Image Restoration

Image Restoration

Jayash SharmaDepartment of Computer Science & Engineering

BMAS Engineering College, AgraEmail: jayash.sharma@gmail.com

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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