Multiple Image Deblurring with High Dynamic-Range Poisson Data

Abstract

An interesting problem arising in astronomical imaging is the reconstruction of an image with high dynamic range, for example a set of bright point sources superimposed to smooth structures. A few methods have been proposed for dealing with this problem and their performance is not always satisfactory. In this paper we propose a solution based on the representation, already proposed elsewhere, of the image as the sum of a pointwise component and a smooth one, with different regularization for the two components. Our approach is in the framework of Poisson data and to this purpose we need efficient deconvolution methods. Therefore, first we briefly describe the application of the Scaled Gradient Projection (SGP) method to the case of different regularization schemes and subsequently we propose how to apply these methods to the case of multiple image deconvolution of high-dynamic range images, with specific reference to the Fizeau interferometer LBTI of the Large Binocular Telescope (LBT). The efficacy of the proposed methods is illustrated both on simulated images and on real images, observed with LBTI, of the Jovian moon Io. The software is available at http://www.oasis.unimore.it/site/home/software.html.

Appendices

Appendix

Regularization functions

In this paper we assume that ${\varvec{f}}$ is a $N \times N$ array extended (when needed) with periodic boundary conditions, i.e., if we set ${\varvec{n}} = (n_1, n_2)$, then ${\varvec{f}}(N+1,n_2) = {\varvec{f}}(1,n_2)$, ${\varvec{f}}(n_1,N+1) = {\varvec{f}}(n_1,1)$ and ${\varvec{f}}(N+1,N+1) = {\varvec{f}}(1,1)$.

For introducing the regularization functions considered in our methods and software we need some notation. We set ${\varvec{n}}_{1 \pm } = (n_1 \pm 1, n_2)$ and ${\varvec{n}}_{2 \pm } = (n_1, n_2 \pm 1)$ and we introduce the square and the modulus of the discrete gradient

$$\begin{aligned} {\varvec{D}}^2({\varvec{n}}) = \big [ {\varvec{f}}({\varvec{n}}_{1+}) - {\varvec{f}}({\varvec{n}})\big ]^2 \!+\! \big [{\varvec{f}}({\varvec{n}}_{2+}) - {\varvec{f}}({\varvec{n}})\big ]^2, \\ \nonumber |{\varvec{D}}({\varvec{n}})| = \sqrt{\big [{\varvec{f}}({\varvec{n}}_{1+}) - {\varvec{f}}({\varvec{n}})\big ]^2 \!+\! \big [{\varvec{f}}({\varvec{n}}_{2+}) - {\varvec{f}}({\varvec{n}}) \big ]^2}. \end{aligned}$$

(28)

Then, the seven regularization functions and the corresponding arrays $U_1,V_1$ are the following:

• Zeroth order Tikhonov (T-0) regularization

  $$\begin{aligned} J_1({\varvec{f}}) = \frac{1}{2} \sum _{\varvec{n}} |{\varvec{f}}({\varvec{n}})|^2~~, \end{aligned}$$

  (29)

  for which (5) holds by setting

  $$\begin{aligned} {\varvec{U}}_1({\varvec{n}}) = 0~~,~~{\varvec{V}}_1({\varvec{n}}) = {\varvec{f}}({\varvec{n}}). \end{aligned}$$

  (30)

• First order Tikhonov (T-1) regularization

  $$\begin{aligned} J_1({\varvec{f}}) = \frac{1}{2} \sum _{{\varvec{n}}} {\varvec{D}}^2({\varvec{n}})~~, \end{aligned}$$

  (31)

  for which (5) holds by setting

  $$\begin{aligned} {\varvec{U}}_1({\varvec{n}}) =&~{\varvec{f}}({\varvec{n}}_{1+})+{\varvec{f}}({\varvec{n}}_{2+})+{\varvec{f}}({\varvec{n}}_{1-})+{\varvec{f}}({\varvec{n}}_{2-})~~, \\ \nonumber {\varvec{V}}_1({\varvec{n}}) =&~4 {\varvec{f}}({\varvec{n}})~~. \end{aligned}$$

