By Cleve Moler
評：本文是Mathworks 公司的首席科學家Cleve Moler寫的一個科普材料。這裡作者說明了採用L1規範是可以恢復原型號的，而採用L2規範則不可以恢復原型號。

When I first heard about compressed sensing, I was skeptical. There were claims that it reduced the amount of data required to represent signals and images by huge factors and then restored the originals exactly. I knew from the Nyquist-Shannon sampling theorem that this is impossible. But after learning more about compressed sensing, I’ve come to realize that, under the right conditions, both the claims and the theorem are true.

The Nyquist-Shannon sampling theo-rem states that to restore a signal exactly and uniquely, you need to have sampled with at least twice its frequency. Of course, this the-orem is still valid; if you skip one byte in a signal or image of white noise, you can’t re-store the original. But most interesting sig-nals and images are not white noise. When represented in terms of appropriate basis functions, such as trig functions or wave-lets, many signals have relatively few non-zero coefficients. In compressed (or compres-sive) sensing terminology, they are sparse.

The Underlying Matrix Problem

Naturally, the aspect of compressed sensing that first caught my attention was the under-lying matrix computation. A raw signal or image can be regarded as a vector f with millions of components. We assume that f can be represented as a linear combination of certain basis functions:

f = Ψc

The basis functions must be suited to a particular application. In our example, Ψ is the discrete cosine transform. We also assume that most of the coefficients c are effectively zero, so that c is sparse. Sampling the signal involves another lin-ear operator,
b = Φf

In our example, b is a few random sam-ples of f, so Φ is a subset of the rows of the identity operator. But more complicated sampling operators are possible.

To reconstruct the signal, we must try to recover the coefficients by solving

Ax = b, where A = ΦΨ

Once we have the coefficients, we can re-cover the signal itself by computing

f ≈ Ψx

Since this is a compression, A is rectan-gular, with many more columns than rows. Computing the coefficients x involves solv-ing an underdetermined system of simulta-neous linear equations, Ax = b. In this situ-ation, there are many more unknowns than equations. The key to the almost magical reconstruction process is to impose a non-linear regularization involving the l1 (that’s “ell-sub-1”) norm.
I described a tiny version of this situa-tion in a 1990 Cleve’s Corner column en-titled “The World’s Simplest Impossible Problem.” I am thinking of two numbers whose average is 3. What are the numbers? After complaining that I haven’t given you enough information, you might answer 2 and 4. If you do, you have unconsciously imposed a kind of regularization that re-quires the result to be two distinct integers.

MATLAB® can provide two different an-swers. Algebraically, the problem is a 1-by-2 system of linear equations with matrix

A = [1/2 1/2]

and right-hand side

b = 3

We want to find a 2-vector y that solves Ay = b. The minimum norm least squares solution is computed by the pseudoinverse,

y = pinv(A)*b

y =

3

3

But backslash generates a different solution:

x = A\b

x =

6

0

Both solutions are valid, but human puzzle-solvers rarely mention them. Notice that the solution computed by backslash is sparse; one of its components is zero.

A Larger Instance of the Same Task

The signal or image restoration problem is a larger instance of the same task. We are given thousands of weighted averages of millions of signal or pixel values. Our job is to re-generate the original signal or image. We have the compressed sample, b, and we need to solve Ax = b. There is a huge number of possible solutions. The basis for picking the right one involves vec-tor norms. The familiar Euclidean distance norm, l2, is

l2: norm(x,2) = sqrt(sum(x.^2))

The “Manhattan” norm, l1, is named after travel time along the square grid formed by city streets.

l1: norm(x,1) = sum(abs(x))

The notation l0 is used for a function that only counts the number of nonzero compo-nents. It’s actually not a norm.

l0: norm0(x) = sum(x~=0)

These norms impose three different non-linear regularizations, or constraints, on our underdetermined linear system.

From all x that satisfy Ax = b, find an x that minimizes lp(x)

For our 20-year-old example, l2(y) is less than l2(x), and l1(y) and l1(x) happen to be equal, but l0(y) is greater than l0(x).

Sparsity and the l1 Norm

The keys to compressed sensing are sparsity and the l1 norm. If the expansion of the orig-inal signal or image as a linear combination of the selected basis functions has many zero coefficients, then it’s often possible to recon-struct the signal exactly. In principle, com-puting this reconstruction should involve counting nonzeros with l0. This is a combi-natorial problem whose computational com-plexity makes it impractical. (It’s NP-hard.) Fortunately, as David Donoho, Emmanuel Candés, and Terence Tao have shown, l0 can be replaced by l1. They explain that “with overwhelming probability” the two prob-lems have the same solution. The l1 compu-tation is practical because it can be posed as a linear programming problem and solved with the traditional simplex algorithm or modern interior point methods.

