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)

5.4 Periodic Noise Reduction by Frequency Domain Filtering
(P. Harikanth)

Bandreject Filters Bandpass Filters Notch 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 Bandreject Filters Bandpass Filters Notch Filters (P. Harikanth)

Periodic Noise Reduction by Frequency Domain 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 Degraded Image d(r,c) D(U,V) Degraded Function h(r,c) Fourier Transform Frequency Domain Filter R(u,v) H(U,V) N(U,V) Noise Model n(r,c) Inverse Fourier Transform Restored Image (P. Harikanth)

Bandreject Filters Bandreject Filters Bandpass Filters Notch 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 Bandreject Filters Bandreject Filters Bandpass Filters Notch Filters (P. Harikanth)

Bandreject 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 Use to eliminate frequency components in some bands. Ideal Band-reject Filter: -D(u,v) =distance from the origin of the centered freq. rectangle -W =width of the band -D0=Radial center of the band. (P. Harikanth)

Bandreject Filters Degraded image DFT Bandreject filter Restored 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 image DFT Periodic noise can be reduced by setting frequency components corresponding to noise to zero. Bandreject filter Restored image (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 Bandreject Filters Bandpass Filters Bandpass Filters Notch Filters (P. Harikanth)

Periodic noise from the previous slide that is Filtered out.
Bandpass 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 Opposite operation of a band-reject filter: Periodic noise from the previous slide that is Filtered out. (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 Bandreject Filters Bandpass Filters Notch Filters Notch Filters (P. Harikanth)

Must appear in symmetric pairs about the origin.
Notch 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 A notch reject filter is used to eliminate some frequency components. Rejects (or passes) frequencies in predefined neighborhoods about a center frequency. Ideal Must appear in symmetric pairs about the origin. Butterworth Gaussian (P. Harikanth)

Notch reject 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 Degraded image Notch filter (freq. Domain) DFT (P. Harikanth) Noise Restored image

Notch reject 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)

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 Bandreject Filters Bandpass Filters Notch Filters (P. Harikanth)

Image Degraded by Periodic Noise
Degraded 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 DFT (no shift) Several pairs of components are present  more than just one sinusoidal component (P. Harikanth) DFT of noise Noise Restored image

5.6 Estimating the degradation Function
(P. Harikanth)

Estimation by Image Observation
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 Estimation by Image Observation Estimation by Experimentation Estimation by Modeling (P. Harikanth)

Estimating the Degradation Function
Degradation model: 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 or Purpose: To estimate h(x,y) or H(u,v) Why? If we know exactly h(x,y), regardless of noise, we can do deconvolution to get f(x,y) back from g(x,y). Methods: 1. Estimation by Image Observation 2. Estimation by Experiment 3. Estimation by Modeling (P. Harikanth)

Estimating the Degradation Function
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 Look for the information in the image itself: search for the small section of image containing simple structure (edge, point) Select a small section from the degraded image Reconstruct an unblurred image of the same size The degradation function can be estimated by : (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 Estimation by Image Observation Estimation by Image Observation Estimation by Experimentation Estimation by Modeling (P. Harikanth)

Original image (unknown) Degraded 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 g(x,y) f(x,y)*h(x,y) f(x,y) Observation Subimage DFT Estimated Transfer function Restoration process by estimation DFT This case is used when we know only g(x,y) and cannot repeat the experiment! Reconstructed Subimage (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 Estimation by Image Observation Estimation by Experimentation Estimation by Experimentation Estimation by Modeling (P. Harikanth)

Estimation by Experiment
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 If we have the equipment used to acquire degraded image we can obtain accurate estimation of the degradation Obtain an impulse response of the degradation using the same system setting A linear space-invariant system is characterized completely by its impulse response (P. Harikanth)

Used when we have the same equipment set up and can repeat the experiment. 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 Response image from the system Input impulse image System H( ) DFT DFT (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 Estimation by Image Observation Estimation by Experimentation Estimation by Modeling Estimation by Modeling (P. Harikanth)

