34 research outputs found
Sampling Sparse Signals on the Sphere: Algorithms and Applications
We propose a sampling scheme that can perfectly reconstruct a collection of
spikes on the sphere from samples of their lowpass-filtered observations.
Central to our algorithm is a generalization of the annihilating filter method,
a tool widely used in array signal processing and finite-rate-of-innovation
(FRI) sampling. The proposed algorithm can reconstruct spikes from
spatial samples. This sampling requirement improves over
previously known FRI sampling schemes on the sphere by a factor of four for
large . We showcase the versatility of the proposed algorithm by applying it
to three different problems: 1) sampling diffusion processes induced by
localized sources on the sphere, 2) shot noise removal, and 3) sound source
localization (SSL) by a spherical microphone array. In particular, we show how
SSL can be reformulated as a spherical sparse sampling problem.Comment: 14 pages, 8 figures, submitted to IEEE Transactions on Signal
Processin
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
Look, no Beacons! Optimal All-in-One EchoSLAM
We study the problem of simultaneously reconstructing a polygonal room and a
trajectory of a device equipped with a (nearly) collocated omnidirectional
source and receiver. The device measures arrival times of echoes of pulses
emitted by the source and picked up by the receiver. No prior knowledge about
the device's trajectory is required. Most existing approaches addressing this
problem assume multiple sources or receivers, or they assume that some of these
are static, serving as beacons. Unlike earlier approaches, we take into account
the measurement noise and various constraints on the geometry by formulating
the solution as a minimizer of a cost function similar to \emph{stress} in
multidimensional scaling. We study uniqueness of the reconstruction from
first-order echoes, and we show that in addition to the usual invariance to
rigid motions, new ambiguities arise for important classes of rooms and
trajectories. We support our theoretical developments with a number of
numerical experiments.Comment: 5 pages, 6 figures, submitted to Asilomar Conference on Signals,
Systems, and Computers Websit
Euclidean Distance Matrices: Essential Theory, Algorithms and Applications
Euclidean distance matrices (EDM) are matrices of squared distances between
points. The definition is deceivingly simple: thanks to their many useful
properties they have found applications in psychometrics, crystallography,
machine learning, wireless sensor networks, acoustics, and more. Despite the
usefulness of EDMs, they seem to be insufficiently known in the signal
processing community. Our goal is to rectify this mishap in a concise tutorial.
We review the fundamental properties of EDMs, such as rank or
(non)definiteness. We show how various EDM properties can be used to design
algorithms for completing and denoising distance data. Along the way, we
demonstrate applications to microphone position calibration, ultrasound
tomography, room reconstruction from echoes and phase retrieval. By spelling
out the essential algorithms, we hope to fast-track the readers in applying
EDMs to their own problems. Matlab code for all the described algorithms, and
to generate the figures in the paper, is available online. Finally, we suggest
directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine
- change of title in the last revisio
FRIDA: FRI-Based DOA Estimation for Arbitrary Array Layouts
In this paper we present FRIDA---an algorithm for estimating directions of
arrival of multiple wideband sound sources. FRIDA combines multi-band
information coherently and achieves state-of-the-art resolution at extremely
low signal-to-noise ratios. It works for arbitrary array layouts, but unlike
the various steered response power and subspace methods, it does not require a
grid search. FRIDA leverages recent advances in sampling signals with a finite
rate of innovation. It is based on the insight that for any array layout, the
entries of the spatial covariance matrix can be linearly transformed into a
uniformly sampled sum of sinusoids.Comment: Submitted to ICASSP201
Convolution on the n-Sphere With Application to PDF Modeling
In this paper, we derive an explicit form of the convolution theorem for functions on an n-sphere. Our motivation comes from the design of a probability density estimator for n-dimensional random vectors. We propose a probability density function (pdf) estimation method that uses the derived convolution result on Sn. Random samples are mapped onto the n -sphere and estimation is performed in the new domain by convolving the samples with the smoothing kernel density. The convolution is carried out in the spectral domain. Samples are mapped between the n-sphere and the n-dimensional Euclidean space by the generalized stereographic projection. We apply the proposed model to several synthetic and real-world data sets and discuss the results
Joint Cryo-ET Alignment and Reconstruction with Neural Deformation Fields
We propose a framework to jointly determine the deformation parameters and
reconstruct the unknown volume in electron cryotomography (CryoET). CryoET aims
to reconstruct three-dimensional biological samples from two-dimensional
projections. A major challenge is that we can only acquire projections for a
limited range of tilts, and that each projection undergoes an unknown
deformation during acquisition. Not accounting for these deformations results
in poor reconstruction. The existing CryoET software packages attempt to align
the projections, often in a workflow which uses manual feedback. Our proposed
method sidesteps this inconvenience by automatically computing a set of
undeformed projections while simultaneously reconstructing the unknown volume.
We achieve this by learning a continuous representation of the undeformed
measurements and deformation parameters. We show that our approach enables the
recovery of high-frequency details that are destroyed without accounting for
deformations
OMP with Unknown Filters for Multipath Channel Estimation
We study a modification of the orthogonal matching pursuit (OMP) for estimating sparse multipath channels. The reflectors that generate the multipath components are not ideal; rather, they act as filters, so that the returned pulses are reshaped and widened. To deal with this, we introduce unknown filters into the OMP, and then search for the best filtered set of atoms, together with the optimal set of filters. Our algorithm extends naturally to unknown pulses lying in any known subspace. We show how this observation allows us to reconstruct sums of Gaussians with unknown widths
Efficient Approximate Scaling of Spherical Functions in the Fourier Domain With Generalization to Hyperspheres
We propose a simple model for approximate scaling of spherical functions in the Fourier domain. The proposed scaling model is analogous to the scaling property of the classical Euclidean Fourier transform. Spherical scaling is used for example in spherical wavelet transform and filter banks or illumination in computer graphics. Since the function that requires scaling is often represented in the Fourier domain, our method is of significant interest. Furthermore, we extend the result to higher-dimensional spheres. We show how this model follows naturally from consideration of a hypothetical continuous spectrum. Experiments confirm the applicability of the proposed method for several signal classes. The proposed algorithm is compared to an existing linear operator formulation