16404857 Digital Image Processing 07 Image Restoration Noise Removal

8/14/2019 16404857 Digital Image Processing 07 Image Restoration Noise Removal

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TDI2131 Digital Image Processing

Image Restoration & NoiseRemoval

Lecture 7

John See

Faculty of Information TechnologyMultimedia University

Some portions of content adapted CYPee's notes, MMU. Most figures from Gonzalez/Woods


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Lecture Outline

Degradation & Restoration Process Models

Noise Models

Restoration in the presence ofnoise only (spatial

filtering) Restoration of Periodic Noise

Restoration in the presence ofdegradation & noise(frequency domain filtering)


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Some Announcements

Reminder: Assignment 1 is due this Friday.


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Image Restoration

Image Restoration To improve the appearance of an image

by application of a restoration process that uses a

mathematical model for image degradation

Types of degradation:

Blurring caused by motion or atmospheric disturbance

Geometric distortion caused by imperfect lenses

Superimposed interference pattern caused by mechanical systems

Noise from electronic sources


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How are these images restored?


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Image Restoration: The Idea

Example

degraded

images

Knowledge of

image creation

process

Developdegradation

model

Developinverse degradation

process

Apply

inverse degradationprocess

Input

imageg(x,y)

Output

image ( , ) f x y


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Degradation/Restoration Process Model

Consists of 2 parts Degradation function & Noise function

General model in spatial domain:

g(x,y): degraded image, h(x,y): degradation function,

f(x,y): original image, n(x,y): additive noise function

),(),(*),(),( yxnyxfyxhyxg +=


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Degradation Model in Freq. Domain

General model in frequency domain:

G(x,y): Fourier transform of degraded image,

H(x,y): Fourier transform of degradation function,F(x,y): Fourier transform of original image,

N(x,y): Fourier transfor of additive noise function

What needs to be done??

FINDDegradation function and Noise model

),(),(),(),( vuNvuFvuHvuG +=


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Noise Models

Noise Any undesired information that contaminates

an image Variety of sources:

Digital image acquisition process, e.g. For CCD (chargedcoupled device) camera, electronics signal fluctuations in

detector, caused by thermal energy and light levels

During image transmission, e.g. Wireless network, may becorrupted by lightning or other atmospheric disturbance


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Noise Models

Assumptions of Noise models in this course:

Noise is independent of spatial coordinates (except forspatially periodic noise)

Noise is uncorrelated w.r.t the image (pixel values)

Spatial Noise Descriptor concern of the statistical behavior

of the intensity values in the noise component of the model

Characterized by probability density function (PDF)


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Noise Models

Typical image noise models that can be modeled:

Gaussian Rayleigh

Erlang (Gamma)

Exponential

Uniform

Impulse (salt-and-pepper)


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Noise Models - PDFs

Gaussian Noise

where

z: gray scale

= mean (average)

= standard deviation

2 2( ) / 2

21( )

2

z p z e

=


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Noise Models - PDFs

Rayleigh Noise

where

mean =

variance =

22

( )exp( ( ) / for( )

0 for

z a z a b z ap z b

z a

=


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Noise Models - PDFs

Uniform Noise

where

mean =

variance =

1for

( )( )

0 elsewhere

a z bb ap z

=

( ) / 2a b+

2( ) /12b a


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Noise Models - PDFs

Uniform Noise

where

==

=

otherwise0

(salt)for(pepper)for

)( bzPazP

zp b

a

b a>


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Example: Noise Models


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Example: Noise Models


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Restoration in Presence of Noise Only

Spatial Filtering Consider an image degraded with only additive noise.

The degradation model is further simplified as

in spatial domain, and

in frequency domain

( , ) ( , ) ( , )g x y f x y n x y= +

( , ) ( , ) ( , )G u v F u v N u v= +


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Noise Removal with Spatial Filters

Spatial filters can effectively remove various types of

noise in digital images Typically operate on small neighborhoods, from 3x3

to 11x11.