Estimation by Modeling
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 Mathematical model of degradation can be for example atmosphere turbulence Hufnagel & Stanley (1964) has established a degradation model due to atmospheric turbulence K is a parameter to be determined by experiments because it changes with the nature of turbulence (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 Used when we know physical mechanism underlying the image formation process that can be expressed mathematically. Original image Severe turbulence Example: Atmospheric Turbulence model k = Mild turbulence Low turbulence k = 0.001 k = (P. Harikanth)

Estimation by Modeling: Motion Blurring
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 Assume that camera velocity is The blurred image is obtained by where T = exposure time. (P. Harikanth)

Motion Blurring - 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 For constant motion Original image Motion blurred image a = b = 0.1, T = 1 (P. Harikanth)

Blur Linear in one direction Horizontal Vertical Diagonal
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 Linear in one direction Horizontal Vertical Diagonal (P. Harikanth)

PSF (Point Spread Function)
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 2D Equivalent to Impulse Response What happen to a single point of light when it passes through a system? PSF describes a LSI system In practise PSF should be estimated (P. Harikanth)

Typical Blur Mask Coefficients

5.7 Inverse Filtering (P. Harikanth)

Inverse 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 (P. Harikanth)

Inverse Filtering Based on properties of the Fourier transforms
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 Based on properties of the Fourier transforms Assume degradation can be expressed as convolution After applying the Fourier transform to Eq. (a), we get An estimate Fˆ(u,v) of the transform of the original image (P. Harikanth)

Inverse 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 (P. Harikanth)

Inverse 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 The degradation can be eliminated using the restoration filter with a transfer function that is inverse to the degradation h. The Fourier transform of the inverse filter is then expressed as H-1(u,v) We obtain the original undegraded image F from its degraded version G Example: (P. Harikanth)

Cutting off values of the ratio outside a radius of 40, 70,85.
Inverse 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 Degradation function Cutting off values of the ratio outside a radius of 40, 70,85. (P. Harikanth)

Restoration Cut-off Frequency
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 Limiting the restoration to a specific frequency about the origin Result: Low-pass image Blurred Ringing (P. Harikanth)

Inverse Filtering - Example

5.8 Minimum Mean Square Error (Wiener) Filtering
(P. Harikanth)

Minimum Mean Square Error (Wiener) 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 The inverse filtration gives poor results, since the information about noise properties is not taken into account. Wiener filtration incorporates a priori knowledge about the noise properties. Restoration by the filter gives an estimate f of the original uncorrupted image f with minimal mean square error : Minimization is easy if the estimate f is a linear combination of the values in the image g; The estimate F of the Fourier transform F of the original image f can be expressed as: (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 Minimum Mean square Estimator (P. Harikanth)

Inverse & Wiener Filtering - Example

Inverse & Wiener Filtering -Example

5.9 Constrained Least Square Filtering
(P. Harikanth)

Constrained Least Squares 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 Vector matrix form is expressed as For g, the size of vector is M X N for H, the size of the vector is MN X MN for f and η the size of vector is MN X 1 (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 Provides a filter that can eliminate some of the artifacts caused by other frequency domain filters Done by smoothing criterion in the filter derivation The result does not have undesirable oscillations (P. Harikanth)

The constraint will be strictly satisfied
By inverse FT where

5.10 Geometric Mean filter (P. Harikanth)

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 (P. Harikanth)

5.11 Image Reconstruction from Projections
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 Introduction Principles of Computed Tomography Projections and the Radon Transform The Fourier-Slice Theorem Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections P. Harikanth

Image Reconstruction Overview
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 To get a better representation of an object, take more pictures from different angles. This principle applied in medical imaging. An accurate image is obtained by combining pictures from different views. In nuclear medicine, the single photon emission computed tomography (SPECT) or positron emission tomography (PET) camera rotate around the patient. Take pictures of radioisotope distribution within the patient from different angles. P. Harikanth

These pictures acquired from the nuclear medicine camera are called “projection” The procedure to put the projections together to obtain a patient ‘s image is called “image reconstruction”. 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

Image Reconstruction Types
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 Two types of algorithms is used in reconstructing images. 1- Analytical algorithm 2- Iterative algorithm P. Harikanth

