Melanie King | Searcharxiv

arXiv2024

Computational refocusing in phase-resolved confocal microscopy

We demonstrate numerical refocusing in coherent confocal laser scanning microscopy based on synthetic optical holography. This physics-based approach implements a computational propagation on the complex signal recovered in synthetic holography consistent with the wave physics and the parameters of the microscope. An experimental demonstration is shown to restore an in-focus image of a test object from data acquired at several focal plane off-sets. Numerical refocusing can provide focused views on samples with large height variation with a potential application in confocal optical surface profiling.

physics.optics↗

arXiv2015

Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles

An integer generalized spline is a set of vertex labels on an edge-labeled graph that satisfy the condition that if two vertices are joined by an edge, the vertex labels are congruent modulo the edge label. Foundational work on these objects comes from Gilbert, Polster, and Tymoczko, who generalize ideas from geometry/topology (equivariant cohomology rings) and algebra (algebraic splines) to develop the notion of generalized splines. Gilbert, Polster, and Tymoczko prove that the ring of splines on a graph can be decomposed in terms of splines on its subgraphs (in particular, on trees and cycles), and then fully analyze splines on trees. Following Handschy-Melnick-Reinders and Rose, we analyze splines on cycles, in our case integer generalized splines. The primary goal of this paper is to establish two new bases for the module of integer generalized splines on cycles: the triangulation basis and the King basis. Unlike bases in previous work, we are able to characterize each basis element completely in terms of the edge labels of the underlying cycle. As an application we explicitly construct the multiplication table for the ring of integer generalized splines in terms of the King basis.

math.RA↗