Total Generalized Variation Based Denoising Models for Ultrasound Images

Abstract

In this paper, we introduce a class of variational models for the restoration of ultrasound images corrupted by noise. The proposed models involve the convex or nonconvex total generalized variation regularization. The total generalized variation regularization ameliorates the staircasing artifacts that appear in the restored images of total variation based models. Incorporating total generalized variation regularization with nonconvexity helps preserve edges in the restored images. To realize the proposed convex model, we adopt the alternating direction method of multipliers, and the iteratively reweighted $\ell _1$ algorithm is employed to handle the nonconvex model. These methods result in fast and efficient optimization algorithms for solving our models. Numerical experiments demonstrate that the proposed models are superior to other state-of-the-art models.

Appendix: The Proof of Theorem 3

Proof

For any fixed $x\in \varOmega ,$ we can deduce from the inequality of arithmetic-geometric mean that for $u > 0$,

$$\begin{aligned} \frac{(f(x)-u)^2}{u} = u + \frac{f(x)^2}{u} - 2f(x) \ge 2|f(x)| - 2f(x). \end{aligned}$$

It follows from the above inequality that

$$\begin{aligned} E(u) \ge \gamma \int _\varOmega \frac{(f-u)^2}{u} \,\,dx \ge \gamma \int _\varOmega 2|f| - 2f \,\,dx, \end{aligned}$$

i.e., E(u) is bounded below. Hence, we can choose a minimizing sequence $\{u^n\}\in L^2(\varOmega )$ for problem (12), and the sequence $\{TGV^2_\alpha (u^n)\}$ is also bounded.

In addition, from the Poincar$\acute{e}$ inequality of $TGV^2_\alpha$, we can obtain that

$$\begin{aligned} \Vert u^n - Pu^n\Vert _2 \le C\cdot TGV^2_\alpha (u^n), \quad \text {for some constant } C, \end{aligned}$$

where $P : L^2(\varOmega ) \rightarrow \ker (TGV_\alpha ^2)$ is a linear projection. It follows that $u^n - Pu^n$ is bounded in $L^2(\varOmega )$ for each n.

We can also easily deduce that for $u > 0$,

$$\begin{aligned} \int _\varOmega \frac{(f-u)^2}{u}\,\,dx= & {} \int _\varOmega u + \frac{f^2}{u} - 2f \,\,dx \ge \Vert u\Vert _1 - \int _\varOmega 2f\,\,dx\\\ge & {} c\cdot \Vert Pu\Vert _1 - \int _\varOmega 2f\,\,dx \ge c\cdot \Vert Pu\Vert _2 - \int _\varOmega 2f\,\,dx, \end{aligned}$$

where c is a positive constant that is independent of n. Since $\{u^n\}$ is a minimizing sequence of problem (12), $\Vert Pu^n\Vert _2$ is bounded from the above inequalities.

Then, from the following inequalities,

$$\begin{aligned} \Vert u^n\Vert _2 \le \Vert u^n - Pu^n + Pu^n \Vert _2 \le \Vert u^n - Pu^n\Vert _2 + \Vert Pu^n\Vert _2, \end{aligned}$$

we can conclude that $\{u^n\}$ is bounded in $L^2(\varOmega )$. Therefore, there exist a subsequence $\{u^{n_k}\}$ and $u^*\in L^2(\varOmega )$ such that the subsequence converges weakly to $u^*$. Since $u^n > 0$ and $E(u^*)$ must be bounded, $u^* > 0$ and $u^*\in S(\varOmega )$.

Because $TGV^2_\alpha$ is lower semicontinuous, E(u) is also lower semicontinuous. By applying Fatou’s Lemma, we have

$$\begin{aligned} E(u^*) \le \liminf _{n\rightarrow \infty } E(u^n) = \inf _{u\in S(\varOmega )} E(u), \end{aligned}$$

which implies that $u^*$ is a solution of problem (12).

If we let $p(t) = \frac{(f(x) -t)^2}{t}$ for any fixed $x\in \varOmega$, then its second derivative is given by $p''(t) = \frac{2f(x)^2}{t^3}$ and thus $p''(t) > 0$ for any $t > 0$. Hence, p is a strictly convex function when $t > 0$. Therefore, problem (12) has a unique minimizer. $\square$