Degradation Model and Diagonalization of circulant and block matrices

1. Digital Image Processing
Presented By,
T. Janani,
II-M.sc(CS&IT)
Nadar Saraswathi College
Of Arts & Science, Theni.

2. Degradation Model :
H – Degradation operator
f(x , y) - Input Image
g(x , y) - Degraded Output

3. Degradation models : noise only
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vuNvuFvuG
yxyxfyxg

 
Noise models
1. Spatial characteristics (independent or dependent)
2. Intensity ( distribution, spectrum)
* Uniform, Gaussian, Rayleigh, Gamma (Erlang),
Exponential, impulse
3. Correlation with the image (additive, multiplicative)
De-noising
1. Spatial filtering
2. Frequency domain filtering

4. Noise models – examples

5. Noise models – periodic noise

6. De-noising
Estimation of noise parameters:
* By spectrum inspection: for periodic noise
* By test image: mean, variance and histogram shape,
if imaging system is available
* By small patches, if only image is available
De-noising :
1. Spatial filtering ( for additive noise)
* Mean filters
* Order-statistics filters
* Adaptive filters
2. Frequency domain filtering (for periodic noise)

7. 

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tsg
mn
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Sts xy
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 
Arithmetic mean filter
Geometric mean filter
De-noising – Gaussian noise example

8. De-noising – evaluation
1. PSNR
2. Visual perception
   

M
i
N
j
ijij SSMN
MSE
1 1
2
'1


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
MSE
PSNR
2
10
255
log10

9. De-noising – evaluation example

10. Degradation models: linear vs non-linear
1. Many types of degradation can be approximated
by linear, space invariant processes
* Can take advantages of the mature
techniques developed for linear systems
2. Non-linear and space variant models are more
accurate
* Difficult to solve
* Unsolvable

11. Linear, space-invariant degradations
 
 
 
 
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),(),,,(),(
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yxddyxhf
yxddyxhf
yxddyxHf
yxddyxfH
yxddyxfH
yxyxfHyxg
ddyxfyxf








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




Sampling theorem -->
Linearity, additivity -->
Linearity, homogeneity -->
Space-invariant -->
(convolution integral)

12. Diagonalization of circulant and block- circulant
matrices
Circulant Matrices:
For MXM circulant matrix :
Define a scalar :

13. Define a vector :
for k=0,1,2…..M-1

15. Block – Circulant Matrix :

18. Effects of Diagonalization on the Degradation Model :

20. In other words, 1-D discrete Formulation of
degradation model is 1D Fourier transform
G(k)=MH(k)F(k)
for k=0.1,2,………….,M-1

21. Thank You