DIGITAL IMAGE PROCESSING

DIGITAL IMAGE PROCESSING
Instructor: P. Harikanth

DIGITAL IMAGE PROCESSING
Chapter 5 - Image Restoration and Reconstruction Instructor: P. Harikanth

Road map of chapter 5 Minimum Mean Square Error (Wiener) Filtering
5.1 5.1 5.2 5.2 5.3 5.3 5.4 5.4 5.5 5.5 5.6 5.8 5.6 5.7 5.7 5.8 Minimum Mean Square Error (Wiener) Filtering Periodic Noise Reduction by Frequency Domain Filtering Noise Models A Model of the Image Degradation/Restoration Process Inverse Filtering Restoration in the Presence of Noise Only-Spatial Filtering Estimating the degradation Function Linear, Position-Invariant Degradations 5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering (P. Harikanth)

Road map of chapter 5 Image Reconstruction from Projections
5.9 5.9 5.10 5.10 5.11 5.11 Image Reconstruction from Projections Geometric Mean Filter Constrained Least Square Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections (P. Harikanth)

Preview Goal of Restoration: Improve Image Quality
Example Degraded Image Develop Degradation Model Develop Inverse Degradation Process Knowledge Of Image Creation Process Input Image d (r,c ) Output Image I(r,c ) Apply Inverse Degradation Process (P. Harikanth)

Preview 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 degradation Stochastic methods try to find the best restoration according to a particular stochastic criterion, e.g., a least square method (P. Harikanth)

5.1 A Model of the Image Degradation/Restoration Process
(P. Harikanth)

A Model of the Image Degradation/Restoration Process
5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections (P. Harikanth)

5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Spatial domain: additive noise The 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 : blurring The degraded image in Frequency domain is Where the terms in capital letters are Fourier transforms. Objective: obtain an estimate of (P. Harikanth)

5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Three types of degradation that can be easily expressed mathematically Relative motion of the camera and object Wrong lens focus Atmospheric turbulence (P. Harikanth)

Noise Models 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 Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Spatial and Frequency Properties of Noise Some Important Noise Probability Density Functions Periodic Noise Estimation of Noise Parameters (P. Harikanth)

The Principal Source 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 Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Noise arise … During Image Acquisition Environment conditions Quality of sensing elements For x. Two factors for CCD: light level and sensor temperature Image Transmission (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Spatial and Frequency Properties of Noise Spatial and Frequency Properties of Noise Some Important Noise Probability Density Functions Periodic Noise Estimation of Noise Parameters (P. Harikanth)

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 Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections 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. (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Spatial and Frequency Properties of Noise Some Important Noise Probability Density Functions Some Important Noise Probability Density Functions Periodic Noise Estimation of Noise Parameters (P. Harikanth)

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 Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections 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. (P. Harikanth)

Most Common PDFs of Noises
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Gaussian noise Are used frequently in practice The PDF of a Gaussian random variable, Z, is given by: Rayleigh noise The PDF of Rayleigh noise: Erlang (Gamma) noise The PDF of Erlang noise : (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Exponential noise The PDF of exponential noise : Uniform noise The 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 unipolar If neither probability is zero (bipolar), and especially if they are approximately equal: salt and pepper noise (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections PDF tells how much each z value occurs. (P. Harikanth)

Noise Factors 5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Gaussian noise: electronic circuit noise and sensors noise due to poor illumination and /or temperature Rayleigh noise: helpful in characterizing noise phenomena in rang imaging Exponential and gamma noise: application in laser imaging Impulse 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)) (P. Harikanth)

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 Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Original image Degraded images Histogram (P. Harikanth)

Image Degradation with Additive Noise
Original image 5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Degraded images Histogram (P. Harikanth)

5.3 Restoration in the Presence of Noise Only - Spatial Filtering
(P. Harikanth)

Restoration in the Presence of Noise Only - Spatial Filtering
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Mean Filters Ordered-Statistic Filters Adaptive Filters (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Mean Filters Mean Filters Ordered-Statistic Filters Adaptive Filters (P. Harikanth)

mn = size of moving window
Mean Filters 5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Performance superior to the filters discussed in Section 3.6 Degradation Model: Arithmetic Mean Filter(Moving Average Filter): Computes the average value of the corrupted image g(x,y) The value of the restored image f To remove this part mn = size of moving window (P. Harikanth)

