1 / 77

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

gabi
Download Presentation

DIGITAL IMAGE PROCESSING

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  3. 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 )

  4. 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. 5.11 Image Reconstruction from Projections ( J.Shanbehzadeh M.Gholizadeh )

  6. 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 )

  7. 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 )

  8. 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 )

  9. 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 )

  10. 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 )

  11. 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 )

  12. 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 )

  13. 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 )

  14. 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 )

  15. 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 )

  16. 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 )

  17. 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 )

  18. 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 )

  19. 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 )

  20. 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 )

  21. 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 )

  22. 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 )

  23. 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 )

  24. 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 )

  25. 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 )

  26. 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 )

  27. 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 )

  28. 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 )

  29. 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 )

  30. 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 )

  31. 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 )

  32. 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 )

  33. 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 )

  34. 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 )

  35. 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 )

  36. 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 )

  37. 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 )

  38. 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 )

  39. 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 )

  40. 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 )

  41. 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 )

  42. 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 )

  43. 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 )

  44. 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 )

  45. 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 )

  46. 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 )

  47. 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 )

  48. 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 )

  49. 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 )

  50. 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 )

More Related