Yair Censor We study the behavior of subgradient projection algorithms for the quasiconvex feasibility problem of finding a point x^* in R^n that satisfies the inequalities f_i(x^*) less or equal 0, for all i=1,2,…,m, where all functions are continuous and quasiconvex. We consider the consistent case when the solution set is nonempty. Since the Fenchel-Moreau subdifferential might … Read more
We study a sequential algorithm for finding the projection of a given point onto the common fixed points set of a semigroup of nonexpansive operators in Hilbert space. The convergence of such an algorithm was previously established only for finitely many nonexpansive operators. Algorithms of this kind have been applied to the best approximation and … Read more
We study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step-sizes. This algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the Bregman distances to the … Read more
The efficient delivery of intensity modulated radiation therapy (IMRT) depends on finding optimized beam intensity patterns that produce dose distributions, which meet given constraints for the tumor as well as any critical organs to be spared. Many optimization algorithms that are used for beamlet-based inverse planning are susceptible to large variations of neighboring intensities. Accurately … Read more
The prescribed goals of radiation treatment planning are often expressed in terms of dose-volume constraints. We present a novel formulation of a dose-volume constraint satisfaction search for the discretized radiation therapy model. This approach does not rely on any explicit cost function. The inverse treatment planning uses the aperture based approach with predefined, according to … Read more
Three-dimensional electron microscopy (3D-EM) is a powerful tool for visualizing complex biological systems. As any other imaging device, the electron microscope introduces a transfer function (called in this field the Contrast Transfer Function, CTF) into the image acquisition process that modulates the various frequencies of the signal. Thus, 3D reconstructions performed with these CTF-affected projections … Read more
We study iterative projection algorithms for the convex feasibility problem of finding a point in the intersection of finitely many nonempty, closed and convex subsets in the Euclidean space. We propose (without proof) an algorithmic scheme which generalizes both the string-averaging algorithm and the block-iterative projections (BIP) method with fixed blocks and prove convergence of … Read more
In this tutorial we discuss modeling issues in intensity modulated radiation therapy, contrasting the continuous model with the fully-discretized one and considering feasibility formulations versus optimization setups. We review briefly some mathematical optimization techniques for IMRT. These include global optimization, multi-objective optimization, linear and mixed integer programming and projection methods. Citation in: J.R. Palta and … Read more
We present a modification of Dykstra’s algorithm which allows us to avoid projections onto general convex sets. Instead, we calculate projections onto either a halfspace or onto the intersection of two halfspaces. Convergence of the algorithm is established and special choices of the halfspaces are proposed. The option to project onto halfspaces instead of general … Read more
A definition of oblique projections onto closed convex sets that use seminorms induced by diagonal matrices which may have zeros on the diagonal is introduced. Existence and uniqueness of such projections are secured via directional affinity of the sets with respect to the diagonal matrices involved. A block-iterative algorithmic scheme for solving the convex feasibility … Read more