The Quest for the Ideal Scintillator for Hybrid Phototubes

In this paper we present the results of extensive studies of scintillators for hybrid phototubes with luminescent screens. The results of the developments of such phototubes with a variety of scintillators are presented. New scintillator materials for such kind of application are discussed. The requirements for scintillators to use in such hybrid phototubes are formulated. It is shown that very fast and highly efficient inorganic scintillators like ZnO:Ga will be ideal scintillators for such kind of application.Comment: 5 pages, 6 figures and 1 table. Submitted to the proceedings of SCINT2007 Conference, Winston-Salem, NC USA, June 4-8, 200

There is no variational characterization of the cycles in the method of periodic projections

The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers of a certain functional. In this paper we answer this question in the negative. Projection algorithms that minimize smooth convex functions over a product of convex sets are also discussed

Asymptotic behavior of compositions of under-relaxed nonexpansive operators

In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces

Generalized Forward-Backward Splitting

This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, our method generalizes it to the case of arbitrary $n$. Our method makes an explicit use of the regularity of $F$ in the forward step, and the proximity operators of the $G_i$'s are applied in parallel in the backward step. This allows the generalized forward backward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of $F$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.Comment: 24 pages, 4 figure

Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This version has updated reference

Four-dimensional Cone Beam CT Reconstruction and Enhancement using a Temporal Non-Local Means Method

Four-dimensional Cone Beam Computed Tomography (4D-CBCT) has been developed to provide respiratory phase resolved volumetric imaging in image guided radiation therapy (IGRT). Inadequate number of projections in each phase bin results in low quality 4D-CBCT images with obvious streaking artifacts. In this work, we propose two novel 4D-CBCT algorithms: an iterative reconstruction algorithm and an enhancement algorithm, utilizing a temporal nonlocal means (TNLM) method. We define a TNLM energy term for a given set of 4D-CBCT images. Minimization of this term favors those 4D-CBCT images such that any anatomical features at one spatial point at one phase can be found in a nearby spatial point at neighboring phases. 4D-CBCT reconstruction is achieved by minimizing a total energy containing a data fidelity term and the TNLM energy term. As for the image enhancement, 4D-CBCT images generated by the FDK algorithm are enhanced by minimizing the TNLM function while keeping the enhanced images close to the FDK results. A forward-backward splitting algorithm and a Gauss-Jacobi iteration method are employed to solve the problems. The algorithms are implemented on GPU to achieve a high computational efficiency. The reconstruction algorithm and the enhancement algorithm generate visually similar 4D-CBCT images, both better than the FDK results. Quantitative evaluations indicate that, compared with the FDK results, our reconstruction method improves contrast-to-noise-ratio (CNR) by a factor of 2.56~3.13 and our enhancement method increases the CNR by 2.75~3.33 times. The enhancement method also removes over 80% of the streak artifacts from the FDK results. The total computation time is ~460 sec for the reconstruction algorithm and ~610 sec for the enhancement algorithm on an NVIDIA Tesla C1060 GPU card.Comment: 20 pages, 3 figures, 2 table

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization