We describe a method for recovering a manifold, intersection-free triangle mesh from the points where edges of a tetrahedral grid pierce a continuous surface. Unlike classic marching cubes or tets, our subgrid marching scheme allows arbitrarily many surface patches within a single cell, capturing fine features and thin sheets. Moreover, it requires neither a well-defined inside/outside (allowing surfaces with boundary), nor consistently-oriented input geometry. Yet we retain the local, parallel nature of classic marching: reconstruction is performed independently per tet, yielding a conforming mesh across tet boundaries. Our key innovation is a generalization of normal coordinates from geometric topology, which encode surface connectivity via arbitrary integer intersection counts along each grid edge. This encoding sidesteps the usual Nyquist-Shannon limit, putting no lower bound on the size of features that can be resolved on a fixed grid. In practice, for similar compute time and equal grid resolution—or even an equal number of output triangles—meshes produced by subgrid marching are far more accurate than those from classic marching. Beyond standard contouring, our method can be used to convert polygon soup into a manifold, intersection-free mesh.

This paper explores the analysis and design of the resting configurations of a rigid body, without the use of physical simulation. In particular, given a rigid body in R 3 , we identify all possible stationary points, as well as the probability that the body will stop at these points, assuming a random initial orientation and negligible momentum. The forward version of our method can hence be used to automatically orient models, to provide feedback about object stability during the design process, and to furnish plausible distributions of shape orientation for natural scene modeling. Moreover, a differentiable inverse version of our method lets us design shapes with target resting behavior, such as dice with target, nonuniform probabilities. Here we find solutions that would be nearly impossible to find using classical techniques, such as dice with additional unstable faces that provide more natural overall geometry. From a technical point of view, our key observation is that rolling equilibria can be extracted from the Morse-Smale complex of the support function over the Gauss map. Our method is hence purely geometric, and does not make use of random sampling, or numerical time integration. Yet surprisingly, this purely geometric model makes extremely accurate predictions of rest behavior, which we validate both numerically, and via physical experiments. Moreover, for computing rest statistics, it is orders of magnitude faster than state of the art rigid body simulation, opening the door to inverse design—rather than just forward analysis.

Ultrasonically-sculpted waveguides provide interesting opportunities for in situ optical imaging in transparent and scattering media. The interference of ultrasonic waves can be designed to form a spatially-varying refractive index pattern in the target medium, acting as a virtual lens to guide light and relay images. The images formed by such lenses are subject to a large amount of spatially-varying blur, which significantly reduces their contrast. To alleviate this issue, the images can be computationally deblurred post experiment to restore the image. First, we demonstrate a brute force deconvolution technique to deblur the relayed images. While effective, this method proves to be computationally intensive due to the spatially-varying blur kernel. The reconfigurability of ultrasonically sculpted waveguides can be leveraged to address this issue by measuring line integrals of the image at multiple angles to form the Radon image of the target, which can be deblurred very efficiently with a simple linear model. We validate this method using simulated and experimental results. The tantalizing notion of using the reconfigurable ultrasonically-sculpted waveguides as part of the imaging systems, demonstrated in this paper, opens new opportunities for hybrid physical-computational optical systems.