digital image processing · image restoration • image restoration: recover an image that has been...

Digital Image ProcessingCOSC 6380/4393

Lecture – 21Mar 30th, 2020Pranav Mantini

Objective

• To introduce a model for image restoration• To study noise that arise in images

Image Restoration

• Similar to enhancement, the goal is to improve an image.

Image Restoration

• Similar to enhancement, the goal is to improve an image.

Enhancement Restoration

Subjective process Objective process

Use psychophysical aspects of HVS to manipulate image

Use prior knowledge of degradation torestore image

Manipulate image to please viewer Recover original image by removing degradation

Image Enhancement

• Enhancement: – Point operations

• Full contrast stretch• Histogram equalization• Histogram flattening

• Restoration:– Removing blur

• By applying a deblurring function

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Image Restoration

• Image restoration: recover an image that has been degraded by using a prior knowledge of the degradation phenomenon.

• Model the degradation and applying the inverse process in order to recover the original image.

A Model of Image Degradation/Restoration Process

𝑓𝑓 𝑥𝑥, 𝑦𝑦


𝑓𝑓 𝑥𝑥, 𝑦𝑦

MotionTurbulence


ℎ 𝑥𝑥, 𝑦𝑦 𝑓𝑓 𝑥𝑥,𝑦𝑦

MotionTurbulence

⊗



MotionTurbulence

Noise:Image acquisition: light, temperature, etc.

Transmission: atmospheric , electronic disturbance

⊗



MotionTurbulence



𝜂𝜂 𝑥𝑥, 𝑦𝑦Additive noise

⊗



MotionTurbulence



𝜂𝜂 𝑥𝑥, 𝑦𝑦Additive noise

⊗

⊗



𝜂𝜂 𝑥𝑥, 𝑦𝑦

⊗

⊗

Ideal image




⊗

⊗

Ideal image⊗

Degradation function




⊗

⊗

Ideal image⊗


𝑔𝑔(𝑥𝑥,𝑦𝑦)




⊗

⊗

Ideal image⊗



Goal of image restoration is to estimate 𝑓𝑓′ 𝑥𝑥,𝑦𝑦that is as close as possible to 𝑓𝑓(𝑥𝑥,𝑦𝑦)




⊗

⊗

Ideal image⊗



Noise

• Source• Types of Noise• Estimation of Noise• Image Restoration

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Noise Sources

• The principal sources of noise in digital images arise during image acquisition and/or transmission

Image acquisition (CCD cameras)e.g., light levels, sensor temperature, etc.

Transmission (interference on channel)e.g., lightning or other atmospheric disturbance in wireless network

Types of Noise

• Spatially independent• Spatially dependent

Types of Noise

• Spatially independent

• Spatially dependent

The noise at location 𝑥𝑥, 𝑦𝑦 , 𝜂𝜂 𝑥𝑥,𝑦𝑦 is defined by a function 𝐻𝐻 that is not dependent on the (𝑥𝑥, 𝑦𝑦)

𝜂𝜂 𝑥𝑥, 𝑦𝑦 → 𝐻𝐻

Types of Noise





The noise at location 𝑥𝑥, 𝑦𝑦 , 𝜂𝜂 𝑥𝑥,𝑦𝑦 is defined by a function 𝐻𝐻 that is dependent on the location (𝑥𝑥,𝑦𝑦)

𝜂𝜂 𝑥𝑥, 𝑦𝑦 → 𝐻𝐻(𝑥𝑥,𝑦𝑦)

Types of Noise





The noise at location 𝑥𝑥, 𝑦𝑦 , 𝜂𝜂 𝑥𝑥,𝑦𝑦 is defined by a function 𝐻𝐻 that is dependent on the location (𝑥𝑥,𝑦𝑦)

𝜂𝜂 𝑥𝑥, 𝑦𝑦 → 𝐻𝐻(𝑥𝑥,𝑦𝑦)

Statistical noise

Periodic noise

Types of Noise


• Statistical noise:– Defined by the function H (which usually is a

probability distribution)– Different H arise in various aspects of imaging.– H is parametric in nature



