Mathematical aspects of shape analysis for object recognition

Abstract

In this paper we survey some of the mathematical techniques that have led to useful new results in shape analysis and their application to a variety of object recognition tasks. In particular, we will show how these techniques allow one to solve a number of fundamental problems related to object recognition for configurations of point features under a generalized weak perspective model of image formation. Our approach makes use of progress in shape theory and includes the development of object-image equations for shape matching and the exploitation of shape space metrices (especially object-image metrics) to measure matching up to certain transformations. This theory is built on advanced mathematical techniques from algebraic and differential geometry which are used to construct generalized shape spaces for various projection and sensor models. That construction in turn is used to find natural metrics that express the distance (geometric difference) between two configurations of object features, two configurations of image features, or an object and an image pair. Such metrics are believed to produce the most robust tests for object identification; at least as far as the object's geometry is concerned. Moreover, these metrics provide a basis for efficient hashing schemes to do identification quickly, and they provide a rigorous foundation for error and statistical analysis in any recognition system. The most important feature of a shape theoretic approach is that all of the matching tests and metrics are independent of the choice of coordinates used to express the feature locations on the object or in the image. In addition, the approach is independent of the camera/sensor position and any camera/sensor parameters. Finally, the method is also independent of object pose or image orientation. This is what makes the results so powerful. Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx SPIE-IS&T/ Vol. 6508 65080E-2 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx SPIE-IS&T/ Vol. 6508 65080E-3 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx SPIE-IS&T/ Vol. 6508 65080E-5 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx SPIE-IS&T/ Vol. 6508 65080E-8 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx

Object-Image Correspondence for Algebraic Curves under Projections

“…One of the essential contributions of [1,2] is the definition of an object/image distance between ordered sets of m points in R 3 and R 2 , such that the distance is zero if and only if these sets are related by a projection. Since, in practice, we are given only an approximate position of points, a "good" object/image distance provides a tool for deciding whether a given set of points in R 2 is a good approximation of a projection of a given set of points in R 3 .…”

“…A rational action of an algebraic group G on R 2 can be prolonged to an action on the n-th jet space J n = R n+2 with coordinates (x, y, y (1) , . .…”

“…The method presented in [15] also uses an additional assumption that a planar curve X ⊂ R 2 has at least two points, whose tangent lines coincide. An alternative approach to the problem in the case when Z and X are finite lists of points under parallel projections was presented in [1,2]. In these articles, the authors establish polynomial relationships that have to be satisfied by coordinates of the points in the sets Z and X in order for a projection to exists.…”

“…, 4, are real parameters of the projection, such that the left 3 × 3 submatrix of 3 × 4 matrix P = (p ij ) has a non-zero determinant. Parameters represent the freedom to choose the center of the projection, the position of the image plane and (in general, non-orthogonal) coordinate system on the image plane 1 . In the case when the distance between a camera and an object is significantly greater than the object depth, a parallel projection provides a good camera model.…”

“…We use the same letter P to denote the 3 × 4 matrix P = (pij) and the map P : R 3 − → R 2 defined by (1). Hence P (Z) = {(x, y) ∈ R 2 | ∃ z ∈ Z such that (1) holds}.3 In the case of parallel projection, when the center is at infinity, the conditions are on the direction of the projection.6 Parallel projections are also called generalized weak perspective projections [1,2]. 7 We will occasionally include a field in the group-notation, e.g.…”