Making Laplacians commute

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data

Disparate View Matching

Matching of disparate views has gained significance in computer vision due to its role in many novel application areas. Being able to match images of the same scene captured during day and night, between a historic and contemporary picture of a scene, and between aerial and ground-level views of a building facade all enable novel applications ranging from loop-closure detection for structure-from-motion and re-photography to geo-localization of a street-level image using reference imagery captured from the air. The goal of this work is to develop novel features and methods that address matching problems where direct appearance-based correspondences are either difficult to obtain or infeasible because of the lack of appearance similarity altogether. To address these problems, we propose methods that span the appearance-geometry spectrum in terms of both the use of these cues as well as the ability of each method to handle variations in appearance and geometry. First, we consider the problem of geo-localization of a query street-level image using a reference database of building facades captured from a bird\u27s eye view. To address this wide-baseline facade matching problem, a novel scale-selective self-similarity feature that avoids direct comparison of appearance between disparate facade images is presented. Next, to address image matching problems with more extreme appearance variation, a novel representation for matchable images expressed in terms of the eigen-functions of the joint graph of the two images is presented. This representation is used to derive features that are persistent across wide variations in appearance. Next, the problem setting of matching between a street-level image and a digital elevation map (DEM) is considered. Given the limited appearance information available in this scenario, the matching approach has to rely more significantly on geometric cues. Therefore, a purely geometric method to establish correspondences between building corners in the DEM and the visible corners in the query image is presented. Finally, to generalize this problem setting we address the problem of establishing correspondences between 3D and 2D point clouds using geometric means alone. A novel framework for incorporating purely geometric constraints into a higher-order graph matching framework is presented with specific formulations for the three-point calibrated absolute camera pose problem (P3P), two-point upright camera pose problem (Up2p) and the three-plus-one relative camera pose problem