Image Restoration for Shift Invariant Blurs-A Survey <s> Projection onto Convex Set <s> We consider the problem of reconstructing remotely obtained images from image-plane detector arrays. Although the individual detectors may be larger than the blur spot of the imaging optics, high-resolution reconstructions can be obtained by scanning or rotating the image with respect to the detector. As an alternative to matrix inversion or least-squares estimation [Appl. Opt. 26, 3615 (1987)], the method of convex projections is proposed. We show that readily obtained prior knowledge can be used to obtain good-quality imagery with reduced data. The effect of noise on the reconstruction process is considered. <s> BIB001 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Projection onto Convex Set <s> The authors address the problem of reconstruction of a high-resolution image from a number of lower-resolution (possibly noisy) frames of the same scene where the successive frames are uniformly based versions of each other at subpixel displacements. In particular, two previously proposed methods, a frequency-domain method and a method based on projections onto convex sets (POCSs), are extended to take into account the presence of both sensor blurring and observation noise. A new two-step procedure is proposed, and it is shown that the POCS formulation presented for the high-resolution image reconstruction problem can also be used as a new method for the restoration of spatially invariant blurred images. Some simulation results are provided. > <s> BIB002
The projection onto convex sets (POCS) technique consists of iterative projection of an initial estimate onto predefined constraint sets BIB001 . Various popular constraint set used are; i) Data fidelity ii) pixel range constraint iii) frequency domain constraint set. constraint set Data fidelity is commonly used as a constraint set to ensure consistency with observations and constrains the difference between the predicted pixel value and the observed pixel value. (33) r( ) is residual given by; (34) S k is the set of pixels in the f(x, y) that contributes to the pixel (x k , y k ), h(x k , y k ; x, y) is the contribution of the pixel f(x, y), and T(x k , y k ) is the threshold value that controls the data fidelity constraint. Selection of threshold should be proportional to the noise power BIB002 . According to data fidelity constraint current estimate is changed when the absolute value of the residual is less than the threshold T(x k , y k ) else it is updated . Fig.5 shows projection operation onto the data fidelity constraint set is formulated as; The pixel range constraint set ensures resulting image pixel values within a certain range. Frequency domain constraint sets is used when a certain frequency range of the true signal is known. Advantage of the POCS technique is the ease of incorporating space-variant PSF into the restoration. The technique guarantees convergence to a solution that is consistent with all the constraint sets when the constraint sets are convex and not disjoint but the solution is not necessarily unique & depend on the initial estimate and the order of the projections.