Convex Hodge Decomposition of Image Flows

The total variation (TV) measure is a key concept in the field of variational image analysis. Introduced by Rudin, Osher and Fatemi in connection with image denoising, it also provides the basis for convex structure-texture decompositions of image signals, image inpainting, and for globally optimal binary image segmentation by convex functional minimization. Concerning vector-valued image data, the usual definition of the TV measure extends the scalar case in terms of the L-norm of the gradients. In this paper, we show for the case of 2D image flows that TV regularization of the basic flow components (divergence, curl) leads to a mathematically more natural extension. This regularization provides a convex decomposition of motion into a richer structure component and texture. The structure component comprises piecewise harmonic fields rather than piecewise constant ones. Numerical examples illustrate this fact. Additionally, for the class of piecewise harmonic flows, our regularizer provides a measure for motion boundaries of image flows, as does the TV-measure for contours of scalar-valued piecewise constant images.