Similarity Domains Machine for Scale-Invariant and Sparse Shape Modeling

We present an approach to extend the functionality and the use of kernel machines in image processing applications. We introduce a novel way to design spatial kernel machines with spatial properties and demonstrate how those newly introduced spatial properties enhance the possibilities of the use of kernel machines in image processing applications as a proof of concept. In this paper, we demonstrate four particular extensions: 1) how to model shapes efficiently with spatially computed kernel parameters in a geometrically scalable way; 2) how to visualize the kernel parameters precisely and intuitively on binary 2D shapes; 3) how to construct a one-class classifier from the binary classifier in a straightforward manner without re-training; and 4) how to use the computed kernel parameters for filtering. The existing literature on kernel machines mostly focuses on estimating the optimal kernel parameters via additional cost function(s). In this paper, instead of employing an additional cost function to estimate the kernel-related parameters, we investigate on an analytical solution to predict the actual kernel parameters locally and show how to build a spatial kernel machine with our analytical approach. Classical kernel machines do not perform well on precise shape modeling with a low number of support vectors as demonstrated in this paper. However, we demonstrate and visualize that our analytical approach provides a natural means to relate the kernel parameters to the 2D shapes for sparse shape modeling, where the shape boundary represents the decision boundary. For that, we incorporate the selected kernel function's geometric properties as an additional constraint into the classifier's optimization problem by defining an easy-to-explain and intuitive concept: similarity domains. In our experiments, we study and demonstrate how the resulting new kernel machine enhances the capabilities of the classical kernel machines with applications on shape modeling, (geometrically) scaling the non-linear decision boundary at various scales and precise visualization of the kernel parameters in 2D images.