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- https://ufdc.ufl.edu/UF00091523/00287
## Material Information- Title:
- Exploring Properties of High-Degree SLEFEs
- Creator:
- McCann, Colin
Peters, Jorg (*Mentor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 2004
- Language:
- English
## Subjects- Genre:
- serial ( sobekcm )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
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- All applicable rights reserved by the source institution and holding location.
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Exploring Properties of High-Degree SLEFEs Colin McCann Computer and Information Sciences and Engineering University of Florida Gainesville, Florida 32607 April 15, 2004 Abstract This paper evaluates SLEFEs (Subdividable Linear Efficient Function Enclo- sures) as an optimal method to bound functions. It investigates the extension to all degrees of the degree three proof in [1] showing that SLEFE widths are at most 7% larger than the widths of the smallest possible linear enclosure. This paper shows that such an extension may be difficult to generalize. However, comparisons to convex hull and minmax box bounding methods indicate that for low-degree func- tions SLEFEs present an attractive alternative. For high-degree functions, SLEFEs with few segments offer decreasing advantages over the other methods. An alter- native formulation of SLEFEs is derived but shows results similar to the original SLEFEs. 1 Introduction 1.1 Previous Work Using high-degree functions is a necessary aspect of many computer graphics and modeling applications. However, these functions prove costly to compute and difficult to manipulate in general. Therefore, linear approximations of these functions can sim- plify t he i r use. SLEFEs (Subdivideable Linear Efficient Function Enclosures), deve- loped by the group headed by Dr. Jdrg Peters in the Computer Engineering department at the University of Florida, are one type of linear enclosure. They provide a piecewise upper and lower bound J and J, respectively, that tightly 'sandwich' a function (Fig. 1). SLEFEs improve upon many other methods of approximation, as they bound functions much more closely at the cost of relatively little runtime computation. Most of the extra computational work done using SLEFEs is done pre-runtime through the tabulation of data tables, which are simply read when calculation is needed. The narrower enclosures SLEFEs provide over other bounding methods mean that SLEFEs ideally find a faster solution to problems. Figure 1: A cubic B6zier segment inside its grey-shaded SLEFE. This SLEFE has three segments (and thus four breakpoints). The work done on degree three SLEFEs by Peters and Wu [1] prove SLEFEs provide a guaranteed narrowing upon subdivision, ensuring the number of subdivisions needed to solve problems within a given tolerance. In addition, in [1] it is proven that degree three SLEFEs are no more than 7% wider than the narrowest possible linear enclosure. This guarantees the maximal error, which other enclosure types do not generally ensure. In short, SLEFEs of degree three are proven to provide an efficient, narrow bound that can be quickly computed at runtime. This research paper aims to explore the ideas in [1] for higher degrees. 1.2 SLEFE Definition Any polynomial can be thought of as being composed of a linear function L plus the weighted sums of some so-called 'antidifference functions' ai. The antidifference functions, chosen such that they can be easily bounded and their weights easily calcu- lated at runtime, are enclosed optimally offline. The contribution of each antidifference function to the polynomial is computed at runtime, and the bounds for a given number of semgents are read from a table, multiplied, and summed to find the SLEFE of the polynomial. The exact formulation, explained in the remainder of the subsection, is not essential to understanding the rest of the paper. The SLEFE of a function f with respect to a domain U is a piecewise linear pair, f, f , of upper and lower bounds that sandwich the function on U: f > f > f . Here, the focus is on piecewise linear f and f and measuring the width, f - f, in the re- cursively applied L' norm: the width is as small as possible where it is maximal - and, having fixed the breakpoint values where the maximal width is taken on (leftmost two breakpoints in Fig. 1), the widths at the remaining breakpoints are recursively minimized subject to matching the already fixed break point values. SLEFEs are based on the two general lemmas in [3] and [4] and the once-and- for-all tabulation of best recursive L' enclosures for a small set of 'antidifference' functions, a := (ai)i 1,...,. The general SLEFE construction is as follows: (1) Choose U, the domain of interest, and the space H of enclosure functions. (2) Choose a difference operator A : B -] R', with ker A = B n H. (3) Compute the antidifference function vector a : ]R' i- B so that Aa is the identity on R' and each antidifference function ai matches the same dim(B n H) additional independent constraints. (4) Compute a, - La. and a. - La. e H. (5) Compute Af+ and Af and assemble f and 7 (Lemma 2). Note that above (1),(2),(3) are precomputed off-line and (4),(5) are cheap, making the computation of SLEFEs efficient. In the SLEFEs used in this paper, let * B be the space of univariate polynomials of degree d, in B6zier form f(u) : E o fkbd (u), bd(u) :- d (1 - u)d k. * H be the space of piecewise linear hat functions h. with break points at p/m, pi {0, ..., m}. The statement hLm E H indicates that there are m linear segments joined to form the upper bound. * Af the s d - 1 second differences of the B6zier coefficients Ai/ - -2/fi +/2 Af := A f /: f1-2/2+ 3 Ad-l fd-2-2d- l+fd (Note that A denotes a second difference while the subscript in A2 picks out the 2nd entry in the vector Af of second differences.) * Lf(u) := fo(l - u) + fdu * U = [0..1]. Figure 2: The degree 4 antidifference function a, with bound. Note the widening bound on the rightmost breakpoint in this 'greedy' method. Figure 3: The degree 4 antidifference function a, with bounds. Although the parallel- bounding method (left) fixes the widening problem of the greedy method, its middle segment is wider than that of the mixed method (right). This mixed method is ulti- mately used for the antidifference function bounds. 