Some can be implemented as convolution masks


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Mean Filters

Arithmetic Mean Filter

Computes the average value of the corrupted image g(x,y)in area defined by Sxy. Sxy represents the set of

coordinates in a rectangular subimage window of size

mxn, centred at point (x,y)

The operation can be implemented using convolutionmask

For random noise

( , )

1( , ) ( , )xys t S

f x y g s t mn

=


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Mean Filters

Geometric Mean Filter

The image restored using a geometric mean filter:

For random noise: lose less detail than Arithmetic Mean

Harmonic Mean Filter

The image restored using a harmonic mean filter:

For salt noise

1

( , )

( , ) ( , )xy

mn

s t S

f x y g s t

=

( , )

( , )1

( , )xys t S

mn f x y

g s t

=


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Mean Filters

Contraharmonic Mean Filter

The image restored with a contraharmonic mean filter:

where Q is called the order of filter

Positive Q: pepper

Negative Q: salt

Q=0: Arithmetic mean

Q=-1: Harmonic mean

1

( , )

( , )

( , )

( , )( , )

xy

xy

Q

s t S

Q

s t S

g s t

f x yg s t

+

=


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Example: Mean Filters


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Example: Mean Filters (Cont'd)


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Example: Mean Filters (Cont'd)


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Order-Statistics Filters

Order Filter based on a specific type of image statistics

called order statistics, sometimes known as Order-Statistics

Filter

Order Statistics: Arranges all the pixels in sequential order

(smallest to largest), based on gray-level value

The selection of the value to be replaced in the center pixel,

is determined by the statistical function used (min, max,

median, etc.)


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Max & Min Filters

Minimum Filter select the smallest value within an ordered

window of pixel values, denoted as

Works best when the noise is primarily of the salt-type

(high value) Maximum Filter select the largest value within an ordered

window of pixel values, denoted as

Works best for pepper-type noise (low value)

( , )

( , ) min { ( , )}xys t S

f x y g s t

=

( , )

( , ) max { ( , )}xys t S

f x y g s t

=


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Median Filters

Median Filter select the middle pixel value within an

ordered window of pixel values, denoted as

Works best with salt-and-pepper noise (both high and low

values) Better noise remover than Averaging Filter, which causes

blurry edges and details in image, thus not effective against

impulse (salt-and-pepper) noise

Preserve line structures

{ }),(),(),(

tsgmedianyxfxySts

=


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Example: Median Filter


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Example: Min & Max Filters


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Midpoint Filter

Midpoint Filter select the average of the maximum and

minimum pixel values within the window, denoted as

Useful for Gaussian and uniform noise

( , )( , )

1( , ) max { ( , )} min { ( , )}2 xyxy s t Ss t S

f x y g s t g s t

= +


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Alpha-trimmed Mean Filter

Alpha-trimmed Mean Filter select the average of the values

within the window, but with some of the endpoint-ranked

values excluded

Suppose we delete d/2 lowest and d/2 highest gray level

values of g(s,t) in the neighborhood Sxy

. Let gr

(s,t) represent

the remaining mn-d pixels, the remaining pixels are

averaged:

Useful for combination noise such as salt-and-pepper with

Gaussian noise

( , )

1( , ) ( , )XY

r

s t S

f x y g s t

mn d

=


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Example: Mean Filters


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Adaptive Filters

Filters that consider behavior changes based on statistical

characteristics of the image inside the filter region.

Capable of superior performance, but causes increase in filter

complexity

Textbook Extra Reading: Adaptive, local noise reduction filter

Adaptive median filter


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Adaptive Filters

Filters that consider behavior changes based on statistical

characteristics of the image inside the filter region.

Capable of superior performance, but causes increase in filter

complexity

Textbook Extra Reading: Adaptive, local noise reduction filter

Adaptive median filter


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Periodic Noise

Periodic Noise can be effectively filtered using frequency

domain techniques

Periodic Noise: Concentrated bursts of energy in the Fourier

transform, at locations corresponding to the frequencies of

the periodic interference

Approach: Use selective filters to isolate noise Bandreject,

Bandpass, Notch filters


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Bandreject Filters

Bandreject Filters Remove noise from a certain location (or band) in

the frequency domain

Image corrupted with additive periodic noise can be easily removed

with a bandreject filter

Bandpass Filter the complement function of the Bandreject filter,

performs the opposite operation.

