9.1 A class of fast algorithms for total variation image restoration  (Page 4/6)

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$\begin{array}{c}\hfill \underset{\mathbf{w},u}{min}\sum _{i}\parallel {\mathbf{w}}_{i}\parallel +\frac{\beta }{2}\sum _{i}\parallel {\mathbf{w}}_{i}-{D}_{i}{u\parallel }^{2}+\frac{\mu }{2}{\parallel Ku-f\parallel }^{2},\end{array}$

where $\beta \gg 0$ is a penalty parameter. It is well known that the solution of ( ) converges to that of ( ) as $\beta \to \infty$ . In the following, we concentrate on problem ( ).

Basic algorithm

The benefit of ( ) is that while either one of the two variables $u$ and $\mathbf{w}$ is fixed, minimizing the objective function with respect to the other has a closed-form formula that we willspecify below. First, for a fixed $u$ , the first two terms in ( ) are separable with respect to ${\mathbf{w}}_{i}$ , and thus the minimization for $\mathbf{w}$ is equivalent to solving

$\begin{array}{c}\hfill \underset{{\mathbf{w}}_{i}}{min}\parallel {\mathbf{w}}_{i}\parallel +\frac{\beta }{2}{\parallel {\mathbf{w}}_{i}-{D}_{i}u\parallel }^{2},\phantom{\rule{1.em}{0ex}}i=1,2,...,{n}^{2}.\end{array}$

It is easy to verify that the unique solutions of ( ) are

$\begin{array}{c}\hfill {\mathbf{w}}_{i}=max\left\{\parallel ,{D}_{i},u\parallel -,\frac{1}{\beta },,,0\right\}\frac{{D}_{i}u}{\parallel {D}_{i}u\parallel },\phantom{\rule{1.em}{0ex}}i=1,...,{n}^{2},\end{array}$

where the convention $0·\left(0/0\right)=0$ is followed. On the other hand, for a fixed $\mathbf{w}$ , ( ) is quadratic in $u$ and the minimizer $u$ is given by the normal equations

$\begin{array}{c}\hfill \left(\sum _{i},{D}_{i}^{\top },{D}_{i},+,\frac{\mu }{\beta },{K}^{\top },K\right)u=\sum _{i}{D}_{i}^{\top }{\mathbf{w}}_{i}+\frac{\mu }{\beta }{K}^{\top }f.\end{array}$

By noting the relation between $D$ and ${D}_{i}$ and a reordering of variables, ( ) can be rewritten as

$\begin{array}{c}\hfill \left({D}^{\top },D,+,\frac{\mu }{\beta },{K}^{\top },K\right)u={D}^{\top }w+\frac{\mu }{\beta }{K}^{\top }f,\end{array}$

where

$\begin{array}{c}\hfill w\triangleq \left(\begin{array}{c}{w}_{1}\\ {w}_{2}\end{array}\right)\in {\mathbb{R}}^{2{n}^{2}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{w}_{j}\triangleq \left(\begin{array}{c}{\left({\mathbf{w}}_{1}\right)}_{j}\\ ⋮\\ {\left({\mathbf{w}}_{{n}^{2}}\right)}_{j}\end{array}\right),\phantom{\rule{0.277778em}{0ex}}j=1,2.\end{array}$

The normal equation ( ) can also be solved easily provided that proper boundary conditions are assumed on $u$ . Since both the finite difference operations and the convolution are notwell-defined on the boundary of $u$ , certain boundary assumptions are needed when solving ( ). Under the periodic boundary conditions for $u$ , i.e. the 2D image $u$ is treated as a periodic function in both horizontal and vertical directions, ${D}^{\left(1\right)}$ , ${D}^{\left(2\right)}$ and $K$ are all block circulant matrices with circulant blocks; see e.g. , . Therefore, the Hessianmatrix on the left-hand side of ( ) has a block circulant structure and thus can be diagonalized by the 2D discreteFourier transform $\mathbf{F}$ , see e.g. . Using the convolution theorem of Fourier transforms, the solution of( ) is given by

$\begin{array}{c}\hfill u={\mathbf{F}}^{-1}\left(\frac{\mathbf{F}\left({D}^{\top },w,+,\left(\mu /\beta \right),{K}^{\top },f\right)}{\mathrm{diag}\left(\mathcal{F},\left(,{D}^{\top },D,+,\left(\mu /\beta \right),{K}^{\top },K,\right)\right)}\right),\end{array}$

where the division is implemented by componentwise. Since all quantities but $w$ are constant for given $\beta$ , computing $u$ from ( ) involves merely the finite difference operation on $w$ and two FFTs (including one inverse FFT), once the constant quantities are computed.

Since minimizing the objective function in ( ) with respect to either $\mathbf{w}$ or $u$ is computationally inexpensive, we solve ( ) for a fixed $\beta$ by an alternating minimization scheme given below.

Algorithm :

• Input $f$ , $K$ and $\mu >0$ . Given $\beta >0$ and initialize $u=f$ .
• While“not converged”, Do
• Compute $\mathbf{w}$ according to ( ) for fixed $u$ .
• Compute $u$ according to ( ) for fixed $w$ (or equivalently $\mathbf{w}$ ).
• End Do

Optimality conditions and convergence results

To present the convergence results of Algorithm "Basic Algorithm" for a fixed $\beta$ , we make the following weak assumption.

Assumption 1 $\mathcal{N}\left(K\right)\cap \mathcal{N}\left(D\right)=\left\{0\right\}$ , where $\mathcal{N}\left(·\right)$ represents the null space of a matrix.

Define

$\begin{array}{c}\hfill M={D}^{\top }D+\frac{\mu }{\beta }{K}^{\top }K\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}T=D{M}^{-1}{D}^{\top }.\end{array}$

Furthermore, we will make use of the following two index sets:

$\begin{array}{c}\hfill L=\left\{i\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}\parallel ,{D}_{i},{u}^{*},\parallel <,\frac{1}{\beta }\right\}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}E=\left\{1,...,{n}^{2}\right\}\setminus L.\end{array}$

Under Assumption 1, the proposed algorithm has the following convergence properties.

Theorem 1 For any fixed $\beta >0$ , the sequence $\left\{\left({w}^{k},{u}^{k}\right)\right\}$ generated by Algorithm "Basic Algorithm" from any starting point $\left({w}^{0},{u}^{0}\right)$ converges to a solution $\left({w}^{*},{u}^{*}\right)$ of ( ). Furthermore, the sequence satisfies

• $\parallel {w}_{E}^{k+1}-{w}_{E}^{*}\parallel \le \sqrt{\rho \left({\left({T}^{2}\right)}_{EE}\right)}\parallel {w}_{E}^{k}-{w}_{E}^{*}\parallel ;$
• $\parallel {u}^{k+1}-{u}^{*}{\parallel }_{M}\le \sqrt{\rho \left({T}_{EE}\right)}{\parallel {u}^{k}-{u}^{*}\parallel }_{M};$

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