# 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