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#CompressiveSensing

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Low Rank Jack

[New Python code: PyNoiselet] About 15 years ago, I wrote a simple set of matlab functions to compute the #Noiselet transform of Coifman et al (R. Coifman, F. Geshwind, and Y. Meyer, "Noiselets", *Applied and Computational Harmonic Analysis*, 10(1):27–44, 2001). The noiselet transform is used in #CompressiveSensing applications as well as in #Sparse signal coding as noiselets have minimally low coherence with wavelet bases (Haar and Daubechies), which is useful for sparse signal recovery.

Today, from a code request received yesterday by email, I decided to quickly rewrite this old code in Python (with the useful help of one LLM I admit).

Here is the result if you need an O(N log N) (butterfly like) algorithm to compute this transformation:

gitlab.com/laurentjacques/PyNo

More information also in this old blog post : laurentjacques.gitlab.io/post/

Feel free to fork it and improve this non-optimized code.

GitLabLaurent Jacques / PyNoiselet · GitLabGitLab.com
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Low Rank Jack

"Phase Transitions in Phase-Only Compressed Sensing," by
Junren Chen, Lexiao Lai, Arian Maleki arxiv.org/abs/2501.11905 #PhaseOnly #CompressiveSensing

arXiv.orgPhase Transitions in Phase-Only Compressed SensingThe goal of phase-only compressed sensing is to recover a structured signal $\mathbf{x}$ from the phases $\mathbf{z} = {\rm sign}(\mathbfΦ\mathbf{x})$ under some complex-valued sensing matrix $\mathbfΦ$. Exact reconstruction of the signal's direction is possible: we can reformulate it as a linear compressed sensing problem and use basis pursuit (i.e., constrained norm minimization). For $\mathbfΦ$ with i.i.d. complex-valued Gaussian entries, this paper shows that the phase transition is approximately located at the statistical dimension of the descent cone of a signal-dependent norm. Leveraging this insight, we derive asymptotically precise formulas for the phase transition locations in phase-only sensing of both sparse signals and low-rank matrices. Our results prove that the minimum number of measurements required for exact recovery is smaller for phase-only measurements than for traditional linear compressed sensing. For instance, in recovering a 1-sparse signal with sufficiently large dimension, phase-only compressed sensing requires approximately 68% of the measurements needed for linear compressed sensing. This result disproves earlier conjecture suggesting that the two phase transitions coincide. Our proof hinges on the Gaussian min-max theorem and the key observation that, up to a signal-dependent orthogonal transformation, the sensing matrix in the reformulated problem behaves as a nearly Gaussian matrix.
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Low Rank Jack

From Julián Tachella @JulianTachella, posted on "Chi":

📰""Learning to reconstruct signals from binary measurements alone"📰

We present theory + a #selfsupervised approach for learning to reconstruct incomplete (!) and binary (!) measurements using the binary data itself. See the first figure and its alt-text.

arxiv.org/abs/2303.08691
with @lowrankjack
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The theory characterizes

- the best approximation of a set of signals from incomplete binary observations
- its sample complexity
- complements existing theory for signal recovery from binary measurements

See the third figure and its alt-text.
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The proposed self-supervised algorithm obtains performances on par with supervised learning and outperforms standard reconstruction techniques (such as binary iterative hard thresholding)

See the second figure and its alt-text.

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Code based on the deepinverse library is available at github.com/tachella/ssbm

Check out the paper for more details!

#SelfSupervisedLearning#CompressiveSensing#Quantization
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Low Rank Jack

Slides and abstract of my talk "Interferometric single-pixel imaging with a multicore fiber" made at ISCS23 (Luxembourg, June 12-14, 2023) now available at laurentjacques.gitlab.io/event and laurentjacques.gitlab.io/publi All this is based on the work of Olivier Leblanc. Long journal preprint version "Interferometric lensless imaging: rank-one projections of image frequencies with speckle illuminations" on arxiv.org /abs/2306.12698 in colaboration with S. Sivankutty, M. Hofer, H. Rigneault, #imaging #lensless #compressivesensing #computationalimaging #speckle #compressedsensing #InverseProblem (toot generated thanks to tootpick.org directly from my website)

Laurent JacquesInterferometric single-pixel imaging with a multicore fiber | Laurent JacquesAbstract: Lensless illumination single-pixel imaging with a multicore fiber (MCF) is a computational imaging technique that enables potential endoscopic observations of biological samples at cellular scale. In this work, we show that this technique is tantamount to collecting multiple symmetric rank-one projections (SROP) of a Hermitian interferometric matrix – a matrix encoding the spectral content of the sample image.
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ISCompSensing

It is our pleasure to announce the title of Prof. Laura Waller's keynote talk, "DiffuserCam: Multi-dimensional Lensless Imaging". @optrickster

Visit #ISCS2023 website for the abstract of this talk.

Submit your papers before March 15, 2023, for platform, poster, or show-and-tell demo presentations.

* Submission link: cmt3.research.microsoft.com/IS

* Conference website: iscs2023.com/

#microscopy#radar#remotesensing
Public
ISCompSensing

The #iscs2023 submission deadline is approaching. Submit your papers before March 15, 2023, for platform, poster, or show-and-tell demo presentations.

* Submission link: cmt3.research.microsoft.com/IS

* Submission guidelines: overleaf.com/latex/templates/i

* Conference website: iscs2023.com/

#microscopy#radar#remotesensing
Public
ISCompSensing

Happy new year! Latex template for the 2-page extended abstract submissions to the 1st edition of the International Symposium on #ComputationalSensing (#iscs2023) in #Luxembourg is now available on iscs2023.com. Submit before March 15, 2023.
Stay tuned for the submission tool, and the upcoming exciting rates for student registration!
#light #electron imaging #radar #remotesensing #astronomical #biomedical #signalprocessing #compressivesensing #electronmicroscopy

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Low Rank Jack

@gabrielpeyre This is indeed a key experiment by Alan V. Oppenheim. Following his results, I realized with T. Feuillen, a few years ago, that this is also true in complex compressive sensing with Gaussian matrices. Paper, slides, demo, and video here:
laurentjacques.gitlab.io/publi
laurentjacques.gitlab.io/event

Laurent JacquesThe Importance of Phase in Complex Compressive Sensing | Laurent JacquesAbstract: We consider the question of estimating a real low-complexity signal (such as a sparse vector or a low-rank matrix) from the phase of complex random measurements. We show that in this phase-only compressive sensing (PO-CS) scenario, we can perfectly recover such a signal with high probability and up to global unknown amplitude if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the complexity level of the signal space.
#CompressiveSensing#Signal#Phase
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Low Rank Jack

@ccanonne I gave a talk on how to "quantize" JL lemma in 2014, slides are available here laurentjacques.gitlab.io/event I recall visually the main principles of the one-bit (sign) case from slide 22. There is a very good review on the topic by Sjoerd Dirksen semanticscholar.org/paper/Quan #OneBit #CompressiveSensing #RandomProjection #JohnsonLindenstrauss

Laurent JacquesWhen Buffon's needle problem helps in quantizing the Johnson-Lindenstrauss Lemma | Laurent JacquesAbstract: In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a ground made of equispaced parallel strips lies on two of them?
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