Stopping Criteria Image Restoration

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Stopping Criteria Image Restoration

• Alfonso Limon
• Claremont Graduate University

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Outline

• Image Restoration
• Noise and Image Details
• Observations using Wavelets
• Some Preliminary Results
• Future Work

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Image Restoration
The goal of image restoration is to improve a
degrade image in some predefined sense.
Schematically this process can be visualized as
where f is the original image, g is a
degraded/noisy version of the original image and
f is a restored version.
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Image Degradation / Restoration
Image restoration removes a known degradation.
If the degradation is linear and
spatially-invariant
where F - original image, H - degradation, N -
additive noise and G - recorded image. Given H,
an estimate of the original image is
Notice that if H 0, the noise will be
amplified.
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Degradation Models
Image degradation can occur for many reasons,
some typical degradation models are
Motion Blur due to camera panning or subject
moving quickly. Atmospheric Blur long
exposure Uniform 2D Blur Out-of-Focus Blur
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Noise Models
Most noise models assume the noise is some known
probability density function. The density
function is chosen based on the underlining
physics.
Gaussian poor illumination. Rayleigh range
image Salt and Pepper faulty switch during
imaging Gamma or Exp laser imaging
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Restoration Model
Many restoration filters can be applied to
recover the degraded image Rudin-Osher-Fatemi,
Wiener, Inverse FFT with threshold, etc. But all
require a stopping criteria based on a noise
measure.
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Stopping Criteria
As was illustrated by S. Osher (on Monday) and M.
Bertero (in the tutorials) whether the
restoration filter is direct or iterative, it
must be stopped before the image begins to
degrade.
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Stopping the Restoration
Noise in the data is amplified as the number of
iterations increase, while the blur decreases as
the number of iteration increase.
This is the prototypical behavior which was
illustrated by Professor Heinz Engl during the
inverse problem tutorials.
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Discrepancy Principle
The discrepancy principle gives a natural
stopping criteria, namely to stop the restoration
process when the residual is of the same order as
the noise.
The drawback to this approach for image
processing is that the discrepancy principle
tends to recover an image that is still too
blurred.
Question How many more times can we iterate
passed the threshold given by the discrepancy
principle and still get an image that is not
predominately noisy.
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Objective
The objective is to highlight fine details in the
image which were suppressed by the blur.
The problem is that enhancement of fine detail
(or edges) is equivalent to enhancement of noise.
The challenge is to enhance details as much as
possible before the noise overtakes the image
details.
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Wavelets and Details
Wavelets provide a nature way to separate signal
scales, especially for 1D signals.
Taking the fine scale wavelet coefficients of the
diagonal of the image provides a measure of the
noise and image details.
The next set of sides show the fine scale
coefficients for a reconstructed version of the
quarter image (to the right) at various
iterations steps.
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Fine Wavelet Coefficients (1)
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Fine Wavelet Coefficients (2)
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Fine Wavelet Coefficients (3)
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Fine Wavelet Coefficients (4)
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Fine Wavelet Coefficients (5)
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Fine Wavelet Coefficients (6)
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Observations
The discrepancy principle gives a good starting
guess for the image details as the noise level is
low.
Noise will eventually overtakes details,
therefore tracking changes in the fine wavelet
coefficient gives a possible way to differential
between image details and noise.
A natural stopping criteria is to stop iterating
when the fine wavelet coefficients that have been
identified as details are of the same order as
the fine wavelet coefficients that have been
identified as noise, i.e., the discrepancy
principle applied to fine wavelet coefficients.
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Stopping Algorithm

1. Choose your initial image according to the
   discrepancy principle.
2. Calculate fine wavelet coefficients corresponding
   to the images diagonal.
3. Calculate the mean of the fine wavelet
   coefficients.
4. Assign the fine wavelet coefficients which are
   larger than the mean to the set details.
5. If there is a previous details set, take the
   intersection of it with the new details set.
6. Repeat steps 2 through 5 until the mean of the
   details set is equal to the mean of the noise
   set noise is the set of wavelet coefficients not
   in details.

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Preliminary Results
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Preliminary Results
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Preliminary Results
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Future Work

• Define an energy measure to compare wavelet
  coefficients instead of using the mean value.
• Apply a moving class to the wavelet coefficients
  to take into account the difference in behavior
  between details and noise.
• Apply different deblurring/noise models to test
  that the stopping criteria is not sensitive to
  the deblurring/noise process.
• Expend the method from a global to local stopping
  criteria.

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Special Thanks to the Organizing Committee and
IPAM Staff.
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References
M. Bertero, Image Deconvolution, Inverse
Problems Computational Methods and Emerging
Applications, IPAM 2003. (http//www.ipam.ucla.edu
/publications/invws1/invws1_3804.pdf) P.
Blomgren and T. F. Chan, Modular Solver for
Constraint Image Restoration Problems Using the
Discrepancy Principle, Numerical Linear Algebra
with Applications,Vol 9, issue 5, pp 347-358.
David Donoho, De-Noising by Soft-Thresholding
(http//www-stat.stanford.edu/donoho/reports.htm
l) H. W. Engl, Inverse Problems 1, Inverse
Problems Computational Methods and Emerging
Applications Tutorials, IPAM 2003.
(http//www.ipam.ucla.edu/publications/invtut/invt
ut_hengl1.pdf) R. Gonzalez and R. Woods, Digital
Image Processing, second edition, Prentice Hall,
Inc., 2002. K. Lee, J. Nagy and L. Perrone,
Iterative Methods for Image Restoration A Matlab
Object Oriented Approach. Iterative Methods for
Image Restoration, May 15, 2002. R. Ramlau,
Morozovs Discrepancy Principle for Tikhonov
regularization of nonlinear operators, Report
01-08, Berichte aus der Technomathematik, July
2001.