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Image Super-Resolution: Historical Overview and Future Challenges
Published in Peyman Milanfar, Super-Resolution Imaging, 2017
Besides the optimization approaches derived from stochastic view as discussed above, another stream of methods is through the well-known Projection onto Convex Sets (POCS) [110]. The POCS methods approach the SR problem by formulating multiple constraining convex sets containing the desired image as a point within the sets. Defining such convex sets is flexible and can incorporate different kinds of constraints or priors, even nonlinear and nonparametric constraints. As an example, we introduce several commonly used convex sets in the POCS methods. The data consistency or reconstruction constraints can be modeled as K convex sets: Ck={X|‖DkHkFkX−Yk‖2≤σ2,1≤k≤K}.
Inverse Problems in Microphone Arrays
Published in Vijay K. Madisetti, The Digital Signal Processing Handbook, 2017
The RAP method is a special case of the projection onto convex sets algorithm. The geometrical interpretation of the above algorithm is given in Figure 32.5. Each equation is modeled as a hyperplane in the solution space. Here, in the figure, it is shown as a line. The initial guess is projected onto the first hyperplane to obtain the second guess. This point is again projected onto the next hyperplane to get the third guess. It can be shown that by repeated projection on to the hyperplanes, the point converges to the solution [4]. λ (0 ≤ λ ≤ – 1) is called the relaxation parameter. It dictates how far we should proceed along the direction of the estimate. It is also a measure of confidence in the estimate, i.e., if the measurements are noisy, then usually λ is given a small value; if the values are relatively less noisy, then a larger value of λ can be used to speed up convergence. The algorithm is guaranteed to converge to the actual solution (if it exists). If a solution does not exist, then the “guess” is guaranteed to converge to the pseudo-inverse solution. The pseudo-inverse solution is the least squares solution which minimizes the energy in the solution vector. The RAP method provides stable estimates at each iteration. Since the method uses only one row at a time, the system can be made adaptive, i.e., as the source moves around in the room, the system response can be varied. For a detailed discussion of adaptive arrays, the reader is referred to [20].
Underwater polarization imaging based on Compressed Sensing theory
Published in Amir Hussain, Mirjana Ivanovic, Electronics, Communications and Networks IV, 2015
We solve the linear programming problem (9) using Projections onto Convex Sets (POCS) algorithm. The target image can be reconstruction by the one-dimensional vector X0 (therefore X ) we solve. minX0l1YI=Φ0X0
Efficient projection onto a low-dimensional ball
Published in Engineering Optimization, 2019
Paul D. Teal, Lakshmi Krishnan, Terence Betlehem
It has been shown (Tomioka, Suzuki, and Sugiyama 2011) that the DALM algorithm can be considered as a proximal minimization algorithm. Frequency domain equalization can be considered as an additional proximity operation in the iteration: proximity being a generalization of the more familiar concept of projection (Youla and Webb 1982; Combettes 1993). The resulting algorithm has some similarity to the method known as Projection Onto Convex Sets (POCS) or alternating projection. Successively projecting onto a set of convex sets has been shown to converge to a point in the intersection of these sets, if this intersection exists (Brègman 1965). POCS is perhaps best known in imaging and image processing (Censor 1988) where it has found many applications. There are many variations on the alternating projection algorithm, most notably Dykstra's algorithm (Boyle and Dykstra 1986), which results not only in a solution in the intersection of the convex sets, but in the projection of the initial point onto this intersection.