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## Multichannel deconvolution

In some applications, the acquisition system is able to provide multiple observations of the original scene. In electron microscopy, for example, many dierently focused version of the same image are acquired during a single experiment, due to an intrinsic trade-o between the bandwidth of the imaging system and the contrast of the resulting image. Besides, pushbroom imaging systems [Song et al., 2017] can acquire hyperspectral images in which each pixel provides local spectral information about a scene of interest across a large number of contiguous bands. In other applications, such as photography or video acquisition, continuous shooting or video capture with the camera provides several images that are blurred in a dierent way since our hand moves randomly (see Figure 2.7). In case of multiple measurements, the restoration algorithm can exploit the redundancy present in the observations and, in principle, it can achieve performance not obtainable from a single measure. The degradation model relating the observations (yk)16k6K to the unknown image x can be expressed as follows:

### Single-image super-resolution

Multiple-image super-resolution algorithms utilize a set of LR images to reconstruct the sharp details of the sought HR image. However, in various real-world applications, several low resolution observations of the same scene is not available. As an extreme but frequent situation, only one single LR image can be provided. One of the proposed approaches to address the single-image SR problem tries to solve the underlying ill-posed problem through the introduction of prior knowledge on the expected high resolution image. This class of methods is termed reconstruction-based methods [Fattal, 2007; Fan and Yeung, 2007; Dai et al., 2009] and aims at reconstructing the missing details of the upscaled image while preserving the continuity and sharpness of the edges [Li and Orchard, 2001; Dai et al., 2007; Fattal, 2007]. This can also be performed by considering regularization functions that preserve strong discontinuities in the spatial domain such as the Total Variation regularization [Aly and Dubois, 2005], or in the frequency domain through wavelet transforms, where the wavelet coecients of the estimated HR image are enforced to be sparse [Sen and Darabi, 2009]. Another class of single-image SR algorithms is referred to as learning-based methods [Mjolsness, 1985; Capel and Zisserman, 2001; Zou and Yuen, 2012] and resorts to trained dictionaries in order to infer ne details. The dictionary contains pairs of HR images/patches and the corresponding degraded LR images/patches. In most learning-based methods, the image is partitioned into (possibly overlapping) square patches of pixels. For each input LR patch, a set of neighboor LR patches from the dictionary is selected. The candidate LR patches are weighted and a reconstruction model is derived from the selected low and high resolution pairs. Afterwards, this model is applied to the input LR patch in order to reconstruct its associated HR patch [Stephenson and Chen, 2006]. Finally, the HR image is obtained by merging all the reconstructed HR patches as illustrated in Figure 2.10.

#### Variable metric forward-backward algorithm

As discussed in the previous section, the forward-backward algorithm is welladapted to minimization problems involving a sum of a dierentiable and convex functions. However, despite its low computational cost, it suers from a slow convergence rate [Chouzenoux et al., 2014]. Thereby, a close attention has been paid to accelerate it. An interesting approach consists in utilizing a variable metric at each iteration of the algorithm thanks to the introduction of preconditioning matrices [Combettes and V~u, 2014a; Chouzenoux et al., 2014]. Hence, let us consider again the minimization problem (3.19) with f and g satisfying the following assumptions:

**Table of contents :**

Resume

**1 Introduction **

Context

Collaborations

Organization of the document

Publications

**2 Background on inverse problems for image and video processing **

2.1 Introduction

2.2 Inverse problems

2.2.1 Introduction

2.2.2 Variational formulation

2.3 Introduction to image deconvolution

2.3.1 Single-channel deconvolution

2.3.2 Multichannel deconvolution

2.4 Introduction to image super-resolution

2.4.1 Multiple-image super-resolution

2.4.2 Single-image super-resolution

2.5 Introduction to video restoration

2.5.1 Video deconvolution

2.5.2 Video super-resolution

2.5.3 Television archives

2.6 Conclusion

**3 Optimization methods **

3.1 Introduction

3.2 Denitions and notation

3.3 Optimization algorithms

3.3.1 Gradient descent algorithm

3.3.2 Proximal point algorithm

3.3.3 Majorize-Minimize strategy

3.3.4 Forward-backward algorithm

3.3.5 Variable metric forward-backward algorithm

3.3.6 Block-coordinate variable metric forward-backward algorithm

3.3.7 Primal-dual algorithms

3.3.8 Dual forward-backward algorithm

3.4 Conclusion

**4 Dual block-coordinate forward-fackward algorithm **

4.1 Introduction

4.2 Non-separable case

4.2.1 Problem statement

4.2.2 Preconditioned dual forward-backward

4.2.3 Link with Dykstra algorithm

4.3 Separable case

4.3.1 Problem statement

4.3.2 Dual block forward-backward algorithm

4.3.3 Simplied form

4.3.4 Particular case when f = 0

4.3.5 Link with the parallel dual forward-backward

4.3.6 Extension to a general metric

4.4 Convergence analysis

4.5 Conclusion

**5 Application to video deconvolution and deinterlacing **

5.1 Introduction

5.2 Observation model

5.2.1 Spatial regularization

5.2.2 Temporal regularization

5.3 Minimization strategy

5.4 Experimental results

5.4.1 Data-set Benchmark

5.4.2 Numerical performance

5.5 Conclusion

**6 Distributed algorithm for computing proximity operators **

6.1 Introduction

6.2 Minimization problem

6.3 Distributed algorithm

6.3.1 Local form of consensus

6.3.2 Derivation of the proposed algorithm

6.3.3 Consensus choice

6.4 A useful special case

6.4.1 Form of the algorithm

6.4.2 Distributed implementation

6.5 Application to video denoising

6.5.1 Observation model

6.5.2 Proposed method

6.5.3 Simulation results

6.6 Conclusion

**7 Alternating proximal method for blind video deconvolution **

7.1 Introduction

7.2 Problem statement

7.2.1 Observation model

7.2.2 Video estimation

7.2.3 Kernel identication

7.3 Optimization method

7.3.1 Minimization strategy

7.3.2 Construction of the majorant

7.3.3 Implementation of the proximity operator of f2

7.3.4 Implementation of the proximity operator for kernel estimation

7.3.5 Convergence analysis

7.4 Experimental results

7.4.1 Blind video deconvolution step

7.4.2 Non-blind video deconvolution step

7.4.3 Real Data

7.5 Conclusion

**8 Conclusion **

List of gures

List of tables

**Bibliography**