1. 9/11/2018
ENEE631 Spring’09 Lecture 7 (2/16/2009)
Image Restoration
Lecture:
xxx – xx slides, about xxx min
Spring ’09 Instructor: Min Wu Electrical and Computer Engineering Department,
University of Maryland, College Park
bb.eng.umd.edu (select ENEE631 S’09)

2. UMCP ENEE631 Slides (created by M.Wu © 2004)
9/11/2018
Overview
Last Time:
Image sharpening and edge detection
Basics on 2-D random signal (random field)
Relation and extension from 1-D random process
(1) Sequences of random variables & joint distributions (2) First two moment functions and their properties (3) Wide-sense stationarity (4) Unique to 2-D case: separable and isotropic covariance function (5) Power spectral density and properties
Today:
Continue on basics on 2-D random field
Image Restoration: inverse filtering, Wiener filter, …
Assignment #2: due Wednesday Feb. 25, 2009
Suggested readings related to implementation issues on 2-D FT and filtering
(Gonzalez 3/e Section 4.6)
UMCP ENEE631 Slides (created by M.Wu © 2004)
Implementation issues: in 3rd edition – 4.6 and some in 4.11; 2nd edition has a collection of implementation discussions in 4.6
M. Wu: ENEE631 Digital Image Processing (Spring'09)

3. Review: Discrete Random Field
9/11/2018
Review: Discrete Random Field
We call a 2-D sequence discrete random field if each of its elements is a random variable
when the random field represents an ensemble of images, we often call it a random image
Mean and Covariance of a complex random field
E[u(m,n)] = (m,n)
Cov[u(m,n), u(m’,n’)] = E[ (u(m,n) – (m,n)) (u(m’,n’) – (m’,n’))* ]
= ru( m, n; m’, n’)
For zero-mean random field, autocorrelation function = cov. function
Wide-sense stationary (or wide-sense homogeneity)
(m,n) =  = constant
ru( m, n; m’, n’) = ru( m – m’, n – n’; 0, 0) = r( m – m’, n – n’ )
also called shift invariant, spatial invariant in some literature
UMCP ENEE631 Slides (created by M.Wu © 2004)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

4. Special Random Fields of Interests
9/11/2018
Special Random Fields of Interests
White noise field
A stationary random field
Any two elements at different locations x(m1,n1) and x(m2,n2) are mutually uncorrelated
rx( m, n ) = x2 ( m, n )
Gaussian random field
Every segment defined on an arbitrary finite grid is Gaussian
i.e. every finite segment of u(m,n) when mapped into a vector have a joint Gaussian p.d.f. of
UMCP ENEE631 Slides (created by M.Wu © 2004)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

5. Properties of Covariance for Random Field
9/11/2018
Properties of Covariance for Random Field
[Recall similar properties of covariance function for 1-D random process]
Symmetry
ru( m, n; m’, n’) = ru*( m’, n’; m, n)
For stationary random field: r( m, n ) = r*( -m, -n )
For stationary real random field: r( m, n ) = r( -m, -n )
Note in general ru( m, n; m’, n’)  ru( m’, n; m, n’)  ru( m’, n; m, n’)
Non-negativity
mnm’n’ x(m, n) ru( m, n; m’, n’) x*(m’, n’)  0
~ Recall for 1-D case, correlation matrix is non-negative definite:
xH R x  0 for all x .
Recall: ru ( m, n; m’, n’) =
E[ (u(m,n) – (m,n)) (u(m’,n’) – (m’,n’))* ]
UMCP ENEE631 Slides (created by M.Wu © 2004)
E[ ( u(m,n) – mu(m,n) ) ( u(m’,n’) – mu(m’.n’))* ] = { E [ ( u(m’,n’) – mu(m’,n’) ) ( u(m,n) – mu(m.n))* ] }*
Non-negativity: in 1-D stationary case, we write r( ) as a correlation matrix and the non-negativity is expressed as the non-negative definiteness of the correlation matrix; and the conjugate symmetry corresponds to the matrix’s Hermitian property.
=> **? Think: physical meaning of non-negativity? Non-negativity can be viewed from Fourier domain that “power” (i.e. p.s.d. func) is non-negative valued
M. Wu: ENEE631 Digital Image Processing (Spring'09)

