Augmented Symmetry Transforms Raif M. Rustamov Department of Mathematics, Purdue University 50 N. University Street, West Lafayette, IN 47907 [email protected]

Abstract “Perfect symmetries are all alike; every imperfect symmetry is imperfect in its own way.” - adapted from Leo Tolstoy’s Anna Karenina. Symmetry has been playing an increasing role in 3D shape processing. Recently introduced Planar Reflective Symmetry Transform(PRST) has been found useful for canonical coordinate frame determination, shape matching, retrieval, and segmentation. Guided by the intuition that every imperfect symmetry is imperfect in its own way, we investigate the possibility of incorporating more information into symmetry transforms like PRST. As a step in this direction, the concept of Augmented Symmetry Transform is introduced; we obtain a family of symmetry transforms indexed by a parameter. While the original PRST measures how much the symmetry is broken, the Augmented PRST also gives some information about how it is broken. Several approaches to calculating the augmented transform are described. We demonstrate that the augmented transform is useful for shape retrieval. Keywords: symmetry transform, 3D shape matching, and retrieval.

1. Introduction Symmetry has been playing an increasing role in 3D shape processing, giving rise to new approaches to problems like range scan reconstruction, canonical coordinate frame determination, shape matching, retrieval, and segmentation. Among these new developments, we will concentrate on the idea of Symmetry Transform (ST), a remarkable example of which is the Planar Reflective Symmetry Transform introduced in [19], and show how STs can be augmented to capture more information. The Planar Reflective Symmetry Transform (PRST) combines two important ideas. First, the classical binary view of symmetry does not offer much descriptive power:

symmetry either exists or does not. A smallest defect destroys an otherwise perfect symmetry. To overcome this difficulty, PRST employs a continuous measure for it. Planes with perfect symmetry still get the perfect score – 1; the others are assigned scores ranging from 0 to 1 depending on what proportion of the object aligns with its reflection about the plane. Second, PRST assembles information about all possible planes: the measure above is calculated for all planes in the bounding volume of the object. This renders PRST a transform conceptually similar to the Fourier or Radon transforms. In fact, the input of PRST is a scalar function in d-dimensional space; the output, is another scalar function, this time defined on the set of all (hyper)planes in the d-space. It is these two ingredients that allow PRST to handle tasks as various as shape alignment, matching and even segmentation. We notice that the continuous symmetry measure above captures how much the symmetry is broken. Can we add some information about how it is broken? Let the term “defect” refer to the parts of an object responsible for breaking the symmetry under consideration; removal of defects results in perfect symmetry. For each plane reflection, PRST tells us what proportion of the object is not defective, and so, it tells us about the defects’ size. What about their spatial distribution? An example would be in place here. Consider the 2D PRST of the two shapes depicted in Figure 1. The areas of circular holes in the second rectangle add up to the area of the hole in the first rectangle. Therefore, PRST for reflection about the dashed line would be the same for both of the shapes. For a given line or plane, PRST takes contribution from all points whose reflections also belong to the shape. Notably, the contexts of the points are not considered. Let us augment the PRST by adding a further restriction: points are not allowed to contribute unless their neighborhoods (here points within some radius r) match as well under the reflection. For a given radius r, we define PRSTr to be this augmented version of PRST. The augmented PRSTs of the rectangles for reflections about the dashed lines are depicted in Figure 1. Actually, the plots of

(Section 3). Second, we provide an algorithm for calculating its discrete version (Section 4). Third, we confirm usefulness of the augmented symmetry for shape retrieval (Section 5).

2. Previous Work 1 0.95

Augmented PRST(squared)

0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

0

0.1

0.2

0.3 Parameter R

0.4

0.5

0.6

Figure 1. Both of the squares depicted above have the same PRST value for the axes shown. However, the Augmented PRST reveals the difference in the spatial distribution of asymmetry, as seen in the plot. The Augmented PRST of the second square decreases faster because the defects are spatially less concentrated.

