This manuscript describes the image degradation model.

1. Image Restoration-2
Subject: Image Procesing & Computer Vision
Dr. Varun Kumar
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 1 / 12

2. Outlines
1 Estimation of degradation model
By observation
By experiment
By mathematical modeling
2 Restoration technique
Inverse filtering
MMSE filtering
Constrained least square filter
3 References
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 2 / 12

3. Degradation model
Degradation model
g(x, y) = f (x, y) ∗ h(x, y) + η(x, y) (1)
and
G(u, v) = F(u, v)H(u, v) + N(u, v) (2)
and
g = Hf + n (3)
⇒ Knowledge of degradation function is essential for acquiring the true image.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 3 / 12

4. Observation, experimentation and mathematical modeling
⇒ In all process, there is a requirement of convolution (blind convolution
) operation. The aim of these process are to find the true estimate.
1 Observation:
gs(x, y) ⇔ Gs(u, v)
fs(x, y) ⇔ Fs(u, v)
Hs(u, v) =
Gs(u, v)
Fs(u, v)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 4 / 12

5. Continued–
2 Experimentation:
⇒ Imaging setup similar to original one.
⇒ We need to find the impulse response of imaging set-up.
⇒ For imaging set-up, there is requirement of simulated impulse response.
Q How do we simulate the impulse response?
Ans. We need a sharp bright spot of beam that can be understand as a
simulated impulse response.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 5 / 12

6. Continued–
F(u, v) = A → F(u, v) ⇔ f (x, y)
and
G(u, v) = H(u, v)F(u, v)
⇒ H(u, v) =
G(u, v)
A
⇒ For simulated impulse response the intensity of the light should be
very high, so that noise effect could be ignored.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 6 / 12

7. Continued–
3 Mathematical modeling:
H(u, v) = e−k(u2+v2)5/6
k → Nature of turbulence
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 7 / 12

8. Basic principle
⇒ f (x, y) → Motion
⇒ x0(t), y0(t) → time varying quantity
⇒ T → Total time, when shutter of the camera is open.
Total observed exposure of light can be expressed as
g(x, y) =
T
0
f x − x0(t), y − y0(t) dt
Also
G(u, v) =
∞
−∞
∞
−∞
g(x, y)e−j2π(ux+vy)
dxdy
or
G(u, v) =
T
0
∞
−∞
∞
−∞
f x − x0(t), y − y0(t) e−j2π(ux+vy)
dxdydt
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 8 / 12

9. Continued–
G(u, v) =
T
0
∞
−∞
∞
−∞
f x − x0(t), y − y0(t) e−j2π(ux+vy)
dxdy
FT→shifting of position
dt
Fourier transform
f (x − x0(t), y − y0(t)) ⇔ F(u, v)e−j2π ux0(t)+vy0(t)
Hence
G(u, v) =
T
0
F(u, v)e−j2π ux0(t)+vy0(t)
dt
⇒ F(u, v)
T
0
e−j2π ux0(t)+vy0(t)
dt
H(u,v)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 9 / 12

10. Continued–
Let x0(t) = at
T and y0(t) = bt
T
H(u, v) =
1
π(ua + vb)
sin π(ua + vb) e−jπ(ua+vb)
In above figure, blurring occurs due to the motion of object. Also
a = 0.1, b = 0.1
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 10 / 12

11. Inverse filtering
G(u, v) = H(u, v)F(u, v)
⇒ ˆF(u, v) =
G(u, v)
H(u, v)
General expression for degradation model
G(u, v) = H(u, v)F(u, v) + N(u, v)
And
ˆF(u, v) = F(u, v) +
N(u, v)
H(u, v)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 23 11 / 12

12. 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 23 12 / 12