A Study of Image Analysis with Tangent Distance

The computation of the geometric transformation between a reference and a target image, known as registration or alignment, corresponds to the projection of the target image onto the transformation manifold of the reference image (the set of images generated by its geometric transformations). It, however, often takes a nontrivial form such that the exact computation of projections on the manifold is difficult. The tangent distance method is an effective algorithm to solve this problem by exploiting a linear approximation of the manifold. As theoretical studies about the tangent distance algorithm have been largely overlooked, we present in this work a detailed performance analysis of this useful algorithm, which can eventually help its implementation. We consider a popular image registration setting using a multiscale pyramid of lowpass filtered versions of the (possibly noisy) reference and target images, which is particularly useful for recovering large transformations. We first show that the alignment error has a nonmonotonic variation with the filter size, due to the opposing effects of filtering on both manifold nonlinearity and image noise. We then study the convergence of the multiscale tangent distance method to the optimal solution. We finally examine the performance of the tangent distance method in image classification applications. Our theoretical findings are confirmed by experiments on image transformation models involving translations, rotations and scalings. Our study is the first detailed study of the tangent distance algorithm that leads to a better understanding of its efficacy and to the proper selection of its design parameters.