PRST2r , where r ranges from 0 to 0.6 are shown. The values of PRST0 for both shapes coincide: indeed, PRST0 is nothing but the original PRST. However, the augmented PRST of the second rectangle decreases faster than of the first. This can be explained as follows: when r is increasing, it is as if we are enlarging the defects – only points distance r away from the defects (and their reflections) contribute to PRSTr . Defects that are split or spread out will grow faster and prevent more points from contributing, than defects that are concentrated. In a similar manner, one can see that the slope of PRSTr at r = 0 is related to the defects’ boundary length. Also notice how PRSTr interpolates between the continuous symmetry measure and the classical binary symmetry when r is increased from zero to infinity. We think that the extra information captured by the augmented planar reflective transform can be useful for the spectrum of applications considered in [19], especially for shape retrieval. The contributions of this paper are threefold. First, we define the augmented symmetry transform

Early approaches concentrated on finding perfect symmetries in images by reducing the problem to substring matching [1, 5, 25]. The reduction was very efficient, but binary by its nature, not allowing detection of imperfect symmetries. Thus, not surprisingly, these first algorithms were highly sensitive to noise. Finding symmetries in 3D objects received attention of many researchers. Methods using graphs [8], octree representation [14], correlation in the Extended Gaussian Image [23], the singular value decomposition [20] were proposed. Recently, Thrun and Wegbreit [24] proposed a hierarchical generate-and-test procedure to detect local symmetries and used them to reconstruct range scans. A robust and efficient approach based on moments was introduced by Martinet et al. [12]. Two other very successful recent approaches are [4, 15]. Symmetry as a continuous property emerged in [27]. The amount of partial symmetry was measured using the concept of symmetry distance. For a given symmetry transformation it is the distance between the given shape and the closest perfectly symmetric shape. In [26] a multiresolution scheme is used to detect symmetric and almost symmetric patterns: symmetries at low resolutions guide the search for high resolutions. Symmetry as a shape descriptor was used in [9, 11], by considering symmetries with respect to all planes or rotations through the center of mass of an object. More recently, all of the planes through the object’s bounding volume were considered [19], and the concept of Planar Reflective Symmetry Transform was introduced. This transform was used for alignment, shape matching, retrieval and even for segmentation [19]. We build upon the work on continuous symmetry measures, to increase the amount of information explicitly captured by them. Notice that our approach does not overlap with the multi-resolution scheme: while [26] considers coarse resolution to make partial symmetries more apparent, we, on the contrary, sacrifice more and more partial symmetries to capture information about the layout of defects. Our augmentation can be applied to any symmetry transform that is based on a continuous symmetry measure. For clarity, we focused on PRST in the introduction. In [19] it is conjectured that PRST is invertible, meaning that given a PRST we may be able to reconstruct the object. This would mean that PRST captures all information about the object,

including the distribution of its defects. However, even if proven possible, the reconstruction will be subtle and most probably sensitive to noise (see [19]). Thus, the augmented PRST will be still useful.

3. Augmented Symmetry Transform Symmetry Transforms: Before formalizing the ideas outlined in the introduction, let us review the definition of a symmetry transform. One considers a set G whose elements are some rigid symmetry transformations of the Euclidean space. An elements of G, i.e. a particular symmetry transformation will be denoted by γ; its action on point x by γx. We consider scalar valued functions f in the space, and define the action of γ on f as follows: γ(f )(x) = f (γ −1 x). A note on terminology would be appropriate: we reserve the word “transformation” for the elements of G – i.e. particular symmetries of Euclidean space. “Transform”, on the other hand, is exclusively used for the construct defined below, and is more like “transform” in “the Fourier transform”. The decision which symmetries to include into G should be made. In [9], G included exclusively reflections about the planes passing through the origin. Afterwards in [19], G is extended to contain reflections about all the planes in the bounding volume of an object. For any object represented by a function f , its Gsymmetry transform (ST) is a real valued function defined on G, ST (f, ·) : G −→ R. The value of ST (f, γ) is defined in terms of the symmetry distance of [27]. Symmetry distance(SD) is the L2 distance between f and the nearest function that is invariant under the transformation γ: SD(f, γ) =

min f − g.

g|γ(g)=g

In case of rotations and reflections, this definition is equivalent to [19] SD(f, γ) =

f − γ(f ) , 2

which is more intuitive and amenable to computation, intuitive because now SD measures the difference(intersection) between the original object and its transformed version, amenable because we no longer need to perform minimization. Podolak et al. [19] define the Planar Reflective Symmetry Transform(PRST) as follows P RST 2 (f, γ) = 1 −

SD2 (f, γ) f − γ(f )2 =1− . 2 f  4f 2

Simplifying and noticing that symmetries under consideration preserve L2 norms, they get P RST 2 (f, γ)

= =

2f 2 − 2f · γ(f ) 4f 2 1 f · γ(f ) + . 2 2f 2

1−

(1)

