DIGITAL IMAGE PROCESSING

DIGITAL IMAGE PROCESSING Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh mhdgholizadeh@gmail.com

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

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

Road map of chapter 5 5.11 5.11 5.9 5.10 5.10 5.9 • 5.9- Constrained Least Square Filtering • 5.10- Geometric Mean Filter • 5.11- Image Reconstruction from Projections • Geometric Mean Filter • Constrained Least Square Filtering • Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

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

Introduction The Fourier-Slice Theorem Restoration in the Presence of Noise Only - Spatial Filtering Principles of Computed Tomography Projections and the Radon Transform Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections • 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 )

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 )

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 • 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 )

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 • 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 )

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 • 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 )

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 • 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 )

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 • 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 )

Introduction The Fourier-Slice Theorem Introduction Restoration in the Presence of Noise Only - Spatial Filtering Principles of Computed Tomography Projections and the Radon Transform Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections • 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 • 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 )

Introduction The Fourier-Slice Theorem Principles of Computed Tomography Restoration in the Presence of Noise Only - Spatial Filtering Principles of Computed Tomography Projections and the Radon Transform Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections • 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 )

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. 1887 ~ 25 May 1956 Johann Radon worked on the Calculus of variations, Differential geometry and Measure theory ( J.Shanbehzadeh M.Gholizadeh )

Digital Image ProcessingThe Nobel Prize in Physiology or Medicine 1979 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 • 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 )

Introduction The Fourier-Slice Theorem Projections and the Radon Transform Restoration in the Presence of Noise Only - Spatial Filtering Principles of Computed Tomography Projections and the Radon Transform Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections • 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 )

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 . • sis 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 )

Introduction The Fourier-Slice Theorem The Fourier-Slice Theorem Restoration in the Presence of Noise Only - Spatial Filtering Principles of Computed Tomography Projections and the Radon Transform Reconstruction Using Parallel-Beam Filtered Backprojections Reconstruction Using Fan-Beam Filtered Backprojections • 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 )

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 )

The Fourier-Slice Theorem Basis for Parallel-Beam Filtered Backprojections Fan-Beam Filtered Backprojections • 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 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 )

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 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 )

DIGITAL IMAGE PROCESSING