Geometry Helps to Compare Persistence Diagrams

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.Comment: 20 pages, 10 figures; extended version of paper published in ALENEX 201

SceneFlowFields: Dense Interpolation of Sparse Scene Flow Correspondences

While most scene flow methods use either variational optimization or a strong rigid motion assumption, we show for the first time that scene flow can also be estimated by dense interpolation of sparse matches. To this end, we find sparse matches across two stereo image pairs that are detected without any prior regularization and perform dense interpolation preserving geometric and motion boundaries by using edge information. A few iterations of variational energy minimization are performed to refine our results, which are thoroughly evaluated on the KITTI benchmark and additionally compared to state-of-the-art on MPI Sintel. For application in an automotive context, we further show that an optional ego-motion model helps to boost performance and blends smoothly into our approach to produce a segmentation of the scene into static and dynamic parts.Comment: IEEE Winter Conference on Applications of Computer Vision (WACV), 201

VoxDet: Voxel Learning for Novel Instance Detection

Detecting unseen instances based on multi-view templates is a challenging problem due to its open-world nature. Traditional methodologies, which primarily rely on 2D representations and matching techniques, are often inadequate in handling pose variations and occlusions. To solve this, we introduce VoxDet, a pioneer 3D geometry-aware framework that fully utilizes the strong 3D voxel representation and reliable voxel matching mechanism. VoxDet first ingeniously proposes template voxel aggregation (TVA) module, effectively transforming multi-view 2D images into 3D voxel features. By leveraging associated camera poses, these features are aggregated into a compact 3D template voxel. In novel instance detection, this voxel representation demonstrates heightened resilience to occlusion and pose variations. We also discover that a 3D reconstruction objective helps to pre-train the 2D-3D mapping in TVA. Second, to quickly align with the template voxel, VoxDet incorporates a Query Voxel Matching (QVM) module. The 2D queries are first converted into their voxel representation with the learned 2D-3D mapping. We find that since the 3D voxel representations encode the geometry, we can first estimate the relative rotation and then compare the aligned voxels, leading to improved accuracy and efficiency. Exhaustive experiments are conducted on the demanding LineMod-Occlusion, YCB-video, and the newly built RoboTools benchmarks, where VoxDet outperforms various 2D baselines remarkably with 20% higher recall and faster speed. To the best of our knowledge, VoxDet is the first to incorporate implicit 3D knowledge for 2D tasks.Comment: 17 pages, 10 figure

Homological mirror symmetry for hypertoric varieties II

In this paper, we prove a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties, and we calculate monodromy autoequivalences of these categories by promoting our result to an equivalence of perverse schobers. We prove our equivalence by matching holomorphic Lagrangian skeleta, on the A-model side, with non-commutative resolutions on the B-model side. The hyperk\"ahler geometry of these spaces provides each category with a natural t-structure, which helps clarify SYZ duality in a hyperk\"ahler context. Our results are a prototype for mirror symmetry statements relating pairs of K-theoretic Coulomb branches.Comment: v2: Section 5 expanded with a discussion of perverse schober

Searching the Sky with CONFIGR-STARS

SyNAPSE program of the Defense Advanced Projects Research Agency (HRL Laboratories LLC, subcontract #801881-BS under DARPA prime contract HR0011-09-C-0001); CELEST, a National Science Foundation Science of Learning Center (SBE-0354378)CONFIGR-STARS, a new methodology based on a model of the human visual system, is developed for registration of star images. The algorithm first applies CONFIGR, a neural model that connects sparse and noisy image components. CONFIGR produces a web of connections between stars in a reference starmap or in a test patch of unknown location. CONFIGR-STARS splits the resulting, typically highly connected, web into clusters, or "constellations." Cluster geometry is encoded as a signature vector that records edge lengths and angles relative to the cluster’s baseline edge. The location of a test patch cluster is identified by comparing its signature to signatures in the codebook of a reference starmap, where cluster locations are known. Simulations demonstrate robust performance in spite of image perturbations and omissions, and across starmaps from different sources and seasons. Further studies would test CONFIGR-STARS and algorithm variations applied to very large starmaps and to other technologies that may employ geometric signatures. Open-source code, data, and demos are available from http://techlab.bu.edu/STARS/

Experimental investigation of flux motion in exponentially shaped Josephson junctions

We report experimental and numerical analysis of expontentially shaped long Josephson junctions with lateral current injection. Quasi-linear flux flow branches are observed in the current-voltage characteristic of the junctions in the absence of magnetic field. A strongly asymmetric response to an applied magnetic field is also exhibited by the junctions. Experimental data are found in agreement with numerical predictions and demonstrate the existence of a geometry-induced potential experienced by the flux quanta in nonuniform width junctions.Comment: 16 pg, 8 figures, Submitted in PRB March