Gaussian Noise

Electronic circuit noiseSensor noise due to poor illuminationSensor noise due to high temperature

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Gaussian Noise

2 2( ) /2

The PDF of Gaussian random variable, z, is given by1 ( )

2z zp z e σ

πσ− −=

where, represents intensity

is the mean (average) value of z is the standard deviation

z

zσ

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Gaussian Noise

2 2( ) /2

The PDF of Gaussian random variable, z, is given by1 ( )

2z zp z e σ

πσ− −=

70% of its values will be in the range

95% of its values will be in the range

[ ])(),( σµσµ +−

[ ])2(),2( σµσµ +−

Rayleigh Noise

Range imaging: Image showing the distance to points in a scene from a specific point (Radar).

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Rayleigh Noise

2( ) /

The PDF of Rayleigh noise is given by2 ( ) for

( )0 for

z a bz a e z ap z b

z a

− − − ≥= <

2

The mean and variance of this density are given by

/ 4(4 )

4

z a bb

ππσ

= +−

=

Erlang (gamma) and Exponential Noise

• Laser imaging: The Laser images are used to get 3D images with the help of laser. The images are captured by sensors which are mounted on laser.

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Erlang (Gamma) Noise

1

The PDF of Erlang noise is given by

for 0 ( ) ( 1)!

0 for

b baza z e z

p z bz a

−−

≥= − <

2 2


/ /

z b ab aσ

=

=

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Exponential Noise

The PDF of exponential noise is given by

for 0 ( )

0 for

azae zp z

z a

− ≥=

<

2 2


1/ 1/

z aaσ

=

=

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Faulty switching results in quick transients

Impulse noise

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Faulty switching results in quick transients

Impulse noise

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Impulse (Salt-and-Pepper) Noise

The PDF of (bipolar) impulse noise is given by for

( ) for 0 otherwise

a

b

P z ap z P z b

== =

If either or is zero, the impulse noise is calleda bP Punipolar

if , gray-level will appear as a light dot, while level will appear like a dark dot.

b a ba

>

Uniform noise

• Not common in practical situations• Basis for numerous random number

generators

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Uniform Noise

The PDF of uniform noise is given by1 for a

( )0 otherwise

z bp z b a

≤ ≤= −

2 2


( ) / 2 ( ) /12

z a bb aσ

= +

= −

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Examples of Noise: Original Image

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Examples of Noise: Noisy Images(1)

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Examples of Noise: Noisy Images(2)

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Periodic Noise

► Periodic noise in an image arises typically from electrical or electromechanical interference during image acquisition.

► It is a type of spatially dependent noise

► Periodic noise can be reduced significantly via frequency domain filtering

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An Example of Periodic Noise

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An Example of Periodic Noise

Estimation of Noise

• Periodic noise:– Produces frequency spikes– Estimated by inspecting the Fourier spectrum.– Can be automated if the spikes are pronounced– Infer periodicity directly from images (only

simplistic cases)

Estimation of Noise

• Statistical noise:– Known partially from the sensor specifications– If imaging system is available, capture a set of flat

images (solid grey board) and study the histogram– If only images are available, select an area of

image with reasonably constant background and study the histogram.

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Estimation of Noise Parameters

The shape of the histogram identifies the closest PDF match

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Rayleigh Noise

2( ) /

The PDF of Rayleigh noise is given by2 ( ) for

( )0 for

z a bz a e z ap z b

z a

− − − ≥= <

2


/ 4(4 )

4

z a bb

ππσ

= +−

=

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Estimation of Noise Parameters

Consider a subimage denoted by , and let ( ), 0, 1, ..., -1, denote the probability estimates of the intensities of the pixels in . The mean and variance of the pixels in :

s iS p z i LS

S

=

1

01

2 2

0

( )

and ( ) ( )

L

i s ii

L

i s ii

z z p z

z z p zσ

−

=

−

=

=

= −

∑

∑

Next

Restoration: How to remove noise

End of Lecture

digital image processing · image restoration • image restoration: recover an image that has been...

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