Heat Kernel Signature

Transcription

1 INTODUCTION Heat Kernel Signature Thomas Hörmann Informatics - Technische Universität ünchen Abstract Due to the increasing computational power and new low cost depth cameras the analysis of 3D shapes has become more popular during the last view years. A special challenge is provided by the comparison of non-rigid shapes. The Heat Kernel Signature is an approach, which is well suited for this kind of problem. Intoduction 3D shape matching is a common field in modern geometry processing. It is based on computational understanding geometry. A lot of methods has been proposed by various scientist to solve this problem, but there is no default choice algorithm yet, like the SIFT algorithm[] in the 2D Image processing. The Heat Kernel Signature (HKS)[2] is one of the concepts that face this challenge. The HKS has received a lot of attention in the last years. In 200 the SHREC benchmark proposed the heat kernel based algorithms as the methods with the best performance[3].. Point Signatures A point signature is a data set that represents specific features of a given point. They can be used to define some feature points on a manifold, detect similarities of two different manifolds or to describe the geometry. In 996 one of the first point signatures was proposed by Chua and Jarvis[4] for effictient feature point mapping. The idea was to directly map two feature point of different 3D shape, instead of the two step process, first detecting all feature points and afterwards map them by a matching algorithm like the RANSAC[5], known from image matching. This point signature describes for each point x a one dimensional function f : [0, 2π] R using the intersection of the manifold and a sphere with constant radius. Unfortunately this representation has some disadvantages. It is not stable against perturbations and it does not describe the shape or the point of the shape in a unique manner. Another approach is the global point signature given by Rustamov[6]. By using the eigenvalues λ i and eigenfunctions φ i of the Laplace Beltrami operator Given the ordering of the eigenvalues the Global Point Signature GPS is defined as GP S(p) = ( φ (p), λ φ i = λ i φ i. () 0 = λ 0 < λ < λ 2 <... (2) λ2 φ 2 (p), λ3 φ 3 (p),...). (3) The GPS has some useful properties like invariance under isometric mappings and uniqueness of the signature for every point in the manifold. But it is not stable under perturbations. This can result in a change of the sign for some eigenfunctions as shown on Figure. The last point signature that is mentioned here is the Wave Kernel Signature[7] that has a lot of similarities the the HKS, but is based on Schrödingers equation ψ (x, t) = i ψ(x, t). (4) t The Wave Kernel Signature is defined as WKS(E, x) = lim T T T 0 ψ E (x, t) 2 (5)

2 2 APPROACH Figure : This figure shows the 9th eigenfunction of the shape Victoria in different poses. The coloring goes from blue negative values to red positive values. We can clearly see the change of the sign of the two models. 2 Approach The heat operator H t is a function that maps an initial heat distribution f : R to the heat distribution for any time t. lim H t(f) = f (6) t 0 The function k t that satisfies the following equation is called the heat kernel. H t f(x) = k t (x, y)f(y)dy (7) In order to find a solution for k t we have to solve the heat equation first. u(x, t) = u(x, t). (8) t First of all we define the scalar product of the function space on manifolds as follows f, g := f(x)g(x)dx. (9) We know that the eigenfunctions of the Laplace Beltrami Operator are orthogonal so we can decompose every function on the manifold as u(x, t) = u(t, ), φ i ( ) φ i (x) (0) Solving the heat equation by using the decomposition we get the following formulas t u(x, t) = t u(t, ), φ i( ) φ i (x) () u(x, t) = u(t, ), φ i ( ) φ i (x) = u(t, ), φ i ( ) λ i φ i (x) (2) Comparing the coefficients brings us to the first order ordinary differential equation t u(t, ), φ i( ) φ i (x) = u(t, ), φ i ( ) λ i φ i (x) (3) 2

3 2 APPROACH Figure 2: The figure shows the heat distribution on a cat for two different initial values f = δ x and f 2 = δ x2 where x x 2 on the left. The right side shows the remaining heat at point x i over time to the correspondent cat from the left. which results in the solution u(t, ), φ i ( ) = d i e λit (4) and a closed form for u u(x, t) = d i e λit φ i (x). (5) To define the unknown d i we use the initial value of f for t 0 = 0. So we get lim u(x, t) = d i φ i (x) = f (6) t 0 d i = f, φ i (7) ( ) φ i (y)f(y)dy e λit φ i (x) = e λit φ i (x)φ i (y)f(y)dy = By comparision we get a solution for the heat kernel k t (x, y) = k t (x, y)f(y)dy (8) e λit φ i (x)φ i (y) (9) The heat kernel itself can be thought of the heat distribution over the manifold with put a single heat source onto the point y at time t 0 = 0. Figure 2 shows two different heat distributions computed with the heat kernel. Also the heat kernel fully characterizes the shape up to isoemetry[8] but holds a lot of redundant information[2]. 3

