Transcript Exact signal and image reconstruction from incomplete

Exact or stable image\signal
reconstruction from incomplete
information
Project guide:
Dr. Pradeep Sen
UNM (Abq)
Submitted by:
Nitesh Agarwal
IIT Roorkee (India)
Images and Signals
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
Digital
Images
(figures from matlab)
Digital Signals (Discrete)
After
Sampling
Analog
Signal
(figures from
Wikipedia)
Digital
Signal
Compressive sampling
Compressive sampling for signals
• Nyquist Theory:
Where B = Bandwidth
= sampling frequency
• Compressive sampling:
a)
b)
c)
Number of samples needed primarily depends upon structural content
rather than its bandwidth.
Uses non linear recovery algorithm which is based on convex
optimization
Can be used if the signal is sparse.
Example:
Original signal consisting of
length 256 and 16 complex
sinusoids
(In frequency
domain)
Acc to Shannon\Nyquist theory
we need at least 256 samples
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
(figures from
http://www.acm.caltech.edu/l1magic/example
s.html)
In frequency domain

Compressive sampling for Images
Sampling in spatial domain
Original image
Real part of fft
25K samples
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:
TV = ((9-10)2+(10-10)2)1/2
and
(figure from matlab)
Recovered images:
Original image 256 X 256
Recovered from 25k Spatial
sampling
Recovered from 25k frequency
sampling
Convex optimization
A 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
t = 0.5
t=2
• Approximation of indicator function
t>0 is a parameter that sets the accuracy of approx.
Now the problem reduces to:
Minimize
Figure from
(http://www.stanford.edu/~bo
yd/cvxbook/bv_cvxbook.pdf)
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 grows
b) When t is large, the fn
is difficult to minimize by
Newton’s method.
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
The quadratic approximation of the functional
around a point z is given by
where
If
= gradient
minimizes
= Hessian matrix
subject to
Then,
Now, with
a)
in hand the step length is chosen such that:
Contd…
b) The function has decreased sufficiently
= user specified parameter (=0.01 for these implementations)
We start with s=1 and decrease by multiples of beta
(= 0.5)
A log barrier algorithm for
SOCPs
The standard log-barrier method transforms a general SOCP problem to:
Subject to
Here
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
• Approach is to minimize each sub problem by
Newton iteration method.
•
is the starting point for the K+1 iteration
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
problem is unique and equal to
The value of the
is given as
for
Where
the minimizer to the
and is valid
= number of spikes in signal
= number of observed frequencies of subset
Contd…
a) If
we recover f perfectly about 80% of the time
b) If
the recovery rate is practically 100%
Figures from
(http://www.acm.caltech.edu/l
1magic/downloads/papers/Can
desRombergTao_revisedNov20
05.pdf)
Where
= number of spikes in signal
= number of observed frequencies of subset
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)
Contd…

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)
Mathematical formulation
The equality constrained TV minimization problem:
Subject to
can be written as the SOCP
Where the inequality functions are defined as
Exact image recovery
a) Sampled along 22 radial lines and 512 samples along each line
in frequency domain.
Original Image
Fourier coefficients are
sampled along 22 radial
lines
Recovered Image
Contd…
b) The original size of image is 256 X 256 and the number of
samples taken is 25000.
Original Image
Recovered Image
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
20k samples
30k samples
15k samples
25k samples
10k samples
b) Sampling in spatial domain
Original image
25k samples
Original image
30k samples
Original image
30k samples
Applications
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In medical imaging
Exact recovery of images and signals from
incomplete data.
Conversion of analog signals to digital signals
Compression of images and signals