sFFT-DT: Sparse Fast Fourier Transform for Exactly and Generally K-Sparse Signals by Downsampling and Sparse Recovery

Performing FFT using the fact that the underlying signal is sparse has been an item of great interest and several researchers are trying to get the faster algorithm they can ( see the FFT tag) . The following paper is trying to push the envelope. I cannot see an implementation of it listed though. Let us note the use of this approach to perform better localization for the GPS signal.

Sparse Fast Fourier Transform for Exactly and Generally K-Sparse Signals by Downsampling and Sparse Recovery by Sung-Hsien Hsieh, Chun-Shien Lu, Soo-Chang Pei

Fast Fourier Transform (FFT) is one of the most important tools in digital signal processing. FFT costs O(N log N) for transforming a signal of length N. Recently, researchers at MIT have proposed Sparse Fast Fourier Transform (sFFT) [1][2] as a breakthrough with computational complexity O(K log N) and O(K log N log N/K) for exactly K-sparse signal (with only K non-zero frequency grids) and generally K-sparse signal (with K significant frequency grids), respectively, to outperform classic FFT.

In this paper, a new sparse Fast Fourier Transform by downsampling in the time domain (sFFT-DT) is proposed, based on the assumption that the distribution of the non-zero frequency grids is uniform. The idea behind sFFT-DT is to downsample the original input signal at the beginning; then, subsequent processing operates under downsampled signals, where signal lengths are proportional to O(K). Downsampling, however, possibly leads to "aliasing". By the shift property of DFT, we recast the aliasing problem as a "moment-preserving problem (MPP)," which is solvable. We prove two theorems related to initializing the downsampling factors under different conditions to have computational complexity, O(K log K) and O(K^(5/4) log K). Moreover, for generally K-sparse signals, solutions to the MPP are inaccurate due to interference from non-significant frequency grids. We observe that significant frequency grids in aliasing are "sparse". This property is exploited, and a new sparse signal recovery algorithm in the context of compressive sensing is presented to refine the solution to MPP. The computational complexity still costs O(K log K) but requires a larger Big-O constant compared to the case of exactly K-sparse signals. We conduct theoretical complexity analyses and simulations to demonstrate that our method (sFFT-DT) outperforms classic FFT and MIT's sFFT.

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Arrival at 67P

Ten years and one day ago, I mentioned the dual use capability of the Rosetta's startracker. Today, Rosetta sent this marvelous up close shot of comet 67P

.@WilliamShatner @esa Comet on screen, Captain. Sensor scans in progress and hailing frequencies open. pic.twitter.com/SZX0sfr7b4
— ESA Rosetta Mission (@ESA_Rosetta) August 6, 2014

 



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WiTrack and RTI Goes Wild: Indoor and Outdoor Radio Tomographic Imaging

Using radio waves to evaluate people's location is featured in two ways today with WiTrack for inside applications (and the ability to distinguish gestures) and with a different application that goes outside.

3D Tracking via Body Radio Reflections by Fadel Adib, Zachary Kabelac, Dina Katabi, Robert C. Miller

This paper introduces WiTrack, a system that tracks the 3D motion of a user from the radio signals reflected off her body. It works even if the person is occluded from the WiTrack device or in a different room. WiTrack does not require the user to carry any wireless device, yet its accuracy exceeds current RF localization systems, which require the user to hold a transceiver. Empirical measurements with a WiTrack prototype show that, on average, it localizes the center of a human body to within a median of 10 to 13 cm in the x and y dmensions, and 21 cm in the z dimension. It also provides coarse tracking of body parts, identifying the direction of a pointing hand with a median of 11.2◦. WiTrack bridges a gap between RF-based localization systems which locate a user through walls and occlusions, and human-computer interaction systems like Kinect, which can track a user without instrumenting her body, but require the user to stay within the direct line of sight of the device.

The project page is here.

