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Divergence-Free Wavelets and High Order Regularization

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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Notes

  1. 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).

  2. 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.

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

  4. \(H^1(\mathbb{R }^d)\) 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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Appendices

Appendix

Fast Divergence-Free Wavelet Transform

To implement numerically the divergence-free wavelet based method of Sect. 5.2, it is necessary to have at hand the fast divergence-free wavelet transform algorithm. For this practical reasons, the divergence-free wavelet bases have to be compactly supported. Such wavelet bases exist and can be fortunately easily constructed. They have been introduced first by Lemarié-Rieusset (1992). The objective here is to detail the anisotropic construction of divergence-free wavelet basis introduce in Sect. 5.1 in order to construct the associated fast wavelet transform algorithm.

Before further detailing this construction, we first give some technical precisions. Since the curl operator introduces derivation operators, it is important to answer to the following questions: what is the derivative of a scaling function? What is the derivative of a wavelet? Does differentiation preserve the \(L^2(\mathbb{R })\) orthogonality property of a wavelet basis?

Using integration by part, to answer the last question we have:

$$\begin{aligned} \int _{\mathbb{R }}\varphi _k^{\prime }(x)\varphi _{\ell }^{\prime }(x)dx =-\int _{\mathbb{R }}\varphi _k(x)\varphi _{\ell }^{\prime \prime }(x)dx. \end{aligned}$$

Thus, if \(\{\varphi _{j,k}:\,k\in \mathbb{Z }\}\) is an orthogonal basis, this property is preserved by differentiation if and only if \(\varphi =-\varphi ^{\prime \prime }\): which is for instance true for sinus or cosinus basis.

The answers to the first two questions are supplied by Kahane and Lemarié-Rieusset (1995). Precisely, let \((\varphi ^1,\tilde{\varphi }^1)\) be a pair of biorthogonal scaling functions associated to biorthogonal wavelets \((\psi ^1,\tilde{\psi }^1)\), with \(\varphi ^1\in \mathcal{C }^{1+\epsilon }(\mathbb{R })\), \(\epsilon >0\). Then there exists another biorthogonal scaling functions \((\varphi ^0, \tilde{\varphi }^0)\) and biorthogonal wavelets \((\psi ^0,\tilde{\psi }^0)\), satisfying Kahane and Lemarié-Rieusset (1995):

$$\begin{aligned} \frac{d}{dx}\varphi ^1(x)=\varphi ^0(x)\,-\,\varphi ^0(x-1), \end{aligned}$$
(51)

and

$$\begin{aligned} \frac{d}{dx}\tilde{\varphi }^0(x)=\tilde{\varphi }^1(x+1) \,-\,\tilde{\varphi }^1(x). \end{aligned}$$
(52)

The associated wavelets verify ( Kahane and Lemarié-Rieusset 1995 ):

$$\begin{aligned} \psi ^1(x)=4\int \limits _{-\infty }^x\psi ^0\quad \text{ and}\quad \tilde{\psi }^0(x)=-4\int \limits _{-\infty }^x\tilde{\psi }^1. \end{aligned}$$
(53)

Hence, according to (51), (52) and (53), the derivative of a scaling function is expressed as a finite difference on neighborhoods of another scaling function and the derivative of a wavelet is another wavelet. Figure 15 shows the plot of an example of these scaling functions and wavelets linked by differentiation and integration.

Fig. 15
figure 15

Example of biorthogonal generators and primal derivatives: case of B-Spline generators \((\varphi ^1,\tilde{\varphi }^1)\) with 3 vanishing moments

Let \((V^1_j)_{j\in \mathbb{Z }}\) and \((V^0_j)_{j\in \mathbb{Z }}\) be one-dimensional multiresolution analyses of \(L^2(\mathbb{R })\) provided by \(\varphi ^1\) and \(\varphi ^0\) respectively, with \(\varphi ^1\) and \(\varphi ^0\) defined by (51). Since we use tensor product construction in higher dimension, each sequence of space, namely:

$$\begin{aligned} V^1_j\otimes V^0_j=\text{ span}\left\{ \varphi ^1_{j,k_1}(x)\varphi ^0_{j,k_2} (y),\,k_1,k_2\in \mathbb{Z }\right\} ,\quad j\in \mathbb{Z }, \end{aligned}$$

or

$$\begin{aligned} V^0_j\otimes V^1_j=\text{ span}\left\{ \varphi ^0_{j,k_1}(x) \varphi ^1_{j,k_2}(y),\,k_1,k_2\in \mathbb{Z }\right\} ,\quad j\in \mathbb{Z }, \end{aligned}$$

form a multiresolution analysis of \(L^2(\mathbb{R }^2)\). Moreover, to compute the fast wavelet decomposition, in the multiresolution generated by \(V^1_j\otimes V^0_j\), it suffices to use the filters of \((\tilde{\varphi }^1,\tilde{\psi }^1)\) in the \(x\) direction and those of \((\tilde{\varphi }^0,\tilde{\psi }^0)\) in the \(y\) direction. The reconstruction is done with the filters of \((\varphi ^1,\psi ^1)\) and \((\varphi ^0,\psi ^0)\) respectively.

