Laplacian Operator, Diffusion Flow and Active Contour on non-Euclidean Images

Our goal is to study image processing techniques for 360-degree images, which are obtained from omni-directional sensors [11, 17]. Having a curved mirror, i.e. hyperbolic, spherical or parabolic, as a lens for the corresponding catadioptric system, we obtain non-Euclidean images. One way of processing such an image is to perform a panoramic projection of this image onto a cylinder. In this way one unfolds the 360-image to the usual 2d image and only after this the usual image processing techniques are performed. We propose here some basic approaches to direct processing of the 360images as we take into consideration the geometry of each one. Obviously, the geometry of each of these images is a consequence of the geometry of the sensor’s mirror used. This technical report is organized as follows: In Section 1 we develop the Laplacian operator on non-Euclidean manifolds. First we start by derivation of Laplacian operator on Riemannian manifolds and than we derive it explicitly for each of the non-Euclidean manifolds of our interest, i.e. hyperboloid, sphere and paraboloid. This allows us to implement the gradient and diffusion flow on hyperbolic and spherical image. For testing this techniques, a synthetic and a real image was used in the case of hyperboloid and sphere respectively. In Section 2 we demonstrate the active contour on non-Euclidean images. First it was derived by directly minimizing the energy functional where the specific geometry of the non-Euclidean image was taken into account. Then the same was proofed through Polyakov action. We give some examples in each of the cases and so derive conclusions about the influence of the geometry for each particular case.