• Second order Tikhonov (T-2) regularization

  $$\begin{aligned} J_1({\varvec{f}}) = \frac{1}{2} \sum _{\varvec{n}}(\varDelta {\varvec{f}})({\varvec{n}})^2~~, \end{aligned}$$

  (32)

  where $\varDelta$ denotes the discrete Laplacian. As remarked in [36], it can be written in the form

  $$\begin{aligned} J_1({\varvec{f}}) = \frac{1}{2} \sum _{\varvec{n}} \big [{\varvec{f}}({\varvec{n}}) -(B{\varvec{f}})({\varvec{n}})\big ]^2~~, \end{aligned}$$

  (33)

  where B is the convolution matrix obtained from the $3 \times 3$ mask with columns (0, 1/4, 0),  (1/4, 0, 1/4) and (0, 1/4, 0). Then (5) holds by setting

  $$\begin{aligned} {\varvec{U}}_1({\varvec{n}}) =&~[(B + B^T) {\varvec{f}}]({\varvec{n}})~~, \\ \nonumber {\varvec{V}}_1({\varvec{n}}) =&~[(I + B^TB){\varvec{f}}]({\varvec{n}})~~. \end{aligned}$$

• Cross-Entropy (CE) regularization [17, 18]

  $$\begin{aligned} J_1({\varvec{f}}) = KL({\varvec{f}}, \bar{\varvec{f}}) = \sum _{{\varvec{n}}} \big \{{\varvec{f}}({\varvec{n}})\ln \left( \frac{{\varvec{f}}({\varvec{n}})}{\bar{\varvec{f}}({\varvec{n}})}\right) + \bar{\varvec{f}}({\varvec{n}}) - {\varvec{f}}({\varvec{n}})\big \}~~, \end{aligned}$$

  (34)

  where $\bar{\varvec{f}}$ is a reference image. When $\bar{\varvec{f}}$ is a constant array, then the cross-entropy becomes the negative Shannon entropy considered, for instance, in [44]. If both ${\varvec{f}}$ and $\bar{\varvec{f}}$ satisfy the constraint (4), then a possible choice for the functions $U_1,~V_1$ is

  $$\begin{aligned} {\varvec{U}}_1({\varvec{n}}) = - \ln \frac{{\varvec{f}}({\varvec{n}})}{c}~~,~~{\varvec{V}}_1({\varvec{n}}) = - \ln \frac{\bar{\varvec{f}}({\varvec{n}})}{c}~~, \end{aligned}$$

  (35)

  where c is the flux constant defined in (4). We remark that, since the background is taken into account by the algorithms, ${\varvec{f}}$ can be zero in some pixels; for this reason in the computation of the gradient we add a small quantity to the values of ${\varvec{f}}$. We also remark that when $\bar{\varvec{f}}$ is a constant, e.g. $c/N^2$, then ${\varvec{V}}_1({\varvec{n}}) = 2 \mathrm{ln}~N$.

• Hypersurface (HS) regularization [20]

  $$\begin{aligned} J_1({\varvec{f}}) = \sum _{{\varvec{n}}} \sqrt{\delta ^2 + {\varvec{D}}^2({\varvec{n}})}~~,~~\delta > 0~~, \end{aligned}$$

  (36)

  for which (5) holds by setting

  $$\begin{aligned} {\varvec{U}}_1({\varvec{n}}) = ~\frac{{\varvec{f}}({\varvec{n}}_{1+}) \!+\! {\varvec{f}}({\varvec{n}}_{2+})}{\sqrt{\delta ^2 \!+\! {\varvec{D}}^2({\varvec{n}})}} \!+\! \frac{{\varvec{f}}({\varvec{n}}_{1-})}{\sqrt{\delta ^2 \!+\! {\varvec{D}}^2({\varvec{n}}_{1-})}} \!+\! \frac{{\varvec{f}}({\varvec{n}}_{2-})}{\sqrt{\delta ^2 \!+\! {\varvec{D}}^2({\varvec{n}}_{2-})}}~,\\ \nonumber {\varvec{V}}_1({\varvec{n}}) = ~\frac{2 {\varvec{f}}({\varvec{n}})}{\sqrt{\delta ^2 \!+\! {\varvec{D}}^2({\varvec{n}})}} + \frac{{\varvec{f}}({\varvec{n}})}{\sqrt{\delta ^2 \!+\! {\varvec{D}}^2({\varvec{n}}_{1-})}} + \frac{{\varvec{f}}({\varvec{n}})}{\sqrt{\delta ^2 \!+\! {\varvec{D}}^2({\varvec{n}}_{2-})}}~. \end{aligned}$$

  The application of SGP to the case of HS regularization is already considered in [15] and [4] for a comparison of its accuracy with that of Total Variation (TV) regularization.