Image Reconstruction Types
Filtered back projection algorithm Image is two-dimensional (2D). It consists of only one point with a certain degree of intensity as fig 2.a . The high of the “pole” indicates the intensity of the point in the object (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 P. Harikanth

Analytical Algorithms
A number of projections are taken from various angles as shown in fig 2.b . How would you reconstruct the image using those projections? 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

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 You look at the projections, you see a spike. The spike is the sum of all activity along the projection path. To reconstruct the image, we must re-distribute the activity in the spike back to its original path. Put equal amounts of activity every where along the path. Do that for all of projections taken from every angle as shown in fig 2.c . What we have done is a standard mathematical procedure called back-projection. If we backproject from all angles over 360º, we will produce an image similar to the one shown in fig 2.d . 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 After back projection, the image is not quite the same as the original image . In order to sharpen the image, we can apply special “filtering” to the projections by introducing negative wings before back projection (fig 3). This image reconstruction algorithm is called “Filtered Back projection Algorithm” Fig. 3. In filtered back projection, negative wings are introduced to eliminate blurring. 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 There are other types of analytical algorithms in which the backprojection is performed first and filtering follows. these types of algorithms are called: Backprojection Filtering Algorithms P. Harikanth

Three-dimensional image reconstruction
It can be formed by stacking slices of 2D images, as shown in fig .4 . This approach does not always work. Fig .5 and 10 show 3D PET and cone-beam imaging geometries. We observe from these figures that there are projection rays that cross multiple image slices. This makes slice-by-slice reconstruction impossible. 3D reconstruction is required. Both filtered backprojection and backprojection filtering algorithms exist for 3D reconstruction. 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 Fig 4. A 3D image can be reconstructed by stacking 2D reconstructions Fig 5. in 3D PET, projection rays that cross slices are used. P. Harikanth

Three-dimensional image reconstruction
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 Such algorithms require parallel plane from various directions as shown in fig 6. Fig 6 . Parallel plane measurements in 3D P. Harikanth

Iterative Reconstruction
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 In nuclear medicine, iterative reconstruction is becoming popular for the following reasons : It is easy to model and handle projection noise, especially when the counts are low. It is easy to model the imaging physics, such as geometry, non-uniform attenuation, scatter , … The basic process of iterative reconstruction is to discretize the image into pixels and treat each pixel value as an unknown. A system of linear equations can be set up according to the imaging geometry and physics. Finally, the system of equations is solved by an iterative algorithm. P. Harikanth

Iterative Reconstruction
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 setup of equations is shown in fig 7. The system of linear equations can be represented in the matrix form as FX =P. each element (Xi) in X is a pixel value . each element (Pi) in P is a projection measurement . (Fij)in F is a coefficient that is the contribution from pixel j to the projection bin i . The image is discredited into pixels and a system of equations is setup to describe the imaging geometry and physics. 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 Is it possible to reconstruct the 3-D volume of an object from projections? Early 20th century: Radon Transform and Fourier Slice Theorem Common methods MRI Noninvasive magnetic field applied. Main function FFT. Positron Emission Tomography Patient injected with radioactive matter. When decay, release radiation which is detected by sensors. Computed Tomography Use x-ray projections of object. Use filtered back-projection to obtain original volume. Contain fine-grained and coarse-grained data parallelism. 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 Introduction Introduction Principles of Computed Tomography Projections and the Radon Transform The Fourier-Slice Theorem Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections P. Harikanth

Introduction 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 Examine the problem of reconstructing an image from projection Focus on X-ray computed tomography (CT) The Earliest and most widely used type of CT One of the principal applications of digital image processing in medicine P. Harikanth

Introduction 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 Flat region showing a simple object, an input parallel beam, and a detector strip. Result of back-projecting the sensed strip data. The beam and detectors rotated by 90 ̊. Back-projection. The sum of (b) and (d) . The intensity where the back-projections intersect is twice the intensity of the individual back-projections. a,b c,d,e P. Harikanth

