Scale-space Processing Using Polynomial Representations

In this study, we propose the application of principal components analysis (PCA) to scale-spaces. PCA is a standard method used in computer vision. The translation of an input image into scale-space is a continuous operation, which requires the extension of conventional finite matrix-based PCA to an infinite number of dimensions. In this study, we use spectral decomposition to resolve this infinite eigenproblem by integration and we propose an approximate solution based on polynomial equations. To clarify its eigensolutions, we apply spectral decomposition to the Gaussian scale-space and scale-normalized Laplacian of Gaussian (LoG) space. As an application of this proposed method, we introduce a method for generating Gaussian blur images and scale-normalized LoG images, where we demonstrate that the accuracy of these images can be very high when calculating an arbitrary scale using a simple linear combination. We also propose a new Scale Invariant Feature Transform (SIFT) detector as a more practical example.