An Example

Our example uses the discrete cosine trans-form (DCT) as the basis. The signal generated by the “A” key on a touch-tone telephone is the sum of two sinusoids with incommensu-rate frequencies,
f(t) = sin(1394πt) + sin(3266πt)
If we sample this tone for 1/8 of a second at 40000 Hz, the result is a vector f of length n = 5000. The upper plot in Figure 1 shows a por-tion of this signal, together with some of the m = 500 random samples that we have taken. The lower plot shows the coefficients c, ob-tained by taking the inverse discrete cosine transform of f, with two spikes at the appro-priate frequencies. Because the two frequen-cies are incommensurate, this signal does not fall exactly within the space spanned by the DCT basis functions, and so there are a few dozen significant nonzero coefficients.

For our example, the condensed signal is a vector b of m random samples of the original signal. We construct a matrix A by extracting m rows from the n-by-n DCT matrix

D = dct(eye(n,n));

A = D(k,:)

where k is the vector of indices used for the sample b. The resulting linear system, Ax = b, is m-by-n, which is 500-by-5000. There are 10 times as many unknowns as equations.

To reconstruct the signal, we need to find the solution to Ax = b that minimizes the l1 norm of x. This is a nonlinear opti-mization problem, and there are several MATLAB based programs available to solve it. I have chosen to use l1-magic, written by Justin Romberg and Emmanuel Candès when they were at Caltech. The upper plot in Figure 2 shows the resulting solution, x. We see that it has relatively few large com-ponents and that it closely resembles the DCT of the original signal. Moreover, the discrete cosine transform of x, shown in the lower plot, closely resembles the original signal. If audio was available, you would be able to hear that the two commands sound(f) and sound(dct(x)) are nearly the same.

For comparison, Figure 3 shows the traditional l2, or least squares, solution to Ay = b, computed by

y = pinv(A)*b

There is a slight hint of the original sig-nal in the plots, but sound(dct(y)) is just a noisy buzz.

It is too early to predict when, or if, we might see compressed sensing in our cell phones, digital cameras, and magnetic res-onance scanners, but I find the underlying mathematics and software fascinating.

References

A twenty-year-old Cleve’s Corner: “The World’s Simplest Impossible Problem,” News & Notes (1990) www.mathworks.com/clevescorner /dec1990

A resource providing over 600 links, including 36 that reference MATLAB:
“Compressive Sensing Resources,”

Rice University

Two survey papers for general technical audiences:

Dana Mackenzie, “Compressed Sensing Makes Every Pixel Count,” What’s Happen-

ing in the Mathematical Sciences 7 (2009), American Math Society www.ams.org/happening-series /hap7-pixel.pdf

Bryan Hayes, “The Best Bits,” American Scientist 97, no.4 (2009) www.americanscientist.org/issues/pub /the-best-bits/7

One of many tutorials:

Richard Baraniuk, Justin Romberg, and Michael Wakin, “Tutorial on Compressive Sensing,” Information Theory and Applications Workshop (2008) www.ece.rice.edu/~richb/talks /cs-tutorial-ITA-feb08-complete.pdf

MATLAB based software:

Justin Romberg, l1-magic www.acm.caltech.edu/l1magic

The two original papers:

Emmanuel Candès, Justin Romberg, and Terence Tao, “Robust uncertainty prin-ciples: Exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, 52(2): 489 - 509, February 2006 www.acm.caltech.edu/~jrom/publications /CandesRombergTao_revisedNov2005.pdf

David Donoho. “Compressed Sensing,”

IEEE Transactions on Information Theory,

52(4): 1289 - 1306, April 2006

www-stat.stanford.edu/~donoho/Reports

/2004/CompressedSensing091604.pdf

An audio demonstration:

Laura Balzano, Robert Nowak, and Jordan Ellenberg, “Compressed Sensing Audio Demonstration” http://sunbeam.ece.wisc.edu/csaudio/

**> Compressed sensing promises, in theory, to reconstruct a signal or

image from surpris-ingly few samples. Discovered just five years ago
by Candès and Tao and by Donoho, the subject is a very active research
area. Practical devices that implement the theory are just now being
developed.

It is important to realize that compressed sensing can be done only by
a compress-ing sensor, and that it requires new record-ing technology
and file formats. The MP3 and JPEG files used by today’s audio
sys-tems and digital cameras are already com-pressed in such a way
that exact reconstruc-tion of the original signals and images is
impossible. Some of the Web postings and magazine articles about
compressed sens-ing fail to acknowledge this fact.**