4 2. Heat Kernel Signature 3 EXPERIENTS 2. Heat Kernel Signature The Heat Kernel Signature is defined as the remaining heat at the point x after time t and initial distribution f = δ x. HKS(x, t) = k t (x, x) = e λit φ 2 i (x) (20) By removing most of the information and only keeping the remaining heat at one point over the time the Heat Kernel Signature stays informative[2]. Because the heat kernel describes the heat diffusion the HKS is also stable against noise and change of topology to some extent. In comparison to the GPS the HKS has additional advantages. The HKS is a weighted sum over the squares of the eigenfunctions, so it is neither sensitive to the order of the eigenfunctions nor the sign. By using only predefined time intervals the HKS can also be used for multiscale matching. This can be easily explained by the fact that the heat only distribute to a finite surface at small time intervals. Figure 3: Color plot of the difference between the HKS using eulcedean distance. The difference increases as the color changes from blue to green to red. left: 0 sampling points right: 50 sampling points Figure 4: Color plot of the difference between the HKS using the scaled HKS. The difference increases as the color changes from blue to green to red. left: 5 sampling points right: 0 sampling points 3 Experiments The eigenfunctions of the Laplace beltrami Operator are invariant under isometric transformations despite their sign. Figure show the 9th eigenfunction of Victoria in two different poses. Since the heat kernel signature is a weighted sum of the square of the eigenfunctions it is not sensitive to the sign or the order of the eigenfunctions. The HKS also represents the curvature of the shape in some sense. The heat distributes slower on regions with positive curvature and faster on regions with negative curvature. So we can approximate for some t the curvature of the shape[2]. 3. Comparing two Signatures In computer science we can compare two signatures by sampling them into discrete function values and compare those vectors by some norm. The first approach is using for the HKS a vector with 4

5 3.2 ulti Scale Feature 4 SUARY equidistant sample values and compare them by a norm like the Euclidean distance. ( N ) d(x, x ) = k ti (x, x) k ti (x, x 2 ), tk+ t k = const (2) Figure 3 shows the comparison of one point on the surface to each other. Since the HKS is a monotonic function and the derivate is decreasing equidistant samplings may not be the best way to compare two signatures. There is less change of the HKS for bigger t and also the difference of two samples decreases, because the HKS converges to lim t k t (x, x) = area() for all x. ( N d(x, x k ti (x, x) k ti (x, x ) 2 ) ) =, k log(tk+ ) log(t k ) = const (22) t(y, y)dy where k t(x, x)dx is called the heat trace and can be computed as k t (x, x)dx = e λit. (23) The function SHKS based on the scaled distance function is called scaled Heat Kernel Signature. SHKS(x, t) = k t(x, x) e λit (24) One of the big advantages of the scaled Heat Kernel Signature is, that we need less sampling points. Figure 5: The figure shows two scaled signatures of of a cat. Figure 5 shows the distance using the scaled HKS from one point to each other. Compared to the Euclidean distance we achieve almost the same results with 5 sampling points of the SHKS against the Euclidean distance with 50 sampling points. 3.2 ulti Scale Feature Another property of the heat kernel signature is, that it can compare multi scale shapes. Since the HKS is based on heat distribution it is well suited for comparing local structure. Using for small t the HKS describes the local structure of the shape. For a partial shape the eigenfunctions can be completely different but the HKS stays the same for a small interval [t, t 2 ]. On Figure 6 we can see that the heat distribution for the full and the partial shape looks almost the same. Also the error plot indicates a rising error only for large t. 4 Summary The HKS can be used for multi scale shape matching among different shapes. It has been shown, that the HKS is powerful tool for multi scale shape matching. For many different types in shape geometry Heat Kernel Signature based methods can be used[3]. However there are still some disadvantages. The HKS is not invariant under different scales of the same shape. Also the eigenfunctions with low eigenvalues are taken more into account, but eigenfunctions with high eigenvalues still carry some informations about the shape. 5

6 REFERENCES REFERENCES Figure 6: The figure shows the heat distribution of the same point for time t = 400. We can see, that the error rises when the heat arrives the edge. left: full shape, middle: only left hand, right: error plot References [] Lowe, D.G.: Distinctive image features from scale-invariant keypoints. International journal of computer vision 60 (2004) 9 0 [2] Sun, J., Ovsjanikov,., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. In: Computer Graphics Forum. Volume 28., Wiley Online Library (2009) [3] Bronstein, A., Bronstein,., Bustos, B., Castellani, U., Crisani,., Falcidieno, B., Guibas, L., Kokkinos, I., urino, V., Ovsjanikov,., et al.: Shrec 200: robust feature detection and description benchmark. Proc. 3DOR 2 (200) 6 [4] Chua, C.S., Jarvis, R.: Point signatures: A new representation for 3d object recognition. International Journal of Computer Vision 25 (996) [5] Bolles, R.C., Fischler,.A.: A ransac-based approach to model fitting and its application to finding cylinders in range data. In: IJCAI. Volume 98. (98) [6] Rustamov, R..: Laplace-beltrami eigenfunctions for deformation invariant shape representation. In: Proceedings of the fifth Eurographics symposium on Geometry processing, Eurographics Association (2007) [7] Aubry,., Schlickewei, U., Cremers, D.: The wave kernel signature: A quantum mechanical approach to shape analysis. In: Computer Vision Workshops (ICCV Workshops), 20 IEEE International Conference on, IEEE (20) [8] Hsu, E.P.: Stochastic analysis on manifolds. Volume 38. American athematical Soc. (2002) 6