Imaging With Nature: Some thoughts on deblurring.

Deblurring is somehow very close to calibration because it is sometimes equivalent to blind deconvolution. The interesting aspect of this procedure are the seemingly different routes used to perform the same task. In [1], Yu Mao  and Jerome Gilles use a regularization method (nonlocal Total Variation) while Jerome Gilles and Stanley Osher [2] use a turbulence model to perform that deconvolution. Florent Couzinie-Devy, Julien Mairal, Francis Bach, Jean Ponce [3] build a dictionary from training patches in low/high resolution and crisp/blurry conditions to eventually perform deblurring.  Finally, Hui Ji , Jia Li, Zuowei Shen, Kang Wang [4] use some heuristic on the wavelet coefficients to perform this same task.

Let us go further and take this to the real word with real sensors and real turbulence. Wouldn't some of these methods perform even better if they took into account that some of the training images used for the eventual deblurring are themselves doing better than the diffraction limit ( as shown in  Turbulence Aided Lucky Imaging ) ? Wouldn't these approaches also help in weak gravitational lensing studies ? Imaging with Nature means that nature sometimes really ...really... wants to help you out....

from [5]

References:

[1]Non Rigid Geometric Distortions Correction Application to Atmospheric Turbulence Stabilization by Yu Mao  and Jerome Gilles. The abstract reads:

 A novel approach is presented to recover image degraded by atmospheric turbulence. Given a sequence of frames affected by turbulence, we construct a variational model to characterize the static image. The optimization problem is solved by the Bregman iteration and operator splitting method. Our algorithm is simple and efficient, and can be easily generalized for different scenarios.

[2] Fried deconvolution by Jerome Gilles and Stanley Osher. The abstract reads:

In this paper we present a new approach to deblur the effect of atmospheric turbulence in the case of long range imaging. Our method is based on an analytical formulation, the Fried kernel, of the atmosphere modulation transfer function (MTF) and a framelet based deconvolution algorithm. An important parameter is the refractive index structure which requires specific measurements to be known. Then we propose a method which provides a good estimation of this parameter from the input blurred image. The final algorithms are very easy to implement and show very good results on both simulated blur and real images.

[3] Dictionary Learning for Deblurring and Digital Zoom by Florent Couzinie-Devy, Julien Mairal, Francis Bach, Jean Ponce. The abstract reads:

This paper proposes a novel approach to image deblurring and digital zooming using sparse local models of image appearance. These models, where small image patches are represented as linear combinations of a few elements drawn from some large set (dictionary) of candidates, have proven well adapted to several image restoration tasks. A key to their success has been to learn dictionaries adapted to the reconstruction of small image patches. In contrast, recent works have proposed instead to learn dictionaries which are not only adapted to data reconstruction, but also tuned for a specific task. We introduce here such an approach to deblurring and digital zoom, using pairs of blurry/sharp (or low-/high-resolution) images for training, as well as an effective stochastic gradient algorithm for solving the corresponding optimization task. Although this learning problem is not convex, once the dictionaries have been learned, the sharp/high-resolution image can be recovered via convex optimization at test time. Experiments with synthetic and real data demonstrate the effectiveness of the proposed approach, leading to state-of-the-art performance for non-blind image deblurring and digital zoom.

[4] Image deconvolution using a characterization of sharp images in wavelet domain by Hui Ji , Jia Li, Zuowei Shen, Kang Wang. The abstract reads:

Image deconvolution is a challenging ill-posed problem when only partial information of the blur kernel is available. Certain regularization on sharp images has to be imposed to constrain the estimation of true images during the blind deconvolution process. Based on the observation that an image of sharp edges tends to minimize the ratio between the `1 norm and the `2 norm of its wavelet frame coe cients, we propose a new characterization of sharp images for image deconvolution. A two-stage method is then developed to solve semi-blind image deconvolution problems. The proposed method is fast, easy to implement and does not require rigorous parameter tune-up. Such a regularization can also be applied to solve non-blind image deconvolution problems and the resulting algorithm achieves good performance without rigorous parameter tune-up.

[5] Mikhail Vorontsov,  Gary W. Carhart, Anisoplanatic imaging through turbulent media: image recovery by local information fusion from a set of short-exposure images, [J. Opt. Soc. Am. A/ Vol. 18, No. 6/June 2001]

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