Persistence-based Interest Point Detection for 3D Deformable Surface

Transcription

1 Pesistence-based Inteest Point Detection fo 3D Defomable Suface Xupeng Wang 1,3, Fedous Sohel 2, Mohammed Bennamoun 3, Yulan Guo 4,3 and Hang Lei 1 1 School of Infomation and Softwae Engineeing, Univesity of Electonic Science and Technology of China, Chengdu, China 2 School of Engineeing and Infomation Technology, Mudoch Univesity, Peth, Austalia 3 School of Compute Science and Softwae Engineeing, Univesity of Westen Austalia, Peth, Austalia 4 College of Electonic Science and Engineeing, National Univesity of Defense Technology, Changsha, China @uestc.std.edu.cn, F.Sohel@mudoch.edu.au, mohammed.bennamoun@uwa.edu.au, yulan.guo@nudt.edu.cn, hlei@uestc.edu.cn Keywods: Abstact: 3D defomable shapes, Inteest point detection, Pesistent homology, Diffusion geomety, Heat kenel signatue function Seveal appoaches fo inteest point detection on igid shapes have been poposed, but few ae available fo non-igid shapes. It is a vey challenging task due to the pesence of the lage degees of local defomations. This pape pesents a novel method called pesistence-based heat kenel signatue (phks). It consists of two steps: scala field constuction and inteest point detection. We popose to use the heat kenel signatue function at a modeately small time scale to constuct the scala field. It has the advantage of being stable unde vaious tansfomations. Based on the pedefined scala field, a 0-dimensional pesistence diagam is computed, and the local geometic and global stuctual infomation of the shape ae captued at the same time. Points with local maxima and high pesistence ae selected as inteest points. We pefom a compehensive evaluation on two popula datasets (i.e., PHOTOMESH and Inteest Points Dataset) to show the effectiveness of ou method. Compaed with existing techniques, ou inteest point detecto achieves a supeio pefomance in tems of epeatability and distinctiveness. 1 INTRODUCTION With the inceasing availability of low-cost 3D sensos (e.g., Micosoft Kinect), thee is a gowing demand fo 3D suface analysis (Biasotti et al., 2015). The epesentation of 3D sufaces is a challenging task due to the pesence of noise, occlusion, clutte and a wide ange of shape tansfomations (Guo et al., 2013b)(Bonstein et al., 2011)(Litman and Bonstein, 2014). A popula appoach to measue the similaities of 3D sufaces is based on a collection of local featues (Guo et al., 2014a)(Guo et al., 2016). Local featuebased appoaches (Guo et al., 2013a)(Guo et al., 2013d)(Guo et al., 2013c)(Guo et al., 2014b)(Wang et al., 2016)(Wang et al., 2015) have been actively investigated fo the past two decades and ae commonly used in many applications including 3D object ecognition, 3D econstuction, 3D shape etieval, egistation and tacking. Local featue based suface desciption geneally consists of two steps: inteest point detection and featue desciption (Tombai et al., 2013). Pominent points on a shape with espect to a paticulaly defined saliency o inteest ae fist detected based on suface analysis. Then, the local suface aound each inteest point is descibed using a 3D suface descipto. Finally, the desciptos aound the inteest points ae popely assembled to map the suface into a featue space. Inteest point detection is a fundamental step because it identifies a collection of 3D stuctues fo futhe suface desciption (Guo et al., 2014a). A numbe of 3D inteest point detectos have been poposed, and most of them ae designed fo igid sufaces (Patikakis et al., 2010)(Mian et al., 2010)(Godil and Wagan, 2011) (see (Guo et al., 2014a) fo a ecent suvey). In ecent yeas, seveal studies (Sun et al., 2009)(Zahaescu et al., 2012) have been poposed fo non-igid shape analysis. In paticula, diffusion geomety achieves a supeio pefomance because of its ability to eflect the intinsic popety of a shape (Bonstein et al., 2011). The heat kenel signatue function, also known as the auto diffusion function, was poposed as a local suface de-