RTI Goes Wild: Radio Tomographic Imaging for Outdoor People Detection and Localization by Cesare Alippi, Maurizio Bocca, Giacomo Boracchi, Neal Patwari, Manuel Roveri

RF sensor networks are used to localize people indoor without requiring them to wear invasive electronic devices. These wireless mesh networks, formed by low-power radio transceivers, continuously measure the received signal strength (RSS) of the links. Radio Tomographic Imaging (RTI) is a technique that generates 2D images of the change in the electromagnetic field inside the area covered by the radio transceivers to spot the presence and movements of animates (e.g., people, large animals) or large metallic objects (e.g., cars). Here, we present a RTI system for localizing and tracking people outdoors. Differently than in indoor environments where the RSS does not change significantly with time unless people are found in the monitored area, the outdoor RSS signal is time-variant, e.g., due to rainfalls or wind-driven foliage. We present a novel outdoor RTI method that, despite the nonstationary noise introduced in the RSS data by the environment, achieves high localization accuracy and dramatically reduces the energy consumption of the sensing units. Experimental results demonstrate that the system accurately detects and tracks a person in real-time in a large forested area under varying environmental conditions, significantly reducing false positives, localization error and energy consumption compared to state-of-the-art RTI methods.

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Oldies but Goodies

• Nobody Cares About You and Your Algorithm
• Negative reviews and the weak minded

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The Visual Microphone: Passive Recovery of Sound from Video

The research that looks at tiny changes in imaging scenes is revealing to be richer in terms of providing information. Here is the latest paper that is making the rounds on the interwebs. Let us note that similar technology used to require interferometry but that, thanks to Moore 's law through the ever redundant supply of more imaging pixels, we can now use that redundant information for what we used to think as technologies that could not exist. Without futher ado:

The Visual Microphone: Passive Recovery of Sound from Video by Abe Davis Michael Rubinstein, Neal Wadhwa, Gautham Mysore, Frédo Durand, William T. Freeman

When sound hits an object, it causes small vibrations of the object’s surface. We show how, using only high-speed video of the object, we can extract those minute vibrations and partially recover the sound that produced them, allowing us to turn everyday objects—a glass of water, a potted plant, a box of tissues, or a bag of chips—into visual microphones. We recover sounds from highspeed footage of a variety of objects with different properties, and use both real and simulated data to examine some of the factors that affect our ability to visually recover sound. We evaluate the quality of recovered sounds using intelligibility and SNR metrics and provide input and recovered audio samples for direct comparison. We also explore how to leverage the rolling shutter in regular consumer cameras to recover audio from standard frame-rate videos, and use the spatial resolution of our method to visualize how sound-related vibrations vary over an object’s surface, which we can use to recover the vibration modes of an object.

The project page is here.


Of related interest:

• Riesz Pyramids for Fast Phase-Based Video Magnification by Neal Wadhwa, Michael Rubinstein, Frédo Durand, William T. Freeman
• Analysis and Visualization of Temporal Variations in Video, Michael Rubinstein, PhD Thesis, MIT Feb 2014

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Review: Low Complexity Regularization of Linear Inverse Problems

Here is a nice review: Low Complexity Regularization of Linear Inverse Problems by Samuel Vaiter, Gabriel Peyré, Jalal M. Fadili 

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem.

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The application of Compressed Sensing for Longitudinal MRI / Multichannel Compressive Sensing MRI Using Noiselet Encoding

Performing some adpative sampling based on the fact that some initial MRI was taken earlier, this is what the first paper proposes. The second paper however is non adaptive but encodes noiselets in order to maximize incoherence.

The application of Compressed Sensing for Longitudinal MRI by Lior Weizman, Yonina C. Eldar, Dafna Ben Bashat

Purpose: The mutual similarity of the follow-up scans in longitudinal studies is exploited on top of the well known sparse transform domains for rapid MRI by reducing the number of k-space measurements.

Theory and Methods: A framework for adaptive Compressed Sensing (CS) MRI that exploits the redundancy of the acquired data in longitudinal studies is proposed. The baseline MR scan is utilized both in the sampling stage, with adaptive CS, and in the reconstruction stage, with weighted CS. In adaptive CS, k-space sampling locations are optimized such that the acquired data is focused on the change between the follow-up MRI and the former one. Weighted CS uses the locations of the nonzero coefficients in the sparse domains as a prior in the recovery process.

Results: Experiments demonstrate that our longitudinal adaptive CS MRI (LACS-MRI) scheme provides reconstruction quality which outperforms traditional CS MRI for rapid MRI. Examples are shown on patients with brain tumors and demonstrate improved spatial resolution and accelerated acquisition for 2D and 3D brain imaging.

Conclusions: Our method provides high quality MRI reconstruction, at 10-fold k-space undersampling. The proposed approach can play a major part and significantly reduce scanning time in many applications that consist of disease follow-up and monitoring of changes.

