Exact or stable image\signal reconstruction from

incomplete information

Project guide:Dr. Pradeep SenUNM (Abq)

Submitted by:Nitesh AgarwalIIT Roorkee (India)

Images and Signals

Digital Images

Digital Signals (Discrete)

AfterSampling

AnalogSignal

Digital Signal

(figures from matlab)

(figures from Wikipedia)

Compressive samplingCompressive sampling for signals

• Nyquist Theory: Where B = Bandwidth

= sampling frequency

• Compressive sampling:

a) Number of samples needed primarily depends upon structural content rather than its bandwidth.

b) Uses non linear recovery algorithm which is based on convex optimization

c) Can be used if the signal is sparse.

Example:

Original signal consisting of length 256 and 16 complex sinusoids

Acc to Shannon\Nyquist theory we need at least 256

samples in time domain

(In frequency domain)

(In time domain)(In time domain)

(figures from

http://www.acm.caltech.edu/l1magic/examples.html)

suppose we have only 80 samples. The observed 80 samples are observed in red color.

It’s known that the signal is sparse. Therefore, We choose the one who’s DFT has

minimum L1 norm; that is, the sum of the magnitudes of the Fourier transform is the

smallest. In doing this, we are able to recover the signal exactly!

In time domain In frequency domain

(figures fromhttp://www.acm.caltech.edu/l1magic/

examples.html)

Compressive sampling for Images

Original image 25K samples

Sampling in spatial domain

Real part of fft

Imaginary part of fft

Steps involved:

• The sampled image is formulated as a convex optimization problem. The given samples act as constraints to solve the problem with the objective of minimizing the total variance of the image. Where, total variance is defined as:

and

TV = ((9-10)2+(10-10)2)1/2

(figure from matlab)

Recovered images:

Original image 256 X 256

Recovered from 25k Spatial sampling

Recovered from 25k frequency sampling

Convex optimizationA mathematical optimization problem has the following form: Minimize

Subject to

= Optimization variable

= Objective function

= Inequality constraints

= Limits or bounds

is the optimal solution of the problem if it has the smallest objective value that satisfy the constraints for any z with

We have

The objective function and the constraint functions are convex

Solving convex optimization problem

• Effectiveness of this algorithm• Interior point method is used for problems having both equality and inequality constraints.

Minimize

Subject to

are convex and twice continuous and differentiable functions • First step:

Minimize

Subject to

Where is the indicator fn for non positive reals

• Approximation of indicator function

t = 0.5

t = 2

t>0 is a parameter that sets the accuracy of approx.Now the problem reduces

to:Minimize

Subject to

is called the logarithmic barrier for the problem

• Trade off between the value of t

a) Quality of the approximation improves as parameter t growsb) When t is large, the fn is difficult to minimize by Newton’s method.

Figure from (http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf)

Convex optimization problems

a) For signal recovery we use:

Subject to

finds the smallest norm that explains the observations b.

b) For image recovery we use:

Subject to

If there exists a with sufficiently few edges (i.e. is non zero for only a small number of indices ) then, will exactly recover .

Newton iteration method

around a point z is given by

where = gradient

= Hessian matrix

If minimizes subject to Then,

The quadratic approximation of the functional

Now, with in hand the step length is chosen such that:

a)

= user specified parameter (=0.01 for these implementations)

We start with s=1 and decrease by multiples of beta (= 0.5)

b) The function has decreased sufficiently

Contd…

A log barrier algorithm for SOCPs

The standard log-barrier method transforms a general SOCP problem to:

Subject toHere are the inequality functions which is either linear:Or, second order cone:

• which implies that as increases • It can be mathematically shown that: = duality

gap

• is the starting point for the K+1 iteration

• Approach is to minimize each sub problem by Newton iteration method.

Complete implementation• Inputs: A feasible starting point , A tolerance

and parameters μ and an initial

• Solve via Newton method

• Use as the initial point for Kth iteration • If ; terminate and return • Else set and go to step 2

Barrier iterations =

Condition for exact recovery Let be discrete signal supported on un known

set, and choose of size randomly. For a given

accuracy parameter M, if

then with probability at least the minimizer to the problem is unique and equal to

The value of the is given as and is valid for

Where = number of spikes in signal

= number of observed frequencies of subset

a) If we recover f perfectly about 80% of the time

b) If the recovery rate is practically 100%

Where = number of spikes in signal = number of observed frequencies of subset

Figures from (http://www.acm.caltech.edu/l1magic/downloads/papers/CandesRombergTao_revisedNov2005.pdf)

Contd…

Exact signal recovery a) N = 512 , T = 20 (number of spikes in time domain), K = 120 (number of samples in frequency domain)

Original signal

Recovered signal

Difference signal of O(10-

5)

b) N = 1024 , T = 50 (number of spikes in time domain), K= 225 (number of samples in frequency domain)

Original signal

Recovered signal

Difference signal of O(10-

5)

Contd…

Mathematical formulationThe equality constrained TV minimization problem:

Subject tocan be written as the SOCP

Where the inequality functions are defined as

Exact image recoverya) Sampled along 22 radial lines and 512 samples along

each line in frequency domain.

Original ImageFourier coefficients are sampled along 22 radial lines

Recovered Image

b) The original size of image is 256 X 256 and the number of samples taken is 25000.

Original ImageRecovered Image

Contd…

Recovery of images signals when number of samples is not on the order of BlogN

For the image:

B is of the order of 40000 and N is of the order of 65000.

So BlogN comes out to be 192000 samples.

For this image the motive is to recover a stable image with a minimal amount error.

a) Sampling in frequency domain

Original image 30k samples 25k samples

20k samples 15k samples 10k samples

b) Sampling in spatial domain

Original image Original image Original image

25k samples 30k samples 30k samples

Applications In medical imaging Exact recovery of images and signals from

incomplete data. Conversion of analog signals to digital signals Compression of images and signals