Integral Invariants for Shape Matching

Abstract

Abstract-For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential counterparts, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (asymptotically), they do not exhibit the noise sensitivity associated with differential quantities and, therefore, do not require presmoothing of the input shape. Our formulation allows the analysis of shapes at multiple scales. Based on integral invariants, we define a notion of distance between shapes. The proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other; its computation results in optimal point correspondence as an intermediate step.Numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts, missing parts and noise. As a quantitative analysis, we report matching scores for shape retrieval from a database.

Shape localization, quantification and correspondence using Region Matching Algorithm

“…As mentioned earlier our method is based on the II scale space, which is inspired from Ref. [24] and eventually provides the basis for the RMA that we develop.…”

“…II, which are fundamentally related to differential invariants, can be used effectively for shape representation [21] and reconstruction, [22] and are robust to noise. [21,23,24] Al-Kadi et al also measured feature robustness under noise presence in medical images and is reported in Refs. [25][26][27] II outperform differential invariants [28,29] for invariance to small perturbations in shapes.…”

Shape matching by integral invariants on eccentricity transformed images

“…Hong defines (circular area) integral invariants in [5] by considering a disc $ å :L; of radius N applied to every point L Ð 5 of a closed contour ò5á that is the boundary of the shape 5. The characteristic function is then given by,…”

“…It is usually performed by producing a shape signature, which ideally is invariant to rigid or isometric transformations, such as, articulations, bending, translation, and rotation. Here, we combine two such techniques, the continuous eccentricity transform [1][2][3][4]12] and integral invariant signatures [5][6]. A detailed review of shape representation [7], matching and description techniques and categorical classification is given in [8].…”

“…Conversely, integral invariants are shape descriptors that are preferred to curvature for their robustness to noise, and have been used effectively for shape matching [5] applications and for dividing shapes into further regions [6] to quantify occlusion and new growth. However, a fundamental problem with integral invariants is that they relate only to the boundary and do not take into account the information from inside the shape.…”