A Variational Approach to Joint Denoising, Edge Detection and Motion Estimation

The estimation of optical flow fields from image sequences is incorporated in a Mumford–Shah approach for image denoising and edge detection. Possibly noisy image sequences are considered as input and a piecewise smooth image intensity, a piecewise smooth motion field, and a joint discontinuity set are obtained as minimizers of the functional. The method simultaneously detects image edges and motion field discontinuities in a rigorous and robust way. It comes along with a natural multi–scale approximation that is closely related to the phase field approximation for edge detection by Ambrosio and Tortorelli. We present an implementation for 2D image sequences with finite elements in space and time. It leads to three linear systems of equations, which have to be iteratively in the minimization procedure. Numerical results underline the robustness of the presented approach and different applications are shown.