– What is image restoration? – Noise and images – Noise models – Noise removal using spatial domain filtering
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What is Image Restoration?
Image restoration attempts to restore images that have been degraded
b/a
2
2
Exponential noise:
ae az p( z ) 0 ＝1 / a;
for
z 0 for z 0
1/ a
2
2
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Noise Models
Uniform noise:
1 （b－a） p( z ) 0 ＝(a b) / 2; if a z b
otherwise
2 (b a) 2 / 12
Impulse noise:
pa p( z ) pb 0
for for
za z b
otherwise
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Noise Models
Impulse noise is also called salt-and-pepper noise. For pa=pb=0.05
Input image
Degraded image
Noise level p=0.05 means that approximately 5% of pixels are contaminated by salt or pepper noise (highlighted by red color)
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Spatial filtering • Inverse filtering • Wiener filtering • >>filter2
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Image Restoration General
One has to have some a priori knowledge about the degragation process. Usually we need to know: The noise in the original image Model for degragation Some information from original image
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Noise Models
There are many different models for the image noise term η(x, y):
– Gaussian
• Most common model
– Rayleigh – Erlang – Exponential – Uniform – Impulse
– Identify the degradation process and attempt to reverse it – Similar to image enhancement, but more objective
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Enhancement vs. Restoration What is Image Restoration?
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What is Image Restoration?
Image Preprocessing
Enhancement
Restoration
Spatial Domain
Frequency Domain
Spatial Domain
Frequency Domain
Point Processing Spatial filtering Filtering • >>imadjust • >>filter2 • >>fft2/ifft2 • >>histeq • >>fftshift
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Noise and Images
The sources of noise in digital images arise during image acquisition (digitization) and transmission
– Imaging sensors can be affected by ambient conditions – Interference can be added to an image during transmission
for for 2
z a z a
＝a b / 4 ;
b( 4 ) 4
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Noise Models
Erlang (Gamma) noise:
p( z ) 0 ＝b / a;
a b z b1 az e （b－1） !
for z 0 for z 0
g ( x, y) h( x, y) * f ( x, y) ( x, y)
In frequency domain representation:
G(u, v) H (u, v) F (u, v) N (u, v)
Where: f(x,y) is the input image, g(x,y) is the degraded image, h(x,y) is the degradation function, and ( x, y ) is the additive noise.
128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 128 128 128 128 128 128 128 128 255 128 128 0 128 128 128 128 128 128 0 128 128 128 255 128 128 128 128 128 128 0 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 255 128 128 128 128 128 128 128 255 128 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 128 128 255 128
MATLAB Command
>Y = IMNOISE(X,'salt & pepper',p)
Notes: Example: impulsenoise.m
The intensity of input images is assumed to be normalized to [0,1]. If X is not double, you need to do normalization first, i.e., X=X/255; If X is uint8, MATLAB would do the normalization automatically The default value of p is 0.05 (i.e., 5 percent of pixels are contaminated) Imnoise function can produce other types of noise as well (you need to change the noise type ‘salt & pepper’)
Unconstrained
• Inverse Filter • Pseudo-inverse Filter
Constrained
• Wiener Filter
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Restoration Model
The objective of restoration is to obtain an estimate f ( x, y) of the original image f(x,y). Generally, more we know about H, and , the the closer f ( x, y) will be to f(x,y). The approach used throughout most of this chapter is based on various types of image restoration filters.
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A Model of Image Degradation and Restoration
The degradation process is modeled as a degradation function that together with an additive noise term.