Variational Image Restoration with Constraints on Noise Whiteness

Abstract

We propose a novel variational framework for image restoration based on the assumption that noise is additive and white. In particular, the proposed variational model uses total variation regularization and forces the resemblance of the residue image to a white noise realization by imposing constraints in the frequency domain. The whiteness constraint constitutes the key novelty behind our approach. The restored image is efficiently computed by the constrained minimization of an energy functional using an alternating directions methods of multipliers procedure. Numerical examples show that the novel approach is particularly suited for textured image restorations.

Appendix

Proof of Proposition (3.1). First, the problem in (3.11) can be equivalently rewritten as follows:

$$\begin{aligned} x^*{\leftarrow }\, \mathrm {arg} \min _{x \in {\mathbb R}^n} \left\{ \, \varphi (x) = \alpha \, \Vert \, x \, \Vert _2 \,{+}\, \frac{\beta }{2} \left\| \, x - q \, \right\| _2^2 \,\right\} \, , \end{aligned}$$

(5.1)

with $q \in {\mathbb R}^n$ defined as in (3.13). In Fig. 8a we give a geometric representation of the problem in the $2$-dimensional case. To prove the proposition statement in (3.12) we consider separately the two cases $\Vert \, q \, \Vert _2 > 0$ and $\Vert \, q \, \Vert _2 = 0$.

Fig. 8

Geometric representation of the problem in (5.1)

Case $\Vert \, q \, \Vert _2 > 0$. First, we prove that the solution $x^*$ of (5.1) lies on the half-line $Oq$ with origin at the $n$-dimensional null vector $O$ and passing through $q$, represented in solid red in Fig. 8a. To this purpose, we demonstrate that for every point $p$ not lying on $Oq$ there always exists a point $p^*$ on $Oq$ providing a lower value of the objective function in (5.1), that is a point $p^*$ such that $\varphi (p) - \varphi (p^*) > 0$. In particular, we define $p^*$ as the intersection point between the half-line $Oq$ and the $n$-dimensional sphere with center in $O$ and passing through $p$, depicted in solid blue in Fig. 8a. Noting that $\Vert p^*\Vert _2 = \Vert p \Vert _2$ by construction, we can thus write:

$$\begin{aligned}&\varphi (p) - \varphi (p^*) = \alpha \left( \Vert \, p \, \Vert _2 - \Vert \, p^* \, \Vert _2 \right) \nonumber \\&+ \frac{\beta }{2} \left( \Vert \, p - q \, \Vert _2^2 - \Vert \, p^* - q \, \Vert _2^2 \right) \nonumber \\&= \frac{\beta }{2} \left( \Vert \, p \, \Vert _2^2 + \Vert \, q \, \Vert _2^2 - 2 \, \langle \, p \, , \, q \rangle - \Vert \, p^* \, \Vert _2^2 - \Vert \, q \, \Vert _2^2\right. \nonumber \\&\left. + 2 \, \langle \, p^* \, , \, q \rangle \right) \nonumber \\&= \beta \, \langle \, p^* - p \, , \, q \, \rangle \nonumber \\&= \beta \, \Vert \, p^* - p \, \Vert _2 \, \Vert \, q \, \Vert _2 \, cos\left( \widehat{O \, p^* p} \right) \, . \end{aligned}$$

(5.2)

Since $\beta > 0$, $\Vert q \Vert _2 >0$ for hypothesis, $p \ne p^*$ by construction, and noting that the angle $\widehat{O \, p^* p}$ is always acute, we can conclude that the expression in (5.2) is positive. Hence, the solution $x^*$ of (5.1) lies on the half-line $Oq$, i.e. $x^* = \xi ^*q$, $\xi ^* \ge 0$.

