Euclid meets Fourier : Applying harmonic analysis to essential matrix estimation in omnidirectional cameras

By combining notions from geometry, signal processing and harmonic analysis, we propose a new method for the estimation of the motion between two omnidirectional cameras. We show that a densely sampled likelihood function can be obtained on the space of essential matrices via a convolution of two signals. The first signal expresses the epipolar geometry of two views, and the second signal encodes the similarity of intensities (or some other measure) between a pixel in one image and a pixel in another image. The proposed method is analogous to a Hough or Radon transform on the space of essential matrices, and is a first step to integrating signal processing and geometry. For computational reasons, we are not aware of researchers attempting a Hough transform on the space of essential matrices, so we are not aware of similar work. Nevertheless, there are some similarities between the proposed method and the recent work of Makadia and Daniilidis [1] and Wexler et al. [2]. In the former case the authors propose rotation estimation using a shift theorem in SO(3), and the latter investigates the estimation of arbitrary epipolar geometries. The breakthrough in this paper is that we can efficiently compute the convolution using spherical and rotational harmonic representations of the signals. Estimation using the proposed method has several advantages: we can automatically represents ambiguities; we are able to estimate multiple motions; and we obtain a framework which can take into account arbitrary, non-Gaussian sensor noise models such as simple blob correspondence.