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On Regularisation of Coherent Imagery with Proximal Methods

F. M. Watson, W. R. B. Lionheart, J. Hellier

TL;DR

It is shown that under certain sufficient conditions the proximal map of such a function G may be calculated as a simple phase correction to that of H, and provided a means to apply practically arbitrary regularisation functions to the magnitude when solving coherent reconstruction problems via proximal optimisation algorithms.

Abstract

In complex-valued coherent inverse problems such as synthetic aperture radar (SAR), one may often have prior information only on the magnitude image which shows the features of interest such as strength of reflectivity. In contrast, there may be no more prior knowledge of the phase beyond it being a uniform random variable. However, separately regularising the magnitude, via some function \(G:=H(|\cdot|)\), would appear to lead to a potentially challenging non-linear phase fitting problem in each iteration of even a linear least-squares reconstruction problem. We show that under certain sufficient conditions the proximal map of such a function \(G\) may be calculated as a simple phase correction to that of \(H\). Further, we provide proximal map of (almost) arbitrary \(G:=H(|\cdot|)\) which does not meet these sufficient conditions. This may be calculated through a simple numerical scheme making use of the proximal map of \(H\) itself, and thus we provide a means to apply practically arbitrary regularisation functions to the magnitude when solving coherent reconstruction problems via proximal optimisation algorithms. This is demonstrated using publicly available real SAR data for generalised Tikhonov regularisation applied to multi-channel SAR, and both a simple level set formulation and total generalised variation applied to the standard single-channel case.

On Regularisation of Coherent Imagery with Proximal Methods

TL;DR

It is shown that under certain sufficient conditions the proximal map of such a function G may be calculated as a simple phase correction to that of H, and provided a means to apply practically arbitrary regularisation functions to the magnitude when solving coherent reconstruction problems via proximal optimisation algorithms.

Abstract

In complex-valued coherent inverse problems such as synthetic aperture radar (SAR), one may often have prior information only on the magnitude image which shows the features of interest such as strength of reflectivity. In contrast, there may be no more prior knowledge of the phase beyond it being a uniform random variable. However, separately regularising the magnitude, via some function \(G:=H(|\cdot|)\), would appear to lead to a potentially challenging non-linear phase fitting problem in each iteration of even a linear least-squares reconstruction problem. We show that under certain sufficient conditions the proximal map of such a function may be calculated as a simple phase correction to that of . Further, we provide proximal map of (almost) arbitrary \(G:=H(|\cdot|)\) which does not meet these sufficient conditions. This may be calculated through a simple numerical scheme making use of the proximal map of itself, and thus we provide a means to apply practically arbitrary regularisation functions to the magnitude when solving coherent reconstruction problems via proximal optimisation algorithms. This is demonstrated using publicly available real SAR data for generalised Tikhonov regularisation applied to multi-channel SAR, and both a simple level set formulation and total generalised variation applied to the standard single-channel case.

Paper Structure

This paper contains 14 sections, 5 theorems, 53 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main result
  3. Application to some common regularisation functions
  4. Vector norms
  5. Total Variation and variants
  6. Restriction to feasible sets
  7. Application to Level Set Reconstruction
  8. Proximal map for more general functions
  9. Application to SAR data
  10. Multi-look SAR with generalised Tikhonov
  11. Level set magnitude reconstruction in complex-valued SAR
  12. Total Generalised Variation for spotlight mode SAR
  13. Conclusion
  14. SAR data model

Key Result

Theorem 1

Let $G(\boldsymbol{\mathbf{z}}) = H(|\boldsymbol{\mathbf{z}}|)$, $G:\mathds{C}^n\rightarrow\mathds{R}$, where $H:\mathds{R}^n\rightarrow \mathds{R}$ is a closed proper convex function. If $\mathop{\mathrm{prox}}\nolimits_H:\mathds{R}^n_{\geq0}\rightarrow \mathds{R}^n_{\geq0}$, then where $\boldsymbol{\mathbf{r}}:=|\boldsymbol{\mathbf{z}}|$, and $\boldsymbol{\mathbf{\Phi}}:=\exp(\mathrm{i}\angle\b

Figures (7)

  • Figure 1: Example magnitude image chips of reconstructions of the Gotcha carpark data, showing (a) the $\mathrm{TV}(|\cdot|)$ regularised result and (b) the result of applying $\mathrm{TV}$ directly to the complex-valued image. Also shown in (c) is the phase of the backprojection image.
  • Figure 2: False colours used for multi-look imagery in s \ref{['fig: multilook']} and \ref{['fig: multilook zoom']}. The major axis of each ellipse shown is oriented towards the azimuth of the centre of each sub-aperture.
  • Figure 3: Multi-aspect images of the Gotcha carpark dataset
  • Figure 4:
  • Figure 5: PaLEnTIR level set reconstruction and back-projection of a sub-scene from the Gotcha dataset. \ref{['fig: level lin']}-\ref{['fig: level bp log']} show the magnitude of these complex-valued images, \ref{['fig: level phase']} the phase of the level set reconstruction, and \ref{['fig: level phase diff']} the phase difference between level set reconstruction and backprojection images
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2
  • proof
  • Remark
  • Remark
  • Lemma 1
  • proof
  • Theorem 2