6. Separable Covariance Functions
9/11/2018
Separable Covariance Functions
Separability
If the covariance function of a random field can be expressed as a product of covariance functions of 1-D sequences
r( m, n; m’, n’) = r1( m, m’) r2( n, n’) ~ Nonstationary case
r( m, n ) = r1( m ) r2( n ) ~ Stationary case
Example:
A separable stationary covariance function often used in image proc for its simplicity
r(m, n) = 2 1|m| 2|n| , |1|<1 and |2|<1
2 represents the variance of the random field; 1 and 2 are the one-step correlations in the m and n directions
UMCP ENEE631 Slides (created by M.Wu © 2004)
Rho^{|m|} represents the autocorrelation function of a zero-mean AR(1) process: x[n] = rho x[n-1] + v[n], v[n] is a white noise proc
M. Wu: ENEE631 Digital Image Processing (Spring'09)

7. Isotropic Covariance Functions
9/11/2018
Isotropic Covariance Functions
Isotropic / circularly symmetric
i.e. the covariance function only changes with respect to the radius (the distance to the origin), and isn’t affected by the angle
Example
A nonseparable exponential function often used as a more realistic model of the covariance function for images
When a1= a2 = a2 , this becomes isotropic: r(m, n) = 2 d
As a function of the Euclidean distance of d = ( m 2 + n 2 ) 1/2
 = exp(-|a|), i.e. correlation is decaying as distance increases
UMCP ENEE631 Slides (created by M.Wu © 2004)
i.e. the covariance has an exponential decay
M. Wu: ENEE631 Digital Image Processing (Spring'09)

8. Estimating the Mean and Covariance Function
9/11/2018
Estimating the Mean and Covariance Function
Approximate the ensemble average with sample average
“ergodicity”
Example: for an M x N real-valued image x(m, n)
UMCP ENEE631 Slides (created by M.Wu © 2004)
Note that similar to the 1-D case in EE624, the cov estimates are biased here in order to achieve smaller variance in estimation and to avoid the possible negativity by unbiased estimate.
Ergodicity: in addition to stationarity, the correlation (or cov) should be decaying fast enough in order to ensure the sample average reflects the ensemble average. O.w. for example, if all samples are perfectly correlated (thus giving a random “constant” img), we only see one identical value instance from all pixels of one instantiation of the process.
Note: similar to the 1-D case, the cov estimates here are biased, in order to achieve smaller variance in estimation and to avoid the possible negative definiteness by unbiased estimate
M. Wu: ENEE631 Digital Image Processing (Spring'09)

9. Spectral Density Function
9/11/2018
Spectral Density Function
The Spectral density function (SDF) is defined as the Fourier transform of the covariance function rx
Also known as the power spectral density (p.s.d.)
( in some text, p.s.d. is defined as the FT of autocorrelation function )
Example: SDF of stationary white noise field with r(m,n)= 2 (m,n)
UMCP ENEE631 Slides (created by M.Wu © 2004)
SDF of stationary white noise field = constant (= sigma^2)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

10. Properties of Power Spectrum
9/11/2018
Properties of Power Spectrum
[Recall similar properties in 1-D random process]
SDF is real: S(1, 2) = S*(1, 2)
Follows the conjugate symmetry of the covariance function
r(m, n) = r *(-m, -n)
SDF is nonnegative: S(1, 2)  0 for 1,2
Follows the non-negativity property of covariance function
Intuition: “power” cannot be negative
SDF of the output from a LSI system w/ freq response H(1, 2)
Sy(1, 2) = | H(1, 2) |2 Sx(1, 2)
DSFT of cross correlation: Syx(1, 2) = H(1, 2) Sx(1, 2)
UMCP ENEE631 Slides (created by M.Wu © 2004)
Note: p.s.d. relation has |H|^2 instead of H because the correlation is defined with y and its conjugate y* at various lags. So filter h[ ] is twice convolved with proper conjugation, giving an overall |H|^2 in Fourier domain.
M. Wu: ENEE631 Digital Image Processing (Spring'09)

