Post on 23-Feb-2016
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DIGITAL IMAGE PROCESSING
Instructors: Dr J. Shanbehzadeh
Shanbehzadeh@gmail.com
M.Gholizadehmhdgholizadeh@gmail.com
DIGITAL IMAGE PROCESSING
Instructors: Dr J. Shanbehzadeh
Shanbehzadeh@gmail.com
M.Gholizadehmhdgholizadeh@gmail.com
Chapter 5 - Image Restoration and Reconstruction
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Road map of chapter 5
5.1 5.3 5.4 5.55.1
5.1- A Model of the Image Degradation/Restoration Process5.2- Noise Models5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering
A Model of the Image Degradation/Restoration Process
5.25.2
Noise ModelsRestoration in the Presence of Noise Only-Spatial Filtering
5.3 5.4
Periodic Noise Reduction by Frequency Domain Filtering
5.5
Linear, Position-Invariant Degradations
5.65.6
Estimating the degradation Function
5.75.7 5.85.8
Inverse FilteringMinimum Mean Square Error (Wiener) Filtering
( J.Shanbehzadeh M.Gholizadeh )
Road map of chapter 5
5.9 5.115.9
5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
Geometric Mean Filter
5.105.10
Constrained Least Square FilteringImage Reconstruction from Projections
5.11
( J.Shanbehzadeh M.Gholizadeh )
Preview
Goal of Restoration: Improve Image Quality
Example Degraded Image
Knowledge Of Image Creation
Process
Develop Degradation
Model
Develop Inverse Degradation
Process
Apply Inverse Degradation
Process
Input Image d (r,c )
Output Image I(r,c )
( J.Shanbehzadeh M.Gholizadeh )
Restoration is an objective process compared to image enhancement: Image restoration is to restore a degraded image back to the original image.Image Enhancement is to manipulate the image so that it is suitable for a specific application.
Contrast stretching is an enhancement technique while debluring function is considered a restoration.Only consider in this chapter a degraded digital image.Restoration can be categorized as two groups:
Deterministic methods are applicable to images with little noise and a known degradationStochastic methods try to find the best restoration according to a particular stochastic criterion, e.g., a least square method
Preview
( J.Shanbehzadeh M.Gholizadeh )
5.1 A Model of the Image Degradation/Restoration Process
( J.Shanbehzadeh M.Gholizadeh )
A Model of the Image Degradation/Restoration Process
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Spatial domain: additive noiseThe degraded image in Spatial domain is
where h(x,y) is a system that causes image distortion and h(x,y) is noise.
Frequency domain : blurringThe degraded image in Frequency domain is
Where the terms in capital letters are Fourier transforms.
Objective: obtain an estimate of
),(),(),(),( yxyxhyxfyxg
),(),(),(),( vuNvuFvuHvuG
A Model of the Image Degradation/Restoration Process
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Three types of degradation that can be easily expressed mathematically
Relative motion of the camera and object
Wrong lens focus
Atmospheric turbulence
UVVTuVUH
)sin(),(
FunctionBesseltheisJararJ
VUH ..)(),( 11
6/5)( 22
),( vuceVUH
A Model of the Image Degradation/Restoration Process
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Noise Models
Spatial and Frequency Properties of Noise
Some Important Noise Probability Density Functions
Periodic Noise
Estimation of Noise Parameters
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
The Principal Source of Noise
Noise arise …During Image Acquisition
Environment conditionsQuality of sensing elementsFor x. Two factors for CCD: light level and sensor temperature
Image Transmission
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Noise Models
Spatial and Frequency Properties of Noise
Some Important Noise Probability Density Functions
Periodic Noise
Estimation of Noise Parameters
Spatial and Frequency Properties of Noise
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Spatial and Frequency Properties of Noise
White noise: The Fourier spectrum of noise is constant.This terminology is a carryover from the physical properties of white light, which contains nearly all frequencies in the visible spectrum in equal properties.
We assume in this chapter: Noise is independent of spatial coordinates.
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Noise Models
Spatial and Frequency Properties of Noise
Some Important Noise Probability Density Functions
Periodic Noise
Estimation of Noise Parameters
Some Important Noise Probability Density Functions
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Noise Probability Density Functions
Noise cannot be predicted but can be approximately described in statistical way using the probability density function (PDF).
The statistical properties of the gray level of spatial noise can be considered random variables characterized by a PDF.
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Most Common PDFs of Noises
Gaussian noiseAre used frequently in practiceThe PDF of a Gaussian random variable, Z, is given by:
Rayleigh noiseThe PDF of Rayleigh noise:
Erlang (Gamma) noise The PDF of Erlang noise :
22 2/)(
21)(
zezp
azfor 0
for )(2)(
/)( 2
azeazbzp
baz
azfor 0
for )!1()(
/)(1
2
azebzp
bazbb za
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Most Common PDFs of Noises
Exponential noiseThe PDF of exponential noise :
Uniform noiseThe PDF of uniform noise is given by:
Impulse noise (Salt and pepper)The PDF of impulse noise is given by:
If b>a gray level b will appear as a light dot; If either Pa or Pb is zero, the impulse is called unipolarIf neither probability is zero (bipolar), and especially if they are approximately equal: salt and pepper noise
azaezp )(
otherwise 0
afor a-b
1)( bzzp
otherwise 0for for
)( bzPazP
zp b
a
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Most Common PDFs of Noises
PDF tells how much each z value occurs.
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Noise Factors
Gaussian noise: electronic circuit noise and sensors noise due to poor illumination and /or temperatureRayleigh noise: helpful in characterizing noise phenomena in rang imagingExponential and gamma noise: application in laser imagingImpulse noise: found in quick transient such as faulty-switching ; is the only one that is visually indicative Uniform noise: basis for random number generator
Difficult to differentiate visually between the five image (Fig 5.4(a) ~Fig5.4(b))
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
Image Degradation with Additive Noise
Degraded imagesOriginal image
Histogram
),(),(),( yxyxfyxg 5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )
),(),(),( yxyxfyxg
Original image
Histogram
Degraded images
Image Degradation with Additive Noise
5.1- A Model of the Image Degradation/Restoration Process
5.2- Noise Models
5.3- Restoration in the Presence of Noise Only-Spatial Filtering5.4- Periodic Noise Reduction by Frequency Domain Filtering5.5 - Linear, Position-Invariant Degradations5.6- Estimating the degradation Function5.7- Inverse Filtering5.8- Minimum Mean Square Error (Wiener) Filtering5.9- Constrained Least Square Filtering5.10- Geometric Mean Filter5.11- Image Reconstruction from Projections