Instructor: Distinguished Professor Gabor T. Herman

Reconstruction from Projections
Thursdays 2:00 -4:00 p.m.
- 3 credits -


The problem of 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.


1. Applications of Image Reconstruction from Projections
2. Overview of the Process of CT
Physical Problems Associated with Data Collection in CT
4. Software for the Development and Testing of CT Algorithms
Data Collection and Reconstruction of a Head Phantom
Basic Concepts of Reconstruction Algorithms
Filtered Backprojection Method for Parallel Beams
Other Transform Methods for Parallel Beams
Filtered Backprojection Method for Divergent Beams
Algebraic Reconstruction Techniques
Qudratic Optimization Methods
Fully Three-Dimensional Reconstruction
Three-Dimensional Display of Organs


Left: Digital x-ray with a horizontal line marking the location of the cross section for which the other two images were obtained. Middle: Projection data of the cross section at multiple angles (corresponding to the vertical axis in the image) around the patient. Right: CT reconstruction from projections reveals details in the interior of the patient.

Displays of bone structures of patients produced from CT reconstructions by software developed by the instructor's research group. The left two images are of facial bones of an accident victim prior to operation and at the time of a one-year postoperative follow-up. The right two images show fractures of a tibia and of a pelvis, respectively.

The course will consist of weekly lectures by the Instructor based on his book Fundamentals of Computerized Tomography: Image Reconstruction from Projections. 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 computerr 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. For a good grade the student will also be expected to attend all lectures from beginning to end.

Student learning outcome:
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, task-oriented 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 fundfamental 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. Discusse the design and application of a large programming systems for image reconstruction, as well as computerized methods for 3D surface detection and display.