IMAGE DEBLURRING USING DERIVATIVE COMPRESSED SENSING FOR OPTICAL IMAGING APPLICATION

IMAGE DEBLURRING USING DERIVATIVE COMPRESSED SENSING FOR OPTICAL IMAGING APPLICATION

The problem of reconstruction of digital images from their blurred and noisy measurements is unarguably one of the central problems in imaging sciences. Despite its ill-posed nature, this problem can often be solved in a unique and stable manner, provided appropriate assumptions on the nature of the images to be recovered. In this paper, however, a more challenging setting is considered, in which accurate knowledge of the blurring operator is lacking, thereby transforming the reconstruction problem at hand into a problem of blind deconvolution. As a specific application, the current presentation focuses on reconstruction of short-exposure optical images measured through atmospheric turbulence. The latter is known to give rise to random aberrations in the optical wavefront, which are in turn translated into random variations of the point spread function of the optical system in use. A standard way to track such variations involves using adaptive optics. Thus, for example, the Shack–Hartmann interferometer provides measurements of the optical wavefront through sensing its partial derivatives. In such a case, the accuracy of wavefront reconstruction is proportional to the number of lens lets used by the interferometer and, hence, to its complexity. Accordingly, in this paper, we show how to minimize the above complexity through reducing the number of the lens lets while compensating for under sampling artifacts by means of derivative compressed sensing. Additionally, we provide empirical proof that the above simplification and its associated solution scheme result in image reconstructions, whose quality is comparable to the reconstructions, obtained using conventional (dense) measurements of the optical wavefront.

Existing System:

The necessity to recover digital images from their distorted and noisy observations is common for a variety of practical applications, with some specific examples including image denoising, super-resolution, image restoration, and watermarking, just to name a few. In such cases, it is conventional to assume that the observed image is obtained as a result of convolution of its original counterpart with a point spread function1 (PSF).

To account for measurement inaccuracies, it is also standard to contaminate the convolution output with an additive noise term, which is usually assumed to be white and Gaussian. Thus, formally While and can be regarded as general members of the signal space of real-valued functions on , the PSF is normally a much smoother function, with effectively band-limited spectrum.
As a result, the convolution with has a destructive effect on the informational content of , in which case typically has a substantially reduced set of features with respect to . This makes the problem of reconstruction of from a problem of significant practical importance.

The knowledge of the PSF may be lacking, which results in the necessity to recover the original image from its blurred and noisy observations alone. Such a reconstruction problem is commonly referred to as the problem of blind deconvolution

Proposed System:
We follow the philosophy of hybrid deconvolution, whose main idea is to leverage any partial information on the PSF to improve the accuracy of image restoration. In particular, in the algorithm described in this project, such partial information is derived from incomplete observations of the partial derivatives of the phase of the generalized pupil function (GPF) of the optical system in use, as detailed in the following.

Optical imaging is unarguably the field of applied sciences from which the notion of image deconvolution has originated. In particular, in short-exposure turbulent imaging, acquired images are blurred with a PSF, which depends on a spatial distribution of the atmospheric refraction index along the optical path connecting an object of interest and the observer. Due to the effect of turbulence, the above distribution is random and time dependent, which implies that the PSF cannot be known in advance.

Among some other factors, the accuracy of phase reconstruction by the SHI depends on the size of its sampling grid, which is in turn defined by the number of lenslets composing the wavefront sensor of the interferometer (see the following). Unfortunately, the grid size and the complexity (and, hence, the cost) of the interferometer tend to increase pro rata, which creates an obvious practical limitation. Accordingly, to overcome this problem, we propose to modify the construction of the SHI through reducing the number of its lenslets.

Although the advantages of such a simplification are immediate to see, its main shortcoming is obvious as well: The smaller the number of lenslets is, the stronger is the effect of under sampling and aliasing.
These artifacts, however, can be compensated for by subjecting the output of the simplified SHI to the derivative compressed sensing (DCS) algorithm of [23]. As will be shown in the following, DCS is particularly suitable for reconstruction of from incomplete measurements of its partial derivatives.

Software Requirements:
.Net
Front End – ASP.Net
Language – C#.Net
Back End – SQL Server
Windows XP

Hardware Requirements:
RAM : 512 Mb
Hard Disk : 80 Gb
Processor : Pentium IV

FUTURE WORK:
While the proposed method offers a practical solution to the problem of phase estimation in AO, some interesting questions about the theoretical aspects of DCS still lay open. In particular, the question of theoretical performance of CS in the presence of side information on the source signal needs to be addressed through future research. For practical purpose one can also take benefit of this algorithm to modify the SHI. Instead of working with the measurements of the phase gradient, their linear combination can be used, e.g., Bernoulli weights. The resulting sensing basis might have smaller coherence with respect to the basis of wavelets, thereby offering the possibility of more accurate and stable reconstruction.

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