Multichannel Compressive Sensing MRI Using Noiselet Encoding by Kamlesh Pawar, Gary F. Egan, Jingxin Zhang

The incoherence between measurement and sparsifying transform matrices and the restricted isometry property (RIP) of measurement matrix are two of the key factors in determining the performance of compressive sensing (CS). In CS-MRI, the randomly under-sampled Fourier matrix is used as the measurement matrix and the wavelet transform is usually used as sparsifying transform matrix. However, the incoherence between the randomly under-sampled Fourier matrix and the wavelet matrix is not optimal, which can deteriorate the performance of CS-MRI. Using the mathematical result that noiselets are maximally incoherent with wavelets, this paper introduces the noiselet unitary bases as the measurement matrix to improve the incoherence and RIP in CS-MRI, and presents a method to design the pulse sequence for the noiselet encoding. This novel encoding scheme is combined with the multichannel compressive sensing (MCS) framework to take the advantage of multichannel data acquisition used in MRI scanners. An empirical RIP analysis is presented to compare the multichannel noiselet and multichannel Fourier measurement matrices in MCS. Simulations are presented in the MCS framework to compare the performance of noiselet encoding reconstructions and Fourier encoding reconstructions at different acceleration factors. The comparisons indicate that multichannel noiselet measurement matrix has better RIP than that of its Fourier counterpart, and that noiselet encoded MCS-MRI outperforms Fourier encoded MCS-MRI in preserving image resolution and can achieve higher acceleration factors. To demonstrate the feasibility of the proposed noiselet encoding scheme, two pulse sequences with tailored spatially selective RF excitation pulses was designed and implemented on a 3T scanner to acquire the data in the noiselet domain from a phantom and a human brain.

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Kernel nonnegative matrix factorization without the curse of the pre-image

This is interesting: Kernel nonnegative matrix factorization without the curse of the pre-image by Fei Zhu, Paul Honeine, Maya Kallas

The nonnegative matrix factorization (NMF) is widely used in signal and image processing, including bio-informatics, blind source separation and hyperspectral image analysis in remote sensing. A great challenge arises when dealing with a nonlinear formulation of the NMF. Within the framework of kernel machines, the models suggested in the literature do not allow the representation of the factorization matrices, which is a fallout of the curse of the pre-image. In this paper, we propose a novel kernel-based model for the NMF that does not suffer from the pre-image problem, by investigating the estimation of the factorization matrices directly in the input space. For different kernel functions, we describe two schemes for iterative algorithms: an additive update rule based on a gradient descent scheme and a multiplicative update rule in the same spirit as in the Lee and Seung algorithm. Within the proposed framework, we develop several extensions to incorporate constraints, including sparseness, smoothness, and spatial regularization with a total-variation-like penalty. The effectiveness of the proposed method is demonstrated with the problem of unmixing hyperspectral images, using well-known real images and results with state-of-the-art techniques.

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Sunday Morning Insight: Physics Driven Sensor Design ?

If you read this Sunday Morning Insight entry two years ago entitled "The Linear Boltzmann Equation and Co-Sparsity" you may remember that the analysis approach as used in compressive sensing (the TV norm minimization falls in that category) is nothing less than an attempt at putting some of the physics back into a signal processing problem. 

Co-sparsity is really a statement of how a specific operator describes field equations in homogenous media. Boundary conditions between homogenous media and sources within homogenous media provide additional constraints within this field approach. The measurement equations eventually further constrain the problem so that only one solution can be physically inferred.

Up until now, I do not think we had many publications that tried to discretize the forward operator of the underlying physics and put it in an analysis approach. Here is a new path in that direction: Brain source localization using a physics-driven structured cosparse representation of EEG signals by Laurent Albera Srđan Kitić Nancy Bertin, Gilles Puy, Rémi Gribonval

Localizing several potentially synchronous brain activities with low signal-to-noise ratio from ElectroEncephaloGraphic (EEG) recordings is a challenging problem. In this paper we propose a novel source localization method, named CoRE, which uses a Cosparse Representation of EEG signals. The underlying analysis operator is derived from physical laws satisfied by EEG signals, and more particularly from Poisson's equation. In addition, we show how physiological constraints on sources, leading to a given space support and fixed orientations for current dipoles, can be taken into account in the optimization scheme. Computer results, aiming at showing the feasability of the CoRE technique, illustrate its superiority in terms of estimation accuracy over dictionary-based sparse methods and subspace approaches.

