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Register a new userDenoising Using Projection Onto Convex Sets (POCS) Based Framework
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Added by
Mohammad Tofighi
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in R^N the corresponding set is also a convex set in R^{N+1}. The iterative optimization approach starts with an arbitrary initial estimate in R^{N+1} and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in total-variation (TV), filtered variation (FV), L_1, and entropic cost functions. A new denoising algorithm using the TV framework is developed. The new algorithm does not require any of the regularization parameter adjustment. Simulation examples are presented.
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Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some non-convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in R^N the corresponding set is a convex set in R^(N+1). The iterative optimization approach starts with an arbitrary initial estimate in R^(N+1) and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in total-variation, filtered variation, l1, and entropic cost functions. It is also experimentally observed that cost functions based on lp, p<1 can be handled by using the supporting hyperplane concept.
A new deconvolution algorithm based on orthogonal projections onto the epigraph set of a convex cost function is presented. In this algorithm, the dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. As the utilized cost function is a convex function in $R^N$, the corresponding epigraph set is also a convex set in $R^{N+1}$. The deconvolution algorithm starts with an arbitrary initial estimate in $R^{N+1}$. At each step of the iterative algorithm, first deconvolution projections are performed onto the epigraphs, later an orthogonal projection is performed onto one of the constraint sets associated with the cost function in a sequential manner. The method provides globally optimal solutions for total-variation, $ell_1$, $ell_2$, and entropic cost functions.
The Frank-Wolfe method solves smooth constrained convex optimization problems at a generic sublinear rate of $mathcal{O}(1/T)$, and it (or its variants) enjoys accelerated convergence rates for two fundamental classes of constraints: polytopes and strongly-convex sets. Uniformly convex sets non-trivially subsume strongly convex sets and form a large variety of textit{curved} convex sets commonly encountered in machine learning and signal processing. For instance, the $ell_p$-balls are uniformly convex for all $p > 1$, but strongly convex for $pin]1,2]$ only. We show that these sets systematically induce accelerated convergence rates for the original Frank-Wolfe algorithm, which continuously interpolate between known rates. Our accelerated convergence rates emphasize that it is the curvature of the constraint sets -- not just their strong convexity -- that leads to accelerated convergence rates. These results also importantly highlight that the Frank-Wolfe algorithm is adaptive to much more generic constraint set structures, thus explaining faster empirical convergence. Finally, we also show accelerated convergence rates when the set is only locally uniformly convex and provide similar results in online linear optimization.
In this paper, an Entropy functional based online Adaptive Decision Fusion (EADF) framework is developed for image analysis and computer vision applications. In this framework, it is assumed that the compound algorithm consists of several sub-algorithms each of which yielding its own decision as a real number centered around zero, representing the confidence level of that particular sub-algorithm. Decision values are linearly combined with weights which are updated online according to an active fusion method based on performing entropic projections onto convex sets describing sub-algorithms. It is assumed that there is an oracle, who is usually a human operator, providing feedback to the decision fusion method. A video based wildfire detection system is developed to evaluate the performance of the algorithm in handling the problems where data arrives sequentially. In this case, the oracle is the security guard of the forest lookout tower verifying the decision of the combined algorithm. Simulation results are presented. The EADF framework is also tested with a standard dataset.
Given a graph, the shortest-path problem requires finding a sequence of edges with minimum cumulative length that connects a source vertex to a target vertex. We consider a generalization of this classical problem in which the position of each vertex in the graph is a continuous decision variable, constrained to lie in a corresponding convex set. The length of an edge is then defined as a convex function of the positions of the vertices it connects. Problems of this form arise naturally in motion planning of autonomous vehicles, robot navigation, and even optimal control of hybrid dynamical systems. The price for such a wide applicability is the complexity of this problem, which is easily seen to be NP-hard. Our main contribution is a strong mixed-integer convex formulation based on perspective functions. This formulation has a very tight convex relaxation and makes it possible to efficiently find globally-optimal paths in large graphs and in high-dimensional spaces.
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