It is clear that the main ingredient in this formula is the dot product. For clarity of presentation, we diverge from the notation of [19], and define the Symmetry Transform to be this dot product. More precisely, for a function normalized by f  = 1, its Symmetry Transform is
ST (f, γ) = f · γ(f ) = f (x)f (γ −1 x)dx. When we write P RST (f, γ), we mean this Symmetry Transform when G is the set of all planar reflections. Notice that the relation to the original PRST is easily established using formula (1). Augmented Symmetry Transforms: For illustration, let us consider the case when f is the characteristic function for an object: it takes value of 1 for points that belong to the object, and value of 0 otherwise. For a given γ, point x contributes to ST (f, γ) if both x and y = γ −1 x belong to the object. Notice that the context of these points is completely neglected. We remedy the situation by looking at the neighborhoods of x and y, and if neighborhoods match as well under γ, then these points are allowed to contribute. We now formalize this idea. Given a point x, define Br (x) to be the set of all points in the space within distance r to x. For a function f , symmetry transformation γ, and a radius r, we need a measure of as to what extent the equality f |Br (x) = γ(f )|Br (x)

(2)

is satisfied at point x. We denote any such measure as Kr (f, γ, x). A simple choice would be to set Kr (f, γ, x) = 1 if equality (2) is satisfied at x, and 0 otherwise. One could also consider smoothly varying alternatives. The Augmented Symmetry Transform is a family of transforms indexed by radius r, and denoted by STr (f, γ). We define it as
STr (f, γ) = f (x)f (γ −1 x)Kr (f, γ, x)dx. The purpose of the term Kr (f, γ, x) is to prevent points from contributing unless equality (2) is satisfied – thereby incorporating local context information into the symmetry measure. Properties: The following discussion will be limited to the characteristic functions of surfaces in 3D. Notice that then

all integrals are surface integrals; for example, integrating f (x) yields the area of the surface. One may think of this differently by considering the surface to have some thickness, then the integrals would reduce to the usual integration in 3D. We concentrate on two properties of the Augmented Symmetry Transform. First, the augmented ST interpolates between the binary and the continuous symmetry transforms. Second, there is a relationship between the slope of the augmented ST at r = 0 and the boundary length of the defective set. A point x on the surface S will be called a defect for γ ∈ G, if γ −1 x does not belong to the surface. We denote the set of all defective points for given γ by D(γ). Let Dr (γ) denote the r neighborhood of D(γ) in S, i.e. all points on the surface that are within the distance r of D(γ). Notice that the usual symmetry transform ST (f, γ) is proportional to the area of the non-defective part of S: ST (f, γ) = 1 −

area of D(γ) . area of S

Augmented symmetry transform STr can accept contributions only from points that are outside of Dr (γ) and γ −1 Dr (γ), because any point within distance r to a defect or its transformed version will not satisfy the equality (2). As a result, we may write STr (f, γ) = 1 −

area of S ∩ (Dr (γ) ∪ γ −1 Dr (γ)) . (3) area of S

One can immediately see that the equality ST (f, γ) = ST0 (f, γ) is satisfied. Indeed, D(γ) = D0 (γ), and since γ −1 D(γ) does not belong to the surface by definition, the formula above becomes ST0 (f, γ) = 1 −

area of S ∩ D(γ) = ST (f, γ). area of S

Moreover, if D(γ) is nonempty, then when r is increased, Dr (γ) grows bigger and eventually it will cover the whole surface. This means that STr (f, γ) converges to the classical binary symmetry transform as r is increased: for perfect symmetries γ, STr (f, γ) will be 1 independently of the radius r; on the other hand, for imperfect symmetries it will decrease to 0. We collect these facts by saying that the augmented symmetry transform interpolates between the binary symmetry transform and continuous ST. The next statement is about the boundary length of the defective set D(γ). It is true generically, i.e it holds for almost every symmetry transformation γ. Exceptions could be those γ that satisfy Dr (γ) ∩ γ −1 Dr (γ) = ∅,

for any r, however small it be.1 On the contrary, if there is some ε such that for every 0 ≤ r < ε the intersection above is empty, then we can write STr (f, γ) = 1−

area of Dr (γ) + area of (Dr (γ) − D(γ)) , area of S

where 0 ≤ r < ε, and we used the fact that as a rigid Euclidean transformation, γ preserves areas. Now, since Dr (γ) contains D(γ), we obtain STr (f, γ) = 1 −

2 × area of Dr (γ) − area of D(γ) . area of S

Consider the difference STr (f, γ)−ST0 (f, γ) = 2×

area of Dr (γ) − area of D(γ) . area of S

Dividing by r, and taking limits as r → 0+ , we get   dSTr (f, γ) boundary length of D(γ) . =2× dr area of S r=0 This formula establishes a relationship between the slope of the augmented ST at r = 0 and the boundary length of the defective set. Of course, this formula is mostly of theoretical interest. However, it reinforces our belief that the Augmented Symmetry Transform captures extra information that is both non-trivial and geometrically relevant.

