image restoration
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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
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
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
Restoration by spatial filtering
Restoration by spatial filtering
Restoration by spatial filtering
Q is the order of filter
Restoration by spatial filtering
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
Restoration by spatial filtering
Restoration by spatial filtering
wrong
Restoration by spatial filtering
-- Repeated passes of median filter tend to blur the image.-- Keep the number of passes as low as possible.
Restoration by spatial filtering
Fig. 8 next page
Restoration by spatial filtering
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
Restoration by spatial filtering
Restoration by spatial filtering
Restoration by spatial filtering
Restoration by spatial filtering
Adaptive filtering
Restoration by spatial filtering
Restoration by spatial filtering
Is Z_med impulse?
Is Z_xy impulse?
Restoration by spatial filtering
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.
Periodic Noise Reduction(Frequency Domain Filtering)
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
Periodic Noise Reduction(Frequency Domain Filtering)
Periodic Noise Reduction(Frequency Domain Filtering)
Band-Pass Filters Opposite operation of a band-reject fiter
1 ( , )bp brH H u v
Periodic Noise Reduction(Frequency Domain Filtering)
Notch Filters Rejects (or passes) frequencies in predefined neighborhoods
about a center frequency
Ideal
Butterworth
Gaussian
Must appear in symmetric pairs about the origin.
Periodic Noise Reduction(Frequency Domain Filtering)
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
Estimation of degradation functions
Estimation of degradation functions
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
Estimation of degradation functions (model bases)
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
Estimation of degradation functions (model bases)
Inverse filtering
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
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
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.
Geometric transformations
Geometric transformations
Geometric transformations
Geometric transformations
Geometric transformations
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