2 scipto (Sun et al., 2009) o a definition of the scala field (Gbal et al., 2009). This function has been successfully applied fo suface matching (Sipian and Bustos, 2013)(Ganapathi-Subamanian et al., 2016), shape etieval (Bonstein et al., 2011) and shape segmentation (Skaba et al., 2010). Ou method takes advantage of the heat kenel signatue function fo its intinsic popety to descibe a suface. Distinctiveness and epeatability ae consideed as two majo chaacteistics of a 3D detecto (Tombai et al., 2013)(Dutagaci et al., 2012). Repeatability measues the capability of an inteest point detecto to find the same set of inteest points unde vaious nuisances, such as senso noise, missing pats and tansfomations. Distinctiveness measues the ability to detect the most salient and epesentative points on the shape fo featue desciption. Since distinctiveness is a athe global popety of a shape, it is quite challenging to effectively achieve this popety by cuent 3D inteest point detectos, because they ae based on the analysis of local sufaces (Tombai et al., 2013). Inspied by the ecent advances in pesistent homology fo the chaacteization of function behavio (Li et al., 2014)(Caièe et al., 2015), we popose a famewok to combine pesistent homology with the heat kenel signatue function fo inteest point detection. Fist, the scala field is constucted using the heat kenel signatue function at a modeately small time scale. The heat kenel signatue function (Sun et al., 2009) is oiginated fom diffusion geomety, and the signatue is obust to isometic tansfomations. When computed at a small time scale, it is diectly elated to suface cuvatue and has been shown to be infomative (Bonstein et al., 2011). Second, a 0-dimensional pesistence diagam is computed to captue the global stuctual infomation of a suface. The saliency of a point is consideed as its pominence fom the view of topological pesistence. Due to the intinsically global popety of the pesistent homology to descibe the suface vaiations, ou poposed inteest point detecto is highly distinctive. The main contibution of this pape is two-fold. Fist, a new inteest point detecto is poposed fo non-igid 3D suface analysis. Second, pesistence homology is used fo inteest point detection to achieve high epeatability and stong distinctiveness. The emainde of this pape is oganized as follows. Section 2 discusses the elated wok and emphasizes on ou contibution to achieve epeatable and distinctive inteest point detection on non-igid sufaces. Section 3 intoduces ou poposed pesistencebased inteest point detecto. Section 4 pesents a compehensive pefomance evaluation of ou poposed method and compaisons with the state-of-theat. Section 5 concludes the pape. 2 RELATED WORK Ove the past decades, a lage numbe of 3D inteest point detection methods have been poposed fo shape analysis. Most existing inteest point detectos fo 3D shapes concentated on the stability unde igid suface tansfomations (Tombai et al., 2013)(Guo et al., 2014a). In ode to deal with shapes undegoing non-igid defomations, detectos that ae invaiant to isometic tansfomations have also been poposed. The simplest inteest point detection method is fastest point sampling in the geodesic metic space, which is quite popula in this field (Auby et al., 2011)(Litman and Bonstein, 2014)(Xie et al., 2016). Howeve, this method fails to detect qualified inteest points, especially in tems of infomativeness. This is because that it gives no consideation to the ichness of the disciminative infomation of the detected inteest points (Guo et al., 2014a). Seveal appoaches have been poposed to extend inteest point detectos developed in the 2D images to the 3D field. Inspied by the SIFT method (Lowe, 2004), Diffeence-of-Gaussians (DOG) was used as a saliency measue fo inteest point detection fo 3D shapes (Zahaescu et al., 2012)(Liu et al., 2016). The scala field on a shape was defined using a photometic o a geometic attibute and was convolved with a set of Gaussian kenels. Then, DOG calculations wee pefomed on the convolution esults and Mesh- DOG inteest points wee selected as the maximum points in the DOG scale space. This method is able to detect a sufficient numbe of epeatable inteest points. Howeve, it is sensitive to vaying mesh esolutions (Guo et al., 2014a). The popula Hais detecto fo 2D image analysis (Hais and Stephens, 1988) was extended to 3D meshes in (Sipian and Bustos, 2011). The Hais 3D detecto fist deives a quadatic suface fom the neighbohood of one point. Deivatives wee calculated by smoothing ove the suface, and these deivatives wee used to calculate the Hais esponse. A fixed faction of points with the lagest esponse wee selected as inteest points. This method is shown to be obust to seveal tansfomations. Howeve, it uses a fixed-scale of the neighbohood, and does not fully exploit the scale infomation encoded in the local geometic stuctues (Guo et al., 2014a). The diffusion geomety has been applied fo inteest point detection and achieved a good pefo-