4. Computation We present three approaches to computing the Augmented Symmetry Transform for a function f rasterized on n×n×n voxel grid. First, given a specific symmetry γ ∈ G we show that the exact value of STr (f, γ) for all radii can be computed in O(n3 ). When this is used for all symmetries in G, which is usually of O(n3 ), we end up with O(n6 ) to calculate the augmented ST. Second, an approximate approach which calculates STr for all symmetries in G and radii ri = ir is described. Third, we show how one can compute STr for all symmetries in G for a fixed radius r. Calculating augmented ST for a polygonal model starts with obtaining a volumetric representation. One can use binary voxelization: set f = 1 for voxels within some distance (usually within one voxel size) of the surface, and f = 0 for the remaining voxels. This may result in sensitivity to noise and small features. More robust approach is to use the Gaussian Euclidean Distance Transform (GEDT) as done in [19]. The result of GEDT is a volumetric function which is 1 for the points on the surface, while its values for remaining points range between 0 and 1, depending on 1 The exact condition is: length of lim −1 D (γ)) = r r→0 (Dr (γ) ∩ γ 0, which is rarely satisfied.

their distance to the surface. To defy sensitivity to noise it is also advisable to use a less stringent function Kr (f, γ, x). A similar effect in the exact algorithm can be achieved by changing the definition of a defective point. The following redefinition seems appropriate: x is a defective point if f (x) > a1 but γ(f )(x) = f (γ −1 x) < a2 . Here a1 and a2 are some thresholds with a1 > a2 .

4.1

Exact Computation

To compute STr (f, γ) for given Euclidean transformation γ ∈ G one can resort to the formula (3) in previous section. One first calculates all of the defective points(voxels) and their images under γ −1 ; then for each voxel we compute the distance to the closest defective voxel or their image. Obviously, a voxel can contribute to STr if and only if this distance is strictly less than r. This is a brute force approach, but it amortizes itself to some extent, because it allows us to calculate STr for all values of r at the same time. Computing the defective set requires O(n3 ) – it can be done in one scan. Now one computes the distance field for the defective set and its image under γ, which can be done in O(n3 ), e.g. using the algorithm of [6, 13]. Now we need one more scan of the grid to get the contributions of voxels to STr . Thus, we are left with total time complexity of O(n3 ). When we we want to compute STr for all γ ∈ G, the size of G will play a decisive role. For example, if we are interested in the augmented PRST, G contains reflections about all O(n3 ) possible planes in the bounding volume of the object. The whole computation will then take O(n6 ) time, which could be prohibitively expensive for most applications.

4.2

Approximate Computation

For the following two algorithms we will use an importance sampling Monte Carlo method, which is especially effective for surfaces because of their sparse nature. Our approach is based on the Monte Carlo algorithm for PRST described in [19]. One first samples points according to their expected significance for the final result (see [19]), then each pair of sampled points “votes” for transformations in G that take the first point to the second. Remember, however, that to calculate the augmented ST one needs to see if the neighborhoods of sampled points also match under the transformation. To check this, one approach is to actually transform the first neighborhood and then see if there is enough overlap with the other neighborhood. Since this will be computationally very expensive, we go for an approximation: the neighborhoods can be compared using a local shape descriptor that is invariant under the symmetries in G. Indeed, we calculate such a descriptor for the

