We discuss several applications of compressive sampling in the area of analog-to-digital conversion and biomedical imaging and review some numerical experiments in new directions. We conclude by exposing the participants to some important open problems.
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-15
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
Conventional wisdom and common practice in acquisition andreconstruction of images or signals from frequency data follows thebasicprinciple of the Nyquist density sampling theory. This principlestates that to reconstruct an image/signal, the number of Fouriersamples weneed to acquire must match the desired resolution of the image/signal, e.g.the...
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2005-12-05
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
Compressed sensing essentially relies on two tenets: the first is that the object we wish to recover is compressible in the sense that it has a sparse expansion in a set of basis functions; the second is that the measurements we make (the sensing waveforms) must be incoherent with these basis functions. This lecture will introduce key results in...
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-06
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We morph compressive sampling into an error correcting code, and explore the implications of this sampling theory for lossy compression and some of its relationship with universal source coding.
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-13
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We will survey the literature on interior point methods which are very efficient numerical algorithms for solving large scale convex optimization problems.
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-14
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We show that accurate estimation from noisy undersampled data is sometimes possible and connect our results with a large literature in statistics concerned with high dimensionality; that is, situations in which the number of observations is less than the number of parameters.
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-12
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
After a rapid and glossy introduction to compressive sampling€“or compressed sensing as this is also called€“the lecture will introduce sparsity as a key modeling tool; the lecture will review the crucial role played by sparsity in various areas such as data compression, statistical estimation and scientific computing.
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-04
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
In many applications, one often has fewer equations than unknowns. While this seems hopeless, we will show that the premise that the object we wish to recover is sparse or compressible radically changes the problem, making the search for solutions feasible. This lecture discusses the importance of the l1-norm as a sparsity promoting functional a...
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-05
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
This lecture will discuss the crucial role played by probability in compressive sampling; we will discuss techniques for obtaining nonasymptotic results about extremal eigenvalues of random matrices. Of special interest is the role played by high- dimensional convex geometry and techniques from geometric functional analysis such as the Rudelson'...
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-08
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We introduce a strong form of uncertainty relation and discuss its fundamental role in the theory of compressive sampling. We give examples of random sensing matrices obeying this strong uncertainty principle; e.g. Gaussian matrices.
Creator:
Candès, Emmanuel J. (California Institute of Technology)
Created:
2007-06-07
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.