This document summarizes popular image restoration techniques including inverse filtering, Wiener filtering, and constrained least squares error filtering. It discusses issues with inverse filtering like unsatisfactory results due to noise. Wiener filtering is described as minimizing the mean square error between the original and reconstructed image using the power spectrum of the image and noise. Constrained least squares error filtering estimates images by minimizing noise variance while imposing a smoothness constraint using the Laplacian. Iterative methods for selecting filter parameters like the regularization parameter gamma are also presented.

1. Popular Image Restoration Technique
Subject: Image Procesing & Computer Vision
Dr. Varun Kumar
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 1 / 13

2. Outlines
1 Inverse filtering (motion blurr)
2 Minimum mean square error (Wiener) filtering
3 Constrained least square error filter
Noise parameter estimation from blurred image
4 Restoration in presence of noise
5 References
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 2 / 13

3. Restoration inverse filtering
Q What is the cause for unsatisfactory results obtained by inverse
filtering ?
Ans
H(u, v) =
T
0
e−j2π[ux0(t)+vy0(t)]
dt
=
T
π(ua + vb)
sin π(ua + vb) e−jπ(ua+vb)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 3 / 13

4. Minimum mean square error (Wiener filtering)
Let f (x, y) and ˆf (x, y) are the original and reconstructed image.
e = E (f − ˆf )2
⇒ Here image intensity and noise intensity are uncorrelated.
ˆF(u, v) =
H∗(u, v)Sf (u, v)
Sf (u, v)|H(u, v)|2 + Sη(u, v)
G(u, v)
H∗(u, v) → Complex conjugate of H(u, v).
Sf (u, v) → Power spectrum of original image.
Sη(u, v) → Power spectrum of noise signal.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 4 / 13

5. Continued–
ˆF(u, v) =
1
H(u, v)
|H(u, v)|2
|H(u, v)|2 +
Sη(u,v)
Sf (u,v)
G(u, v)
⇒ In general,
Sη(u,v)
Sf (u,v) = k
ˆF(u, v) =
1
H(u, v)
|H(u, v)|2
|H(u, v)|2 + k
G(u, v)
This k can be adjusted manually.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 5 / 13

6. Continued–
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 6 / 13

7. Constrained least square estimation
⇒ Wiener filter estimate at constant value of k.
⇒ In constrained least square method, only the noise PDF is required.
⇒ Let the mean and variance of noise are mη and σ2
η.
As per the degradation model,
g = Hf + n
Here, the value of H is very sensitive to the noise.
⇒ We should adopt the optimality criteria for image smoothness.
⇒ Second order derivative and Laplacian operation removes the
irregularity of an image.
C =
M−1
x=0
N−1
y=0
2
f (x, y)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 7 / 13

8. Continued–
⇒ Here, optimality criteria is based on the Laplacian. Hence,
g − H ˆf 2
= n 2
⇒ ˆf is the reconstructed image.
Frequency domain representation through LS method
ˆF(u, v) =
|H∗(u, v)|2
|H(u, v)|2 + γ|P(u, v)|2
G(u, v)
where
p(x, y) =


0 1 0
1 −4 1
0 1 0

 ⇒ Laplacian Mask
Note: Since, the image size is M × N, hence the Fourier transform of
mask should be of the same order.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 8 / 13

9. Continued–
⇒ Here, γ should be adjusted manually.
Let r be the residual vector, where
g = r − H ˆf
Let a function φ(γ) is defined in such a way that
φ(γ) = rT
r = r 2
(1)
Based on (1), let
r 2
= n 2
± a → accuracy factor
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 9 / 13

10. Iterative selection of γ
1 Select initial value of γ.
2 Compute φ(γ) = r 2 .
3 Stop if r 2= n 2 ± a else proceed to 4.
4 Increase γ if r 2< n 2 − a or
Decrease γ if r 2> n 2 + a
5 Use new value of γ to recompute
ˆF(u, v) =
|H∗(u, v)|2
|H(u, v)|2 + γ|P(u, v)|2
G(u, v)
6 Go to 2
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 10 / 13

11. Performance comparison
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 11 / 13

12. Noise parameter estimation
σ2
η =
1
MN
M−1
x=0
N−1
y=0
η(x, y) − mη
2
and
mη =
1
MN
M−1
x=0
N−1
y=0
η(x, y)
and
η 2
= MN σ2
η − m2
η
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 12 / 13

13. References
M. Sonka, V. Hlavac, and R. Boyle, Image processing, analysis, and machine vision.
Cengage Learning, 2014.
D. A. Forsyth and J. Ponce, “A modern approach,” Computer vision: a modern
approach, vol. 17, pp. 21–48, 2003.
L. Shapiro and G. Stockman, “Computer vision prentice hall,” Inc., New Jersey,
2001.
R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using
MATLAB. Pearson Education India, 2004.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 24 13 / 13