By setting $x = \xi q$, $\xi \ge 0$, the $n$-dimensional minimization problem in (5.1) can be transformed into the following $1$-dimensional problem:

$$\begin{aligned}&\xi ^* {\leftarrow }\, \mathrm {arg} \min _{\xi \ge 0} \left\{ \, \alpha \, \Vert \, \xi q \, \Vert _2 \,{+}\, \frac{\beta }{2} \left\| \, \xi q - q \, \right\| _2^2 \,\right\} \nonumber \\&{\leftarrow }\, \mathrm {arg} \min _{\xi \ge 0} \left\{ \, \alpha \, \Vert \, q \, \Vert _2 \xi \,{+}\, \frac{\beta }{2} \left\| \, q \, \right\| _2^2 \left( \xi - 1 \right) ^2 \,\right\} \nonumber \\&{\leftarrow }\, \mathrm {arg} \min _{\xi \ge 0} \left\{ \, \alpha \xi \,{+}\, \frac{\beta }{2} \left\| \, q \, \right\| _2 \left( \xi - 1 \right) ^2\,\right\} \nonumber \\&{\leftarrow }\, \mathrm {arg} \min _{\xi \ge 0} \left\{ \, g(\xi ) = \left( \frac{\beta }{2} \left\| \, q \, \right\| _2 \right) \xi ^2 \,{+}\, \left( \alpha - \beta \, \left\| \, q \, \right\| _2 \right) \xi \,\right\} \, . \nonumber \\ \end{aligned}$$

(5.3)

Since the coefficient $\frac{\beta }{2} || \, q \, ||_2$ is strictly positive (in fact, we are assuming that $\beta > 0$ and $\Vert q \Vert _2 > 0$), the function $g(\xi )$ in (5.3) represents a strictly convex parabola passing through the origin and having its unconstrained minimum at the abscissa of its vertex:

$$\begin{aligned} \xi _{v} = 1 - \frac{\alpha }{\beta \Vert \, q \, \Vert _2} \, . \end{aligned}$$

(5.4)

Hence, for what concern the constrained minimization in (5.3) we have the two cases illustrated in Fig. 8b. In formulas:

$$\begin{aligned} \xi ^* = \left\{ \begin{array}{lll} \xi _v &{} \, \mathrm {if} \, &{} \xi _v \ge 0 \\ 0 &{} \, \mathrm {if} \, &{} \xi _v < 0 \end{array} \right. \,\, , \,\; \mathrm {that} \, \mathrm {is:} \quad \, \xi ^* = \max \left\{ \xi _v \, , \, 0 \right\} \, . \end{aligned}$$

(5.5)

Therefore, from (5.4) and (5.5) the solution $x^* = \xi ^* q$ of the $n$-dimensional problem in (5.1) is:

$$\begin{aligned} x^* = \xi ^* q = \max \left\{ 1 - \frac{\alpha }{\beta \Vert \, q \, \Vert _2} \, , \, 0 \right\} q \, , \end{aligned}$$

(5.6)

thus proving case a) of the proposition statement in (3.12).

Case $\Vert \, q \, \Vert _2 = 0$. Since $q$ is the $n$-dimensional null vector, the objective function minimized in (5.1) depends on $x$ only through its norm $\Vert x \Vert _2$. Hence, the minimizers will be all the vectors $x^*$ belonging to the $n$-dimensional sphere $\Vert x \Vert _2 = r^*$ with radius $r^* \ge 0$ given by the solution of the constrained $1$-dimensional minimization problem obtained from (5.1) by setting $q = 0$ and $r = \Vert x \Vert _2$, that is

$$\begin{aligned} r^* = \mathrm {arg} \min _{r \ge 0} \left\{ \, \alpha \, r \,{+}\, \frac{\beta }{2} r^2 \,\right\} \, . \end{aligned}$$

(5.7)

The minimization problem in (5.7) is very similar to problem (5.3) and the solution can be analogously computed as follows:

$$\begin{aligned} r^* = \max \left\{ \, -\frac{\alpha }{\beta } \, , \, 0 \, \right\} . \end{aligned}$$

(5.8)

Therefore, if $\Vert q \Vert _2 = 0$ the solution of (5.1) is given by:

$$\begin{aligned} x^* = 0 \quad \quad \mathrm {if} \quad \quad \alpha \ge 0 \, , \end{aligned}$$

(5.9)

while we get an infinite number of solutions

$$\begin{aligned} x^* \,{\in }\,\, \left\{ \, x \in {\mathbb R}^n : \,\, \Vert \, x \, \Vert _2 = r^* \,\right\} \quad \quad \mathrm {if} \quad \quad \alpha < 0 \, , \end{aligned}$$

(5.10)

with $r^*$ defined in (5.8). This proves cases b) and c) of the proposition statement in (3.12). $\square$