Convex Hodge Decomposition and Regularization of Image Flows

Authors:
Jing Yuan;Christoph Schnörr;Gabriele Steidl
Affiliations:
Dept. Math. & Comp. Sci., University of Heidelberg, Heidelberg, Germany;Dept. Math. & Comp. Sci., University of Heidelberg, Heidelberg, Germany;Dept. Math. & Comp. Sci., University of Heidelberg, Heidelberg, Germany
Venue:
Journal of Mathematical Imaging and Vision
Year:
2009

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Abstract

The total variation (TV) measure is a key concept in the field of variational image analysis. In this paper, we focus on vector-valued data and derive from the Hodge decomposition of image flows a definition of TV regularization for vector-valued data that extends the standard componentwise definition in a natural way. We show that our approach leads to a convex decomposition of arbitrary vector fields, providing a richer decomposition into piecewise harmonic fields rather than piecewise constant ones, and motion texture. Furthermore, our regularizer provides a measure for motion boundaries of piecewise harmonic image flows in the same way, as the TV measure does for contours of scalar-valued piecewise constant images.