We previously mentioned a Data Driven or Zero Knowledge Sensor Design approach. Here though, we now have a whole new world of Physics Driven Sensor Design where measurement ensembles, instead of being RIP admissible or gaussian, are required to fit Physics based set of constraints (instantiated, in part, through the appearance of sharp phase transitions). Back in 1996, this work pretty much convinced many people that dictionary learning was physics based/biologically relevant. In the same way, I am sure both approaches (Data Driven and Physics based designs), instead of looking diametrically opposed, will eventually converge. More on that later.

Thanks to Srđan Kitić for the heads-up in this conversation.


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January-July 2014: Seven Months of Reproducible Research

Summer generally allows for some downtime at which point I can update other pages such as the Reproducible Research ( implementations ) page, the Big Picture in Compressive Sensing or the Advanced Matrix Factorization Jungle page. Here are the implementations I will add shortly to these pages. They include all implementations listed here on the Nuit Blanche in Review starting January 2014 till yesterday's entry:

Compressive Sensing

• Adaptive-Rate Compressive Sensing Using Side Information - implementation -
• NLCS : Non-Local Compressive Sampling Recovery - implementation -
• Multidimensional Compressed Sensing and their Applications - implementation -
• The SwAMP Thing! Sparse Estimation with the Swept Approximated Message-Passing Algorithm -implementation -
• On the Convergence of Approximate Message Passing with Arbitrary Matrices
• NLR-CS : Compressive Sensing via Nonlocal Low-rank Regularization - implementation -
• Matrix-free Interior Point Method for Compressed Sensing Problems / A Second order Method for Compressed Sensing with Coherent and Redundant Dictionaries - implementation -
• emd_flow : The Constrained Earth Mover Distance Model,with Applications to Compressive Sensing
• Tree_approx : A Fast Approximation Algorithm for Tree-Sparse Recovery
• CGIHT, ASD and ScaledASD: Compressed Sensing and Matrix Completion
• GAGA: GPU Accelerate Greedy Algorithms for Compressed Sensing - new version -
• RCoS : Image Compressive Sensing Recovery via Collaborative Sparsity
• GrAMPA: Generalized Approximate Message Passing for Analysis Compressive Sensing
• SparseFHT: A Fast Hadamard Transform for Signals with Sub-linear Sparsity in the Transform Domain - implementation -
• EMaC : Robust Spectral Compressed Sensing via Structured Matrix Completion
• Summary of Contribution: Compressed sensing with linear correlation between signal and measurement noise
• STSBL-EM : Spatiotemporal Sparse Bayesian Learning with Applications to Compressed Sensing of Multichannel Physiological Signals
• Sparse Spikes Deconvolution with Continuous Basis-Pursuit
• Bayesian Pursuit Algorithms
• Image Compressive Sensing Recovery Using Adaptively Learned Sparsifying Basis via L0 Minimization

Regression

• Confidence Intervals and Hypothesis Testing for High-Dimensional Regression
• Path Thresholding: Asymptotically Tuning-Free High-Dimensional Sparse Regression

Optimization solvers

• UnLocBox / PyUnLocBox (Matlab and Python convex optimization toolbox)
• PNOPT: Proximal Newton-type methods for minimizing composite functions

Image restoration

• Image Restoration Using Joint Statistical Modeling in Space-Transform Domain

Compressive Detection

• FCT : Fast Compressive Tracking
• Fast Tracking via Spatio-Temporal Context Learning

Sparse Approximation/Reconstruction/Denoising

• SASIR: LOFAR Sparse Image Reconstruction - implementation -
• GHP : Gradient Histogram Estimation and Preservation for Texture Enhanced Image Denoising
• Group-based Sparse Representation for Image Restoration
• WESNR : Mixed Noise Removal by Weighted Encoding with Sparse Nonlocal Regularization

Blind Deconvolution

• Blind Image Deblurring by Spectral Properties of Convolution Operators
• Blind Image Deblurring with Unknown Boundaries Using the Alternating Method of Multipliers
• Blind Identification via Lifting

Matrix Factorization (other than NMF)

