Local All-Pass Geometric Deformations

This paper deals with the estimation of a deformation that describes the geometric transformation between two images. To solve this problem, we propose a novel framework that relies upon the brightness consistency hypothesis-a pixel's intensity is maintained throughout the transformation. Instead of assuming small distortion and linearizing the problem (e.g. via Taylor Series expansion), we propose to interpret the brightness hypothesis as an all-pass filtering relation between the two images. The key advantages of this new interpretation are that no restrictions are placed on the amplitude of the deformation or on the spatial variations of the images. Moreover, by converting the all-pass filtering to a linear forward-backward filtering relation, our solution to the estimation problem equates to solving a linear system of equations, which leads to a highly efficient implementation. Using this framework, we develop a fast algorithm that relates one image to another, on a local level, using an all-pass filter and then extracts the deformation from the filter-hence the name "Local All-Pass" (LAP) algorithm. The effectiveness of this algorithm is demonstrated on a variety of synthetic and real deformations that are found in applications, such as image registration and motion estimation. In particular, when compared with a selection of image registration algorithms, the LAP obtains very accurate results for significantly reduced computation time and is very robust to noise corruption.