From relations (51) and (52), one can derive two interesting properties of biorthogonal multiresolution analyses \((V^1_j,\tilde{V}^1_j)\) and \(( V^0_j,\tilde{V}^0_j)\) (Kahane and Lemarié-Rieusset 1995):

$$\begin{aligned} \frac{d}{dx}V^1_j=V^0_j, \quad \quad \quad \quad \tilde{V}^0_j=\int \limits _{-\infty }^x\tilde{V}^1_j. \end{aligned}$$
(54)

The interest of relations (54) appears in the numerical implementation of fast divergence-free wavelet transform. This relation allows to build a multiresolution analysis of \((L^2(\mathbb{R }^2))^2\) that preserves the divergence-free property (Lemarié-Rieusset 1992).

As stated in (19), the space \(\mathcal{H }_{div}(\mathbb{R }^2)\) corresponds to the curl of \(H^1(\mathbb{R }^2)\) scalar potential. Then, taking the curl of any multiresolution analysis of \(H^1(\mathbb{R }^2)\) will provide a multiresolution analysis of \(\mathcal{H }_{div}(\mathbb{R }^2)\). However, let us consider a “regular” scalar multiresolution analysis of \(H^1(\mathbb{R }^2)\) generated by spaces \( V^a_j\otimes V^b_j\), with \(V^a_j\ne V^b_j\). Taking the curl of a such multiresolution analysis, we get:

$$\begin{aligned} \mathbf{curl}[V^a_j\otimes V^b_j]=\left( \begin{array}{ll} V^a_{j}\otimes {(V^b_{j})}^{\prime }\\ \, \\ - {(V^a_{j})}^{\prime }\otimes V^b_{j} \end{array} \right). \end{aligned}$$
(55)

Then, to deal with the divergence-free wavelets contained in the spaces \( \mathbf{curl}[V^a_j\otimes V^b_j]\), we have to manipulate four different types of biorthogonal wavelet filter banks associated respectively to the one-dimensional BMRAs that appear in (55): \(V^a_j, (V^a_j)^{\prime }, V^b_j\) and \((V^b_j)^{\prime }\). To overcome this problem, the two-dimensional scalar multiresolution analysis that we will consider is generated by spaces \(V^1_j\otimes V^1_j\) and using Lemarié-Rieusset’s results (54), one can easily prove that:

$$\begin{aligned} \mathbf{curl}\left(V^1_j\otimes V^1_j\right)\subset \left(V^1_j\otimes V^0_j\right)\times \left(V^0_j\otimes V^1_j\right)={\mathbf{V}}_j. \end{aligned}$$
(56)

Accordingly, the divergence-free scaling functions and wavelets of Sect. 5.1 verify:

$$\begin{aligned} {\varvec{\Phi }}^{div}_{j,{{{{{\varvec{k}}}}}}}=\mathbf{curl} [\varphi ^1_{j,k_1}\otimes \varphi ^1_{j,k_2}] =\left( \begin{array}{ll} \varphi ^1_{j,k_1}\otimes {(\varphi ^1_{j,k_2})}^{\prime }\\ \, \\ - {(\varphi ^1_{j,k_1})}^{\prime }\otimes \varphi ^1_{j,k_2} \end{array} \right) \in {\mathbf{V}}_j, \end{aligned}$$

and

$$\begin{aligned} {\varvec{\Psi }}^{div}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}} \!=\!\left( \begin{array}{ll} 2^{j_2\!+\!2}\psi ^1_{j_1,k_1}\otimes \psi ^0_{j_2,k_2}\\ \, \\ \!-\! 2^{j_1\!+\!2}\psi ^0_{j_1,k_1}\otimes \psi ^1_{j_2,k_2} \end{array} \right) \in \left(W^1_{j_1}\!\otimes \! W^0_{j_2}\right)\\ \times \left(W^0_{j_1}\otimes W^1_{j_2}\right). \end{aligned}$$

Fast Divergence-Free Wavelet Transform

By construction, we have seen that the vector spaces \(\mathbf{V}_j=\left(V^1_j\otimes V^0_j\right)\times \left(V^0_j\otimes V^1_j\right)\) constitute a multiresolution analysis of \((L^2(\mathbb{R }^2))^2\) and this multiresolution analysis preserves the divergence-free constraint (Lemarié-Rieusset 1992). From a standard anisotropic vector wavelet decomposition associated to \(\mathbf{V}_j\), the objective of this section is to describe how to compute in practice the anisotropic divergence-free wavelet decomposition of any \(\mathbf{u}\in \mathcal{H }_{div}(\mathbb{R }^2)\).

