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Exploring Properties of High-Degree SLEFEs

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Exploring Properties of High-Degree SLEFEs
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McCann, Colin
Peters, Jorg ( Mentor )
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Gainesville, Fla.
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University of Florida
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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