Mean Filters Geometric Mean Filter:
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Geometric Mean Filter: Smooth comparable to the arithmetic mean filter, but it tends to loss less detail. Harmonic Mean Filter: Work well for salt noise and Gaussian noise, but fails for pepper noise Works well for salt noise but fails for pepper noise (P. Harikanth)

Mean Filters Contra harmonic Mean Filter: Positive Q is suitable for
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Contra harmonic Mean Filter: Positive Q is suitable for eliminating pepper noise. Negative Q is suitable for eliminating salt noise. Q = the filter order For Q = 0, the filter reduces to an arithmetic mean filter. For Q = -1, the filter reduces to a harmonic mean filter. (P. Harikanth)

Arithmetic Mean Filter - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections (P. Harikanth)

Geometric Mean Filter - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image Corrupted by AWGN Original Image Image obtained using a 3x3 arithmetic mean filter Image obtained using a 3x3 geometric mean filter AWGN: Additive White Gaussian Noise (P. Harikanth)

Geometric Mean Filter - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image corrupted by pepper noise with prob. = 0.1 Image corrupted by salt noise with prob. = 0.1 Image obtained using a 3x3 contraharmonic mean filter With Q = 1.5 Image obtained using a 3x3 contra-harmonic mean filter With Q=-1.5 (P. Harikanth)

Contraharmonic Filters: Incorrect Use
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image corrupted by pepper noise with prob. = 0.1 Image corrupted by salt noise with prob. = 0.1 Image obtained using a 3x3 contraharmonic mean filter With Q=-1.5 Image obtained using a 3x3 contraharmonic mean filter With Q=1.5 (P. Harikanth)

Contra harmonic Filters - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections The Contraharmonic mean filter works well for images containing salt or pepper type noise (P. Harikanth)

Contra harmonic Filters - Example

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Mean Filters Ordered-Statistic Filters Ordered-Statistic Filters Adaptive Filters (P. Harikanth)

Order-Statistic Filters: Revisit
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Subimage Original image Statistic parameters Mean, Median, Mode, Min, Max, Etc. Moving Window (P. Harikanth) Output image

Order-Statistics Filters
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Median filters: Are particularly effective in the presence of both bipolar and unipolar impulse noise Max filters (Fig. 5.8) Max filter: reduce low values caused by pepper noise Min filters (Fig. 5.8) Min filter: reduce high values caused by salt noise Reduce “dark” noise (pepper noise) Reduce “bright” noise (salt noise) (P. Harikanth)

Order-Statistics Filters
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Midpoint filter Combines order statistics and averaging (P. Harikanth)

Median Filter: How it works
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections A median filter is good for removing impulse, isolated noise Salt noise Pepper noise Median Moving Window Degraded image Sorted Array Salt noise Pepper noise Filter output Normally, impulse noise has high magnitude and is solated. When we sort pixels in the moving window, noise pixels are usually at the ends of the array. (P. Harikanth) Therefore, it’s rare that the noise pixel will be a median value.

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections 1 2 Image corrupted by salt and pepper noise with pa=pb= 0.1 3 4 Images obtained using a 3x3 median filter (P. Harikanth)

Max and Min Filters - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image corrupted by salt noise with prob. = 0.1 Image corrupted by pepper noise with prob. = 0.1 Image obtained using a 3x3 min filter Image obtained using a 3x3 max filter (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Various Windows size for Max and Min Filters: (P. Harikanth)

Alpha -trimmed Mean Filter
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Formula: where gr(s,t) represent the remaining mn-d pixels after removing the d/2 highest and d/2 lowest values of g(s,t). This filter is useful in situations involving multiple types of noise such as a combination of salt-and-pepper and Gaussian noise. (P. Harikanth)

Alpha-trimmed Mean Filter - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image corrupted by additive uniform noise Image additionally corrupted by additive salt-and- pepper noise 1 2 Image 2 obtained using a 5x5 arithmetic mean filter Image 2 obtained using a 5x5 geometric mean filter (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image additionally corrupted by additive salt-and- pepper noise Image corrupted by additive uniform noise 1 2 Image 2 obtained using a 5x5 alpha- trimmed mean filter with d = 5 Image 2 obtained using a 5x5 median filter (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image obtained using a 5x5 arithmetic mean filter Image obtained using a 5x5 geometric mean filter Image obtained using a 5x5 alpha- trimmed mean filter with d = 5 Image obtained using a 5x5 median filter (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Mean Filters Ordered-Statistic Filters Adaptive Filters Adaptive Filters (P. Harikanth)