11. Z-Transform Expression of Power Spectrum
9/11/2018
Z-Transform Expression of Power Spectrum
The Z transform of ru
Known as the covariance generating function (CGF) or the ZT expression of the power spectrum
UMCP ENEE631 Slides (created by M.Wu © 2004)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

12. UMCP ENEE631 Slides (created by M.Wu © 2004)
9/11/2018
2-D Z-Transform
The 2-D Z-transform is defined by
The space represented by the complex variable pair (z1, z2) is 4-D
Unit surface
If Region of Convergence (ROC) include unit surface =>
Transfer function of 2-D discrete LSI system
UMCP ENEE631 Slides (created by M.Wu © 2004)
To discuss in later lectures. See Lim-Oppenheim adv topic book as reference.
M. Wu: ENEE631 Digital Image Processing (Spring'09)

13. UMCP ENEE631 Slides (created by M.Wu © 2004)
9/11/2018
Stability
Recall for 1-D LTI system
Stability condition in bounded-input bounded-output sense (BIBO) is that the impulse response h[n] is absolutely summable
=> ROC of H(z) includes the unit circle
H(z) for a causal and stable system should have all poles inside the unit circle
2-D Stable LSI system
Requires the 2-D impulse response is absolutely summable
=> i.e. ROC of H(z1, z2) must include the unit surface |z1|=1, |z2|=1
UMCP ENEE631 Slides (created by M.Wu © 2004)
For z on unit circle: |H(z)| <= sum_n{ |h[n]| |z^-n| } = sum_n{ |h[n]| }
For causal 1-D system, ROC is outside the outmost pole.
Evaluation of the stability condition for 2-D: check if the H(z1, z2) is finite on unit surface. Note 2-D polynomial is not always possible to be written in product of lower order polynomials.
M. Wu: ENEE631 Digital Image Processing (Spring'09)

14. UMCP ENEE631 Slides (created by M.Wu © 2004)
9/11/2018
Image Restoration
UMCP ENEE631 Slides (created by M.Wu © 2004)
Ref: Jain’s boo ; Ray Liu’s EE624 lecture notes on Multi-dim chapter
M. Wu: ENEE631 Digital Image Processing (Spring'09)

15. UMCP ENEE631 Slides (created by M.Wu © 2001)
9/11/2018
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Matlab ImageToolbox Documentation pp12-4
M. Wu: ENEE631 Digital Image Processing (Spring'09)

16. Imperfectness in Image Capturing
9/11/2018
Imperfectness in Image Capturing H
u(n1, n2 )
v(n1, n2 )
N(n1, n2 )
Blurring ~ linear spatial-invariant filter model w/ additive noise
Impulse response h(n1, n2) & H(1, 2)
Point Spread Function (PSF) ~ non-negative I/O
[No blur] h(n1, n2) = (n1, n2)
[Linear translational motion blur]
local average along motion direction
[Uniform out-of-focus blur]
local average in a circular neighborhood
Atomspheric turbulence blur, etc.
UMCP ENEE631 Slides (created by M.Wu © 2001)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

17. Fourier Transform of PSF for Common Distortions
9/11/2018
Fourier Transform of PSF for Common Distortions
From Bovik’s Handbook Sec.3.5 Fig.2&3
UMCP ENEE631 Slides (created by M.Wu © 2001)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

18. Undo Linear Spatial-Invariant Distortion
9/11/2018
Undo Linear Spatial-Invariant Distortion
Assume noiseless and PSF of distortion h(n1, n2) is known
Restoration by Deconvolution / Inverse-Filtering
Often used for deblurring
Want to find g(n1, n2) satisfies h(n1, n2)  g(n1, n2) = (n1, n2)
 h(k1, k2) g(n1 -k1, n2-k2) = (n1, n2) for all n1, n2
Easy to decouple & solve in spectrum domain
Convolution  Multiplication
H(1, 2) G(1, 2) = 1
Interpretation: choose G to compensate distortions from H H
u(n1, n2)
v(n1, n2)
(n1, n2)=0 G
u’(n1, n2)
w(n1, n2)
UMCP ENEE631 Slides (created by M.Wu © 2001)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

