Nonlocal ultrasound image despeckling via improved statistics and rank constraint

Abstract

Ultrasound images are often contaminated by speckle noise during the acquisition process, which influences the performance of subsequent applications. Hence, it is necessary to design an effective algorithm for despeckling to obtain a clearer ultrasound image. According to the low-rank property of ultrasound images and the statistical property of similar image patch matrices, a nonlocal low-rank model with an improved data fidelity function (LRDF) is introduced in this paper, which integrates the weighted nuclear norm minimization (WNNM) and an improved data fidelity term. The advantage of WNNM is that it can adaptively assign weights on different singular values to preserve more details in restored images. The fidelity term deduced from log-compressed images fits better to ultrasonic data. We adopt the alternating direction method of multipliers (ADMM) to solve this nonconvex optimization problem. The experimental results on simulated images and real medical ultrasound images verify the reweighting strategy is helpful in this application and demonstrate the excellent performance of the proposed method compared with other five state-of-the-art methods.

Appendix: proof of theorem 2

Proof

Let $U_{t}\varSigma _{t}V^{T}_{t}$ denote the SVD of the patch matrix $\frac{Z^{t}}{\mu _{t}}+X^{t+1}$ in the $(t+1)$-th iteration, where $\varSigma _{t}=\{diag(\sigma ^{1}_{t},\sigma ^{2}_{t},\ldots ,\sigma ^{m}_{t})\}$. One can see $L^{t+1}=U_{t}{\mathcal {S}}_{\frac{\lambda w}{\mu ^{t}}}\left( \varSigma _{t}\right) V^{T}_{t}$ according to the conclusion (3), where ${\mathcal {S}}_{\frac{\lambda w}{\mu ^{t}}}(\varSigma _{t})_{ii}=\max \left( \sigma ^{i}_{t}-\frac{\lambda w_{i}}{\mu ^{t}},0\right)$. So with respect to (14), it is reasonable to deduce that

$$\begin{aligned} \left\| Z^{t+1}\right\| _{F}&=\left\| Z^{t}+\mu ^{t}(X^{t+1}-L^{t+1})\right\| _{F}\nonumber \\&=\mu ^{t}\left\| \frac{Z^{t}}{\mu ^{t}}+X^{t+1}-L^{t+1}\right\| _{F}\nonumber \\&=\mu ^{t}\left\| U_{t}\varSigma _{t}V^{T}_{t}-U_{t}{\mathcal {S}}_{\frac{\lambda w}{\mu ^{t}}}\left( \varSigma _{t}\right) V^{T}_{t}\right\| _{F}\nonumber \\&=\mu ^{t}\left\| \varSigma _{t}-{\mathcal {S}}_{\frac{\lambda w}{\mu ^{t}}}(\varSigma _{t})\right\| _{F}\nonumber \\&\le \mu ^{t}\sqrt{\sum _{i}\left( \frac{\lambda w_{i}}{\mu ^{t}}\right) ^{2}}\nonumber \\&=\sqrt{\sum _{i}\left( \lambda w_{i}\right) ^{2}}, \end{aligned}$$

(A.1)

which implies that $\{Z^{t}\}$ is bounded.

Then, we analyze the boundedness of ${\mathcal {L}}(X^{t+1}, L^{t+1}, Z^{t}, \mu ^{t})$. Since X and L are the globally optimal solutions of (11) and (13), we can hold that

$$\begin{aligned} {\mathcal {L}}(X^{t+1}, L^{t+1}, Z^{t}, \mu ^{t}) \le {\mathcal {L}}(X^{t}, L^{t}, Z^{t}, \mu ^{t}). \end{aligned}$$

(A.2)

Combining (10) and (14), it is not hard to show

$$\begin{aligned}&{\mathcal {L}}(X^{t}, L^{t}, Z^{t}, \mu ^{t})\nonumber \\&\quad ={\mathcal {L}}(X^{t}, L^{t}, Z^{t-1}, \mu ^{t-1}) +\langle Z^{t}-Z^{t-1},X^{t}-L^{t}\rangle \nonumber \\&\qquad +\frac{\mu ^{t}-\mu ^{t-1}}{2}\left\| X^{t}-L^{t}\right\| \nonumber \\&\quad ={\mathcal {L}}(X^{t}, L^{t}, Z^{t-1}, \mu ^{t-1})+\frac{\mu ^{t} +\mu ^{t-1}}{2(\mu ^{t-1})^{2}}\left\| Z^{t}-Z^{t-1}\right\| ^{2}_{F}, \end{aligned}$$