Introduction 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 This image is a cross section of 3-D region of human body. Round object is a tumor Pass a thin, flat beam of X-rays from left to right No way to determine from single projection whether deal with a single object along the path of the beam P. Harikanth

Introduction 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

Introduction 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 As the number of projections increases, the strength of non-intersecting backprojections decreases relative to the strength of regions in which multiple backprojections intersect. Brighter regions will dominate the result. Backprojections with few or no intersections will fade into the background as the image is scaled for display. Figure 5.33(f) formed from 32 projections. The image is blurred by a “halo” effect, the formation of which can be seen in progressive stages in fig 5.33. For example in e appears as a “star” whose intensity is lower that that of object, but higher than the background. 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 Introduction Principles of Computed Tomography Principles of Computed Tomography Projections and the Radon Transform The Fourier-Slice Theorem Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections P. Harikanth

Principles of Computed Tomography (CT)
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 Goal: Obtain a 3-D representation of the internal structure of an object by x-raying the object from many different directions. The basic feature of the method is that the X-ray tube, in a definite pattern of movement, permits the rays to sweep in many directions through a cross-section of the body or the organ being examined. The X-ray film is replaced by sensitive crystal detectors, and the signals emitted by amplifiers when the detectors are struck by rays are stored and analyzed mathematically in a computer. The computer is programmed to rapidly reconstruct an image of the examined cross-section by solving a large number of equations including a corresponding number of unknowns. P. Harikanth

History of CT 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 J Radon Predicted that through his mathematical projections a three dimensional image could be produced Godfrey Hounsfield Credited with the invention of computed tomography in 1970 – 1971 Built the first scanner on a lathe bed Took nine days to produce the first image P. Harikanth

History of CT 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 Johann Radon, Czech , 16 Dec ~ 25 May 1956 Johann Radon worked on the Calculus of variations, Differential geometry and Measure theory P. Harikanth

Digital Image Processing The Nobel Prize in Physiology or Medicine 1979
Allan M. Cormack Godfrey N. Hounsfield 1/2 of the prize 1/2 of the prize 1924~1998 1919~2004 Tufts University ，Medford, MA, USA Central Research Laboratories, EMI London, United Kingdom Physicist Electrical Engineer This year's Nobel Prize in physiology or medicine has been awarded to Allan M Cormack and Godfrey N Hounsfield for their contributions toward the development of computer-assisted tomography, a revolutionary radiological method, particularly for the investigation of diseases of the nervous system. Computer-Assisted Tomography ＝CAT; CT=Computed Tomography; Tomography = the Greek tomos, a cut, and graph, written P. Harikanth

What is projection? 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 Shadow gram obtained by illuminating an object by penetrating radiation Each pixel on the projected image represents the total absorption of the X-ray along its path from source to detector Rotate the source-detector assembly around the object – projection views for several different angles can be obtained. Reconstructing a cross-section of an object from several images of its transaxial projections –an important problem to be addressed in image processing Goal of image reconstruction is to obtain an image of a cross-section of the object from these projections. P. Harikanth

CT Scanners 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 Imaging systems that generate such slice views are called computerized tomography scanners Resolution lost along path of X-rays while obtaining projections CT restores this resolution by using information from multiple projections. Special case of image restoration P. Harikanth

First Generation Scanners
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

Second Generation Scanners
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

Third Generation Scanners
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

CT Chest Images 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

X-ray Projection Image records transmission of X-rays through object
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 records transmission of X-rays through object The integral is a line-integral or a “projection” through object - X-ray attenuation coefficient, a tissue property, a function of electron density,… 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 Introduction Principles of Computed Tomography Projections and the Radon Transform Projections and the Radon Transform The Fourier-Slice Theorem Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections P. Harikanth

Projections and the Radon Transform
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 straight line by its normal representation: P. Harikanth

Radon Transform Assume f (x, y) is a two dimensional (2-D) real function in the x-y coordinate system. Assume a ray L (s, θ) is defined by two parameters, s and θ . s is the distance from the origin to the ray . θ is the angle between the y-axis and the ray. the ray is described by the following equation : x cos θ+y sinθ = s. 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 ray L (s,θ ) in x-y coordinate system P. Harikanth

