Scalable Hypergraph Structure Learning with Diverse Smoothness Priors

In graph signal processing, learning the weighted connections between nodes from a set of sample signals is a fundamental task when the underlying relationships are not known a priori. This task is typically addressed by finding a graph Laplacian on which the observed signals are smooth. With the extension of graphs to hypergraphs - where edges can connect more than two nodes - graph learning methods have similarly been generalized to hypergraphs. However, the absence of a unified framework for calculating total variation has led to divergent definitions of smoothness and, consequently, differing approaches to hyperedge recovery. We confront this challenge through generalization of several previously proposed hypergraph total variations, subsequently allowing ease of substitution into a vector based optimization. To this end, we propose a novel hypergraph learning method that recovers a hypergraph topology from time-series signals based on a smoothness prior. Our approach addresses key limitations in prior works, such as hyperedge selection and convergence issues, by formulating the problem as a convex optimization solved via a forward-backward-forward algorithm, ensuring guaranteed convergence. Additionally, we introduce a process that simultaneously limits the span of the hyperedge search and maintains a valid hyperedge selection set. In doing so, our method becomes scalable in increasingly complex network structures. The experimental results demonstrate improved performance, in terms of accuracy, over other state-of-the-art hypergraph inference methods; furthermore, we empirically show our method to be robust to total variation terms, biased towards global smoothness, and scalable to larger hypergraphs.

Causes and Corrections for Bimodal Multipath Scanning with Structured Light

Structured light illumination is an active 3-D scanning technique based on projecting/capturing a set of striped patterns and measuring the warping of the patterns as they reflect off a target object's surface. As designed, each pixel in the camera sees exactly one pixel from the projector; however, there are exceptions to this when the scanned surface has a complicated geometry with step edges and other discontinuities in depth or where the target surface has specularities that reflect light away from the camera. These situations are generally referred to multipath where a given camera pixel receives light from multiple positions from the projector. In the case of bimodal multipath, the camera pixel receives light from exactly two positions from the projector which occurs when light bounce back from a reflective surface or along a step edge where the edge slices through a pixel so that the pixel sees both a foreground and background surface. In this paper, we present a general mathematical model and address the bimodal multipath issue in a phase measuring profilometry scanner to measure the constructive and destructive interference between the two light paths, and by taking advantage of this interesting cue, separate the paths and make two separated depth measurements. We also validate our algorithm with both simulation and a number of challenging real cases.

Structured Light Phase Measuring Profilometry Pattern Design for Binary Spatial Light Modulators

Structured light illumination is an active 3-D scanning technique based on projecting/capturing a set of striped patterns and measuring the warping of the patterns as they reflect off a target object's surface. In the case of phase measuring profilometry (PMP), the projected patterns are composed of a rolling sinusoidal wave, but as a set of time-multiplexed patterns, PMP requires the target surface to remain motionless or for scanning to be performed at such high rates that any movement is small. But high speed scanning places a significant burden on the projector electronics to produce contone patterns inside of short exposure intervals. Binary patterns are, therefore, of great value, but converting contone patterns into binary comes with significant risk. As such, this paper introduces a contone-to-binary conversion algorithm for deriving binary patterns that best mimic their contone counterparts. Experimental results will show a greater than 3 times reduction in pattern noise over traditional halftoning procedures.