XY-Separable Scale-Space Filtering by Polynomial Representations and Its Applications

Abstract

In this paper, we propose the application of principal component analysis (PCA) to scale-spaces. PCA is a standard method used in computer vision. Because the translation of an input image into scale-space is a continuous operation, it requires the extension of conventional finite matrix-based PCA to an infinite number of dimensions. Here, we use spectral theory to resolve this infinite eigenvalue problem through the use of integration, and we propose an approximate solution based on polynomial equations. In order to clarify its eigensolutions, we apply spectral decomposition to Gaussian scale-space and scale-normalized Laplacian of Gaussian (sLoG) space. As an application of this proposed method, we introduce a method for generating Gaussian blur images and sLoG images, demonstrating that the accuracy of such an image can be made very high by using an arbitrary scale calculated through simple linear combination. Furthermore, to make the scale-space filtering efficient, we approximate the basis filter set using Gaussian lobes approximation and we can obtain XY-Separable filters. As a more practical example, we propose a new Scale Invariant Feature Transform (SIFT) detector.