3 (a) Input shape (b) Scala field definition (c) Pesistence diagam computation (d) detected keypoints Figue 1: An illustation of the poposed pesistence-based inteest point detecto. (a) An input 3D model of Amadillo. (b) The scala field geneated fo the shape using the auto diffusion function with a paticula time scale. (c) The 0-dimensional pesistence diagam of the pedefined eal-valued function. (d) Inteest points extacted using ou poposed inteest point detecto. mance. It is invaiant to isometic tansfomations and emains stable unde suface petubations. In (Sun et al., 2009), the heat kenel function was esticted to the tempoal domain and used the local maximum of the function to find inteest points. The local maximum is obtained by compaing each point with its 2-ing neighbohoods. This method is able to detect highly distinctive inteest points, but it depends on the mesh esolution (Guo et al., 2014a). Ou method exploits the heat kenel signatue function by combining it with the concept of pesistent homology, which is moe obust than the method poposed in (Sun et al., 2009). All these inteest point detectos ae poposed based on the analysis of the local suface, and they tend to detect points with local maximums using a saliency measue. As a esult, they ae not highly distinctive (Tombai et al., 2013). Ou wok is motivated by the ecent advances in topological data analysis, i.e., the theoy of pesistent homology (Edelsbunne and Hae, 2010)(Chazal et al., 2013). Pesistence homology summaizes the stuctue of a topological space in a compact and povably stable way using a pesistence diagam (Caièe et al., 2015). This appoach has been poved to be stable (Chazal et al., 2009) and has been successfully applied to clusteing (Chazal et al., 2013), shape segmentation (Skaba et al., 2010), shape matching and etieval (Caièe et al., 2015)(Gao and Giachetti, 2016). 3 PROPOSED METHOD Assume M is a compact 2D manifold embedded in R 3 without bounday and mesh M is a discete epesentation of M. M consists of n V vetices, n E edges and n F convex polygons (i.e. facets). M can be viewed as a connectivity gaph G= (V,E), whee the set of vetices V = {v 1,v 2,...,v nv } epesents samples on the manifold, E = {(v i,v j )} epesents the elationship between adjacent vetices. Each vetex v i is associated with a 3D point in the Euclidean space, i.e., v i R 3. M may undego a non-igid tansfomation. Ou goal is to develop an inteest point detecto which identifies 3D inteest points fom M with a high epeatability, stong distinctiveness and a good invaiance to isometic defomations. An illustation of the poposed 3D inteest point detection method is shown in Fig. 1. Ou inteest point detection method follows a typical pipeline (see (Zahaescu et al., 2012) fo example) and compises two steps: scala field constuction and inteest point detection. Fo an input shape, the fist and impotant step is the scala function definition, since its accuacy diectly affects the subsequent pocessing steps. Since ou method is designed fo defomable suface analysis, this definition should be stable unde isometic defomations. We use the heat kenel signatue function (Gbal et al., 2009) with a modeately small time scale fo its emakable esistence to extinsic and intinsic shape vaiations. Figs. 1(a) and (b) epesent the oiginal shape and the coesponding scala field, espectively. Then the 0-dimensional pesistence diagam of this eal-valued function is computed, whee the pominence of the salient points is encoded, as shown in Fig. 1(c). Finally, inteest points ae selected as points with a local maximum value and a high pesistence (as shown in Fig. 1(d)). 3.1 Scala Field Constuction Heat Diffusion Pocess Imagine that thee is an initial heat distibution ove M, and then heat stats to popagate. This heat diffu-

