Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

This papers deals with the restoration of images corrupted by a non-invertible or ill-conditioned linear transform and Poisson noise. Poisson data typically occur in imaging processes where the images are obtained by counting particles, e.g., photons, that hit the image support. By using the Anscombe transform, the Poisson noise can be approximated by an additive Gaussian noise with zero mean and unit variance. Then, the least squares difference between the Anscombe transformed corrupted image and the original image can be estimated by the number of observations. We use this information by considering an Anscombe transformed constrained model to restore the image. The advantage with respect to corresponding penalized approaches lies in the existence of a simple model for parameter estimation. We solve the constrained minimization problem by applying a primal-dual algorithm together with a projection onto the epigraph of a convex function related to the Anscombe transform. We show that this epigraphical projection can be efficiently computed by Newton's methods with an appropriate initialization. Numerical examples demonstrate the good performance of our approach, in particular, its close behaviour with respect to the $I$-divergence constrained model

A CT-study of the Cranial Suture Morphology and its Reorganization during the Obliteration

Obliteration of the cranial sutures is an age-dependent process. Its premature occurrence (craniosynostosis) causes different craniofacial deformations, dependent on the affected suture(s). The understanding of the suture morphology and the remodeling processes during the obliteration is essential for early diagnosis and treatment of the premature closure. This study aimed to investigate the morphology of open and obliterated sutures and to perform comparison analysis on the 3D images obtained by both industrial and medical computed tomography (CT) systems with various resolutions. A segment of the sagittal suture of dry skulls of known age and sex was scanned using Nikon XTH 225, an industrial CT system, developed by Nikon Metrology. The same section of the sagittal suture was observed on patients undergoing CT scanning with a multislice system Toshiba Aquilion 64 with 0.5 mm slice thickness. For 3D visualization, VGStudioMax 2.2 were used. The suture morphology was observed in coronal section on sequential 2D slices. Micro-CT (μCT) scanning of dry skulls enabled calculation of the morphometric parameters and visualization of the microarchitecture of the suture and its reorganization during the obliteration, unlike the CT imaging of patients, where the sutures were scarcely discernable. In the entirely open sections of the suture the bone edges were separated by a gap of various widths. As the obliteration proceeded, the gap gradually reduced and the bone edges got into a contact. In the fi nal stages, the traces from the contact faded away and the sutural area became a homogenous structure of increased integrity. The μCT scanning of dry bones is a powerful non-destructive technique for examination of the suture morphology. Remodeling of the suture during the obliteration leads to gradually diminishing of the gap between the bone edges to their entire coalescence

Rational Approximations in Robust Preconditioning of Multiphysics Problems

Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface blocks of the preconditioners are fractional Laplacians. At the discrete level, we propose to replace the inverse of the fractional Laplacian with its best uniform rational approximation (BURA). The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the proposed preconditioners have optimal computational complexity O(N), where N is the number of unknowns (degrees of freedom) of the coupled discrete problem. The main theoretical contribution is the condition number estimates of the BURA-based preconditioners. It is important to note that the obtained estimates are completely analogous for both positive and negative fractional powers. At the end, the analysis of the behavior of the relative condition numbers is aimed at characterizing the practical requirements for minimal BURA orders for the considered Darcy–Stokes and 3D–1D examples of coupled problems