DIGITAL IMAGE PROCESSING

DIGITAL IMAGE PROCESSING
Instructors: Dr J. Shanbehzadeh M.Gholizadeh

Chapter 5 - Image Restoration and Reconstruction
DIGITAL IMAGE PROCESSING Chapter 5 - Image Restoration and Reconstruction Instructors: Dr J. Shanbehzadeh M.Gholizadeh ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

5.11 Image Reconstruction from Projections
( J.Shanbehzadeh M.Gholizadeh )

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 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 ( J.Shanbehzadeh M.Gholizadeh )

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 Suppose you are touring Yazd and you have discovered a unique example of architecture and you would like to share your experience with your friends back home. Take some pictures . To get a better representation of it, 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. ( J.Shanbehzadeh M.Gholizadeh )

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 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”. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 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). ( J.Shanbehzadeh M.Gholizadeh )

Analytical Algorithms
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 number of projections are taken from various angles as shown in fig 2.b . How would you reconstruct the image using those projections? ( J.Shanbehzadeh M.Gholizadeh )

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 . ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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  ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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,… ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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: ( J.Shanbehzadeh M.Gholizadeh )

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 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. A ray L (s,θ ) in x-y coordinate system ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 . ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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.” ( J.Shanbehzadeh M.Gholizadeh )

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).” ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh ) 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 ( J.Shanbehzadeh M.Gholizadeh )

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 Solution Generalized delta functions Practically ( J.Shanbehzadeh M.Gholizadeh ) Band-limits the ramp filter

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 Hamming window Hann window When c = 0.54 When c = 0.5 ( J.Shanbehzadeh M.Gholizadeh )

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 + Hamming window Ringing was reduced in the windowed ramp (the ratios of the peak to trough in respective Figs). 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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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: ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh ) 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: ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

Reconstruction Using Fan-Beam Filtered Backprojections
Why Fan-Beam Filtered Backprojection Although Parallel-Beam is simple nad 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 dnote 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. ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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. ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

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 ( J.Shanbehzadeh M.Gholizadeh )

DIGITAL IMAGE PROCESSING

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