The standard anisotropic vector wavelet associated to \(\mathbf{V}_j\) are:

$$\begin{aligned} {\varvec{\Psi }}^{1}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}} =\left( \begin{array}{ll} \psi ^1_{j_1,k_1}\otimes \psi ^0_{j_2,k_2}\\ 0 \end{array} \right), \end{aligned}$$

and

$$\begin{aligned} {\varvec{\Psi }}^{2}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}} =\left( \begin{array}{ll} 0\\ \psi ^0_{j_1,k_1}\otimes \psi ^1_{j_2,k_2} \end{array} \right). \end{aligned}$$

Since \(\mathbf{u}\!=\!(u_1,u_2)\) belongs to \((L^2(\mathbb{R }^2))^2\) and \(({\varvec{\Psi }}^{1}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}}, {\varvec{\Psi }}^{2}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}})_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}\in \mathbb{Z }^2}\) is a wavelet basis of \((L^2(\mathbb{R }^2))^2\), we get:

$$\begin{aligned} \mathbf{u}=\sum _{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}\in \mathbb{Z }^2}\mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^1 {\varvec{\Psi }}^{1}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}} +\sum _{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}\in \mathbb{Z }^2}\mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^2 {\varvec{\Psi }}^{2}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}}. \end{aligned}$$
(57)

Through an easy calculation, by identification one can show that:

$$\begin{aligned} u_1=\sum _{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}\in \mathbb{Z }^2}\mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^1\,\psi ^1_{j_1,k_1}\otimes \psi ^0_{j_2,k_2}, \end{aligned}$$

and

$$\begin{aligned} u_2=\sum _{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}\in \mathbb{Z }^2}\mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^2\,\psi ^0_{j_1,k_1}\otimes \psi ^1_{j_2,k_2}. \end{aligned}$$

Following Deriaz and Perrier (2006) and using the relation:

$$\begin{aligned} {\varvec{\Psi }}^{div}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}} =2^{j_2+2}{\varvec{\Psi }}^{1}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}}-2^{j_1+2} {\varvec{\Psi }}^{2}_{{{{{{\varvec{j}}}}}},{{{{{\varvec{k}}}}}}}, \end{aligned}$$
(58)

we find:

$$\begin{aligned} \mathbf{d}^{div}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}=\frac{2^{j_2+2}}{4^{j_1+2}+4^{j_2+2}} \mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^1-\frac{2^{j_1+2}}{4^{j_1+2}+4^{j_2+2}} \mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^2, \end{aligned}$$
(59)

and

$$\begin{aligned} \mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^1=2^{j_2+2}\mathbf{d}^{div}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}, \quad \quad \quad \quad \mathbf{d}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}^2=-2^{j_1+2}\mathbf{d}^{div}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}. \end{aligned}$$
(60)

Therefore, decomposition and reconstruction associated to divergence-free wavelets is simply performed using scalar wavelet filter banks. Finally, the algorithm is of low complexity and its structure remains identical to the scalar case.

The algorithm is summarized then as follows. Starting with \(\mathbf{u}=(u_1,u_2)\), to get the divergence-free wavelet coefficients \(\mathbf{d}^{div}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\):

  1. 1a.

    Compute \(\mathbf{d}^{1}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\)  associated to \(u_1\) in \(V^1_j\otimes V^0_j\).

  2. 2a.

    Compute \(\mathbf{d}^{2}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\)  associated to \(u_2\) in \(V^0_j\otimes V^1_j\).

  3. 3a.

    Compute \(\mathbf{d}^{div}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\)  from \(\mathbf{d}^{1}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\)  and \(\mathbf{d}^{2}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\) using (59).

For the reconstruction:

  1. 1b.

    Compute \(\mathbf{d}^{1}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\) and \(\mathbf{d}^{2}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\) from \(\mathbf{d}^{div}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\) using (60).

  2. 2b.

    Compute \(u_1\) from \(\mathbf{d}^{1}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\) in \(V^1_j\otimes V^0_j\).

  3. 3b.

    Compute \(u_2\) from \(\mathbf{d}^{2}_{{{{{\varvec{j}}}}},{{{{\varvec{k}}}}}}\) in \(V^0_j\otimes V^1_j\).

Steps 1a and Step 2a correspond to a two dimensional fast wavelet transform. Step 3a is a change of basis which theoretical complexity is linear, thus the theoretical complexity of the decomposition phase is about \(O(N)\). As the same, Step 3b is a change of basis, Steps 2b and Step 3b correspond to an inverse two dimensional fast wavelet transform, the theoretical complexity of recomposition phase is also about \(O(N)\).

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Kadri-Harouna, S., Dérian, P., Héas, P. et al. Divergence-Free Wavelets and High Order Regularization. Int J Comput Vis 103, 80–99 (2013). https://doi.org/10.1007/s11263-012-0595-7

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Keywords

  • Divergence-free wavelets
  • High order derivatives regularization
  • Optic-flow estimation