EEE F435 (2023-24-I)
Digital Image Processing
(Image Restoration-II)
BITS Pilani
K K Birla Goa Campus
Ashish Chittora
Image Restoration
• Estimation of Degradation Model
• Inverse Filter
• Wiener Filter
• Geometric Mean Filter
• Constrained Least Squares Filter
Estimation of Degradation Model
Degradation model:
g ( x , y ) f ( x , y ) h ( x, y ) ( x , y )
or
G(u, v) F (u, v) H (u, v) N (u, v)
Purpose:
To estimate h(x,y) or H(u,v)
Why
If we know exactly h(x,y), regardless of noise, we can do
deconvolution to get f(x,y) back from g(x,y).
Estimation of Degradation Model
The process of restoring an image by using a degradation function
that has been estimated in some way sometimes is called ‘blind
de-convolution’, due to the fact that true degradation function is
seldom known completely.
Methods:
1. Estimation by Image Observation
2. Estimation by Experiment
3. Estimation by Modeling
1. Estimation by Image Observation
• Estimate the function by gathering information from the
image itself.
• We look for areas of strong signal content. Using simple
gray levels of the object and background, we can
construct an unblurred image of the same size and
characteristics as the observed subimage.
• Assuming the effect of noise is negligible because of
our choice of a strong signal area.
Gs (u , v)
H s (u , v)
Fˆs (u , v)
• We then deduce the complete function H(u, v).
Original image (unknown) Degraded image
f(x,y) f(x,y)*h(x,y) g(x,y)
Observation
Estimated Transfer Function: DFT Subimage
Gs (u, v ) g s ( x, y )
Gs (u , v) Restoration
H (u , v) H s (u , v) process by
Fˆs (u , v) estimation
DFT Reconstructed
This case is used when we Fˆs (u, v )
know only g(x,y) and cannot Subimage
repeat the experiment! fˆs ( x, y )
2. Estimation by Experiment
• Used when we have the same equipment set up and can repeat
the experiment.
• Images similar to the degraded image can be acquired with
various system settings until they are degraded as closely as
possible to the image we wish to restore.
• Then the idea is to obtain the impulse response of the
degradation by imaging an impulse (small dot of light) using
the same system settings.
• An impulse is simulated by a bright dot of light, as bright as
possible to reduce the effect of noise.
Input impulse image Response image from
the system
System
H( )
A ( x, y ) g ( x, y )
DFT DFT
DFT A ( x, y ) A G(u, v)
G ( u, v )
H ( u, v )
A
3. Estimation by Modeling
• Used when we know physical mechanism underlying the
image formation process that can be expressed mathematically.
• In some cases, the model can even take into account
environmental conditions that cause degradations.
Example:
A degradation model proposed by Hufnagel and Stanley[1964].
Its based on the physical characteristics of atmospheric
turbulence.
k ( u 2 v 2 )5 / 6
H ( u, v ) e
• Similar to the Gaussian LPF
• ‘k’ is a constant that depends on the nature of the turbulence.
Original image Severe turbulence
k = 0.0025
k = 0.001 k = 0.00025
Mild turbulence Low turbulence
Example: Motion Blurring
Image f(x,y) undergoes a planar motion.
Assume that time varying components of motion in x- and y- directions
The blurred image is obtained by ( x0 (t ), y0 (t ))
T
Where T = exposure time.
g ( x, y ) f ( x x0 (t ), y y0 (t ))dt
0
g(x, y) is the blurred image.
Taking Fourier transform of the above equation, we get
j 2 ( ux vy )
G (u, v) g ( x , y ) e dxdy
T
j 2 (uxvy )
f ( x x0 (t ), y y0 (t ))dt e dxdy
0
T
f ( x x0 (t ), y y0 (t ))e j 2 ( ux vy )
dxdydt
0
T
G (u, v) f ( x x0 (t ), y y0 (t ))e j 2 ( ux vy )
dxdydt
0
T
F (u, v)e j 2 (ux0 (t )vy0 (t )) dt
0
T
F (u, v) e j 2 (ux0 (t ) vy0 (t ))dt F (u, v) H (u, v)
0
Then we get, the motion blurring transfer function:
T
H (u, v ) e j 2 ( ux0 ( t )vy0 ( t ))dt
0
For constant/uniform motion: ( x0 (t ), y0 (t )) (at, bt)
T
1
H (u, v) e j 2 ( ua vb )
dt sin( (ua vb))e j (ua vb )T
0
(ua vb)
Motion blurred image
Original image
a = b = 0.1, T = 1
Inverse Filter
From degradation model:
G(u, v) F (u, v) H (u, v) N (u, v)
after we obtain H(u,v), we can estimate F(u,v) by the inverse filter:
ˆ G ( u, v ) N ( u, v )
F ( u, v ) F ( u, v )
H ( u, v ) H (u, v )
• The expression tells that even if we know the degradation
function we can’t recover the undegraded image exactly because
N(u, v) is a random function whose Fourier Transform is not
known.