4 sion pocess ove M is govened by the heat equation: ( M + )u(x,t) = 0, (1) t whee the solution u(x, t) epesents the amount of heat at a point x M at time t, and M denotes the positive semi-definite Laplace-Beltami opeato on M, a Riemannian equivalent of the Laplacian. Given an initial heat distibution f : M R, the heat opeato applied to f descibes the distibution of the heat ove M at time t. That is, H t f = u(,t) with u(,0) = f, whee H t epesents the heat opeato. Fo a squae integable function f, thee always exists a function h t (x,y) : R + M M R satisfying H t f (x) = h t (x,y) f (y)dy, (2) M whee dy is the diffeential fom of y M. The minimum function h t (x,y) satisfying Eq. 2 is called the heat kenel. It measues the amount of heat that gets tansfeed fom x to y at time t with a unit heat souce initially located at x. Since the heat opeato is compact, self-adjoint and positive semi-definite (Dey et al., 2010), it has a discete spectum 1 = λ 0 λ with H t ϕ i = λ i ϕ i. Accoding to the Spectal Theoem, the heat kenel has the following eigen-decomposition: h t (x,y) = i 1 e λ it Φ i (x)φ i (y), (3) whee λ i and Φ i ae the ith eigenvalue and eigenfunction of the heat kenel, espectively. The heat diffusion kenel function has a numbe of popeties making it suitable fo suface desciption as a point signatue (Sun et al., 2009). Fist, it is isometically invaiant due to the invaiance of the Laplace-Beltami opeato. It can theefoe be used to analyze shapes unde non-igid defomations. Second, it is infomative as it captues all the intinsic geometic infomation of a shape. Consequently it is able to fully chaacteize shapes up to isomety. Thid, it captues the geometic infomation aound an inteest point with multiple scales. Since heat diffuses pogessively to lage neighbohoods, the time paamete povides an intuitive notion of scale to descibe the local suface. Specifically, the heat kenel h t (x, ) with a small t encodes the local popeties of the suface aound x. As heat dissipates fom the inteest point to the est of the shape, the scale of the neighbohood is inceased Heat Kenel Signatue To make the heat kenel concise and measuable, the heat kenel function was esticted to the tempoal domain only, that is: HKS(x,t) = i 1 e λ it Φ 2 i (x), (4) which is known as the heat kenel signatue (Sun et al., 2009). This signatue has also been poved to inheit many useful popeties fom the heat kenel. Paticulaly, thee is an asymptotic expansion of the heat kenel signatue function as t appoaches 0, that is: HKS(x,t) = (4πt) d/2 a i t i, (5) i=0 whee a 0 = 1 and a 1 = 1 6s(x) with s(x) being the scala cuvatue at point x. Fo a 2D manifold, s(x) is efeed to as the Gaussian cuvatue at point x. This fomula coesponds to the well-known popety of the heat diffusion pocess, that is, heat tends to diffuse faste at points with low cuvatue, and slowe at points with high cuvatue. Moeove, the heat diffusion pocess at an ealy stage is dominated by the intinsic cuvatue of the manifold. Besides, the heat kenel signatue paameteized by a small t povides a meaningful notion of cuvatue fo the local suface aound x Scala Field Constuction We use the heat kenel signatue with a small time scale to constuct the scala field. Examples ae shown in the top ow of Fig. 3. Note that, aeas with high and low Gaussian cuvatues coespond to lage and small values of HKS(x, t), espectively. In addition, the scala field emains stable on the two hoses with non-igid tansfomations. Note that this scala field constuction appoach is also suitable fo incomplete and patial shapes (Dey et al., 2010)(Ovsjanikov et al., 2010), since the heat kenel signatue with small time scales epesents the popeties of a local suface. To futhe enhance its stability, edge points and outlies with extemely high values compaed to its neighbohood ae emoved Computation In pactice, the undelying manifold is unknown as a sample of the shape, and it is usually given in the fom of a tiangula mesh. In ode to obtain the heat kenel signatue, the Finite Elements Method (Reute et al., 2006) is used to compute the eigenvalues and eigenvectos of the Laplace-Beltami opeato associated with the shape.

