Emmanuel Candès, 2004 ($500k) 

• by coincidence (Shepp–Logan phantom)
• l1 minimalization
• “It was as if you gave me the first three digits of a 10-digit bank account number — and then I was able to guess the next seven,”

So what and why?

Here’s how it would work with a photograph.

Compressive sensing

• magic of notion called sparsity
• pic made of solid blocks of color or wiggly lines is sparse, 
• a screenful of random and chaotic dots is not
• out of all the bazillion possible reconstructions, the simplest, or sparsest, image is almost always the right one or very close to it
• how to find the sparsest image quickly?

R example

library(R1magic)
N <- 100 
# Sparse components 
K <- 4 
#  Up to Measurements  > K LOG (N/K)
M <- 40
# Measurement Matrix (Random Sampling Sampling)
phi <- GaussianMatrix(N,M)
# R1magic generate random signal
xorg <- sparseSignal(N, K, nlev=1e-3)
y <- phi %*% xorg ;# generate measurement
T <- diag(N) ;# Do identity transform
p <- matrix(0, N, 1) ;# initial guess

# R1magic Convex Minimization
ll <- solveL1(phi, y, T, p)
x1 <- ll$estimate

plot( 1:100, seq(0.011,1.1,0.011), type = "n",xlab="",ylab="")
title(main="Random Sparse Signal Recovery",
      xlab="Signal Component",ylab="Spike Value")
lines(1:100, xorg , col = "red")
lines(1:100, x1, col = "blue", cex = 1.5)