Image Restoration
Jayash SharmaDepartment of Computer Science & Engineering
BMAS Engineering College, AgraEmail: [email protected]
What is Image Restoration:
Image restoration aim to improve an image in some predefined sense.
What about image enhancement?
Image enhancement also improves an image by applying filters.
Difference:
Image Enhancement --- Subjective process
Image Restoration --- Objective Process
Restoration tries to recover / restore degraded image by using a prior knowledge of the degradation phenomenon.
Restoration techniques focuses on:
1. Modeling the degradation
2. Applying inverse process in order to recover the original image.
Model of the Image Degradation / Restoration Process
Degradation function along with some additive noise operates on f(x, y) to produce degraded image g(x, y)
Given g(x, y), some knowledge about the degradation function H and additive noise η(x, y), objective of restoration is to obtain estimate f’(x, y) of the original image.
If H is linear, position invariant process then degraded image in spatial domain is given by:
h(x, y) = Spatial representation of H * indicates convolution
Since convolution in Spatial domain = multiplication in Frequency Domain
We Assume that H is identity operator
We deal only with degradation due to Noise
Noise Models: Noise in digital image arises during
1. Image Acquisition
2. Transmission
During Image Acquisition
Environmental conditions (Light Levels) Quality of sensing element
During Transmission
Interference during transmission
Spatial Properties of Noise:
1. With few exception we consider that noise is independent of spatial coordinates.
2. We assume that noise is uncorrelated with respect to the image itself (There is no correlation between image pixels and the values of noise components)
Fourier Properties of Noise:
Refers to the frequency contents of noise in the Fourier sense.
If Fourier spectrum of noise is Constant, the noise is usually called WHITE NOISE
Some Noise Probability Density Functions (PDFs):
Gaussian Noise Rayleigh Noise Erlang (Gamma) Noise Exponential Noise Uniform Noise Impulse (Sal & Pepper Noise) Periodic Noise
Spatial Noise Descriptor
Statistical behavior of the gray level values in the noise component.
Can be considered as random variables
Characterized by Probability Density Functions (PDFs)
Gaussian / Normal Noise Model
1. Most frequently used.
2. PDF of Gaussian random variable z is given by:
z Gray level
µ Mean of average value of z
σ Standard Deviation of z
σ2 Variance of z
When z is defined by this equation then
About 70% of its values will be in the range [(µ - σ),(µ + σ)] and
About 95% of its values will be in the range [(µ - 2σ),(µ + 2σ)]
Plot of function
Rayleigh Noise Model
PDF of Rayleigh Noise is given by:
z Gray level
µ Mean of average value of z
σ2 Variance of z
Basic shape of this density is skewed to the right.
Quite useful for approximating skewed histograms.
Plot of function
Erlang (Gamma) Noise Model
PDF of Erlang Noise is given by:
z Gray level
µ Mean of average value of z
σ2 Variance of z
Above equation is also called Erlang Density
If denominator is Gamma function then it is called Gamma density
Plot of function
a > 0 b = positive integer
Exponential Noise Model
PDF of Exponential Noise is given by:
z Gray level µ Mean of average value of z σ2 Variance of z a > 0
Special case of Erlang Density Where b=1
Plot of function
Uniform Noise Model
PDF of Uniform Noise is given by:
z Gray level µ Mean of average value of z σ2 Variance of z
Plot of function
Impulse (Salt & Pepper) Noise Model
PDF of Uniform Noise is given by:
z Gray level
If b > a then b light dot and a dark dot
If either Pa or Pb = 0 Unipolar Impulse Noise otherwise Bipolar Impulse Noise.
If Neither probability is 0 and approximately equal then noise values will resemble salt & pepper granules randomly distributed over the image.
Also referred as Shot and Spike Noise
Plot of function
This test pattern is well-suited for illustrating the noise models, because it is composed of simple, constant areas that span the grey scale from black to white in only three increments. This facilitates visual analysis of the characteristics of the various noise components added to the image.
Example
Restoration using Spatial Filtering
We can use spatial filters of different kinds to remove different kinds of noise.
Arithmetic Mean Filter
Let Sxy represents the set of coordinates in a rectangular sub image window of size m x n centred at (x, y).
This filter computes the average value of the corrupted image in the area defined by Sxy.
xySts
tsgmn
yxf),(
),(1
),(ˆ1/9 1/9 1/91/9 1/9 1/91/9 1/9 1/9
Implemented as simple smoothing filter
Well Suited for Gaussian / Uniform Noise
Geometric Mean Filter
Each restored pixel is given by the product of the pixels in the sub image window, raised to the power 1/mn.
