Image Restoration

Image Restoration

Digital Image Processing

Content

Introduction Image degradation/restoration model Noise models Restoration by spatial filtering Estimation of degradation functions Inverse filtering Wiener filtering Geometric transformation

Introduction

Objective of image restoration to recover a distorted image to the original

form based on idealized models. The distortion is due to

Image degradation in sensing environment e.g. random atmospheric turbulence

Noisy degradation from sensor noise. Blurring degradation due to sensors

• e.g. camera motion or out-of-focus

Geometric distortion• e.g. earth photos taken by a camera in a satellite

Introduction

Enhancement Concerning the

extraction of image features

Difficult to quantify performance

Subjective; making an image “look better”

Restoration Concerning the

restoration of degradation

Performance can be quantified

Objective; recovering the original image

Image degradation / restoration model

Noise models

Assuming degradation only due to additive noise (H = 1)

Noise from sensors Electronic circuits Light level Sensor temperature

Noise from environment Lightening Atmospheric disturbance Other strong electric/magnetic signals

Noise models

Assuming that noise is independent of spatial coordinates, and uncorrelated with respect to the image content

Noise models

Noise models

Other common noise modelsRayleigh noiseGamma noiseExponential noiseUniform noise

Noise Models

Rayleigh Noise

Gamma(Erlang) Noise

Exponential Noise

2( ) /2( ) ( ) for

0 for

z a bp z z a e z a

bz a

( ) for 0

=0 for 0

azp z ae z

z

Noise models

-3-levels-simple constant areas(spans from black to white)

paper

salt

Additive Noise

Histograms

Periodic Noise

Noise components

Are generated due to electrical or electromechanical interference during image acquisition

Periodic noise can be reduced in via frequency domain

Restoration by spatial filtering

Noise is unknown

Spatial filtering is appropriate when only additive noise is present

Restoration by spatial filtering is the set of coordinates in

a rectangular subimage window

of size centered at point

( , )

xyS

m n

x y


Q is the order of filter


Noise level is Mean =0Variance = 400

Restoration by spatial filtering Mean filters (noise reduced by blurring)

Arithmetic mean filter and geometric mean filter are well suited for random noise such as Gaussian noise

Contraharmonic mean filter is well suited for impulse noise• Disadvantage: must know pepper noise or salt noise in

advance


wrong


-- Repeated passes of median filter tend to blur the image.-- Keep the number of passes as low as possible.


Fig. 8 next page


Pepper noise Salt noise

-High level of noise large filter

-Median and alpha-trimmed filter performed better- Alpha-trimmed did better than median filter

Restoration by spatial filtering Filters discussed so far

Do not consider image characteristics Adaptive filters to be discussed

Behaviors based on statistical characteristics of the subimage under a filter window

Better performance More complicated Adaptive, local noise reduction filter Adaptive median filter


Adaptive filtering


Is Z_med impulse?

Is Z_xy impulse?

Periodic Noise Reduction(Frequency Domain Filtering)

Band-Reject Filters Ideal Band-reject Filter

0

0 0

0

( , ) 1 if ( , )2

0 if ( , )2 2

1 if ( , )2

WH u v D u v D

W WD D u v D

WD u v D

-D(u,v) =distance from the origin of the centered freq. rectangle-W =width of the band-D0=Radial center of the band.


Butterworth Band-Reject Filter of order n

Gaussian Band-Reject Filter

2

2 20

1( , )

( , )1

( , )

nH u vD u v W

D u v D

22 20( , )1

2 ( , )( , ) 1

D u v D

D u v WH u v e


Band-Pass Filters Opposite operation of a band-reject fiter

1 ( , )bp brH H u v


Notch Filters Rejects (or passes) frequencies in predefined neighborhoods

about a center frequency

Ideal

Butterworth

Gaussian

Must appear in symmetric pairs about the origin.


Notch FiltersIdeal

1 0 2 0( , ) 0 if ( , ) or ( , )

1 otherwise

H u v D u v D D u v D

1/ 22 21 0 0

1/ 22 22 0 0

( , ) ( / 2 ) ( / 2 )

and

( , ) ( / 2 ) ( / 2 )

D u v u M u v N v

D u v u M u v N v

Center frequency components

Shift with respect to the center

Horizontal lines of the noise pattern I can be seen

Notch pass filter

Optimum Notch Filtering

Several pairs of components are present more than just one sinusoidal component

Optimum Notch Filtering

Estimation of degradation functions

Estimation of degradation functions (model bases)

Mathematical model for uniform linear motion between the image and the sensor during image acquisition Let x0(t) and y0(t) denote time varying components of

motion in the x- and y-directions Degradation model

0 00( , ) ( ), ( ) where ( , ) is the blurred image

Tg x y f x x t y y t dt g x y

0 0

0 0

2 ( ) ( )

0

2 ( ) ( )

0

( , ) ( , ) exp where

( , ) is degradation function

T j ux t vy t

T j ux t vy t

G u v F u v dt

H u v e dt


Uniform Linear motion in the x and y direction x0(t)=at/T and y0(t)=bt/T where the image has been displaced by a total distance a in the x-direction and b in the y-direction

( )( , ) sin ( )( )

j ua vbTH u v ua vb e

ua vb

Inverse filtering

Inverse filtering5 / 62 2( / 2) ( / 2)

( , )k u M v N

H u v e Degradation function

Cutting off values of the ratio outside a radius of 40, 70,85.

Cur

tain

of

nois

e

Wiener filtering

Wiener filtering

( , ) is the degradation functionH u v

Wiener filtering

White noinse

Wiener filtering

Geometric transformations Objective: to eliminate geometric

distortion that occurs when an image is captured

Examples of geometric distortion Pincushion distortion (associated with

zoom lenses)

Geometric transformations


Two steps in geometric transformationSpatial transformation: rearrangement

of pixels on the image planeGray-level interpolation: assignment of

gray levels to pixels in the spatially transformed image

Geometric transformations • Solution

To formulate the spatial relocation of pixels by the use of the corresponding tiepoints

Tiepoints: a subset of pixels whose locations in the input (distorted) and output (restored) images are known.

Image Restoration

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