Deconvolution, Deblurring and Restoration

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Deconvolution, Deblurring andRestoration

• T-61.182, Biomedical Image Analysis
• Seminar Presentation 14.4.2005
• Seppo Mattila Mika Pollari

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Overview (1/2)

• Linear space-invariant (LSI) restoration filters
• - Inverse filtering
• - Power spectrum equalization
• - Wiener filter
• - Constrained least-squares restoration
• - Metz filter
• Blind Deblurring

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Overview (2/2)

• Homomorphic Deconvolution
• Space-variant restoration
• Sectioned image restoration
• Adaptive-neighbourhood deblurring
• The Kalman filter
• Applications
• - Medical
• - Astronomical

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Introduction

• Find the best possible estimate of the original
  unknown image from the degraded image.
• One typical degradation process has a form

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Image Restoration General

• One has to have some a priori knowledge about the
  degragation process.
• Usually one needs 1) model for degragation, some
  information from 2) original image and 3) noise.
• Note! Eventhough one doesnt know the original
  image some information such as power spectral
  density (PSD) and autocorreletion function (ACF)
  are easy to model.

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Linear-Space Invariant (LSI) Restoration Filters

• Assume linear and shift-invariant degrading
  process
• Random noise statistically indep.
  of image-generating process
• Possible to design LSI filters to
  restore the image
  

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Inverse Filtering

• Consider degrading process in matrix form
• Given g and h, estimate f by minimising the
  squared error between observed image (g) and
  
• where and are approximations of f and
  g
• Set derivative of ?2 to zero

(if noise present)
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Inverse Filtering Examples

• Works fine if no noise but...
• H(u,v) usually low-pass function.
• N(u,v) uniform over whole spectrum.
• High-freq. Noise amplified!!

0.4x
0.2x
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Power Spectrum Equalization (PSE)

• Want to find linear transform L such that
• Power spectral density (PSD) FT(Autocorrelation
  function)

i.e.
. . .
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The Wiener Filter (1/2)

• Degradation model
• Assumtions Image and noise are
  second-order-stationary random processes and they
  are statistically independent
• Optimal mean-square error (MSE) criterion Find
  Wiener filter (L) which minimize MSE

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The Wiener Filter (2/2)

• Minimizing the criterion we end up to optimal
  Wiener filter.
• The Wiener filter depends on the autocorrelation
  function (ACF) of the image and noise (This is no
  problem).
• In general ACFs are easy to estimate.

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Comparison of Inverse Filter, PSE, and Wiener
Filter
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Constrained Least-squares Restoration

• Minimise with constraint
  where L is a linear filter
  operator
• Similar to Wiener filter but does not require the
  PSDs of the image and noise to be known
• The mean and variance of the noise needed to set
  optimally. If 0 inverse filter

. . .
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The Metz Filter

• Modification to inverse filter.
• Supress the high frequency noise instead of
  amplyfying it.
• Select factor so that mean-square error (MSE)
  between ideal and filtered image is minimized.

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Motion Deblurring Simple Model

• Assume simple in plane movement during the
  exposure
• Either PSF or MTF is needed for restoration

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Blind Deblurring

• Definition of deblurring.
• Blind deblurring models of PSF and noise are not
  known cannot be estimated separately.
• Degragated image (in spectral domain) consist
  some information of PSF and noise but in combined
  form.

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Method 1 Extension to PSE

• Broke image to M x M size segment where M is
  larger than dimensions of PSF then
• Average of PSD of these segments tend toward the
  true signal and noise PSD
• This is combined information of blur function
  and noise which is needed in PSE
• Finaly, only PSD of image is needed

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Extension to PSE Cont...
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Method 2 Iterative Blind Deblurring

• Assumptation MTF of PSF has zero phase.
• Idea blur function affects in PSD but phase
  information preserves original information from
  edges.

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Iterative Blind Deblurring Cont...

• Fourier transform of restored image is
• Note that smoothing operator S has small effect
  to smooth functions (PSF). This leads to
  iterative update rule

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Examples of Iterative Blind Deblurring
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Homomorphic deconvolution

• Start from
• Convert convolution operation to addition
• Complex cepstrum
• Complex cepstra related
• Practical application, however, not simple...

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Steps involved in deconvolution using complex
cepstrum
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Space-variant Image Restoration

• So far we have assumed that images are spatially
  (and temporaly) stationary
• This is (generally) not true at the best images
  are locally stationary
• Techniques to overcome this problem
• Sectioned image restoration
• Adaptive neighbourhood deblurring
• The Kalman filter (the most elegant approach)

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Sectioned Image Restoration

• Divide image into small P x P rectangular,
  presumably stationary segments.
• Centre each segment in a region, and pad the
  surrounding with the mean value.
• For each segment apply separately image
  restoration (e.g. PSE or wiener).

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Adaptive-neighborhood deblurring (AND)

• Grow adaptive neighborhood regions
• Apply 2D Hamming window to each region
• Estimate the noise spectrum

Pixel locations within the region
Centered on (m,n)
A is a freq. domain scale factor that depends on
the spectral characterisics of the region grown
etc.
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AND segmentation
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Adaptive-neighborhood deblurring (AND) Cont

• Frequency-domain estimate of the uncorrupted
  adaptive-neighborhood region
• Obtain estimate for deblurred adaptive
  neigborhood region m,n(p,q) by FT-1
• Run for every pixel in the input image g(x,y)

Deblurred image
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Comparison of Sectioned and AND-technique
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Kalman Filter

• Kalman filter is a set of mathematical equations.
  
• Filter provides recursive way to estimate the
  state of the process (in non-stationary
  environment), so that mean of squared errors is
  minimized (MMSE).
• Kalman filter enables prediction, filtering, and
  smoothing.

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Kalman Filter State-Space

• Process Eq.
• Observation Eq.
• Innovation process

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Kalman Filter in a Nutshell (1/2)

• Data observations are available
• System parameters are known
• a(n1,n), h(n), and the ACF of driving and
  observation noise
• Initial conditions
• Recursion

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Kalman Filter in a Nutshell (2/2)

1. Compute the Kalman gain K(n)
2. Obtain the innovation process
3. Update
4. Compute the ACF of filtered state error
5. Compute the ACF of predicted state error

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Wiener Filter Restoration of Digital Radiography
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Astronomical applications

• Images blurred by atmospheric turbulence
• Observing above the atmosphere very expensive
  (HST)
• Improve the ground-based resolution by
• Suitable sites for the observatory (_at_ 4 km
  height)
• Real time Adaptive optics correction
• Deconvolution

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Point Spread Function (PSF) in Astronomy
Iobserved Ireal ?PSF

• Easy to measure and model from several stars
  usually present in astro-images
• Determines the spatial resolution of an image
• Commonly used for image matching and
  deconvolution

Ideal PSF if no atmosphere FWHM 1.22x?/D
Atmospheric turbulence broadens the PSF
Gaussian PSF with FWHM 1"
lt 0.1" (8m telescope)
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Richardson-Lucy deconvolution

• Used in both fields astronomy medical imaging
• Start from Bayes's theorem, end up with
• Takes into account statistical fluctuations in
  the signal, therefore can reconstruct noisy
  images!
• In astronomy the PSF is known accurately
• From an initial guess f0(x) iterate until converge

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Astro-examples
Observed PSF
Inverse filter Richardson-Lucy