DIP -Unit 3 ppt.pptx

Model of image degradation/restoration process
Noise models.
Restoration in the Presence of Noise Only– Spatial
Filtering.
 Periodic Noise Reduction by Frequency Domain
Filtering.
 Linear Position– Invariant Degradations
 Inverse filtering.
Image Restoration

Image restoration is to restore a degraded image
back to the original image while image
enhancement is to manipulate the image so that it
is suitable for a specific application.

Preview
■ Goal of Image Restoration
■ Improve an image in some predefined sense
■ Difference with image enhancement ?
■ Features
■ Image restoration v.s image enhancement
■ Objective process v.s. subjective process
■ A prior knowledge v.s heuristic process
■ A prior knowledge of the degradation
phenomenon is considered
■ Modeling the degradation and apply the
inverse process to recover the original
image

Degradation/Restoration Process
Degradation function H, and some knowledge about the additive noise
term the objective of restoration is to obtain an estimate of the original
image. f(x,y) is an input g(x,y)is an degraded and n(x,y) is a noisy image
* ->convolution
g(x,y)=f(x,y)*h(x,y)+n(x,y)
G(u,v)=F(u,v)H(u,v)+N(u,v)

degradation/restoration
process

Noisemodels
■ Source of noise
■ Image acquisition (digitization)
■ Image transmission
■ Spatial properties of noise
■ Statistical behavior of the gray-level values
of pixels
■ Noise parameters, correlation with the image
■ Frequency properties of noise
■ Fourier spectrum

Noise probability density
functions
■ Noises are taken as random variables
■ Random variables
■ Probability density function (PDF)
■ a probability density function (PDF), density of a
continuous random variable lying between a
specific range of values

Gaussian noise
■ In spatial and frequency
domain.
 Due to factors such as electronic
circuit noise and sensor noise due to
poor illumination and/or high
temperature.

Erlang (gamma) Noise
Eq. (5.2-5) often is referred to as the gamma density, or the Erlang
density

Impulse (salt-and-pepper)
noise
• If intensity b > a then b will appear as a light dot in the image.
• If a >b a will appear like a dark dot.
• If either is zero, the impulse noise is called unipolar.
• Bipolar impulse noise also is called salt and- pepper noise. Data-
drop-out and spike noise.

• The Rayleigh density -noise phenomena in range imaging
• Exponential and gamma densities-laser imaging
• Impulse noise -faulty switching, take place during imaging.
• Uniform density - Random number generators that are used in
simulations

PDFs of some important noise
models

Test for noise behavior
■ Test pattern
Its histogram:
0 255

Periodic noise
■ Arise from electrical or
electromechanical interference
during image acquisition
■ Can be reduced using frequency
domain

Sinusoidal noise:
frequency domain

Estimation of Noise Parameters
The parameters of periodic noise-inspection of the
Fourier spectrum of the image.
 Automated analysis - visual analysis.
Attempt to infer the periodicity of noise
components directly from the image, in
simplistic cases.

Mean and variance
Mean and variance of intensity levels from the data
from the image strips.
Consider a strip (subimage) denoted by S, and
denote the probability estimates of the intensities
of the pixels in S, where L is the number of possible
intensities in the entire image (e.g., 256 for an 8-bit
image).

Additive noise only
g(x,y)=f(x,y)+ (x,y)
G(u,v)=F(u,v)+N(u,v)

Restoration in the presence of noise
only – spatial filtering
When the only degradation present in an image
is noise, Eqs. (5.1-1) and (5.1-2)

Mean filter
1. Arithmetic mean filter
• Let Sxy represent the set of coordinates in a
rectangular subimage window (neighborhood) of
size mxn centered at point (x, y).
• The arithmetic mean filter computes the
average value of the corrupted image g(x,y)in
the area defined by Sxy .
• A mean filter smooths local variations in an
image, and noise is reduced with blurring.

Mean filter
2. Geometric mean filter
• Each restored pixel is given by the product of the
pixels in the subimage window, raised to the power
1/mn.
• Good smoothing comparable to the arithmetic mean
filter, but lose less image details.

Mean filter
3. Harmonic mean filter
• The harmonic mean filter works well for salt
noise, but fails for pepper noise.
• Works well for Gaussian noise.

Mean filter
3. Contraharmonic mean filter
• Q is called the order of the filter.
• Useful in reducing effects of salt-and-pepper noise.
• Q is positive , eliminates pepper noise.
• Q is negative eliminates salt noise.
• It cannot do both simultaneously.
• arithmetic mean filter if Q = 0,
• harmonic mean filter if Q = -1.

