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International Journal of Computer Vision

Erschienen in:

01.05.2013

Divergence-Free Wavelets and High Order Regularization

verfasst von: S. Kadri-Harouna, P. Dérian, P. Héas, E. Mémin

Erschienen in: International Journal of Computer Vision | Ausgabe 1/2013

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Abstract

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.

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In the following, we will restrict ourselves to the study of DFD equation, but the approach remains valid for any other integrated data model. Indeed, for other image modalities, many other brightness evolution models have been proposed in the literature to link the image intensity function to the sought velocity field (Liu and Shen 2008).

In practice we use a quasi-newton method combined to the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method Nocedal and Wright (1999) to approximate the Hessian matrix. Thus, we can optimize the parameter $\alpha $ which is the acceptable stepsize in the direction found in the first step according to a Wolfe condition.

For $d=2$, we define $\mathbf{curl}(\chi ):=(\partial _y\chi ,-\partial _x\chi )$.

$H^1(\mathbb{R }^d)$ denotes the classical Sobolev space:

$$\begin{aligned} \Vert f\Vert ^2_{H^1(\mathbb{R }^d)}=\Vert f\Vert ^2_{L^2(\mathbb{R }^d)}+\Vert \nabla f\Vert ^2_{L^2(\mathbb{R }^d)} \end{aligned}$$

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Titel: Divergence-Free Wavelets and High Order Regularization
verfasst von: S. Kadri-Harouna
P. Dérian
P. Héas
E. Mémin
Publikationsdatum: 01.05.2013
Verlag: Springer US
Erschienen in: International Journal of Computer Vision / Ausgabe 1/2013
Print ISSN: 0920-5691
Elektronische ISSN: 1573-1405
DOI: https://doi.org/10.1007/s11263-012-0595-7

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