Radon Transform 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 Since s and can be any real values, the integral of f(x,y) along the ray L(s,θ) defines a 2-D function, denoted as P(s,θ).We call P(s,θ) the Radon transform of f(x, y) P(s,θ) = ʃ f(x,y)ds With the help of the 2-D distribution: this transform can be expressed as P. Harikanth

Radon Transform 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 When is a constant, the 2-D function of P (s, θ) becomes a one-variable function of s, denoted by Pθ(s). Because Pθ(s) represents a collection of integrals along a set of parallel rays, Pθ(s) is also called parallel projections of P (s, θ) at view (see this Fig). the Radon transform also can be expressed as P. Harikanth

Radon Transform 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 Projection of a parallel-ray beam may be modeled by a set of lines . P. Harikanth

Sinogram 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 When the radon transform is displayed as an image with ρ and θ as rectilinear coordinates, the result is called a sinogram . Sinograms can be readily interpreted for simple regions . This is for simulating the absorption of tumor in brain P. Harikanth

Main Idea of CT 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 key objective of CT is to obtain a 3-D representation of volume from its projections. How we can do that? Back-project each projection . Sum all the back-projections to generate one image. 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 Introduction Principles of Computed Tomography Projections and the Radon Transform The Fourier-Slice Theorem The Fourier-Slice Theorem Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections P. Harikanth

The Fourier-Slice Theorem
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 Fourier Transform of 1-D Projection of 2-D Image = Slice of 2-D Fourier Transform of Image Formula can be rearranged as filtered backprojection. P. Harikanth

Basis 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 Parallel-Beam Filtered Backprojections Fan-Beam Filtered Backprojections Expressions: 1-D Fourier Transform of a projection with respect to ρ and given value of Ѳ : Substituting the following equation in above one, we will have 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 By letting The above expression will be recognized as the 2-D transform of f(x,y) evaluated at the values of u and v indicated. Where F(u,v) denotes the 2-D Fourier Transform of f(x,y) Definition: The equation is known as the Fourier-slice theorem (or the projection-slice theorem) which states that “the Fourier transform of a projection is a slice of the 2-D Fourier transform of the region from which the projection was obtained.” 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 reason behind the terminology: As it is shown in above figure, “The 1-D Fourier transform of any arbitrary projection is obtained by extracting the values of F(u,v) along a line oriented at the same angle as the angle used in generating the projection. In principle, we could obtain f(x,y) simply by obtaining the inverse Fourier transform of F(u,v).” 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 Introduction Principles of Computed Tomography Projections and the Radon Transform The Fourier-Slice Theorem Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections P. Harikanth

Reconstruction Using Parallel-Beam Filtered Backprojections
Problem caused by ordinary Backprojections: Unacceptably Blurred Result Solution: Filtering the projection before computing the back-projections We know form the previous that the Inverse Fourier Transform is: By letting and Inverse Fourier Transform P. Harikanth Inverse Fourier Transform in polar coordinates

Using FST, we have By splitting integral in two ranges, we have and Then we can express With respect to ω , Then we have 0 −∝ 𝜋 ∝ [|𝜔|𝐺(𝜔,𝜃) 𝑒 𝑗2𝜋𝜔𝜌 𝑑𝜔] 𝑑𝜃 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 Inverse 1-D Fourier Transform Ramp filter + Problem One-dimensional filter function Undefined Theoretically Generalized delta functions Solution Practically P. Harikanth Band-limits the ramp 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 + Ringing was reduced in the windowed ramp (the ratios of the peak to trough in respective Figs). Hamming window Because the width of the central lobe in Fig 5.42(e) is slightly wider than in Fig 5.42(b), we would expect backprojections based on using a Hamming window to have less ringing but slightly more blurred. P. Harikanth

How to band-limit a function? Use a box in a frequency domain Problem: A box has undesirable ringing properties Solution: Using a smooth window instead Reason: As we expected, using a box window for band-limiting results in noticeable ringing in the spatial domain. We also know that filtering in spatial filtering domain is equivalent to convolution in the spatial domain, so spatial filtering with a function that exhibits ringing will produce a result corrupted by ringing, too. How does smooth window help? An M-point discrete function used frequently for implementation with the 1-D FFT is given by P. Harikanth Hamming window Hann window When c = 0.54 When c = 0.5