Mean filter
Corrupted with additive Gaussian noise of zero
mean and variance of 400

Image is corrupted now by pepper noise with
probability of 0.1.

Wrong sign in contra-harmonic filter
Q=-1.5 Q= 1.5

5.3.2 Order-Statistic Filters
1. Median filter
• median of the intensity levels in the neighborhood
of that pixel:
• Reduce bipolar and unipolar impulse noise

2. Max and min filters
• Useful for finding the brightest points in an
image.
• Pepper noise reduced by this filter

2. Max and min filters
• Finding the darkest points in an image.
• Reduces salt noise

3. Midpoint filter
• Midpoint between the maximum and minimum
values.
• combines order statistics and averaging.
• It works best for randomly distributed noise, like
Gaussian or uniform noise.

4. Alpha-trimmed mean filter

5.3.3 Adaptive Filters
• The filters are independent of image characteristics
vary from one point to another
• Adaptive, local noise reduction filter
 Depend on Mean and Variance
 Mean is a -- of average gray level in the region,
 variance gives a measure of contrast in that region.
'mean' ---- contribution of individual pixel intensity for the entire
image
Variance - how each pixel varies from the neighbouring pixel
(or centre pixel)

On a local region Sxy it should satisfy four
quantities:
(a) g(x,y)the value of the noisy image at (x, y);
(b) The variance of the noise corrupting
f(x,y)to g(x,y)
(c) mL local mean of the pixels in Sxy
(d) The local variance of the pixels in Sxy

2.If the local variance is high relative to the filter should
return a value close to g(x, y).
3.If the two variances(local variance and noise variance)
are equal return the arithmetic mean value of the pixels in
Sxy.
An adaptive expression is
Estimation required

Adaptive median filter
Suitable for higher rate of Impulse Noise > 0.2 by preserving the details of an image

Three purposes
1. Remove salt-and-pepper (impulse) noise
2. Reduce distortion, such as excessive thinning
or thickening of object boundaries.
Adaptive median filter

Periodic Noise Reduction by Frequency
Domain Filtering
The three types of selective filters
5.4.1 Bandreject Filters
• A band reject filter is useful when the general location of the noise in
the frequency domain is known.
• A band reject filter blocks frequencies within the chosen range
and lets frequencies outside of the range pass through.

5.4.2 Bandpass Filters
• A bandpass filter performs the opposite operation of a
bandreject.
• Enhance edges (suppressing low frequencies) while
reducing the noise at the same time (attenuating high
frequencies).
• Given as a fraction of the highest frequency
represented in the Fourier domain image.

5.4.3 Notch Filters
• A notch filter rejects (or passes) frequencies
in predefined neighborhoods about a center
frequency.

5.4.4 Optimum Notch Filtering
• It minimizes local variances of the restored
estimate .

Extract the principal frequency components
of the interference pattern
HNP(u, v), is the noise with the interference
pattern, then the Fourier transform of the
interference noise pattern is given by the
expression
G(u,v) is Fourier transform of the corrupted image.

• The effect of components not present in the
estimate of can be minimized instead by
subtracting g(x,y) from a weighted portion of
to obtain an estimate of f(x,y):
• is the estimate of f(x,y) and w(x,y)is to be
determined.
• The function w(x,y) is called a weighting or
modulation function, and the objective of the
procedure is to select this function so that the result
is optimized.

5.5 Linear, Position-Invariant
Degradations
The input-output relationship in restoration stage is
expressed as

is called the impulse response of H
If in Eq. (5.5-1) then is the response of H to
an impulse at coordinates (x, y). The impulse becomes a point
of light and is point spread function (PSF). This
which is called the superposition (or Fredholm) integral of the
first kind,from the linear system theory. If the response of H to
an impulse is known, the response to any input can be
calculated by means of Eq. (5.5-11)

This expression is the convolution integral. This integral tells us that
knowing the impulse response of a linear system allows us to compute
its response, g, to any input f. The result is simply the convolution of
the impulse response and the input function.
In the presence of additive noise

5.7 Inverse Filtering
An estimate of the transform of the original
image simply by dividing the transform of the
degraded image G(u,v)by the degradation function
Substituting the right side of Eq. (5.1-2) for G(u,v)in
Eq. (5.7-1) yields

if we know the degradation function we cannot
recover the undegraded image [the inverse
Fourier transform of F(u,v) exactly because is
N(u,v)is not known.
If the degradation function has zero or very
small values, then the ratio could
easily dominate the estimate

DIP -Unit 3 ppt.pptx

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