19. Problems With Inverse Filtering Under Noise
9/11/2018
Problems With Inverse Filtering Under Noise
Zeros in H(1, 2)
Interpretation: distortion by H removes all info. in those freq.
Inverse filter tries to “compensate” by assigning infinite gains
Amplifies noise:
W(1, 2) = H (1, 2) U (1, 2)
U’ (1, 2) = (W+N) / H  N/H if W=0
Solutions ~ Pseudo-inverse Filtering
Assign zero gain for G at spectrum nulls of H
Interpretation: not bother to make impossible compensations
UMCP ENEE631 Slides (created by M.Wu © 2001)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

20. Examples of Inverse & Pseudo-inverse Filtering
9/11/2018
Examples of Inverse & Pseudo-inverse Filtering
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Jain Fig.8.10
M. Wu: ENEE631 Digital Image Processing (Spring'09)

21. Handling Spectrum Nulls Via High-Freq Cut-off
9/11/2018
Handling Spectrum Nulls Via High-Freq Cut-off
UMCP ENEE631 Slides (created by M.Wu © 2004)
 Limit the restoration to lower frequency components to avoid amplifying noise at spectrum nulls esp. in high-frequency range
Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 5)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

22. Handling Noise in Deconvolution
9/11/2018
Handling Noise in Deconvolution
Inverse filtering is sensitive to noise
Does not explicitly model and handle noise
Try to balance between deblurring vs. noise suppression
Minimize MSE between the original and restored
e = E{ [ u(n1, n2) – u’(n1, n2) ] 2 }, where u’(n1, n2) is a func. of {v(m1, m2) }
Best estimate is conditional mean E[ u(n1 , n2) | all v(m1 , m2) ]
see EE621; but usually difficult to solve for general restoration (need conditional probability distribution, and estimation is nonlinear in general)
Get the best linear estimate instead  Wiener filtering
Consider the (desired) image and noise as random fields
Produce a linear estimate from the observed image to minimize MSE
UMCP ENEE631 Slides (created by M.Wu © 2001/2004) H
u(n1, n2)
v(n1, n2)
(n1, n2) G
u’(n1, n2)
w(n1, n2)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

23. EE630 Review: Wiener Filtering
9/11/2018
EE630 Review: Wiener Filtering
Want to know the actual values (of the current/interested realization) of a random process {d[n]}, but cannot directly observe it.
What we can observe is a process {x[n]} that is statistically related to d[n]. We want to estimate d[n] from x[n].
M. Wu: ENEE631 Digital Image Processing (Spring'09)

24. EE630 Review: Principle of Orthogonality
9/11/2018
EE630 Review: Principle of Orthogonality
“Orthogonal” in a statistical sense: i.e. the optimal error signal and each observation sample used in the filtering (and also their combinations) are statistically uncorrelated
plugging e[n] into the orthogonality principle leads to the normal equation.
M. Wu: ENEE631 Digital Image Processing (Spring'09)

25. UMCP ENEE631 Slides (created by M.Wu © 2001, 2007)H u v  G u’ w
9/11/2018
Wiener Filtering
Get the best linear estimate minimizing MSE
Assume: spatial-invariant restoration filter u’(n1, n2) = g (n1, n2) v(n1, n2) ;
wide-sense stationarity for original signal and noise;
noise is zero-mean and uncorrelated with original signal.
Solutions
Principle of orthogonality E{ [ u(n1, n2) – u’(n1, n2) ] v*(m1, m2) }=0
=> E[ u(n1,n2) v*(m1,m2) ] = E[ u’(n1,n2) v*(m1,m2) ] => Ru v (k,l) = Ru’ v(k,l)
i.e. the restored image should have similar stochastic properties as the original.
Find out expressions of the two cross-correlation functions:
Extend 1-D: for y(n1,n2) = x(n1,n2) + w(n1,n2) => Ruy (k,l) = Rux(k,l) + Ruw(k,l) ;
if x(n1,n2) and w(n1,n2) uncorrelated => Ryy (k,l) = Rx x(k,l) + Rw w(k,l) .
Ru’ v(k,l) = g(k,l)  Rvv(k,l) = g(k,l)  [ Rww(k,l) + R(k,l) ]
Ru v(k,l) = Ruw(k,l) + Ru(k,l) = h*(k,l)  Ruu(k,l) + 0
UMCP ENEE631 Slides (created by M.Wu © 2001, 2007)
u(m,n) => [ H ] => (+ noise) => v(m,n)
v = conv( h, u ) + noise; u’ = conv( g, v )
Note we have S_uv instead of S_vu in the solution, requiring the use of H* instead of H
M. Wu: ENEE631 Digital Image Processing (Spring'09)

