelet threshold to image denoising, in: Proceedings of the ICICIC, vol. 2, 2006, pp. 693. PAGE 45 45 BIOGRAPHICAL SKETCH IGil Kim obtained his B.S. degree in the Depa rtment of Electronics Engineering from Hong Ik University, Korea, in 1999. After gra duation from the university, he moved to the United States to pursue his graduate studies He received his M.S degree in Electrical Engineering from University of Southern Californi a in 2002. He served as a software engineer at the digital media R&D center in Samsung, Korea, from 2003 to 2005. He was admitted to the University of Florida in the fall of 2005 and has worked on numerous projects in the fields of image processing and pattern r ecognition in the department of electrical and computer engineering while completing his engineering degree. 