Computational Photography

Compressing Imaging

by dr. Francho Melendez

today's schedule

• Homework for XMAS
• Light Field ++
• Compressing Imaging

list of papers

Review Form

January 12th

• Presentation 10 minutes
• 5 minutes question
• send your presentation to me by the 11th

objectives

• identify main contribution
• identify limitations
• present the content

compressive imaging

compressive sensing?

also compressive sampling

compressing sensing?

a novel sampling paradigm that asserts that one can recover certain signals from far fewer samplers than traditional methods

Nyquist–Shannon Sampling

If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

A sufficient sample-rate is therefore 2B samples/second, or anything larger. Equivalently, for a given sample rate fs, perfect reconstruction is guaranteed.

Nyquist–Shannon Sampling

For some signals like images that are not naturally bandlimited, the sampling rate is dictated not by the Shannon theorem but by the desired temporal or spatial resolution.

compressing sensing

sparsity of a signal

many signals have some structure

incoherence

sensing modality

sparse signals

images are sparse in some domain

other domain

images are sparse in some domain

JPEG

images are sparse in some domain

sampling as linear transformation

compressed sensing, much less measurments

solving a Ax = b problem (b measurements, x unknown signal)

information preservation

no in general

infinite x for the given y (linear subspace)

information preservation

in the sparse case, yes!

information preservation

easy! if you know the signal.

but we don't know the signal... :(

a desing problem

I want to design a $\phi$ matrix, that if I take K colums of it, they are full rank. (rank K)

And I want them to be close to orthogonal.

Restricted Isometry Property (RIP)

linear subspaces

• k-sparse signal in space
• we want to find a $\phi$ that projects the signal into a lower dimensional space where it's still sparse
• dimensional reduction: we want to preserve distances

N-P Problem

$\phi$ is still very large

check the rank of every possible selection of K

random matrices are RIP

they found that this matrices are quite common, in fact random matrices usually work well

M is the number of samples, only grow logaritmically

examples

single pixel camera

plus l1 minimization

applications

• measuring might me expensive
• takes long time
• 2013 Used for light fields

conclusion

• many signals in nature are sparse
• we can take advantage of that when sampling
• taking random samples works with many signals
• given the correct basis
• and reconstructed with $l_1$ norm
• this is called compressed sensing

announcements

18.15 SALA 25

January 13th

• Tell me the paper you choose: ASAP
• 5 minutes question
• send your presentation to me by the 12th