• Markov random field (MRF) regularization [27]

  $$\begin{aligned} J_1({\varvec{f}}) = \frac{1}{2}\sum _{{\varvec{n}}} \sum _{{\varvec{n}}' \in \mathcal{N}({\varvec{n}})}\sqrt{\delta ^2 + \left( \frac{{\varvec{f}}({\varvec{n}})-{\varvec{f}}({\varvec{n}}')}{\epsilon ({\varvec{n}}')}\right) ^2}~~, \end{aligned}$$

  (37)

  where $\delta >0$, $\mathcal{N}({\varvec{n}})$ is a symmetric neighborhood made up of the eight first neighbors of ${\varvec{n}}$ and $\epsilon ({\varvec{n}}')$ is equal to 1 for the horizontal and vertical neighbors and equal to $\sqrt{2}$ for the diagonal ones; thanks to the symmetry of $\mathcal{N}({\varvec{n}})$, Eq. (5) holds by setting

  $$\begin{aligned} {\varvec{U}}_1({\varvec{n}}) =&~\sum _{{\varvec{n}}' \in \mathcal{N}({\varvec{n}})}\frac{{\varvec{f}}({\varvec{n}}')}{\epsilon ({\varvec{n}}')\sqrt{\delta ^2 + \left( \frac{{\varvec{f}}({\varvec{n}}')-{\varvec{f}}({\varvec{n}}')}{\epsilon ({\varvec{n}}')}\right) ^2}}~~, \\ \nonumber {\varvec{V}}_1({\varvec{n}}) =&~\sum _{{\varvec{n}}' \in \mathcal{N}({\varvec{n}})}\frac{{\varvec{f}}({\varvec{n}})}{\epsilon ({\varvec{n}}')\sqrt{\delta ^2 + \left( \frac{{\varvec{f}}({\varvec{n}})-{\varvec{f}}({\varvec{n}}')}{\epsilon ({\varvec{n}}')}\right) ^2}}~~. \end{aligned}$$

• MISTRAL regularization (MIST) [38]

  $$\begin{aligned} J_1({\varvec{f}}) = \sum _{{\varvec{n}}} \left\{ |{\varvec{D}}({\varvec{n}})| -\delta \ln \left( 1 + \frac{|{\varvec{D}}({\varvec{n}})|}{\delta }\right) \right\} ~~,~~\delta > 0~~, \end{aligned}$$

  (38)

  for which (5) holds by setting

  $$\begin{aligned} {\varvec{U}}_1({\varvec{n}}) =&~\frac{{\varvec{f}}({\varvec{n}}_{1+}) + {\varvec{f}}({\varvec{n}}_{2+})}{\delta + |{\varvec{D}}({\varvec{n}})|} +\frac{{\varvec{f}}({\varvec{n}}_{1-})}{\delta + |{\varvec{D}}({\varvec{n}}_{1-})|} + \frac{{\varvec{f}}({\varvec{n}}_{2-})}{\delta + |{\varvec{D}}({\varvec{n}}_{2-})|}~~, \\ \nonumber {\varvec{V}}_1({\varvec{n}}) =&~\frac{2{\varvec{f}}({\varvec{n}})}{\delta + |{\varvec{D}}({\varvec{n}})|} + \frac{{\varvec{f}}({\varvec{n}})}{\delta + |{\varvec{D}}({\varvec{n}}_{1-})|} + \frac{{\varvec{f}}({\varvec{n}})}{\delta + |{\varvec{D}}({\varvec{n}}_{2-})|}~~. \end{aligned}$$