(A.3)

which yields

$$\begin{aligned}&{\mathcal {L}}(X^{t+1}, L^{t+1}, Z^{t}, \mu ^{t})\nonumber \\&\quad \le {\mathcal {L}}(X^{1}, L^{1}, Z^{0}, \mu ^{0}) +\sum ^{t}_{k=1}\frac{\mu ^{k}+\mu ^{k-1}}{2(\mu ^{k-1})^2} \left\| Z^{k}-Z^{k-1}\right\| ^{2}_{F}\nonumber \\&\quad ={\mathcal {L}}(X^{1}, L^{1}, Z^{0}, \mu ^{0})+\sum ^{t}_{k=1} \frac{\rho +1}{2\mu ^{k-1}}\left\| Z^{k}-Z^{k-1}\right\| ^{2}_{F}\nonumber \\&\quad \le {\mathcal {L}}(X^{1}, L^{1}, Z^{0}, \mu ^{0})+\sum ^{t}_{k=1} \frac{\tau (\rho +1)}{2\mu ^{0}\rho ^{k-1}}, \end{aligned}$$

(A.4)

where $\tau$ represents the upper bound of sequence $\Vert Z^{k}-Z^{k-1}\Vert ^{2}_{F}$. One can see that $\left\{ \frac{\tau (\rho +1)}{2\mu ^{0}\rho ^{k-1}}\right\}$ is a geometrical sequence and $\frac{1}{\rho }<1$ is the common ratio, which means $\lim \limits _{t\rightarrow \infty }\sum ^{t}_{k=1} \frac{\tau (\rho +1)}{2\mu ^{0}\rho ^{k-1}}$ converges. Hence, we can conclude that ${\mathcal {L}}(X^{t+1}, L^{t+1}, Z^{t}, \mu ^{t})$ is upper bounded.

Based on (10) and (14), there is

$$\begin{aligned} G(X^{t})+\lambda \Vert L^{t}\Vert _{w,*}&={\mathcal {L}}(X^{t}, L^{t}, Z^{t-1}, \mu ^{t-1})\nonumber \\&\quad -\frac{1}{2\mu ^{t-1}}\left( \Vert Z^{t}\Vert ^{2}_{F} -\Vert Z^{t-1}\Vert ^{2}_{F}\right) , \end{aligned}$$

(A.5)

where $G(X^{t})=\sum \frac{(X^{t}_{ij}-Y_{ij})^{2}}{X^{t}_{ij}}$. Considering the function $G(X^{t})=\frac{(X^{t}-Y)^{2}}{X^{t}}$ for $X^{t}>0$, it is easily to prove that $G(X^{t})$ is a nonnegative convex function. This property together with the fact that all terms on the right side of (A.5) are bounded means that $\{X^{t}\}$ and $\{L^{t}\}$ are bounded and $X^{t}\nrightarrow 0$. Since $\{Z^{t}\}$, $\{X^{t}\}$ and $\{L^{t}\}$ all are bounded and infinite sequences, according to the Weierstrass’s accumulation point theorem, there exists at least one accumulation point for $\{Z^{t}, X^{t}, L^{t}\}$, which is a feasible solution to (10). Then, it follows that

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\Vert X^{t+1}-L^{t+1}\Vert _{F} =\lim \limits _{t\rightarrow \infty }\frac{1}{\mu ^{t}}\Vert Z^{t+1}-Z^{t}\Vert _{F}=0. \end{aligned}$$

(A.6)

For the optimization function $X^{t+1}=\mathrm {arg} \min _{X}G(X^{t})+\frac{\mu ^{t}}{2}\left( \frac{Z^{t}}{\mu ^{t}}+X^{t}-L^{t}\right) ^{2}$ with $G(X^{t})=\frac{(X^{t}-Y)^{2}}{X^{t}}$, its first-order optimality is in the form of

$$\begin{aligned} \nabla G(X^{t+1})+\mu ^{t}\left( \frac{Z^{t}}{\mu ^{t}}+X^{t+1}-L^{t}\right) =0, \end{aligned}$$