neighborhoods of sampled points, and let points x and y contribute only if the local shape descriptor at x is within a threshold of the descriptor at y. Of course, we will have to calculate these descriptors for neighborhoods of varying sizes as dictated by the value of r. Using SHD: We explain how to calculate STr for a set of equally spaced radii ri = ir, where i = 1, 2, ..., n, using Spherical Harmonic Descriptors(SHD) of [10]. For each sampled point we calculate SHD for spheres centered at the point with radii ri . To proceed, we pick up a pair of sampled points, and compare the SHDs around these points to see up to what radius ri their neighborhoods can be considered to match. Transformations in G taking one point to the other are determined, and these points contribute to the augmented ST for all matching radii. for sampled points x: calculate SHDri (x) for sampled points x: for sampled points x
: γ = transformation(s) taking x to x
for i = 0 to n if SHDri (x) − SHDri (x
) < thresholdi STri (γ)+ = w(x, x
, γ) else break Here w(x, x
, γ) is the weight of contribution of pair (x, x
) for transformation γ. This will include a term accounting for the importance sampling that was performed, and may contain another ingredient arising from change-ofvariables, see [19], if one is involved. Euclidean distance can be used to compare the SHDs. Thresholdi is the maximal difference between descriptors that allows us to consider the neighborhoods of radii ri to be similar. Notice how the hierarchical character of the SHD is used here. The most inner loop is terminated as soon as we reach the radius at which the SHD’s differ more than the threshold. To give an idea about timing: it was reported in [21] that calculating SHDs using 16 frequencies on 32 shells for 2000 sampled points took about 80 seconds. In this context, it would also be interesting to apply the approach of [16] to compute the Spherical Harmonic Transform, which we leave to the future work. Using Zernike Descriptors: 3D Zernike descriptors were introduced by [3], and were popularized for 3D shape retrieval in [17, 18]. These descriptors can capture object coherence in radial direction and are more concise than SHD. Moreover, their computation does not require radial sampling, which makes them more invariant to rotations, especially for small radii. However, computing Zernike descriptors at every sampled point would have been very expensive. We propose a solution to this difficulty by using the Fast Fourier Transform. In addition, we explain how adaptive

thresholds are possible when deciding whether two neighborhoods match. Notice that since Zernike descriptors do not have the hierarchical structure of SHD, we limit our attention to calculating STr for a given radius r. Let Z(x) be the vector of Zernike descriptors calculated for the ball of radius r around the point x. The previous algorithm gets modified as follows. for all grid points x: calculate Z(x) and threshold(x) for sampled points x: for sampled points x
: if Z(x) − Z(x
) < threshold(x) γ = transformation(s) taking x to x
STr (γ)+ = w(x, x
, γ) Now we will explain how to calculate the Zernike descriptors for the ball of radius r centered at every grid point, and how to choose the thresholds adaptively. A short review of Zernike descriptors will be useful. Zernike descriptors are similar to Spherical Harmonic Descriptors in that they are also based on projecting the given 3D function onto a carefully chosen basis and combining the coefficients to obtain rotational invariants. The basis used is Zernike-Canterakis basis consisting of functions m (r, θ, φ) = Rnl (r)Ylm (θ, φ) Znl with −l < m < l, 0 ≤ l ≤ n, and n − l even. Here Ylm (θ, φ) are spherical harmonics, and Rnl (r) are radial functions defined by Canterakis. 3D Zernike moments of f (x) are defined as the coefficients of the expansion in this orthonormal basis, i.e. by the formula
3 m Ωm = f (x)Z nl (x)dx. nl 4π |x|≤1 These are combined as follows   l  2 Fnl =  (Ωm nl ) m=−l

to get rotationally invariant 3D Zernike descriptors Fnl . Index n is called the order of the descriptor. Zernike descriptors can be calculated in Cartesian coordinates remarkably easily. Indeed, one first computes the geometric moments of f (x),
f (x)xr y s z t dx, Mrst = |x|≤1

where x = (x, y, z) ∈ R3 . These moments are combined to obtain the Zernike moments:  3 Ωm χrst nl = nlm Mrst , 4π r+s+t≤n

