The Beltrami flow over implicit manifolds

In many medical computer vision tasks the relevant data is attached to a specific tissue such as the colon or the cortex. This situation calls for regularization techniques which are defined over surfaces. We introduce in this paper the Beltrami flow over implicit manifolds. This new regularization technique overcomes the over-smoothing of the L₂ flow and the staircasing effects of the L₁ 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 implicitly defined non flat surfaces. It is shown that once again the Beltrami flow interpolates between the L₂ and L₁ 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.