Majorization–Minimization approach for real-time enhancement of sparsity-driven SAR imaging

Abstract

The earlier works in the context of sparsity-driven SAR imaging have shown significant improvement in the reconstruction process due to admitting sparsity as a prior. In spite of the importance of real-time processing requirement in the remote sensing (RS) applications, most of the works have not focused on real-time procedures and reducing the computational burden, but rather enhancing the quality of formed image. To address this weakness, this paper presents a problem-driven algorithm, which relies on Majorization–Minimization (MM) procedure. Using MM in our solutions, a simpler surrogate optimization problem is solved instead of the difficult original form. To show the efficacy of MM algorithm in real-time applications experimental results based on simulated and real data along with a performance analysis are presented. All results validate the superiority of the proposed MM-based method in terms of computational load and processing time as compared with previous works.

Appendices

Appendix A (Proof of Eq. (13)):

To reach the solution of (13), we first resort the complex-valued derivative rules and obtain the first order derivation of ${S}_{t}\left(\mathbf{f}\right)$, given by.

$$\frac{\partial {S}_{t}\left(\mathbf{f}\right)}{\partial \mathbf{f}}=-{\left({\mathbf{H}}^{H}\mathbf{y}\right)}^{*}+{\left({\mathbf{H}}^{H}\mathbf{H}\right)}^{T}{\mathbf{f}}^{*}+\frac{{\lambda }_{1}^{2}}{2}\left({ {\varvec{\Lambda}}}_{1k}^{-1}\right){\mathbf{f}}^{*}+{\lambda }_{2}^{2}{\mathbf{D}}^{T}{{\varvec{\Lambda}}}_{2k}^{-1}\mathbf{D}$$

(A-1)

Remarkably, both terms of $\frac{{\uplambda }_{1}^{2}}{2}{\Vert {\mathbf{f}}_{k}\Vert }_{1}$ and $\frac{{\uplambda }_{2}^{2}}{2}{\Vert \mathbf{D}\left|{\mathbf{f}}_{k}\right|\Vert }_{1}$ in (12) depend on $\mathbf{f}$ at previous iteration and thus they are fixed with respect to $\mathbf{f}$ at current iteration. Consequently, they don't have any impacts on the derivation procedure.

By equating (A-1) to zero the solution of (13) is simply resulted.

Appendix B (Proof of Eq. (19))

The cosine terms in (18) can be incorporated to a single cosine through phasor addition rule. To do so, the phasor expressions of the terms $\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}\right)$ and $\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}-\frac{\uppi }{2}\right)$ within (18) are given as follows

$$\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}\right)\equiv 1\boldsymbol{\measuredangle }0$$

(B-1)

$$\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}-\frac{\uppi }{2}\right)\equiv 1\boldsymbol{\measuredangle }-\frac{\uppi }{2}=-\mathbf{j}$$

(B-2)

Using (B-1) and (B-2) the term ${\left(\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}\right)\right)}^{T}.\mathfrak{N}{e}\left\{{\mathbf{B}}^{H}{\mathbf{H}}^{H}\mathbf{y}\right\}+{\left(\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}-\frac{\uppi }{2}\right)\right)}^{T}.\mathfrak{I}\mathrm{m}\left\{{\mathbf{B}}^{H}{\mathbf{H}}^{H}\mathbf{y}\right\}$ within (18) is equivalent to the following expression.

$${\left(\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}\right)\right)}^{T}.\mathfrak{N}{e}\left\{{\mathbf{B}}^{H}{\mathbf{H}}^{H}\mathbf{y}\right\}+{\left(\mathbf{c}\mathbf{o}\mathbf{s}\left({\varvec{\Phi}}-\frac{\uppi }{2}\right)\right)}^{T}.\mathfrak{I}\mathrm{m}\left\{{\mathbf{B}}^{H}{\mathbf{H}}^{H}\mathbf{y}\right\}\equiv {1}^{T}.{\varvec{\zeta}}-{\mathbf{j}}^{T}.{{\varvec{\zeta}}}^{{{\prime}}}$$

(B-3)

where two vectors of ${\varvec{\zeta}}\in {\mathbb{R}}^{I\times 1}$ and ${{\varvec{\zeta}}}^{{{\prime}}}\in {\mathbb{R}}^{I\times 1}$ are, respectively, defined as ${\varvec{\zeta}}\triangleq \mathfrak{N}{e}\left\{{\mathbf{B}}^{H}{\mathbf{H}}^{H}\mathbf{y}\right\}$ and ${{\varvec{\zeta}}}^{{{\prime}}}\triangleq \mathfrak{I}\mathrm{m}\left\{{\mathbf{B}}^{H}{\mathbf{H}}^{H}\mathbf{y}\right\}$.

By expanding the above formula, we have

$${1}^{T}.{\varvec{\zeta}}-{\mathbf{j}}^{T}.{{\varvec{\zeta}}}^{{{\prime}}}=\sum_{i=1}^{I}{\zeta }_{i}-j{\zeta }_{i}^{{{\prime}}}$$

(B-4)

By finding the magnitude and the phase of each term within (B-4), the above equation can be re-expressed in new format as follows.

$${\sum }_{i=1}^{I}\sqrt{{\zeta }_{i}^{2}+{{\zeta }_{i}^{{{\prime}}}}^{2}}\cdot\mathrm{cos}\underset{{\Omega }_{i}}{{\left({\phi }_{1}+{\mathrm{tan}}^{-1}\left(-\frac{{\zeta }_{i}^{{{\prime}}}}{{\zeta }_{i}}\right)\right)}}$$

(B-5)

To minimize (19), each of cosine term within (B-5) should be maximized and accordingly we must have ${\left.{\Omega }_{i}\right|}_{i=1}^{I}=0$. By doing so, each ${\left.{\phi }_{i}\right|}_{i=1}^{I}$ can be estimated as below.

$${\left.{\widehat{\phi }}_{i}\right|}_{i=1}^{I}=-{\mathrm{tan}}^{-1}\left(-\frac{{\zeta }_{i}^{{{\prime}}}}{{\zeta }_{i}}\right)$$

(B-6)

By defining the vector of $\widehat{{\varvec{\Phi}}}\triangleq {\left[{\widehat{\phi }}_{1},{\widehat{\phi }}_{2},\dots ,{\widehat{\phi }}_{I}\right]}^{T}$, comprised of the entire phase values, (B-6) can be reformulated as.

$$\widehat{{\varvec{\Phi}}}=-{\mathbf{tan}}^{-1}\left( -{\varvec{\zeta}}^{\prime}. \left/ {\varvec{\zeta}} \right. \right)$$

(B-7)