(A.7)

where $\nabla G(X^{t+1})=1-\frac{Y^{2}}{(X^{t+1})^{2}}$. Integrating with (14), it is simplified as

$$\begin{aligned} X^{t+1}-X^{t}=-\frac{\nabla G(X^{t+1})}{\mu ^{t}}-\frac{Z^{t}}{\mu ^{t}} -\frac{\rho (Z^{t}-Z^{t-1})}{\mu ^{t}}. \end{aligned}$$

(A.8)

Next we have

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\Vert X^{t+1}-X^{t}\Vert _{F}&\le \lim \limits _{t\rightarrow \infty }\frac{1}{\mu ^{t}} \left\| \nabla G(X^{t+1})\right\| _{F}\nonumber \\&\quad +\frac{1}{\mu ^{t}} \left\| Z^{t}\right\| _{F}+\frac{\rho }{\mu ^{t}} \left\| Z^{t}-Z^{t-1}\right\| _{F}. \end{aligned}$$

(A.9)

Since $\{X^{t}\}$ is bounded, $X^{t}>0$ and $X^{t}\nrightarrow 0$, $\nabla G(X^{t+1})$ is bounded. In addition, $\lim \limits _{t\rightarrow \infty }\frac{1}{\mu ^{t}} =\lim \limits _{t\rightarrow \infty }\frac{1}{\mu ^{0}\rho ^{t}}=0$. Therefore, we can immediately claim that

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\Vert X^{t+1}-X^{t}\Vert _{F}=0. \end{aligned}$$

(A.10)

Eventually, according to (14), we can prove $\lim \limits _{t\rightarrow \infty }\Vert L^{t+1}-L^{t}\Vert _{F}=0$ as follows:

$$\begin{aligned}&\lim \limits _{t\rightarrow \infty }\Vert L^{t+1}-L^{t}\Vert _{F}\nonumber \\&\quad =\lim \limits _{t\rightarrow \infty }\left\| X^{t+1}-\frac{Z^{t+1}}{\mu ^{t}} +\frac{Z^{t}}{\mu ^{t}}-L^{t}\right\| _{F}\nonumber \\&\quad \le \lim \limits _{t\rightarrow \infty }\left\| \frac{Z^{t-1}}{\mu ^{t-1}}+X^{t} -L^{t}\right\| _{F}+\left\| X^{t+1}-X^{t}\right\| _{F}\nonumber \\&\qquad +\left\| \frac{Z^{t}}{\mu ^{t}}-\frac{Z^{t+1}}{\mu ^{t}} -\frac{Z^{t-1}}{\mu ^{t-1}}\right\| _{F}\nonumber \\&\quad =\lim \limits _{t\rightarrow \infty }\left\| \varSigma _{t-1}-{\mathcal {S}}_{\frac{\lambda w}{\mu ^{t-1}}}(\varSigma _{t-1})\right\| _{F}+\left\| X^{t+1}-X^{t}\right\| _{F}\nonumber \\&\qquad +\left\| \frac{Z^{t}}{\mu ^{t}}-\frac{Z^{t+1}}{\mu ^{t}} -\frac{Z^{t-1}}{\mu ^{t-1}}\right\| _{F}\nonumber \\&\quad \le \lim \limits _{t\rightarrow \infty }\frac{1}{\mu ^{t-1}}\sqrt{\sum _{i} \left( \lambda w_{i}\right) ^{2}}+\left\| X^{t+1}-X^{t}\right\| _{F}\nonumber \\&\qquad +\left\| \frac{Z^{t}}{\mu ^{t}}-\frac{Z^{t+1}}{\mu ^{t}} -\frac{Z^{t-1}}{\mu ^{t-1}}\right\| _{F}\nonumber \\&\quad =0, \end{aligned}$$

(A.11)

where $U_{t-1}\varSigma _{t-1}V^{T}_{t-1}$ denotes the SVD of the patch matrix $\frac{Z^{t-1}}{\mu ^{t-1}}+X^{t}$ in the t-th iteration, and then $L^{t}=U_{t-1}{\mathcal {S}}_{\frac{\lambda w}{\mu ^{t-1}}} \left( \varSigma _{t-1}\right) V^{T}_{t-1}$. $\square$