Image Restoration and Denoising

1
Image Restoration and Denoising

• Chapter Seven Part II

2
Image Restoration Techniques

• Inverse of degradation process
• Depending on the knowledge of degradation, it can
  be classified into

Deterministic Random
If prior knowledge about degradation is known If not known
Linear Non-linear
Restore the image by a filter e.g. Inverse Filtering Drawback ringing artifacts near edges Nonlinear function is used Ringing artifacts is reduced
3
Image Restoration Model
f(x,y)
? (x, y)
4
Image Restoration Model

• In this model,
• f(x,y) ? input image
• h(x,y) ? degradation
• f'(x,y) ? restored image
• ?(x,y) ? additive noise
• g(x,y) ? degraded image

5
Image Restoration Model

• In spatial domain,
• g(x,y) f(x,y) h(x,y)
• and in frequency domain
• G(k,l) F(k,l). H(k,l)
• Where G,F and H are fourier transform of g,f and h

6
Linear Restoration Technique

• They are quick and simple
• But limited capabilities
• It includes
• Inverse Filter
• Pseudo Inverse Filter
• Wiener Filter
• Constrained Least Square Filter

7
Inverse Filtering

• If we know exact PSF and ignore noise effect,
  this approach can be used.
• In practice PSF is unknown and degradation is
  affected by noise and hence this approach is not
  perfect.
• Advantage - Simple

8
Inverse Filtering

• From image restoration model
• For simplicity, the co-ordinate of the image are
  ignored so that the above equation becomes
• Then the error function becomes

9
Inverse Filtering

• We wish to ignore ? and use to approximate
  under least square sense. Then the error function
  is given as
• To find the minimum of , the above
  equation is differentiated wrt and equating
  it to zero

10
Inverse Filtering

• Solving for , we get
• Taking fourier transform on both sides we get
• The restored image in spatial domain is obtained
  by taking Inverse Fourier Transform as

11
Inverse Filtering
12
Inverse Filtering

• Advantages
• It requires only blur PSF
• It gives perfect reconstruction in the absence of
  noise
• Drawbacks
• It is not always possible to obtain an inverse
  (singular matrices)
• If noise is present, inverse filter amplifies
  noise. (better option is wiener filter)

13
Inverse Filtering with Noise
14
Pseudo-Inverse Filtering

• For an inverse filter,
• Here H(k,l) represents the spectrum of the PSF.
• The division of H(k,l) leads to large
  amplification at high frequencies and thus noise
  dominates over image

15
Pseudo-Inverse Filtering

• To avoid this problem, a pseudo-inverse filter is
  defined as
• The value of e affects the restored image
• With no clear objective selection of e, the
  restored images are generally noisy and not
  suitable for further analysis

16
Pseudo-Inverse Filtering
17
Pseudo-Inverse Filtering with e 0.2
18
Pseudo-Inverse Filtering with e 0.02
19
Pseudo-Inverse Filtering with e 0.002
20
Pseudo-Inverse Filtering with e 0
21
SVD Approach for Pseudo-Inverse Filtering

• SVD stands for Singular Value Decomposition
• Using SVD any matrix can be decomposed into a
  series of eigen matrices
• From image restoration model we have
• The blur matrix is represented by H
• H is decomposed into eigen matrices as
• where U and V are unitary and D is diagonal
  matrix

22
SVD Approach for Pseudo-Inverse Filtering

• Then the pseudo inverse of H is given by
• The generalized inverse is the estimate is the
  result of multiplying H with g
• R indicates the rank of the matrix
• The resulting sequence estimation formula is
  given as

23
SVD Approach for Pseudo-Inverse Filtering

• Advantage
• Effective with noise amplification problem as we
  can interactively terminate the restoration
• Computationally efficient if noise is space
  invariant
• Disadvantage
• Presence of noise can lead to numerical
  instability

24
Wiener Filter

• The objective is to minimize the mean sqaure
  error
• It has the capability of handling both the
  degradation function and noise
• From the restoration model, the error between
  input image f(m,n) and the estimated image
  is given by

25
Wiener Filter

• The square error is given by
• The mean square error is given by

26
Wiener Filter

• The objective of the Wiener filter is top
  minimize
• Given a system we have
• yhx v
• h-blur function
• x - original image
• y observed image (degraded image)
• v additive noise

27
Wiener Filter

• The goal is to obtain g such that
• is the restored image that minimizes mean
  square error
• The deconvolution provides such a g(t)

28
Wiener Filter

• The filter is described in frequency domain as
• G and H are fourier transform of g and h
• S mean power of spectral density of x
• N mean power of spectral density of v
• - complex conjugate

29
Wiener Filter

• Drawback It requires prior knowledge of power
  spectral density of image which is unavailable in
  practice