5 p q q p s s Figue 2: An illustation of the 0-dimensional pesistence diagam computation. Left: a smooth function with thee local maximums. Right: 0-dimensional pesistence diagam of the function. 3.2 Inteest Point Detection Pesistence Diagam Let f be a eal-valued function defined on the nodes of G, f : V R. A node v V is called a peak if its function value is a local maximum in its local neighbohood, i.e., fo all u with (u,v) E, f (v) f (u). The 0-dimensional pesistence diagam encodes the topological changes of the sub-level sets F α = f 1 ([α,+ )) as α deceases fom + to. Let C(v,α) be a connected subgaph, C(v,α) F α G, whee α f (v) and f (v) is the lagest value. The subgaph is claimed to be bon at v, o v epesents the component. The infimum of α with which f (v) emains to be the global maximum in the component is called the death of C(v,α). As α continues to decease, the subgaph C(v,α) is meged into anothe component. The lifespan of a component is thus detemined by its bith f (v) and death values f (u) f (v). This allows us to poject each component onto a point ( f (u), f (v)) on a 2D plane. Pesistence o pominence of the component is defined as its lifespan, P = f (v) f (u) 0. The 0-dimensional pesistence diagam of f is computed by collecting all these points togethe. In the diagam, all points live in the half space above x 1 = x 2, and thei pesistence can easily be calculated as the vetical distance fom the point ( f (u), f (v)) to this diagonal line (Edelsbunne and Hae, 2010). Figue 2 shows an illustation of the 0- dimensional pesistence diagam computation. It can be seen that the topological infomation of a smooth function can be captued by the 0-dimensional pesistence diagam Inteest Point Detection Simila to existing 3D featue detectos, inteest points ae selected by consideing local extema of a saliency measue. The saliency measue detemines the type of local suface to be selected by the detecto, and it is significantly impotant to the distinctiveness and epeatability of the detecto. We conside that the saliency of a point is not only detemined by its absolute saliency measue, but also by its elative significance as compaed to its neighboing points. Fom the view of topological data analysis, inteest points ae detemined as those points with local maxima and in paticula, high pesistence. Since each component is epesented by a peak, the 0-dimensional pesistence diagam can be explained in anothe way (Li et al., 2014). That is, the pesistence diagam encodes the elative pominence of diffeent peaks of a given function by consideing the connectivity infomation in the domain, as shown in Fig. 2. Based on the pedefined scala function (i.e., heat kenel signatue with a small time scale), the pesistence diagam is able to select seveal peaks with a high pesistence, all of them ae distinctive points on a shape (as shown in Fig. 1(d)). In addition, the pesistence diagam has seveal impotant popeties and it is theefoe highly suitable fo inteest point detection. Fist, it is stable unde petubations of the scala function (Caièe et al., 2015)(Chazal et al., 2009). Consequently, the pesistence diagam is highly obust to noise. Besides, it is invaiant to tanslation, otation, scaling and nonigid tansfomations (as shown in Fig. 3), as long as the scala functions ae consistent acoss shapes Computation In pactice, the 0-dimensional pesistence diagam can be computed using the Union-Find algoithm (Comen, 2009). It anks the nodes of the gaph with espect to thei scala function values, and keeps tack of the evolution of the coesponding connected components. In this pape, DIPHA (Baue et al., 2014) is used to compute the pesistence diagam.