We remember from the following equation that 1-D Fourier Transform Which is a single projection obtained at a fixed angle, Ѳ . How about Complete, back-projected Image? The following equation will state how a complete, back-projected image f(x,y) is obtained. Steps: Compute the 1-D Fourier transform of each projection Multiply each Fourier transform by the filter |w| which has been multiplied by a suitable (e.g., Hamming) window. Obtain the inverse 1-D Fourier transform of each resulting filtered transform Integrate (sum) all the 1-D inverse transforms from last step P. Harikanth

Another Concern: Sampling Rates Effects: The selection of sampling rates has a profound influence on image processing results. Considerations: The number of rays used, which determine the number of samples in each projection The number of rotation angle increments, which determines the number of reconstructed images The difference between unfiltered and filtered projection 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 An Example: Image reconstruction using filtered backprojections Techniques: Unfiltered backprojections Filtered backprojections with ramp filter Filtered backprojections with ramp filter modified by Hamming window Properties: Image size: 600 * 600 pixles Increments of rotation: 0.5 degree Number of Rays: 849 rays Angle of rotation: 45 – 135 degrees 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 key difference between the Hamming and Hann window is that the end points are zero. The difference between the two is imperceptible in image processing applications. P. Harikanth

Results of Backprojections: Ramp Filter: The absence of any visually detectable bluring. Ringning is present, visible as faint line, especially around the corner of the rectangle. Ram Filter with Hamming window: Helped considerably with the ringing problem Slight blurring Smother image Unfiltered: Blurred image Image without quality Backprojections without any filtering Filtered backprojections with ramp filter, and Hamiing windowed ramp filter P. Harikanth

Another Approach: We obtain filtered backprojections via an FFT implementation Another approach: Equivalnt results can be obtained using spatial convolution Inverse Fourier transform of the product of two frequency domain functions According to the convolution theorem, we know: Inverse Fourier transform of the product of two frequence domain functions is equal to the convolution of the spatial representation of these two functions. Therefore by letting denote the inverse Fourier Transformation of , then the above equation: P. Harikanth

What do they mean? Individual backprojections at an angle can be obtained by convolving the corresponding projection, , and the inverse Fourier transform of the ramp filter, What did we do before? As before, the complete back-projected images is obtained by integrating (summing) all the individual back-projected images. Note: There is no need to store all the back-projected images during reconstruction. Instead, a single sum is updated with the latest back-projected image. At the end of the procedure, the running sum will be equal the sum of all the backprojections. Some Notes: Both approach are equal. (with the exception of round-off differences in computation) In pratical CT implementations, Convolution generally turns out to be more efficient computationally P. Harikanth Both approach are equal. (with the exception of round-off differences in comptation)

An Issue: Because the ramp filter (even when it is windowed) zeroes the dc term in the frequency domain, each backprojection image will have a zero average vale. This means that each backprojection image will have negative and positive pixels. When all the backprojections are added to form the final image, some negative locations may become positive and the average value may not be zero, but typically, the final image will still have negative pixels. Solutions : The simplest approach, when there is no knowledge regarding what the average values should be, is to accept the fact that negetive values are inherent in the approach and scale the result using following equaltions: P. Harikanth

When knowledge about what a “typical: average value should be is available: That value can be added to the filter in the frequency domain, thus offsetting the ramp and preventing zeroing the dc term When working in the spatial domain with convolution, the very act of truncating the length of the spatial filter (inverser Fourier transform of the ramp) prevents it from having a sero average value, thus avoiding the zeroing problrm altogether. P. Harikanth

Problems 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 In most application of CT (especially in medicine), ringing effect is very important. Minimizing them is very important. Tuning the filtering algorithms and using a large number of detectors can help reduce these effects. 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 Introduction Principles of Computed Tomography Projections and the Radon Transform The Fourier-Slice Theorem Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections P. Harikanth

