# Compressed Sensing

### Compressive Imaging and Characterization of Sparse Light Deflection Maps

Abstract: Light rays incident on a transparent object of uniform refractive index undergo deflections, which uniquely characterize the surface geometry of the object. Associated with each point on the surface is a deflection map (or spectrum) which describes the pattern of deflections in various directions.

### Quantization and Compressive Sensing

Abstract: Quantization is an essential step in digitizing signals, and, therefore, an indispensable component of any modern acquisition system. This chapter explores the interaction of quantization and compressive sensing and examines practical quantization strategies for compressive acquisition systems.

### Testing a Quasi-Isometric Quantized Embedding

It took me a certain time to do it. Here is at least a first attempt to test numerically the validity of some of the results I obtained in “A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon’s Needle” (arXiv) I have decided to avoid using the too conventional matlab environment.

### Testing a Quasi-Isometric Quantized Embedding

It took me a certain time to do it. Here is at least a first attempt to test numerically the validity of some of the results I obtained in “A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon’s Needle.

### When Buffon's needle problem meets the Johnson-Lindenstrauss Lemma

Quasi-isometric embeddings of vector sets with quantized sub-Gaussian projections | Le Petit Chercheur Illustré - Apr 2, 2015 […] explained in my previous post on quantized embedding and the funny connection with Buffon’s needle problem, I have recently noticed that for finite […]

### When Buffon's needle problem meets the Johnson-Lindenstrauss Lemma

[caption id=“attachment_346” align=“alignnone” width=“640”] (left) Picture of [8, page 147] stating the initial formulation of Buffon’s needle problem (Courtesy of E. Kowalski’s blog) (right) Scheme of Buffon’s needle problem.[/caption](This post is related to a paper entitled “A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon’s Needle” (arxiv, pdf) that I have recently submitted for publication.

### When Buffon''s needle problem meets the Johnson-Lindenstrauss Lemma

Last July, I read the biography of Paul Erdős written by Paul Hoffman and entitled “The Man Who Loved Only Numbers”. This is really a wonderful book sprinkled with many anecdotes about the particular life of this great mathematician and about his appealing mathematical obsessions (including prime numbers).

### A useless non-RIP Gaussian matrix

Recently, for some unrelated reasons, I discovered that it is actually very easy to generate a Gaussian matrix $\Phi$ that does not respect the restricted isometry property (RIP) [1]. I recall that such a matrix is RIP if there exists a (restricted isometry) constant $0<\delta<1$ such that, for any $K$-sparse vector $w\in \mathbb R^N$,

### A useless non-RIP Gaussian matrix

Recently, for some unrelated reasons, I discovered that it is actually very easy to generate a Gaussian matrix $$\Phi$$ that does not respect the restricted isometry property (RIP) [1]. I recall that such a matrix is RIP if there exists a (restricted isometry) constant $$0<\delta<1$$ such that, for any $$K$$-sparse vector $$w \in \mathbb R^N$$, \[(1-\delta)\|w\|^2\leq \|\Phi w\|^2 \leq (1+\delta)\|w\|^2.

### Compressive Acquisition of Sparse Deflectometric Maps

Abstract: Schlieren deflectometry aims at measuring deflections of light rays from transparent objects, which is subsequently used to characterize the objects. With each location on a smooth object surface a sparse deflection map (or spectrum) is associated.