Reconstruction from Projections
Reconstruction From Projections
Reconstruction from projections has important applications in many areas of science, engineering and medicine. Several Nobel prices have been awarded in recent decades for work associated with reconstruction from projections. The use of computers in such work has been essential but nontrivial and, therefore, deserving to be covered in a graduate course in Computer Science.
The problem of image reconstruction from projections has arisen independently in a large number of scientific fields. An important version of the problem in medicine is that of obtaining the density distribution within the human body from multiple x-ray projections. This process is referred to as computerized tomography (CT); it has revolutionized diagnostic radiology over the last 40 years. The 1979 Nobel prize in medicine was awarded for work on CT.
This course is devoted to the fundamentals of this field. Its subject matter is the computational and mathematical procedures underlying the data collection, image reconstruction, and image display in the practice of CT. It is aimed at the practitioner: points of implementation are carefully discussed and illustrated. The major emphasis is on reconstruction algorithms; these are studied in depth.
The course will consist of weekly lectures by the Instructor based on his book Fundamentals of Computerized Tomography: Image Reconstruction from Projections, Second Edition, Springer, 2009.
1. Applications of Image Reconstruction from Projections
2. Overview of the Process of CT
3. Physical Problems Associated with Data Collection in CT
4. Software for the Development and Testing of CT Algorithms
5. Data Collection and Reconstruction of a Head Phantom
6. Basic Concepts of Reconstruction Algorithms
8. Filtered Backprojection Method for Parallel Beams
9. Other Transform Methods for Parallel Beams
10. Filtered Backprojection Method for Divergent Beams
11. Algebraic Reconstruction Techniques
12. Quadratic Optimization Methods
13. Fully Three-Dimensional Reconstruction
14. Three-Dimensional Display of Organs
After completing this course the student will be able to:
1. Describe how projection data are obtained and the resulting reconstructions are used in science and medicine, focusing on x-ray data but also covering other fields such as electron microscopy, nuclear medicine, ultrasound, materials science and nondestructive testing.
2. Present a comparative evaluation of reconstruction methods, their accuracy under ideal and realistic circumstances, computational costs, taskoriented performance, and general applicability.
3. Implement and rigorously evaluate reconstruction algorithms, including filtered backprojection, Fourier and linogram reconstruction methods, algebraic reconstruction techniques, and quadratic optimization.
4. Explain fundamental computational and mathematical concepts such as basis functions, functions to be optimized, norms, generalized inverses, least squares and maximum entropy solutions, and most likely estimates.
5. Discuss the design and application of a large programming systems for image reconstruction, as well as computerized methods for 3D surface detection and display.
There will be a final exam, but the grade will be determined to a larger extent by the quality of the material submitted by the students on computer projects that will provide them with opportunity to implement reconstruction algorithms and to evaluate them in a rigorous fashion. These will be carried out using the SNARK09 software, which will be introduced in early lectures.