A Revision Restoring Projection after Nipple Reconstruction by Burying Four Triangular Dermal Flaps

Background Numerous techniques have been used to achieve long-term projection of the nipple following nipple-areola reconstruction. However, the reconstructed nipple loses projection over time. We describe a technique that uses local flaps to improve the lost projection of reconstructed nipples. Methods Between November 2013 and March 2015, nine patients (11 nipples) underwent revisional nipple reconstruction for lost projection. Only C-H nipple reconstructions were included in this study. The medical history of each patient was reviewed and photographs were taken in front and lateral views. All patients attended routine follow-up visits. Deepithelialized triangular flaps were made on all four sides of the nipple and buried in the opposite corners in order to augment the volume of the nipple. Anchoring sutures were used to attach each triangular flap on the side opposite their point of origin, and the resulting defects were closed directly. Results This procedure was used successfully in nine patients (11 nipples). Adequate projection was achieved in all patients and no complications occurred. The average nipple height was 3 mm before operation, 7 mm one day after operation, 5 mm at the six-month follow-up, and 5 mm at the 12-month follow-up. The average nipple-areolar angle was 164 degrees before the operation, 111 degrees one day after the operation, 130 degrees at the six-month follow-up, and 133 degrees at the 12-month follow-up. Conclusions The method described provides a solution to the loss of projection in reconstructed nipples. We recommend this technique because it leads to better projection, greater volume, and a more natural shape

A direct comparison of high-speed methods for the numerical Abel transform

The Abel transform is a mathematical operation that transforms a cylindrically symmetric three-dimensional (3D) object into its two-dimensional (2D) projection. The inverse Abel transform reconstructs the 3D object from the 2D projection. Abel transforms have wide application across numerous fields of science, especially chemical physics, astronomy, and the study of laser-plasma plumes. Consequently, many numerical methods for the Abel transform have been developed, which makes it challenging to select the ideal method for a specific application. In this work eight transform methods have been incorporated into a single, open-source Python software package (PyAbel) to provide a direct comparison of the capabilities, advantages, and relative computational efficiency of each transform method. Most of the tested methods provide similar, high-quality results. However, the computational efficiency varies across several orders of magnitude. By optimizing the algorithms, we find that some transform methods are sufficiently fast to transform 1-megapixel images at more than 100 frames per second on a desktop personal computer. In addition, we demonstrate the transform of gigapixel images.Comment: 9 pages, 5 figure

Constrained Consensus

We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimate of each agent is restricted to lie in a different constraint set. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed ``projected consensus algorithm'' in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed ``projected subgradient algorithm'' which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set.Comment: 35 pages. Included additional results, removed two subsections, added references, fixed typo