Fast image alignment in the Fourier domain

In this paper we propose a framework for gradient descent image alignment in the Fourier domain. Specifically, we propose an extension to the classical Lucas & Kanade (LK) algorithm where we represent the source and template image´s intensity pixels in the complex 2D Fourier domain rather than in the 2D spatial domain. We refer to this approach as the Fourier LK (FLK) algorithm. The FLK formulation is especially advantageous, over traditional LK, when it comes to pre-processing the source and template images with a bank of filters (e.g., Gabor filters) as: (i) it can handle substantial illumination variations, (ii) the inefficient pre-processing filter bank step can be subsumed within the FLK algorithm as a sparse diagonal weighting matrix, (iii) unlike traditional LK the computational cost is invariant to the number of filters and as a result far more efficient, (iv) this approach can be extended to the inverse compositional form of the LK algorithm where nearly all steps (including Fourier transform and filter bank pre-processing) can be pre-computed leading to an extremely efficient and robust approach to gradient descent image matching. We demonstrate robust image matching performance on a variety of objects in the presence of substantial illumination differences with exactly the same computational overhead as that of traditional inverse compositional LK during fitting.