Reconstruction Using Fan-Beam Filtered Backprojections
Why Fan-Beam Filtered Backprojection Although Parallel-Beam is simple and intuitive, but modern CT systems use Fan-Beam geometry. A basic fan-beam imaging geometry in which the detectores are arranged on a circular arc and the angular increments of the source are assumed to be equal. Expressions: Let denote a fan-beam projection, where is the angular position of a particular detector measure with respect to the center ray, and is the angular displacement of the source, measured with respect to the y-axis. Fan-beam ray representation as a line: A ray in the fan beam can be represented as a line, , in normal form, which is the approach we used to present a ray in parallel-beam imaging geometry. This allows us to utilize parallel-beam results as the starting point for deriving the corresponding equations for the fan-beam geometry. P. Harikanth

We begin by noticing that the parameters of Line are related to the parameters of a fan-beam ray by The distance from the center of the source t othe orgin of the xy plane We remember from the parallel-beam imaging geometry that the convolution backprojection formula Without loss of generality, suppose that we focuse attention on objects that are encompassed within a circular area of radius T about the origin of the plane. Then and Where we used the fact that projections 180o apart are mirror images of each other. In this way, the limits of the outer integral are made to span a full circle, as required by a fan-beam arrangement in which the detectors are arranged in a circle. P. Harikanth

Due to our interest in integrating with respect to and , we change the current coordinates to polar one By letting It follows that We will have: Parallel-beam reconstruction formula written in polar ocrdinates By transforming coordinates using We will have: where P. Harikanth

Simplification of formula The limits to for span the entire range of 360o. Because all fuctions of are priodic, with periodic , the limits of the outer integral can be replace by 0 and , respectively. , corresponding to , beyond which , so we can replace the limits of the inner integral by and , respectively. A raysum of a fan beam along the line must be equal the raysum of the parallel beam along the same line. A raysum is a sum of all values along a line, so the result must be the same for a given ray, regardless of the coordinate system is which it is expressed. This is true of any raysum for corresponding values of and Thus, letting denote a fan-beam projections, it follows that And from P. Harikanth

Incorporating these observation in following equation results in the expression 2 This the fundamental fan-beam reconstruction formula based on filtered backprojections Put in more familiar convolution form It can be shown that 1 The distance from the source to an arbitrary point in a fan ray The angle between the ray and the center ray Substituting 1 into 2 yields 3 P. Harikanth

It can also be shown that Using this expression, we can write equation 3 as Convolution expression A weighting factor inversely proportional to distance from the source where and Implementating Equation : To convert a fan-beam geometry to a parallel-beam geometry using following equations: Use the parallel-beam reconstruction approach. P. Harikanth

Relation between Parallel and fan beam projections As we noted earlier, a fan-beam projection, , taken at angle has a corresponding parallel-beam projection, , taken at a corresponding angle and, therefore, Let denote the angular increment between successive fan-beam projections Let be angular increment between rays, which determines the number of samples in each projection We impose the restriction that Then, and for some integer values of m and n, we can write as Conclusion This equation indicates that the nth ray in the mth radial projection is equal to the nth ray in the (m + n)th parallel projection. The term implies that parallel projections converted from fan-beam projections are not sampled uniformly, an issue that can lead to blurring, ringing, and aliasing artifacts if samoling intervals and are too coarse. P. Harikanth

An Example: Image reconstruction using fan-beam backprojections Techniques: Fan-beam projections Parallel-beam projections by converting fan-beam projections Filtered backprojections with Hamming window Properties: Increments of rotation: 0.5o, 0.25o, 0.125o P. Harikanth

Results of Backprojections: Angle of 1o: Clearly obvious that 1o increments are too coarse, as blurring and ringing are quite evident. Angle of 0.5o: The result is interesting, in the sense that it compares poorly with same parallel-beam projection, which was generated the same angle increment of 0.5o. Angle of 0.25o: Even with angle increments of 0.25o the reconstruction still is not as good as parallel-beam projection. Angle of 0.125o: Angle increments of 0.125o make the results comparable. 720 samples vs. 849 samples P. Harikanth

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