6 Figue 3: An illustation of the scala fields on thee shape modes fom the TOSCA dataset (two hoses with non-igid tansfomations and a cat), and thei coesponding 0-dimensional pesistence diagam. Top: Heat kenel signatue with a small and fixed time scale computed on these models. Note that values of the heat kenel signatue function incease while the colo on the shape changes fom blue to geen and then to ed. Bottom: The coesponding pesistence diagam computed fom the eal-valued function. 4 EXPERIMENTS shown in Fig. 4. A compehensive evaluation is pefomed to test ou inteest point detecto unde two diffeent scenaios. Section 4.1 shows its pefomance evaluation on the PHOTOMESH dataset (Zahaescu et al., 2012) in tems of epeatability. In section 4.2, the pefomance of ou poposed inteest point detecto is compaed with seveal state-of-the-at methods on the Inteest Points dataset (Dutagaci et al., 2012) in tems of distinctiveness, i.e., to measue how much the extacted points ae compatible with human peception Evaluation Methodology 4.1 whee gd(v, p) epesents the geodesic distance between v and p. Suppose the gound-tuth one-to-one coespondence (MN = GT (MTi )) is known as a pio, a point v extacted fom a tansfomed model MTi is consideed to be coectly detected if its coesponding point GT (v) is located within the geodesic ball defined by N (v ), whee v is an inteest point detected on MN. Repeatability is calculated as the pecentage of coectly detected inteest points. In this expeiment, we used the same setting as (Zahaescu et al., 2012), whee was set to be 1% of the suface aea. We used the fist 100 eigenvalues and eigenvectos of the Laplace-Beltami opeato on each shape to compute the heat kenel signatue, and the time paamete was set to 0.1. Five inteest points with the highest pominence on two hands, feet and head wee detected, which wee consideed as epesentatives of the human shape by ou method. Fo a fai compaison, the esults achieved by the MeshDOG appoach Pefomance on The PHOTOMESH Dataset Dataset The PHOTOMESH dataset consists of thee base shapes, so called null shapes. The simulated tansfomations applied to these null shapes can be classified into two categoies: photometic tansfomations (Scala-noise and Scala-shot noise) and geometic tansfomations (noise, shot noise, otation, scale, local scale, sampling, holes, mico-holes, topology and isomety). Five levels of stength ae applied to each tansfomation. Totally, 65 shapes ae poduced fo each null shape, and thee ae 135 shapes in the dataset. Since ou method is designed to encode the geometic infomation, the shapes with photometic tansfomations wee not consideed in this expeiment. Examples of all possible tansfomations ae Expeiments wee conducted on this dataset to test the epeatability of each featue detecto. The pefomance was quantitatively measued by compaing the inteest points detected on each tansfomed shape MTi and its coesponding null shape MN. Fo a point v on a shape M, its local neighbohood within geodesic adius is defined as N (v) = {p M gd(v, p) }, (6)

7 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figue 4: Examples of all possible tansfomations of a null shape on the PHOTOMESH dataset. (a) Scala-noise (b) Scalashot noise (c) Noise (d) Shot noise (e) Holes (f) Mico-holes (g) Isomety (h) Local scale (i) Rotation (j) Sampling (k) Scale (l) Topology. (Zahaescu et al., 2012) ae also epoted since this method achieves a good pefomance in tems of epeatability on the PHOTOMESH dataset. The Mesh- DOG inteest point detecto was implemented with thee diffeent scala fields defined ove the manifolds, including colo intensity, mean cuvatue and Gaussian cuvatue Results and Discussion Compaative esults on the epeatability of theses detectos ae pesented in Tables 1-4. In the cases of noise, shot noise and sampling, the pefomance of MeshDOG is deceased as the level of tansfomation inceases, especially fo the Mesh- DOG detectos geneated with geometic scala fields (i.e., mean cuvatue and Gaussian cuvatue). In contast, ou method achieves bette esults. The epeatability achieved by ou phks method emains to be 1 unde all levels of noise, shot noise and sampling tansfomations. This is because the scala field used in ou method is moe stable than the