• Further, if the degradation has zero or very small values, then
the ratio N(u, v) / H(u, v) could dominate the estimate.
• To avoid this problem, we limit the analysis to frequencies
near the origin, we reduce the probability of encountering
zero values.
• To avoid the side effect of enhancing noise, we can apply this
formulation to freq. component (u,v) with in a radius D0 from
the center of H(u,v).
• Practically, the direct inverse filter is not popularly used.
• Following filters are the improvement over the direct inverse
filtering.
Example: Inverse Filter
Result of applying Result of applying
Original image the full filter the filter with D0=40
Blurred image Result of applying Result of applying
Due to Turbulence the filter with D0=70 the filter with D0=85
0.0025( u 2 v 2 )5 / 6
H ( u, v ) e
Wiener Filter
• Also known as ‘Minimum Mean Square Error’ Filter.
• It incorporates both the degradation function and the statistical
characteristics of noise into the restoration process.
• It considers images and noise as random variables and the
objective is to find an estimate of the uncorrupted image such
that the mean square error between them is minimized.
This error measure is given by:
e2 E ( f fˆ )2
where E{..} is the expected value of the argument.
It is also assumed that the noise and the image are
uncorrelated; that one or the other has zero mean; and that the
gray levels in the estimate are a linear function of the levels in
the degraded image.
Wiener Filter [1942] Formula:
H *
( u , v ) S ( u , v )
Fˆ (u, v ) f
G ( u , v )
S f (u, v ) H (u, v ) S (u, v )
2
H * ( u, v )
G ( u , v )
H (u, v ) S (u, v ) / S f (u, v )
2
1 H ( u, v )
2
G ( u , v )
H (u, v ) H (u, v ) S (u, v ) / S f (u, v )
2
where H(u,v) = Degradation function
S(u,v) = Power spectrum of noise
Sf(u,v) = Power spectrum of the undegraded image
Note that if the noise reduces to zero, it reduces to inverse filter.
In wiener filter formula:
1 H ( u , v )
2
Fˆ (u, v ) G(u, v )
H (u, v ) H (u, v ) S (u, v ) / S f (u, v )
2
Difficult to estimate
Approximated Formula: When we are dealing with spectrally
white noise, the power spectrum of noise is constant, which
simplifies things considerably.
1 H ( u , v )
2
Fˆ (u, v) G(u, v)
H (u, v) H (u, v) K
2
Practically, K is chosen manually to obtained the best visual result.!
Example: Wiener Filter
Original image
Result of the Result of the inverse Result of the
full inverse filter filter with D0=70 Wiener filter
Original image
Blurred image Result of the
Due to Turbulence Wiener filter
Image Result of the Result of the
degraded inverse filter Wiener filter
by motion
blur +
AWGN s2=650
s2=325
Note: K is
chosen
manually
s2=130
Geometric Mean Filter
This filter represents a family of filters combined into a
single expression
1
H ( u, v )
* *
H ( u, v )
ˆ
F ( u, v ) G ( u, v )
2
H ( u , v ) S ( u , v )
H ( u, v )
2
S f (u, v )
= 1 the inverse filter
= 0 the Parametric Wiener filter
=0.5 Geometric mean filter
= 0, = 1 the standard Wiener filter
= 1, < 0.5 More like the inverse filter
= 1, > 0.5 More like the Wiener filter
= 1, = 0.5 spectrum equalization filter
ADAPTIVE FILTERS:A brief introduction
• Filters selected so far are applied to an image without regard for
how image characteristics vary from one point to another.
• Adaptive filters are a class of filters whose behaviour changes
based on the statistical characteristics of the image inside the filter
region defined by the m*n rectangular window Sxy .
• They are capable of a superior performance to the filters discussed
so far.
• Price paid for improved filtering power is an increase in filter
complexity
Thank you