• GreBsmo: Greedy Bilateral Sketch, Completion & Smoothing - implementation -
• Tight convex relaxations for sparse matrix factorization - implementation -
• Fast matrix completion without the condition number - implementation -
• Dynamic MR image reconstruction–separation from undersampled (k,t)-space via low-rank plus sparse prior - implementation -
• TGA: Grassmann Averages for Scalable Robust PCA
• Spectral redemption: clustering sparse networks
• Sparse Randomized Kaczmarz for Multiple Measurement Vectors
• Fast and Robust Archetypal Analysis for Representation Learning
• Probabilistic Archetypal Analysis
• ROSL : Robust Orthonormal Subspace Learning: Efficient Recovery of Corrupted Low-rank Matrices - implementation -
• CGIHT, ASD and ScaledASD: Compressed Sensing and Matrix Completion
• PSPG: Efficient Algorithms for Robust and Stable Principal Component Pursuit Problems
• CPCP: Scalable Robust Matrix Recovery: Frank-Wolfe Meets Proximal Methods
• ADMIP: An Alternating Direction Method with Increasing Penalty for Stable Principal Component Pursuit
• OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage
• Scalable methods for nonnegative matrix factorizations of near-separable tall-and-skinny matrices
• Introduction into cross approximation / Fast multidimensional convolution in low-rank formats via cross approximation - implementation -
• LRRSC : Subspace Clustering by Exploiting a Low-Rank Representation with a Symmetric Constraint - implementation -
• WNNM: Weighted Nuclear Norm Minimization with Application to Image Denoising
• Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach
• Low-Rank Modeling of Local k-Space Neighborhoods (LORAKS): Implementation and Examples for Reproducible Research
• IRNN: Generalized Nonconvex Nonsmooth Low-Rank Minimization
• HASI: Probabilistic Low Rank Matrix Completion with Adaptive Spectral Regularization Algorithms
• Interest Zone Matrix Approximation

NMF

• Random Projections for Non-negative Matrix Factorization
• Enhancing Pure-Pixel Identification Performance via Preconditioning - implementation -

Nonlinear Compressive Sensing

• Nonlinear Basis Pursuit
• Nonlinear estimation of material abundances in hyperspectral images with L1-norm spatial regularization

One-Bit

• One-bit compressive sensing with norm estimation
• Binary Compressed Imaging - implementation -

Phase Retrieval

• Phase Retrieval via Wirtinger Flow: Theory and Algorithms - implementation -
• PR-GAMP : Compressive Phase Retrieval via Generalized Approximate Message Passing
• Balancing Sparsity and Rank Constraints in Quadratic Basis Pursuit - implementation -

Randomized Numerical Linear Algebra

• Randomized Numerical Linear Algebra for Large Scale Data Analysis ( libSkylark: Sketching based Matrix computations for Machine Learning )
• Randomized LU Decomposition

Tensor

• Stable, Robust and Super Fast Reconstruction of Tensors Using Multi-Way Projections - implementation -

Machine Learning

• Mondrian Forests: Efficient Online Random Forests - implementation -
• Understanding Random Forests: From Theory to Practice - implementation -
• SPAL: Sparse Projection-based Adaptive Learning 
• Active Subspace: Towards Scalable Low-Rank Learning
• A Riemannian approach to low-rank algebraic Riccati equations - implementation -
• KNIFE: Automatic Feature Selection via Weighted Kernels and Regularization
• Avoiding pathologies in very deep networks
• Kernel LMS algorithm with forward-backward splitting for dictionary learning
• MISO: Incremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning

Subspace Learning

• LSR: Robust and Efficient Subspace Segmentation via Least Squares Regression
• PETRELS: Parallel Subspace Estimation and Tracking by Recursive Least Squares from Partial Observations

Applications

• Randomized Comments: Code for GWAS and CS study, a new blog, Geometric theory of information
• PURIFY: a new algorithmic framework for next-generation radio-interferometric imaging - implementation -
• REMODE: Probabilistic, Monocular Dense Reconstruction in Real Time
• HODLR and george: Fast Direct Methods for Gaussian Processes and the Analysis of NASA Kepler Mission Data - implementation -
• CaSPIAN: A Causal Compressive Sensing Algorithm for Discovering Directed Interactions in Gene Networks - implementation -
• Saturday Morning Video: SVO: Fast Semi-Direct Monocular Visual Odometry - implementation -
• SUGAR : Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection
• Selecting thresholding and shrinking parameters with generalized SURE for low rank matrix estimation
• The Flutter Shutter Code Calculator - implementation -

Credit photo: OSIRIS narrow angle camera view of 67P/C-G from a distance of 1950 km on 29 July 2014. Credits: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA

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