Deformation Based Curved Shape Representation

Abstract

Abstract-In this paper, we introduce a deformation based representation space for curved shapes in R n . Given an ordered set of points sampled from a curved shape, the proposed method represents the set as an element of a finite dimensional matrix Lie group. Variation due to scale and location are filtered in a preprocessing stage, while shapes that vary only in rotation are identified by an equivalence relationship. The use of a finite dimensional matrix Lie group leads to a similarity metric with an explicit geodesic solution. Subsequently, we discuss some of the properties of the metric and its relationship with a deformation by least action. Furthermore, invariance to reparametrization or estimation of point correspondence between shapes is formulated as an estimation of sampling function. Thereafter, two possible approaches are presented to solve the point correspondence estimation problem. Finally, we propose an adaptation of k-means clustering for shape analysis in the proposed representation space. Experimental results show that the proposed representation is robust to uninformative cues, e.g. local shape perturbation and displacement. In comparison to state of the art methods, it achieves a high precision on the Swedish and the Flavia leaf datasets and a comparable result on MPEG-7, Kimia99 and Kimia216 datasets.

Deformation-Based Abnormal Motion Detection using 3D Skeletons

“…Subsequently, we use a deformation-based distance function, as defined in [18], [19], for measuring the difference between two given motion representations; a brief description of the considered distance function is given below.…”

“…Thereafter, following [18], [19], the distance between two curve representations C a,ϕ and C b,ϕ is defined as…”

“…such that R i and v i are rotation matrices and translation vectors in R 3 , respectively. Hence, the distance between two curves G(M a ) and G(M b ) that are represented by C a,ϕ and C b,ϕ , respectively, can be computed using (6), see [18], [19] for further details.…”

“…We model a normal motion sequence using the sample mean of the training dataset. Consequently, we compute the sample mean of a curve representation dataset [18], [19], as…”

“…The latter are then used to compute a series of sub-distances. This approach is different from a simple linearization as proposed in [19]. We define the motion-quality distance such that the variation in time is emphasized, while the optimal alignment proposed in [19] emphasizes invariance to variation in time.…”