scala field used in MeshDOG. Fo the tansfomations of otation, scale, local scale and isomety, both MeshDOG and phks inteest point detectos can successfully find the same set of inteest points. Those esults clealy show that MeshDOG and phks methods ae invaiant to igid and non-igid tansfomations. Both holes and mico holes tansfomations decease the pefomance of MeshDOG linealy. In contast, ou method poduces good esults since the constucted scala field is able to handle patial and incomplete shapes. In some cases, ou appoach cannot poduce a full epeatability of 0 because seveal inteest points ae missing due to the pesence of holes. In the pesence of topological tansfomations, the pefomance of MeshDOG is significantly deceased. Fo example, the epeatability achieved by MeshDOG with a Gaussian cuvatue is 5 unde the fist level of topological tansfomation. Its epeatability is then deceased to 4 unde the fifth level of topological tansfomation. In contast, ou appoach achieves a vey high epeatability, as shown in Table 4. This is mainly due to the definition of saliency of inteest point, which is not only a local maximum but also has a lage pesistence, i.e., a stong contast as compaed to its neighbohood in a connected topological space. Theefoe, although the whole shape is tansfomed into seveal stuctues, the peaks emain the same unde ou scala field constuction. The inteest points ae successfully selected as long as thei pesistences ae not geatly changed. In some cases (e.g., with the thid level of topological tansfomation), the esult is not satisfactoy because the pesistences of

8 some peaks ae destoyed by the topological tansfomations. Table 1: Repeatability of MeshDOG (photometic). Stength Tansfom. 1 < 2 < 3 < 4 < 5 Noise Shot Noise Rotation Scale Local Scale Sampling Holes Mico-Holes Topology Isomety Table 2: Repeatability of MeshDOG (mean cuvatue). Stength Tansfom. 1 < 2 < 3 < 4 < 5 Noise Shot Noise Rotation Scale Local Scale Sampling Holes Mico-Holes Topology Isomety Table 3: Repeatability of MeshDOG (Gaussian cuvatue). Stength Tansfom. 1 < 2 < 3 < 4 < 5 Noise Shot Noise Rotation Scale Local Scale Sampling Holes Mico-Holes Topology Isomety Pefomance on The Inteest Points Dataset Dataset This dataset consists of two sub-datasets (dataset A and B) (Dutagaci et al., 2012). It is epoted that Table 4: Repeatability of the poposed method (phks). Stength Tansfom. 1 < 2 < 3 < 4 < 5 Noise Shot Noise Rotation Scale Local Scale Sampling Holes Mico-Holes Topology Isomety the pefomance of existing algoithms is consistent acoss the two datasets (Dutagaci et al., 2012), so only dataset B is used in this pape as it is much lage. In dataset B, thee ae 43 models manually maked by 16 humans. Gound-tuth points ae constucted fom the human-maked points based on two citeia: adius of an inteest egion σ and the numbe of uses n who maked a point within the adius. In addition to the location, the pominence of a gound-tuth point is included as the numbe of humans who have maked it within its local neighbohood. An example of inteest points detected on a model fom dataset B ae shown in Fig Evaluation Methodology On this dataset, evaluation was pefomed to test the compatibility of each algoithm with human peception. This expeiment is designed to test an inteest point detecto in tems of distinctiveness, which demonstates the ability of an inteest point to detect epesentative and chaacteistic points on a suface. Ou evaluation was pefomed on each single instance of the model using human geneated goundtuth. Thee measues ae used as in (Dutagaci et al., 2012), i.e., false positive eos (FPE), false negative eos (FNE) and weighted miss eo (WME). Let G M (n,σ) be the set of gound-tuth points on a model M and D M be the set of inteest points detected by an algoithm. A point v is consideed to be coectly detected if thee is a detected point within the geodesic ball N (v), whee the paamete is the localization eo toleance. Given the numbe of gound-tuth points N GT, the points detected by an algoithm N D and the coectly detected points N C, FNE, FPE and WME ae defined as follows: FNE() = 1 N C N GT, (7) FPE() = N D N C N D, (8)

