DIGITAL IMAGE PROCESSING

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