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Deconvolution, Deblurring and Restoration

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Kalman filter enables prediction, filtering, and smoothing. Kalman Filter State-Space Process Eq. Observation Eq. Innovation process: ... – PowerPoint PPT presentation

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Title: Deconvolution, Deblurring and Restoration


1
Deconvolution, Deblurring andRestoration
  • T-61.182, Biomedical Image Analysis
  • Seminar Presentation 14.4.2005
  • Seppo Mattila Mika Pollari

2
Overview (1/2)
  • Linear space-invariant (LSI) restoration filters
  • - Inverse filtering
  • - Power spectrum equalization
  • - Wiener filter
  • - Constrained least-squares restoration
  • - Metz filter
  • Blind Deblurring

3
Overview (2/2)
  • Homomorphic Deconvolution
  • Space-variant restoration
  • Sectioned image restoration
  • Adaptive-neighbourhood deblurring
  • The Kalman filter
  • Applications
  • - Medical
  • - Astronomical

4
Introduction
  • Find the best possible estimate of the original
    unknown image from the degraded image.
  • One typical degradation process has a form

5
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.

6
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

7
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)
8
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
9
Power Spectrum Equalization (PSE)
  • Want to find linear transform L such that
  • Power spectral density (PSD) FT(Autocorrelation
    function)

i.e.
. . .
10
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

11
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.

12
Comparison of Inverse Filter, PSE, and Wiener
Filter
13
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

. . .
14
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.

15
Motion Deblurring Simple Model
  • Assume simple in plane movement during the
    exposure
  • Either PSF or MTF is needed for restoration

16
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.

17
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

18
Extension to PSE Cont...
19
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.

20
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

21
Examples of Iterative Blind Deblurring
22
Homomorphic deconvolution
  • Start from
  • Convert convolution operation to addition
  • Complex cepstrum
  • Complex cepstra related
  • Practical application, however, not simple...

23
Steps involved in deconvolution using complex
cepstrum
24
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)

25
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).

26
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.
27
AND segmentation
28
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
29
Comparison of Sectioned and AND-technique
30
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.

31
Kalman Filter State-Space
  • Process Eq.
  • Observation Eq.
  • Innovation process

32
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

33
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

34
Wiener Filter Restoration of Digital Radiography
35
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

36
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)
37
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

38
Astro-examples
Observed PSF
Inverse filter Richardson-Lucy
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