# Close Encounters of the Binary Kind: Signal Reconstruction Guarantees for Compressive Hadamard Sampling With Haar Wavelet Basis

@article{Moshtaghpour2019CloseEO, title={Close Encounters of the Binary Kind: Signal Reconstruction Guarantees for Compressive Hadamard Sampling With Haar Wavelet Basis}, author={Amirafshar Moshtaghpour and Jos{\'e} M. Bioucas-Dias and Laurent Jacques}, journal={IEEE Transactions on Information Theory}, year={2019}, volume={66}, pages={7253-7273} }

We investigate the problems of 1-D and 2-D signal recovery from subsampled Hadamard measurements using Haar wavelet as a sparsity inducing prior. These problems are of interest in, e.g., computational imaging applications relying on optical multiplexing or single-pixel imaging. However, the realization of such modalities is often hindered by the coherence between the Hadamard and Haar bases. The variable and multilevel density sampling strategies solve this issue by adjusting the subsampling…

## 26 Citations

### Non-uniform Recovery Guarantees for Binary Measurements and Infinite-Dimensional Compressed Sensing

- Computer ScienceArXiv
- 2019

The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case.

### Non-uniform Recovery Guarantees for Binary Measurements and Infinite-Dimensional Compressed Sensing

- Computer ScienceJournal of Fourier Analysis and Applications
- 2021

The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case.

### Improved recovery guarantees and sampling strategies for TV minimization in compressive imaging

- Computer ScienceSIAM J. Imaging Sci.
- 2021

Focusing on two important imaging modalities -- namely, Fourier imaging and structured binary imaging via the Walsh--Hadamard transform -- it derive uniform recovery guarantees asserting stable and robust recovery for arbitrary random sampling strategies.

### Recovering wavelet coefficients from binary samples using fast transforms

- Computer ScienceSIAM J. Sci. Comput.
- 2022

This work derives an algorithm, which bypasses the NM storage requirement and the OpNMq computational cost of matrix-vector multiplication with this matrix when using Walsh-Hadamard samples and wavelet reconstruction.

### WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions

- Computer ScienceArXiv
- 2021

This work provides a first-order method: Weighted, Accelerated and Restarted Primal-dual (WARPd), based on primal-duAL iterations and a novel restart-reweight scheme, and shows how to unroll WARPd as neural networks.

### An 81.92Gpixels/s Fast Reconstruction of Images from Compressively Sensed Measurements

- Computer Science, Mathematics2022 IEEE International Symposium on Circuits and Systems (ISCAS)
- 2022

A sparse sensing matrix consisting mainly of zero-vectors is adopted, which accelerates the state-of-the-art method by 65 × and achieves 81.92Gpixels/s reconstruction of Compressed Sensing.

### Fast Scalable Image Restoration Using Total Variation Priors and Expectation Propagation

- Computer ScienceIEEE Transactions on Image Processing
- 2022

The simulation results illustrate that the Expectation Propagation framework can provide a posteriori estimates on par with those obtained via sampling methods but at a fraction of the computational cost, and does not exhibit strong underestimation of posteriori variances, in contrast to variational Bayes alternatives.

### Compressive Sampling Using a Pushframe Camera

- Computer ScienceIEEE Transactions on Computational Imaging
- 2021

A strongly performing static binarized noiselet compressive sampling mask design is developed, tailored to pushframe hardware, allowing both a single exposure per motion time-step, and retention of 2D correlations in the scene.

### WARPd: A Linearly Convergent First-Order Primal-Dual Algorithm for Inverse Problems with Approximate Sharpness Conditions

- Computer ScienceSIAM Journal on Imaging Sciences
- 2022

This work provides a first-order method: weighted, accelerated, and restarted primal-dual (WARPd), based on primal- DUal iterations and a novel restart-reweight scheme, which achieves stable linear convergence to the desired vector under a generic approximate sharpness condition.

### The troublesome kernel: why deep learning for inverse problems is typically unstable

- Computer ScienceArXiv
- 2020

This paper presents a comprehensive mathematical analysis explaining the many facets of the instability phenomenon in DL for inverse problems, and demonstrates a counterintuitive phenomenon: training a neural network may generically not yield an optimal reconstruction method for an inverse problem.

## References

SHOWING 1-10 OF 89 REFERENCES

### Performance of Compressive Sensing for Hadamard-Haar Systems

- Computer Science
- 2019

In this work, an explicit sample-complexity bound for Hadamard-Haar systems is computed; a seemingly missing result in the related literature is computed.

### Stable and Robust Sampling Strategies for Compressive Imaging

- Computer ScienceIEEE Transactions on Image Processing
- 2014

The local coherence framework developed in this paper implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis-as opposed to bounded maximal coherence-as long as the sampling strategy is adapted accordingly.

### On oracle-type local recovery guarantees in compressed sensing

- Computer ScienceArXiv
- 2018

Improved sampling complexity bounds for stable and robust sparse recovery in compressed sensing are presented and the potential of this theory for devising adaptive sampling strategies in sparse polynomial approximation is shown.

### Non-uniform Recovery Guarantees for Binary Measurements and Infinite-Dimensional Compressed Sensing

- Computer ScienceArXiv
- 2019

The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case.

### Coherence estimates between Hadamard matrices and Daubechies wavelets

- Mathematics
- 2016

Traditionally the compressive sensing theory have been focusing on the three principles of sparsity, incoherence and uniform random subsampling. Recent years research have shown that these principles…

### Sampling from binary measurements - on reconstructions from Walsh coefficients

- Mathematics2017 International Conference on Sampling Theory and Applications (SampTA)
- 2017

Reconstructing infinite-dimensional signals from a limited amount of linear measurements is a key problem in many applications such as medical imaging [35], single-pixel and lensless cameras [27],…

### Variable density compressed image sampling

- Computer Science2009 17th European Signal Processing Conference
- 2009

This paper addresses the design of a novel variable density sampling strategy, where the “a priori” information about the statistical distributions that natural images exhibit in the wavelet domain is exploited.

### Compressed sensing with local structure: uniform recovery guarantees for the sparsity in levels class

- Computer ScienceApplied and Computational Harmonic Analysis
- 2019

### Compressed sensing with structured sparsity and structured acquisition

- Computer ScienceApplied and Computational Harmonic Analysis
- 2019