Rotation Invariant Kernels and Their Application to Shape Analysis

• Published in: IEEE transactions on pattern analysis and machine intelligence 2009-11, Vol.31 (11), p.1985-1999
• Main Authors: Hamsici, O.C., Martinez, A.M.
• Format: Article
• Language: English
• Subjects: Algorithms; Applied sciences; Artificial Intelligence; Computer science; control theory; systems; Computer vision; Exact sciences and technology; Face recognition; handshape; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Imaging, Three-Dimensional - methods; Invariance; Invariants; Kernel; kernel functions; Kernels; LB1; Least squares approximation; Machine learning; Mathematical analysis; Mathematical models; Multi-stage noise shaping; Nonlinearity; Object recognition; Parameter estimation; Pattern Recognition, Automated - methods; Pattern recognition. Digital image processing. Computational geometry; Probability density function; Reproducibility of Results; Rotation; rotation invariance; Sensitivity and Specificity; Shape; Shape analysis; spherical-homoscedastic distributions; Studies; Translations; Vectors (mathematics)
• ISSN: 0162-8828; 1939-3539
• DOI: 10.1109/TPAMI.2008.234

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 modeling 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 modeling and classification problems demonstrates the accuracy of this proposed approach.