26. Wiener Filter in Frequency-Domain Representation
9/11/2018
Wiener Filter in Frequency-Domain Representation
Ru v (k, l) = Ru’ v(k, l)
Ru’ v(k,l) = g(k,l)  Rvv(k,l) = g(k,l)  [ Rww(k,l) + R(k,l) ]
Ru v(k,l) = Ruw(k,l) + Ru(k,l) = h*(k,l)  Ruu(k,l) + 0
Take DFT to get representation in power spectrum density
UMCP ENEE631 Slides (created by M.Wu © 2007)
u(m,n) => [ H ] => (+ noise) => v(m,n)
v = conv( h, u ) + noise; u’ = conv( g, v )
Note we have S_uv instead of S_vu in the solution, requiring the use of H* instead of H H u v  G u’ w
M. Wu: ENEE631 Digital Image Processing (Spring'09)

27. More on Wiener Filtering
9/11/2018
More on Wiener Filtering
Balancing between two jobs for deblurring noisy image
HPF filter for de-blurring (undo H distortion)
LPF for suppressing noise
Noiseless case ~ S = 0 (inverse filter)
Wiener filter becomes pseudo-inverse filter for S  0
No-blur case ~ H = 1 (Wiener Smoothing Filter)
Zero-phase filter to attenuate noise according to SNR at each freq.
UMCP ENEE631 Slides (created by M.Wu © 2001)
In practice, we often approx SNR as a constant to walk around the need of estimating image p.s.d
Note the phase response of Wiener is the same as inverse filter 1 / H( ), i.e. not compensate phase distortion due to noise
M. Wu: ENEE631 Digital Image Processing (Spring'09)

28. UMCP ENEE631 Slides (created by M.Wu © 2001)
9/11/2018
Comparisons
UMCP ENEE631 Slides (created by M.Wu © 2001)
Wiener: sort of a band-pass effect to balance between LP noise smoothing and HP/sharpening inverse filter
From Jain Fig.8.11
M. Wu: ENEE631 Digital Image Processing (Spring'09)

29. Example: Wiener Filtering vs. Inverse Filtering
9/11/2018
Example: Wiener Filtering vs. Inverse Filtering
UMCP ENEE631 Slides (created by M.Wu © 2004)
Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 5)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

30. Example (2): Wiener Filtering vs. Inverse Filtering
9/11/2018
Example (2): Wiener Filtering vs. Inverse Filtering
UMCP ENEE631 Slides (created by M.Wu © 2004)
Top row: noise variance = 650; 2nd row: noise variance ~ 65; 3rd row: noise variance 10^-4
Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 5)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

31. Wiener Filter: From Theory to Practice
9/11/2018
Wiener Filter: From Theory to Practice
Recall: assumed p.s.d. of image and noise random fields, and frequency response of distortion filter are known
Why make the assumptions?
Are these reasonable assumptions?
What do they imply in our implementation of Wiener filter?
UMCP ENEE631 Slides (created by M.Wu © 2004)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

32. Summary of Today’s Lecture
9/11/2018
Summary of Today’s Lecture
Image restoration
Deconvolution / Inverse filtering
Pseudo inverse filtering to cope with spectral nulls
Wiener filtering
Next time
Basic compression techniques
Readings
Gonzalez’s book
For further reading: Woods’ book 7.1, 7.2, (7.7); 3.1, 3.2, 3.5.0
Jain’s book ; Bovik’s Handbook Sec.3.5 (subsection-4, pp136)
“Blind Image Deconvolution” by Kundur et al, IEEE Sig. Proc. Magazine, vol.13(3), 1996
UMCP ENEE631 Slides (created by M.Wu © 2004)
Jain’s book
M. Wu: ENEE631 Digital Image Processing (Spring'09)