Achieves smoothing comparable to Arithmetic Mean Filter but tends to lose less image details in the process.
Well Suited for Gaussian / Uniform Noise
mn
Sts xy
tsgyxf
1
),(
),(),(ˆ
Harmonic Mean Filter
Works well for salt noise but fails for pepper noise.
Also does well for Gaussian Noise
xySts tsg
mnyxf
),( ),(1
),(ˆ
Example:
OriginalImage
ImageCorrupted By Gaussian Noise
After A 3*3Geometric Mean Filter
After A 3*3Arithmetic
Mean Filter
Order Statistic Filter
Result is based on the ranking / ordering of the pixels contained in the image area encompassed by the filter.
Median Filter
)},({),(ˆ),(
tsgmedianyxfxySts
Effective for both uni-polar and bipolar impulse noise.
Excellent at noise removal, without the smoothing effects that can occur with other smoothing filters
Max Filter Good for Pepper Noise
Min Filter Good for Salt Noise
)},({max),(ˆ),(
tsgyxfxySts
)},({min),(ˆ),(
tsgyxfxySts
Mid Point Filter Good for Gaussian / Uniform Noise
)},({min)},({max
2
1),(ˆ
),(),(tsgtsgyxf
xyxy StsSts
ImageCorrupted
By Salt AndPepper Noise
Result of 1 Pass With A 3*3 MedianFilter
Result of 2Passes With
A 3*3 MedianFilter
Result of 3 Passes WithA 3*3 MedianFilter
Example:
ImageCorruptedBy Pepper
Noise
ImageCorruptedBy SaltNoise
Result Of Filtering Above With A 3*3 Min Filter
Result OfFiltering
AboveWith A 3*3Max Filter
Example:
ImageCorrupted
By UniformNoise
Image FurtherCorruptedBy Salt andPepper Noise
Filtered By5*5 Arithmetic
Mean Filter
Filtered By5*5 Median
Filter
Filtered By5*5 GeometricMean Filter
Filtered By5*5 Alpha-TrimmedMean Filter
Example (Combined):
Periodic Noise
Typically arises due to electrical / Electro-mechanical interference during image acquisition.
Spatially dependent noise.
Can be reduced significantly via Frequency Domain Filtering.
Parameters can be estimated by inspecting the Frequency Spectrum of the image.
Periodic noise tend to produce frequency spikes
Image corrupted by Sinusoidal noise
Spectrum (Each pair of conjugate impulses
corresponds to one sine wave)
Periodic Noise Reduction by Frequency Domain Filtering
Removing periodic noise form an image involves removing a particular range of frequencies from that image
Bandreject FilterBandpass FiltersNotch Filter
Bandreject Filter
Removes / Attenuates a band of frequencies about the origin of the Fourier Transform.
Ideal Bandreject Filter:
2),( 1
2),(
2 0
2),( 1
),(
0
00
0
WDvuDif
WDvuD
WDif
WDvuDif
vuH
D(u, v) = Distance of point from the origin
W = Width of the band
D0 = Radial Centre
Butterworth Bandreject Filter:
Gaussian Bandreject Filter:
Ideal BandReject Filter
ButterworthBand Reject
Filter (of order 1)
GaussianBand Reject
Filter
Image corrupted by sinusoidal noise
Fourier spectrum of corrupted image
Butterworth band reject filter
Filtered image
Example:
Inverse Filtering
An approach to restore an image.
Compute an estimate F’( u, v) of the transform of the original image by:
Divisions are made between individual elements of the functions.
Inverse Filtering…
Above Equation concludes that:
Even if we know degradation function, we can not recover the undegraded image [Inverse Fourier Transform of F(u, v)] exactly because
N(u, v) is random function whose Fourier Transform is not known.
If degradation has ZERO or less value then N(u, v) / H(u, v) dominates the estimated F’(u, v).
No explicit provision for handling Noise.
Maximum Mean Square Error (Wiener) Filtering
Incorporates both degradation function and statistical characteristics of noise into restoration process.
Considers images and noise as random process.
Find an estimate f’ of the uncorrupted image f such that mean square error between them is minimized. Error measure is given by:
E{.} = Expected value of the argument
Assumptions: image and noise are uncorrelated. One or other has Zero mean Gray levels in the estimate are a linear function of levels in the
degraded image.
Maximum Mean Square Error (Wiener) Filtering…
Based on these conditions:
Singularity & Ill-condition?
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