Divergence-Free Wavelets and High Order Regularization

Expanding on a wavelet basis the solution of an inverse problem provides several advantages. First of all, wavelet bases yield a natural and efficient multiresolution analysis which allows defining clear optimization strategies on nested subspaces of the solution space. Besides, the continuous representation of the solution with wavelets enables analytical calculation of regularization integrals over the spatial domain. By choosing differentiable wavelets, accurate high-order derivative regularizers can be efficiently designed via the basis's mass and stiffness matrices. More importantly, differential constraints on vector solutions, such as the divergence-free constraint in physics, can be nicely handled with biorthogonal wavelet bases. This paper illustrates these advantages in the particular case of fluid flow motion estimation. Numerical results on synthetic and real images of incompressible turbulence show that divergence-free wavelets and high-order regularizers are particularly relevant in this context.