33. UMCP ENEE631 Slides (created by M.Wu © 2004)
9/11/2018
To Explore Further 
UMCP ENEE631 Slides (created by M.Wu © 2004)
Ref: Jain’s boo ; Ray Liu’s EE624 lecture notes on Multi-dim chapter
M. Wu: ENEE631 Digital Image Processing (Spring'09)

34. Review: Handling Noise in Deconvolution
9/11/2018
Review: Handling Noise in Deconvolution
Inverse filtering is sensitive to noise
Does not explicitly modeling and handling noise
Try to balance between deblurring vs. noise suppression
Consider the (desired) image and noise as random fields
Minimize MSE between the original and restored
e = E{ [ u(n1, n2) – u’(n1, n2) ] 2 } where u’(n1, n2) is a func. of {v(m1, m2) }
Best estimate is conditional mean E[ u(n1 , n2) | all v(m1 , m2) ]
Get the best linear estimate  Wiener filtering
Produce a linear estimate from the observed image to minimize MSE H
u(n1, n2)
v(n1, n2)
(n1, n2) G
u’(n1, n2)
w(n1, n2)
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

35. Wiener Filter: From Theory to Practice
9/11/2018
Wiener Filter: From Theory to Practice
Recall: assumed p.s.d. of image and noise random fields, and frequency response of distortion filter are known
Why make the assumptions?
Are these reasonable assumptions?
What do they imply in our implementation of Wiener filter?
UMCP ENEE631 Slides (created by M.Wu © 2004)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

36. Wiener Filter: Issues to Be Addressed
9/11/2018
Wiener Filter: Issues to Be Addressed
Wiener filter’s size
Theoretically has infinite impulse response ~ require large-size DFTs
Impose filter size constraint: find the best FIR that minimizes MSE
Need to estimate power spectrum density of orig. signal
Estimate p.s.d. of blurred image v and compensate variance due to noise
Estimate p.s.d. from a set of representative images similar to the images to be restored
Or use statistical model for the orig. image and estimate parameters
Constrained least square filter ~ see Gonzalez Sec.5.9
Optimize smoothness in restored image (least-square of the rough transitions)
Constrain differences between observed blurred image and blurred version of reconstructed image
Estimate the restoration filter w/o the need of estimating p.s.d.
Unknown distortion H ~ Blind Deconvolution
UMCP ENEE631 Slides (created by M.Wu © 2001)
Constrained least square filtering: Min | conv( HPF, u’ ) |, subject to | v – conv( H, u’ ) | < epsilon
M. Wu: ENEE631 Digital Image Processing (Spring'09)

37. Basic Ideas of Blind Deconvolution
9/11/2018
Basic Ideas of Blind Deconvolution
Three ways to estimate H: observation, experimentation, math. modeling
Estimate H via spectrum’s zero patterns
Two major classes of blur (motion blur and out-of-focus)
H has nulls related to the type and the parameters of the blur
Maximum-Likelihood blur estimation
Each set of image model and blur parameters gives a “typical” blurred output; Probability comes into picture because of the existence of noise
Given the observation of blurred image, try to find the set of parameters that is most likely to produce that blurred output
Iteration ~ Expectation-Maximization approach (EM)
Given estimated parameters, restore image via Wiener filtering
Examine restored image and refine parameter estimation
Get local optimums
 To explore more: Bovik’s Handbook Sec.3.5 (subsection-4, pp136)
“Blind Image Deconvolution” by Kundur et al, IEEE Sig. Proc. Magazine, vol.13(3), 1996
UMCP ENEE631 Slides (created by M.Wu © 2001/2004)
M. Wu: ENEE631 Digital Image Processing (Spring'09)

38. Filtering Through Transform Domain Operation
9/11/2018
Filtering Through Transform Domain Operation
E.g.1 Realize Wiener in DFT domain
E.g.2 Use zonal mask in transform domain
Realize “ideal” LPF/BPF/HPF
Computation complexity for transform could be high for large image
UMCP ENEE631 Slides (created by M.Wu © 2001)
From Jain Fig.7.31&7.32
M. Wu: ENEE631 Digital Image Processing (Spring'09)