# Convergence Gain in Compressive Deconvolution: Application to Medical Ultrasound Imaging
## 1 Introduction  
Ultrasound imaging has become a very important medical imaging scheme, as it is noninvasive, harmless, and cost-effective compared with computed tomography (CT) and nuclear magnetic resonance (MRI). Moreover, ultrasound requires little energy, which makes ultrasound imaging a good candidate for hand-held applications. An extremely promising application scenario for ultrasound imaging is breast cancer localization, where cancer regions have statistically different acoustic properties compared with benign areas [1]. Compressive sensing (CS) can accelerate the acquisition rate without decreasing the reconstructed signal quality and maintain the image quality with fewer data. Based on CS theory, one can effectively implement the reconstruction under the condition of the restricted isometry property (RIP) [2; 3]. The applicability of CS, such as 2D and 3D ultrasound imaging [4] or duplex Doppler [5], has attracted an increasing number of researchers to propose new theories and methods. Among them, several ultrasound imaging devices [6; 7; 8; 9; 10] have been proposed in which acquisition and compression are processed simultaneously. The main idea is to combine CS and deconvolution into ultrasound imaging and form the compressive deconvolution problem as follows:  
\[y=\Phi Hx+\
u, \tag{1}\]where \(y\in\mathbb{R}^{M}\) involves \(M\) linear measurements acquired by projecting one RF image \(Hx\in\mathbb{R}^{N}\) onto the CS acquisition matrix \(\Phi\in\mathbb{R}^{M\times N}\), with \(M\ll N\). \(H\in\mathbb{R}^{N\times N}\) represents a block circulant with a circulant block (BCCB) matrix modeling the 2D convolution between the 2D point spread function (PSF) of the ultrasound system, and the tissue reflectivity function (TRF), \(\
u\in\mathbb{R}^{M}\), represents a zero-mean additive white Gaussian noise. Some sequential approaches such as YALL1 [11] are prone to first reformulate (1) into an unconstrained optimization model:  
\[a=\arg\min_{a}\{\|y-\Phi\Psi a\|_{2}^{2}+2\mu\|a\|_{1}\}. \tag{2}\]  
When the blurred RF image \(\bar{r}=\Psi a\) is restored, the TRF \(x\) will be inferred by minimizing:  
\[\min_{x\in\mathbb{R}^{N}}\alpha\|x\|_{1}+\|\bar{r}-Hx\|_{2}^{2}. \tag{3}\]  
The above two-step manipulation is the most intuitive way but inevitably yields larger estimation errors as illustrated in [12]. As such, [PERSON] et al. [13] investigate the following model by solving CS and deconvolution problems simultaneously:  
\[\min_{x,a\in\mathbb{R}^{N}} \|w\|_{1}+\alpha\|x\|_{q}^{q}+\frac{1}{2\mu}\|y-Aa\|_{2}^{2}\] (4) s.b. \[Hx=\Psi a,w=a,A=\Phi\Psi\]  
where \(q\) denotes the shape parameter of the generalized Gaussian distribution. Define \(A_{1}=\left( \begin{array}{c}I_{N}\\ \Psi\end{array} \right)\), \(B_{1}=\left( \begin{array}{cc}-I_{N}&0\\ 0&-H\end{array} \right)\), \(\lambda=\left( \begin{array}{c}\lambda_{1}\\ \lambda_{2}\end{array} \right)\) and \(p=\left( \begin{array}{c}w\\ x\end{array} \right)\), Equation (4) can be solved by alternatively solving three easier sub-problems based on alternating direction method of multipliers (ADMM):  
\[w^{k+1}=prox_{\|\cdot\|_{1}/\beta}\left(a^{k}-\frac{\lambda^{k}}{\beta}\right), \tag{5a}\] \[x^{k+1}\approx prox_{x\gamma\|\cdot\|_{p}^{p}/\beta}\left(x^{k}-\gamma H^{ \mathrm{T}}(Hx^{k}+\frac{\lambda_{2}^{k}}{\beta}-\Psi a^{k})\right),\] (5b) \[a^{k+1}=(\tfrac{1}{\mu}A^{\mathrm{T}}A+\beta I_{N}+\beta\Psi^{ \mathrm{T}}\Psi)^{-1}(\tfrac{1}{\mu}A^{\mathrm{T}}y+\lambda_{1}^{k}+\Psi^{ \mathrm{T}}\lambda_{2}^{k}+\beta w^{k+1}+\beta\Psi^{\mathrm{T}}Hx^{k+1}),\] (5c) \[\lambda^{k+1}=\lambda^{k}-\beta(A_{1}a^{k+1}-B_{2}p^{k+1}) \tag{5d}\]  
where \(h(x)=\tfrac{1}{2}\|\Psi a^{k}-Hx-\lambda_{2}^{k}\|_{2}^{2}\), \(\gamma\) is a Lipschitz parameter, and \(prox\) represents the proximal operator [14]. For the above three sub-problems, \(w^{k+1}\) has a closed solution and \(x^{k+1}\) has an approximate closed solution, but \(a^{k+1}\) is not easy to compute, even though the effective Sherman-Morrison-Woolbury inversion manipulation can be introduced.