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The Beltrami Flow over Manifolds

In many medical computer vision tasks, the relevant data is attached to a specific tissue such as the cortex or the colon. This situation calls for regularization techniques which are defined over non flat surfaces. We introduce in this paper the Beltrami flow over manifolds. This new regularization technique overcomes the over-smoothing of the L_2 flow and the staircasing effects of the L_1 flow, that were recently suggested via the harmonic map methods. The key of our approach is first to clarify the link between the intrinsic Polyakov action and the implicit Harmonic energy functional and then use the geometrical understanding of the Beltrami Flow to generalize it to images on explicitly and implicitly defined non flat surfaces. It is shown that once again the Beltrami flow interpolates between the L_2 and L_1 flows on non-flat surfaces. The implementation scheme of this flow is presented and various experimental results obtained on a set of various real images illustrate the performances of the approach as well as the differences with the harmonic map flows. This extension of the Beltrami flow to the case of non flat surfaces opens new perspectives in the regularization of noisy data defined on manifolds.