Adaptive Filters Global Concept:
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Global Concept: Can apply to an image with regard to how image characteristics vary from one point to another Filter behavior depends on statistical characteristics of local areas inside m×n moving window More complex but superior performance compared with “fixed” filters Statistical characteristics: Local mean: Local Variance: Noise variance (P. Harikanth)

Adaptive, Local Noise Reduction Filters
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections The response of the filter based on four quantities: The value of the noise image The variance of the noise corrupting f(x,y) to form g(x,y) The local means of the pixel in Sxy The local variance of the pixels in Sxy The filter can be proceeded as follows: The variance of the noise sh2 is zero (zero noise): g(x,y)=f(x,y) High local variance (edges) relative to sh2 : edges that should be preserved; return a value close to g(x,y) The two variances are equal: return the arithmetic mean value of the pixels in Sxy (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Purpose: Want to preserve edges Concept: 1. If sh2 is zero,  No noise the filter should return g(x,y) because g(x,y) = f(x,y) 2. If sL2 is high relative to sh2,  Edges (should be preserved), the filter should return the value close to g(x,y) 3. If sL2 = sh2,  Areas inside objects the filter should return the arithmetic mean value mL Formula: (P. Harikanth)

5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections The variance of the overall noise needs to be estimated. The only quantities that needs to be known or estimated: The variance of the overall noise then the ratio (Eq ) is set to 1: Prevent negative gray levels If negative values occur, rescale the gray values at the end (lost of the dynamic range) Similar noise removal results compared to other mean filter (P. Harikanth)

Adaptive Noise Reduction Filter - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image corrupted by additive Gaussian noise with zero mean and s2=1000 Image obtained using a 7x7 arithmetic mean filter Image obtained using a 7x7 adaptive noise reduction filter Image obtained using a 7x7 geometric mean filter (P. Harikanth)

Adaptive Noise Reduction Filter - Example

Adaptive Median Filter
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Median filter perform well as long as spatial density of the impulse noise is not vary large (Pa and Pb <0.2 ) The adaptive filter Can handle impulse noise with probabilities even larger than Pa and Pb >0.2 Preserve detail while smoothing non-impulse noise Change (increase) Sxy during filter operation (P. Harikanth)

Adaptive Median Filter
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Purpose: Want to remove impulse noise while preserving edges Algorithm: Level A: A1= zmedian – zmin A2= zmedian – zmax If A1 > 0 and A2 < 0, go to level B Else increase window size If window size <= Smax repeat level A Else return zxy Level B: B1= zxy – zmin B2= zxy – zmax If B1 > 0 and B2 < 0, return zxy Else return zmedian where zmin = minimum gray level value in Sxy zmax = maximum gray level value in Sxy zmedian = median of gray levels in Sxy zxy = gray level value at pixel (x,y) Smax = maximum allowed size of Sxy (P. Harikanth)

Adaptive Median Filter: How it works
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Level A: A1= zmedian – zmin A2= zmedian – zmax Determine whether zmedian is an impulse or not If A1 > 0 and A2 < 0, go to level B Else  Window is not big enough increase window size If window size <= Smax repeat level A Else return zxy (P. Harikanth)

Adaptive Median Filter: How it works
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Level B:  zmedian is not an impulse B1= zxy – zmin B2= zxy – zmax If B1 > 0 and B2 < 0,  zxy is not an impulse return zxy  to preserve original details Else return zmedian  to remove impulse Determine whether zxy is an impulse or not (P. Harikanth)

Adaptive Median Filter - Example
5.1- A Model of the Image Degradation/Restoration Process 5.2- Noise Models 5.3- Restoration in the Presence of Noise Only-Spatial Filtering 5.4- Periodic Noise Reduction by Frequency Domain Filtering 5.5 - Linear, Position-Invariant Degradations 5.6- Estimating the degradation Function 5.7- Inverse Filtering 5.8- Minimum Mean Square Error (Wiener) Filtering 5.9- Constrained Least Square Filtering 5.10- Geometric Mean Filter 5.11- Image Reconstruction from Projections Image corrupted by salt-and-pepper noise with pa=pb= 0.25 Image obtained using a 7x7 median filter Image obtained using an adaptive median filter with Smax = 7 More small details are preserved (P. Harikanth)

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