9 (a) (b) (c) (d) (e) (f) (g) Figue 5: An illustation of inteest points detected by diffeent algoithms on Amadillo fom Inteest Points Dataset B (a) gound-tuth points (b) Mesh saliency (c) Salient points (d) 3D-Hais (e) SD-cones (f) HKS (g) phks WME() = 1 1 N GT N GT i=1 n n i δ, (9) i i=1 whee δ i is set to 1 if the gound-tuth point is detected. Othewise, it is set to 0. n i is defined as the pominence of an inteest point v i, which is equal to the numbe of humans voting fo that point. In this expeiment, we used the fist 100 eigenvalues and coesponding eigenvectos of the Laplace- Beltami opeato to compute heat kenel signatue and its time paamete was set to 01. Fo compaison, we pesent the esults achieved by five state-ofthe-at methods including mesh saliency (Lee et al., 2005), salient points (Godil and Wagan, 2011), 3D- Hais (Patikakis et al., 2010), SD-cones (Novatnack and Nishino, 2007) and HKS (Sun et al., 2009) Results and Discussion The compaative esults ae pesented in Fig. 6. It can be seen fom the esults that mesh saliency, salient points, 3D-Hais, SD-cones and HKS ae able to achieve low FNE and WME measues at the cost of a high FPE. Paticulaly, when σ = 3 and n = 8, these methods can find almost 80% of the goundtuth points. Howeve, 90% of thei detected points ae false. In contast, the aveage FNE and WME measues achieved by HKS ae extemely high, which means that a lage pecentage of the gound-tuth inteest points ae not detected by the HKS method. On the othe hand, HKS pefoms the best on FPE. Fo example, when σ = 5 and n = 2, the FNE and WME scoes of HKS ae about, and the aveage FPE is as low as. This is because HKS tends to select the extemities of the models, which ae often of geat inteest to humans. Ou appoach poduces an oveall good pefomance as a tade-off is achieved between FNE, FPE and WME. As the localization eo toleance inceases, ou poposed phks can detect about 80% of the whole gound-tuth points with fa less false positives as compaed to the othe methods. This means that the set of points detected by phks is moe concise and effective. This esult coesponds to the fact that humans usually mak inteest points coesponding to extemities of 3D shapes (Dutagaci et al., 2012). Although the points detected by HKS usually have high pominence, the numbe of detected inteest points is too limited. That is because HKS selects points with lage values and as local maxima, and the scale of points has to be pedefined (Sun et al., 2009). Ou phks method is able to detect moe points, because we select inteest points with a stong contast to thei neighbohood and the scale of the neighbohood is adaptively detemined using topological data analysis. 5 CONCLUSION In this pape, we pesent a method fo inteest point detection on 3D non-igid shapes using topological pesistence. Ou appoach simultaneously captues the local geometic and global stuctual infomation of the suface. It extacts inteest points with a high epeatability and a stong distinctiveness. Ou expeiments on two public datasets have clealy demonstated the effectiveness of ou appoach. In the futue, we will combine ou inteest point detecto with a local suface descipto and test it on applications such as suface egistation, shape tacking and shape etieval of non-igid objects. ACKNOWLEDGEMENTS This eseach is suppoted by a China Scholaship Council (CSC) scholaship (No ), Austalian Reseach Council gants (DE , DP and DP ), the National Natual Science Foundation of China (Nos and ) and the Innovative Postdoctoal Talent Pogam of China (No ).

10 Aveage FNE Aveage WME (a) σ = 3 and n = 2 Aveage FPE Mesh saliency Salient points 3D-Hais SD-cones HKS phks Aveage FNE Aveage WME (b) σ = 3 and n = 8 Aveage FPE Mesh saliency Salient points 3D-Hais SD-cones HKS phks Aveage FNE Aveage WME (c) σ = 5 and n = 2 Aveage FPE Mesh-saliency Salient points 3D-Hais SD-cones HKS phks Avaage FNE Aveage WME Aveage FPE Mesh saliency Salient points 3D-Hais SD-cone HKS phks (d) σ = 5 and n = 8 Figue 6: Pefomance on Dataset B (43 Models, 16 Subjects) of the Inteest Points Dataset. False negative eo (fist column), weighted miss eo (second column), and false positive eo (thid column) ae shown in the figue. The settings fo gound-tuth geneation ae given unde the plots, whee stands fo the localization eo toleance.

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