# Compressive Sampling

Monday, December 5, 2005 - 10:30am - 11:30am

EE/CS 3-180

Emmanuel Candès (California Institute of Technology)

Conventional wisdom and common practice in acquisition and

reconstruction of images or signals from frequency data follows the

basic

principle of the Nyquist density sampling theory. This principle

states that to reconstruct an image/signal, the number of Fourier

samples we

need to acquire must match the desired resolution of the image/

signal, e.g.

the number of pixels in the image.

This talk introduces a newly emerged sampling theory which shows that

this conventional wisdom is inaccurate. We show that perhaps

surprisingly, images or signals of scientific interest can be

recovered accurately and sometimes even exactly from a limited number

of nonadaptive random measurements. In effect, the talk introduces a

theory suggesting the possibility of compressed data acquisition

protocols which perform as if it were possible to directly acquire

just the important information about the image of interest. In other

words, by collecting a comparably small number of measurements rather

than pixel values, one could in principle reconstruct an image with

essentially the same resolution as that one would obtain by measuring

all the pixels, a phenomenon with far reaching implications.

The reconstruction algorithms are very concrete, stable (in the sense

that they degrade smoothly as the noise level increases) and

practical; in fact, they only involve solving convenient convex

optimization programs.

reconstruction of images or signals from frequency data follows the

basic

principle of the Nyquist density sampling theory. This principle

states that to reconstruct an image/signal, the number of Fourier

samples we

need to acquire must match the desired resolution of the image/

signal, e.g.

the number of pixels in the image.

This talk introduces a newly emerged sampling theory which shows that

this conventional wisdom is inaccurate. We show that perhaps

surprisingly, images or signals of scientific interest can be

recovered accurately and sometimes even exactly from a limited number

of nonadaptive random measurements. In effect, the talk introduces a

theory suggesting the possibility of compressed data acquisition

protocols which perform as if it were possible to directly acquire

just the important information about the image of interest. In other

words, by collecting a comparably small number of measurements rather

than pixel values, one could in principle reconstruct an image with

essentially the same resolution as that one would obtain by measuring

all the pixels, a phenomenon with far reaching implications.

The reconstruction algorithms are very concrete, stable (in the sense

that they degrade smoothly as the noise level increases) and

practical; in fact, they only involve solving convenient convex

optimization programs.