An adaptive level set method for nondifferentiable constrained image recovery.

The formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals. In previous years, nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges. They also arise naturally in minimax inconsistent set theoretic recovery problems.

Image restoration subject to a total variation constraint.

Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex programming framework.

Hybrid projection-reflection method for phase retrieval.

The phase-retrieval problem, fundamental in applied physics and engineering, addresses the question of how to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently we identified two important methods for phase retrieval, namely, Fienup's basic input-output and hybrid input-output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the hybrid projection-reflection (HPR) algorithm, for solving phase-retrieval problems featuring nonnegativity constraints in the object domain.

Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization.

The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success.

Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections.

Solving a convex set theoretic image recovery problem amounts to finding a point in the intersection of closed and convex sets in a Hilbert space. The projection onto convex sets (POCS) algorithm, in which an initial estimate is sequentially projected onto the individual sets according to a periodic schedule, has been the most prevalent tool to solve such problems. Nonetheless, POCS has several shortcomings: it converges slowly, it is ill suited for implementation on parallel processors, and it requires the computation of exact projections at each iteration.

Signal recovery by best feasible approximation.

The objective of set theoretical signal recovery is to find a feasible signal in the form of a point in the intersection of S of sets modeling the information available about the problem. For problems in which the true signal is known to lie near a reference signal r, the solution should not be any feasible point but one which best approximates r, i.e.

[The removal of gastrin secreting tissue in the treatment of Zollinger-Ellison syndrome (author's transl)].

In the light of 24 cases, routine total gastrectomy in the treatment of Zollinger-Ellison's syndrome should be abandoned in favour of the removal of gastrin-secreting tissue. This treatment which gives just as good results on the ulcer disease, also has the advantage of removal of the tumour tissue malignant in almost 2/3 of cases. Removal of the gastrin secreting tissue thus becomes the operation of choice.