2 Research and Discussion 2.1 Antidifference Function Bounds The original 'greedy' method of generating SLEFE antidifference function bound tables, used in the degree three case in [1], finds the segment where the second deriva- tive of the antidifference function is largest and computes a "parallel bound" there. This is done by setting the upper segment's endpoints equal to the function value at those points, and then setting the lower bound equal to the line segment tangent to the function with the same slope as the upper bound. After the first segment is computed, the upper bound values are set equal to the function values at the breakpoints. A lower bound segment's values are computed by fixing the already-solved end and minimizing the width of the other end, making a line between them tangent to the antidifference function. In some high-degree cases, this greedy method computed a bound that had values increasing across segments (Fig. 2). The properties of SLEFEs proven for degree three depend on having bound widths that are non-increasing from one segment to the next. To correct this problem, initially a method of parallel-bounding each segment and taking the wider bound at each endpoint was used. While providing the needed decreasing bound widths, the widths were clearly larger than was necessary (Fig. 3, left). Mixing the two methods seems to be the best solution. The greedy method is used whenever possible, but if an increase in the bound width is detected, that segment is in- stead parallel-bounded (Fig. 3, right). The greedy method is then continued. Numerous tests showed that this balanced method of bounding always equaled or outperformed either other bounding method. Independently developed bound values from [2] were also examined. These bounds gave different answers for the analyses presented below, though the same general be- haviors were observed. This indicates that while the numerical ratios are highly bound- dependent, the properties of the SLEFE bounds hold for all bounds with similar prop- erties. This underscores the need for the tightest possible bounds for antidifference functions, and also the validity of examining only one type of bound to determine gen- eral SLEFE properties. 2.2 Comparison to Optimal Bound The main goal in investigating higher-degree functions was to obtain an upper bound on the ratio of the infinity-norm breakpoint widths of SLEFEs to that of the narrowest possible linear enclosure. This 'optimal bound' was found by taking the parallel-bound of the piece of the function with the largest curvature, i.e. the largest second derivative. In [1] only three-piece SLEFEs were examined and found to have a maximal ra- tio of 7%. Investigating other numbers of segments revealed that higher ratios were present. The maximum ratio found was 50% in the two-piece degree three SLEFE. In addition, the ratios generally decreased as the number of segments increased, but fluctuations were present. However, the maxima always occurred where the second differences of the function were equal, facilitating statistical sampling of maximums for a given number of segments. The focus of higher degree comparison was degree four, the simplest unexamined function type. Eight unique types of degree four function are possible based on the combination of second differences. Each of these was examined graphically and found to be similar. It was decided to investigate in detail only the simplest case, where the second derivative of the function is always positive. Analysis of the degree-four SLEFE to optimal bound ratios shows that the maxima occur in between the extremes of function curvature, unlike in degree three, where the Degree 4 Optimal Difference Ratio for m = 5 max = 20 8% at (0 43, 0 57) S15 S10, 5 Beta 0 0 2 04 06 08 1 Alpha Figure 4: Plot of the SLEFE width to optimal width ratio for degree four, five seg- ments. Alpha, Beta, and (1-Alpha-Beta) are the barycentric second differences of the function. Note that the maximum does not occur where the second differences are equal (Alpha=Beta=0.33). This is unlike the degree three case, where the maximum occurs where the second differences are equal. The location of this maximum is segment- dependent. maximums occur at the extremes of function curvature (Fig. 4). As in degree three, the locations of the maxima are highly dependent on the number of segments used in the bound and generally decrease as the number of segments increases. However, the maximum ratio for a given number of segments is higher than the corresponding value for degree three (36% vs. 7% for three segments). Higher-degree examinations focused on functions with only positive second differ- ences. These functions' ratios increase as the degree increases, indicating that there may be no ultimate upper bound on the SLEFE to optimal bound ratio that holds for all degrees. No method was found to predict the maximal SLEFE to optimal bound ratio other than statistical sampling. 2.3 Comparison to Convex Hull and Minmax Box A statistical comparison, using 1000 data points per degree and segment, of the one- norm of SLEFEs to the one-norms of the convex hull and minmax box of a function was carried out. While such an approach shows that SLEFEs certainly outperform both enclosures in the low-degree or high-segment case (Fig. 5), the comparison showed little one-norm performance benefits in the case of high degree and low number of segments. Figure 5: A cubic B6zier segment with coefficients 0, -1, 1, 0. The 3-piece SLEFE (left) is much narrower than both the minmax box (middle) and the convex hull (right). 