Rotation Invariant Kernels and Their Application to Shape Analysis

Abstract

Shape analysis requires invariance under translation, scale and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors can be made rotational invariant by modelling the resulting data using complex scalar rotation invariant distributions defined on the complex hypersphere, e.g., using the complex Bingham distribution. However, the use of these distributions is hampered by the difficulty in estimating their parameters and the nonlinear nature of their formulation. In the present paper, we show how a set of kernel functions, that we refer to as rotation invariant kernels, can be used to convert the original nonlinear problem into a linear one. As their name implies, these kernels are defined to provide the much needed rotation invariance property allowing one to bypass the difficulty of working with complex spherical distributions. The resulting approach provides an easy, fast mechanism for 2D & 3D shape analysis. Extensive validation using a variety of shape modelling and classification problems demonstrates the accuracy of this proposed approach.

Bayesian estimation of a subspace

“…There exist only a few distributions on the Stiefel or Grassmann manifolds [7], [8], the most popular being the Bingham or von Mises Fisher (vMF) distributions which have proven to be relevant in a number of applications including meteorology, biology, medicine, image analysis (see [7] and references therein), modeling of multipath communications channels [9] and shape analysis [10]. These distributions depend on a matrix whose range space is "close" to that of U , along with a concentration parameter that rules the distance between the subspaces.…”

Complex representations for learning statistical shape priors

“…The problem of shape representation has created a wealth of research in the fields of computer vision and pattern recognition [1], [2], [3], [4], [5]. Seminal first works included analysis of shapes via the use of non-linear morphological operators [1], [2].…”

“…In the early 2000s, the seminal shape representation for matching was the, so-called, shape context [3], which employs a histogram-based manner in order to describe the coarse arrangement of the shape with respect to a point that lies either inside or on the boundary of the shape. Recently, a very interesting method was proposed [4] that represents the shape as a non-linear surface, learned via the application of one-class Support Vector Machine (SVM), as well as a method to design rotation-invariant kernels for shape matching [5]. Additionally, state-of-the-art discriminative shape representations learned via the application of deep learning strategies [6] have been currently employed.…”

“…To the best of our knowledge, the complex shape representations have only been used in order to achieve rotation invariance for the task of shape matching [5] and fitting [11]. In these works, in order to achieve rotation invariance the shape feature vectors need to be described using the complex Bingham distribution [12].…”

A Geometric Invariant Shape Descriptor Based on the Radon, Fourier, and Mellin Transforms

“…Contour-based shape descriptors are extracted from the shape contour exploiting its boundary information. Important extraction techniques include 1D Fourier transform [2], curvature scale-space [3], 2D histogram of neighboring contour pixels [4], inner-distance in the shape silhouette [5], and rotation invariant kernel [6]. In spite of their popularity, contour-based shape descriptors are applicable only to certain kinds of application due to several limitations.…”