Useful for isolating noise pattern for analysis


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Example: Bandreject Filter


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Notch Filters

Notch Filter Rejects or passes frequencies in predefined

neighborhoods about a center frequency

Due to symmetry of the Fourier Transform, notch filters must

appear in symmetric pairs about the origin in order to obtain

meaningful results Available as Notch Pass and Notch Reject (one a complement

of the other)

Useful for removing periodic noise (horizontal, vertical,

diagonal periodic lines in images) which are concentrated on

one small spot in the frequency spectrum


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Example: Notch (Reject) Filters

l h l


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Example: Notch Filter

i d l


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Restoration Process Model

Degraded image

g(x,y)

Degradation function

h(x, y)

Noise modeln(x, y)

Fourier

Transform

Frequency

Domain

Filter

R(u,v)

G(u,v)

H(u,v)

N(u,v)

Inverse

FourierTransform

Restored Image ( , ) f x y

R i P


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Restoration Process

Mathematical model:

G(u,v) = H(u,v)F(u,v) + N(u,v)

where:

G(u,v) = Fourier transform of degraded image

H(u,v) = Fourier transform of degradation function

F(u,v) = Fourier transform of original imageN(u,v) = Fourier transform of additive noise function

To obtain the restored image:

where:

= the restored image, an approximation of

= the inverse Fourier transform

R(u,v) = the restoration (frequency domain) filter

( , ) f x y

1 1 ( , ) [ ( , )] [ ( , ) ( , )]f x y F u v F R u v G u v = =

1[]

D d ti F ti ?


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Degradation Function?

Question: How do we estimate the degradationfunction?

Image Observation

Experimentation

Mathematical Modeling

I Filt i


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Inverse Filtering

Uses the same model, with assumption of no noise.Fourier transform of degraded image:

Fourier transform of the original image will be:

To find the original image, take the inverse Fourier

transform of F(u,v):

( , ) ( , ) ( , ) 0G u v H u v F u v= +

( , ) 1

( , ) ( , )( , ) ( , )

G u v

F u v G u vH u v H u v= =

1 1

1

( , )( , ) [ ( , )]

( , )

1( , )

( , )

G u vf x y F F u v F

H u v

F G u v

H u v

= =

=

E l I Filt i


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Example: Inverse Filtering

50 50 25

( , ) 20 20 20

20 35 22

H u v

=

=

221

351

201

201

201

201

251

501

501

),(1

vuH

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , )

G u v H u v F u v N u v N u vF u v F u v

H u v H u v H u v H u v= = + = +

In erse Filtering Some Problems


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Inverse Filtering: Some Problems

If any points in H(u,v) are zero division by zero

Solution: Do not take zero-points of H(u,v) into account

In presence of noise:

As the value of H(u,v) becomes very small, the second term

becomes very large, and it overshadows the F(u,v)

Limit the restoration to a specific radius about the origin inthe spectrum the restoration cutoff frequency

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , )

G u v H u v F u v N u v N u vF u v F u v

H u v H u v H u v H u v= = + = +

Example: Inverse Filter


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Example: Inverse Filter

original image

Image blurred withan 11 x 11 gaussian

convolution mask

Inverse filter, with cutofffrequency = 40,

histogram stretched with 3%

low and high clipping to show

detail

Inverse filter, with cutoff

frequency = 60, histogram

stretched

Wiener Filter


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Wiener Filter

Also known as minimum mean-square error (MMSE) filter

Attempt to model the error in the restored image through

the use of statistical characteristics of noise

The average error is mathematically minimized, resulting in

the equation for Wiener filter:

where

2*

2 2( , ) ( , )

( , ) ( , )

( , )( , ) 1( , )

( , )( , ) ( , )n nl l

w S u v S u v

S u v S u v

H u vH u vR u v

H u v H u v H u v= =

+ +

*

2

2

( , ) complex conjugate of ( , )

( , ) ( , ) power spectrum of the noise

( , ) ( , ) power spectrum of the original image

n

l

H u v H u v

S u v N u v

S u v F u v

=

= =

= =

Example: Wiener Filter


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Example: Wiener Filter

Image blurred with

an 11 x 11 gaussian

convolution mask

Image with gaussian noise

variance = 5; mean = 0

Inverse filter, with cutoff


stretched with 3 % low and

high clipping to show detail

Wiener filter, with cutoff


stretched

Example: Motion Blur + Additive Noise


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Example: Motion Blur + Additive Noise


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Recommended Readings


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Recommended Readings

Digital Image Processing (3rd Edition), Gonzalez &

Woods,

Chapter 4: Image Restoration

5.1 5.4, 5.6 5.10 (Week 7)

Chapter 9: Morphological Image Processing

9.1 9.4 (Week 8)

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16404857 Digital Image Processing 07 Image Restoration Noise Removal