300 3500 250 3000 200 2500 o 150 \ H \2000 100 o 5\ 1500 50 -50 5 ---- 500 l-2_ 1 2 3 4 5 6 7 82 3 4 5 6 7 Degree Degree Figure 6: The average 1-norm percent difference between convex hulls and SLEFEs of a function (higher numbers indicates better SLEFE performance). For two SLEFE segments (left), the SLEFE becomes worse than the convex hull at degree five, whereas for six SLEFE semgents (right), the SLEFE has a better norm up through degree eight. As in the optimal bound comparison, the ratios only worsened as the degree in- creased, meaning that no maximum difference for all degrees could be found. Al- though, statistically, relatively small average differences were found, there were rare instances where the SLEFEs did not perform well at all compared to convex hulls or minmax boxes. This suggests that for low degree functions, SLEFEs will generally result in a performance boost over other enclosures, but for higher degree functions the benefit of using SLEFEs with a low number of segments is negligible. A good metric is to use a number of segments in the SLEFE equal to the degree of the function being bounded. At least up through degree eight, a convex hull is on average at least three times as large as the SLEFE, and a minmax box is on average at least six times as large. SLEFEs thus generally perform far better in this case. 800 8000 7000 600 6000 S400 5000 I 4000 200 30 \00 2000 1000 -201 2 3 4 5 6 7 8 01 2 3 4 5 6 7 8 Degree Degree Figure 7: The average 1-norm percent difference between minmax boxes and SLEFEs of a function. For two SLEFE segments (left), the SLEFE becomes worse than the minmax box at degree seven, whereas for six SLEFE semgents (right), the SLEFE performs better through degree eight. 2.4 Alternative Definition of SLEFEs The maximal ratios in the optimal bound comparisons are correlated to where the contributions of the most curved antidifference functions are greatest. A different for- mulation of SLEFE was explored to correct this problem. This new formulation was designed so that this problematic curvature minimizes, instead of maximizing, the ratio. As an additional benefit, this new SLEFE formulation only adds one new antidifference function per increase in degree as opposed to a number equal to the degree minus one, so fewer tables of antidifference function bounds are necessary. The alternative SLEFE definition is d d f: = ftb =L(f) + Fifa j= 0 i= 2 where the ith functional Fj is the (scaled) average of ith differences of f: 1 Zd iA'f, AX F df 0 d Eko Af, A1, :k k 1(-1)'f+l ) The ith antidifference function is of degree i and satisfies Fak a 1 ifi = k 0 else. Analysis of these new SLEFEs gave results similar to the original formulation. In comparing the SLEFE bounds to the optimal, the maxima once again depends on the number of segments, albeit in a different location. In addition, the ratio to the optimal bound is slightly smaller (27% opposed to 36% for three segments). Statistical sampling results comparing the convex hull and minmax boxes were nearly identical to the original formulation. The runtime bounds using the new antidifference functions are more difficult to calculate, likely overshadowing the small computational improvements in this new for- mulation. In addition, this formulation lacks the proofs of the original definition. Com- bined with the marginal improvement in bound width, this definition was ultimately discarded in favor of the original. 3 Conclusion The guaranteed ratio of SLEFE widths to the optimal bound width is proven for degree three, but this proof resists extension to higher degrees. This research shows that the ratio of higher-degree SLEFE width to the optimal width may increase without bound, therefore statistical calculation of the error for a particular degree and number of segments was undertaken. SLEFEs perform well compared to minmax boxes and convex hulls for low degrees, as well as for a high number of segments. As the degree increases or the number of segments used decreases, SLEFE performance degrades in comparison. For the cases where they do perform well, SLEFEs can potentially perform far better than convex hulls or minmax boxes. SLEFEs that use a number of segments equal to the degree of the function have a much smaller one-norm than either other bounding method. 3.1 Possible Future Work It is possible that a different formulation of SLEFEs or another method of creating bounds could succeed in guaranteeing for all degrees the properties of SLEFEs dis- cussed in this paper. The programs created in this project could be easily modified to create other bounds if another method were found, and the comparison programs could quickly verify or invalidate the properties of new bounds. 3.2 Acknowledgements This paper was made possible in part by a grant from the University Scholars Pro- gram. The guidance of the author's faculty mentor, Dr. J6rg Peters, was instrumental in developing the ideas and techniques used in this research. References [1] Peters, Jorg and X. Wu. "On the Optimality of Piecewise Linear Max-Norm En- closures Based on Slefes." 30 Dec. 2002: http://www.cise.ufl.edu/research/SurfLab/papers [2] Wu, X., Peters, J., 2003. The SubLiME package (Subdividable Linear Maximum- norm Enclosure). Downloadable from http://www.cise.ufl.edu/research/SurfLab/SubLiME/. [3] Lutterkort, D., Envelopes for Nonlinear Geometry, PhD thesis, Purdue University, May 2000. [4] Lutterkort, D. and J. Peters, "Optimized refinable enclosures of multivariate poly- nomial pieces", Comp. Aided Geom. Design, 2002, 851-863. [5] Peters, Jorg and X. Wu. "SLEVEs for Planar Spline Curves." 30 Mar. 2004: http://www.cise.ufl.edu/research/SurfLab/papers |