2D Affine and Projective Shape Analysis

Abstract

Current techniques for shape analysis tend to seek invariance to similarity transformations (rotation, translation, and scale), but certain imaging situations require invariance to larger groups, such as affine or projective groups. Here we present a general Riemannian framework for shape analysis of planar objects where metrics and related quantities are invariant to affine and projective groups. Highlighting two possibilities for representing object boundaries-ordered points (or landmarks) and parameterized curves-we study different combinations of these representations (points and curves) and transformations (affine and projective). Specifically, we provide solutions to three out of four situations and develop algorithms for computing geodesics and intrinsic sample statistics, leading up to Gaussian-type statistical models, and classifying test shapes using such models learned from training data. In the case of parameterized curves, we also achieve the desired goal of invariance to re-parameterizations. The geodesics are constructed by particularizing the path-straightening algorithm to geometries of current manifolds and are used, in turn, to compute shape statistics and Gaussian-type shape models. We demonstrate these ideas using a number of examples from shape and activity recognition.

Bayesian Active Contours with Affine-Invariant, Elastic Shape Prior

“…The work of Bryner et al [4] incorporates recent simplifications for elastic shape analysis provided in [15] for a computational speed-up. The works of [8] and [4] only formulate an intrinsic similarityinvariant shape prior rather than allowing for an affineinvariant, elastic shape model, which has been developed in [3] and [2].…”

“…After reviewing the current literature, we find that (i) while item (1) was introduced in both [8] and [4], these papers did not have items (2) and (3); (ii) item (2) -elastic, affine-invariant shape analysis -was introduced in [3] and developed further in [2], but those papers did not have items (1) or (3); (iii) item (3) was proposed in [17] but without either (1) or (2). To reiterate, there is no paper currently in the literature that performs even two of the three items together, which underlines the novelty of our approach.…”

Characteristic Number: Theory and Its Application to Shape Analysis

“…An affine transformation is also acceptable if objects are distant from the camera compared with the focal length of the camera. Therefore, invariants to projective and affine transformations are quite important to computer vision problems, especially for shape analysis, including shape representation and matching [2,14]. Figure 1.…”

“…Researchers also build robust constraints upon these invariants in order to match geometric primitives between images, such as points [5,9], lines [10,11] and closed contours [12,13]. In a recent work, Bryner et al derive novel metrics on geometric invariants to affine and projective groups in a general Riemannian framework and develop shape analysis algorithms for both point sets and parametric curves [14]. In the context of facial analysis, Riccio and Dugelay devise features for recognition based on 2D/3D geometric invariants [15].…”

Shape Projectors for Landmark-Based Spline Curves

“…They are popular for interactive applications, where a user manually enforces landmark correspondence to increase robustness. Other applications, where invariance to parameterization is required have been studied in [17]- [19]. …”