where χrst nlm are some constants, see [17, 18]. Therefore, calculating the Zernike descriptors reduces to calculating the geometric moments. We compute geometric moments using convolutions of the object grid with ball grids (of radius r) that contain values corresponding to the geometric moments being computed. These convolutions can be efficiently calculated using the Fast Fourier Transform. The radius for which the augmented ST is computed, r, is assumed to be given as number of voxels it spans. First, the n × n × n object grid is padded with zeroes to become (n + r) × (n + r) × (n + r); this is done to avoid aliasing. To calculate Mrst , a ball of radius r is rasterized onto a grid of the same size (n + r) × (n + r) × (n + r). The center of the ball is assumed to be the origin, and the voxel with coordinates (x, y, z) inside the ball is assigned the value xr y s z t . The Fourier Transforms of both grids are caclulated, multiplied and inverted to give the convolution. The resulting convolution contains the geometric moments that can be plugged into the formula for Zernike moments. In terms of computation time, if we use Zernike descriptors up to the order n = 5, which gives a total of 12 descriptors, we need to make 55 convolutions. In our experiments with ball radius equal to 10 voxels in a 64 × 64 × 64 grid this took only 4 seconds on a low-end PC. The time could be easily reduced (almost halved) by precomputing the FFTs of geometric moment balls. The knowledge of local Zernike descriptors at every voxel allows using adaptive thresholds when comparing two neighborhoods. Keeping the threshold constant across the shape may not be the best option: the rate at which shape descriptors change under small deformations depends on the initial shape. Thus, we would want to tolerate more difference in descriptors for some neighborhoods, and less for others. We provide a solution to this problem by setting the threshold at every point adaptively. The values of descriptors at adjacent voxels give us a gauge of how much difference is tolerable. We set the threshold at a given voxel to the maximal value of descriptor difference between the given voxel and its neighboring voxels.

5. Experimental Results for Shape Retrieval Augmented symmetry transforms were introduced with the PRST as a guiding example. However, augmentation can be applied to any symmetry transform. In fact, to test the performance of the PRST, Podolak et al. [19] manually registered all models into a common coordinate frame. To avoid manual registration, we conducted our experiments on the Translational Symmetry Transform (TST) – a Symmetry Transform for which G contains rigid translations within the bounding volume of an object. We also consider its augmented version. For both of these symmetry transforms, a rotationally invariant shape descriptor is obtained

Table 1. The statistics of 3D model retrieval using TST and the augmented TST.

TST TST10

NN 72.4% 70.6%

FT 32.2% 29.9%

ST 43.6% 40.3%

E-M 30.5% 27.5%

DCG 66.3% 64.0%

Overall, the original symmetry transform outperforms the augmented one. However, for about 30% of the models the augmented TST performs better. Figure 2 shows the difference in the DCG values for all of the models in the benchmark. The results for the nearest neighbor classification are similar: for seven out of 45 classes, the augmented transform is better; for 15 classes the original transform is better; for the remaining classes there is no difference in performance. The augmented transform performed better for the following classes: Bracket like Parts, Pulley Like Parts, U shaped parts, Machined Plates, and all of the three Miscellaneous classes. The worst performance is observed for Machined Blocks and Container Like Parts. To summarize, these results demonstrate that the extra information captured by the augmented ST can enhance the results of shape retrieval. Of course, computing and storing the augmented version along with the original is more expensive, but the extra cost may worth the improved performance for some applications.

60

Augmented is Better 40

Difference in DCG (%)

by using the spherical harmonic representation as described in [10]. We expect translation symmetries to be relevant in the realm of engineering shapes, and this dictated our choice of the benchmark. A public engineering shape benchmark [7] comparable to Princeton Shape Benchmark [22] in size is available from Purdue Research and Education Center for Information Systems in Engineering (PRECISE). The Engineering Shape Benchmark(ESB) provided by PRECISE contains 801 3D CAD models classified into 42 categories of similar parts such as “Discs”, “T-shaped parts”, “Bracketlike parts” and etc. In addition, there are 66 other models classified under the name “Miscellaneous” (actually, there are three different classes with this name). We compare the performance of the original TST with its augmented version, TST10 . The neighborhood radius for augmented version is set to 10 voxels for 64 × 64 × 64 grid; the approach based on Zernike descriptors is used. We conduct a series of “leave-one-out” experiments: every model in the benchmark is queried against all other models. The ranked result lists generated by the queries are used to compute five retrieval statistics as suggested in [22]: nearest neighbor(NN), first tier (FT), second tier (ST), E-Measure (E-M), and discounted cumulative gain (DCG). The resulting statistics are listed in Table 1.

20

0

−20

Objects in the benchmark

−40

Original is Better −60

Figure 2. The DCG difference between TST and augmented TST for all objects in the benchmark.

6. Summary and Future Work We have defined the Augmented Symmetry Transform, which captures information about the spatial distribution of defects – parts that break the symmetries under consideration. We have described several approaches to computing this augmented ST. Our experiments show utility of the augmented ST for shape retrieval. In future work we plan to investigate the effect of augmentation for other geometry processing tasks including alignment, segmentation and viewpoint selection.

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