Structure from sound with incomplete data

In this paper, we consider the problem of jointly localizing a microphone array and identifying the direction of arrival of acoustic events. Under the assumption that the sources are in the far field, this problem can be formulated as a constrained low-rank matrix factorization with an unknown column offset. Our focus is on handling missing entries, particularly when the measurement matrix does not contain a single complete column. This case has not received attention in the literature and is not handled by existing algorithms, however it is prevalent in practice. We propose an iterative algorithm that works with pairwise differences between the measurements eliminating the dependence on the unknown offset. We demonstrate state-of-the-art performance both in terms of accuracy and versatility

Shapes from Echoes: Uniqueness from Point-to-Plane Distance Matrices

We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the notable "structure from sound" approach to microphone localization with unknown sources. In our earlier work we proposed computational methods for localization from point-to-plane distances and noted that such localization suffers from various ambiguities beyond the usual rigid body motions; in this paper we provide a complete characterization of uniqueness. We enumerate equivalence classes of configurations which lead to the same distance measurements as a function of the number of planes and points, and algebraically characterize the related transformations in both 2D and 3D. Here we only discuss uniqueness; computational tools and heuristics for practical localization from point-to-plane distances using sound will be addressed in a companion paper.Comment: 13 pages, 13 figure

Sampling the Multiple Facets of Light

The theme of this thesis revolves around three important manifestations of light, namely its corpuscular, wave and electromagnetic nature. Our goal is to exploit these principles to analyze, design and build imaging modalities by developing new signal processing and algorithmic tools, based in particular on sampling and sparsity concepts. First, we introduce a new sampling scheme called variable pulse width, which is based on the finite rate of innovation (FRI) sampling paradigm. This new framework enables to sample and perfectly reconstruct weighted sums of Lorentzians; perfect reconstruction from sampled signals is guaranteed by a set of theorems. Second, we turn to the context of light and study its reflection, which is based on the corpuscular model of light. More precisely, we propose to use our FRI-based model to represent bidirectional reflectance distribution functions. We develop dedicated light domes to acquire reflectance functions and use the measurements obtained to demonstrate the usefulness and versatility of our model. In particular, we concentrate on the representation of specularities, which are sharp and bright components generated by the direct reflection of light on surfaces. Third, we explore the wave nature of light through Lippmann photography, a century-old photography technique that acquires the entire spectrum of visible light. This fascinating process captures interferences patterns created by the exposed scene inside the depth of a photosensitive plate. By illuminating the developed plate with a neutral light source, the reflected spectrum corresponds to that of the exposed scene. We propose a mathematical model which precisely explains the technique and demonstrate that the spectrum reproduction suffers from a number of distortions due to the finite depth of the plate and the choice of reflector. In addition to describing these artifacts, we describe an algorithm to invert them, essentially recovering the original spectrum of the exposed scene. Next, the wave nature of light is further generalized to the electromagnetic theory, which we invoke to leverage the concept of polarization of light. We also return to the topic of the representation of reflectance functions and focus this time on the separation of the specular component from the other reflections. We exploit the fact that the polarization of light is preserved in specular reflections and investigate camera designs with polarizing micro-filters with different orientations placed just in front of the camera sensor; the different polarizations of the filters create a mosaic image, from which we propose to extract the specular component. We apply our demosaicing method to several scenes and additionally demonstrate that our approach improves photometric stereo. Finally, we delve into the problem of retrieving the phase information of a sparse signal from the magnitude of its Fourier transform. We propose an algorithm that resolves the phase retrieval problem for sparse signals in three stages. Unlike traditional approaches that recover a discrete approximation of the underlying signal, our algorithm estimates the signal on a continuous domain, which makes it the first of its kind. The concluding chapter outlines several avenues for future research, like new optical devices such as displays and digital cameras, inspired by the topic of Lippmann photography