1. Case Study Computed Tomography (CT) Non-invasive medical imaging techniques

2. 最传统的方法  Standard X-ray  Lost spatial information  Very limited use

3. X-ray

4. Ultrasound (B 超 )  Use very high frequency sound (MHz)  Safe  Image in real time  Equipment is not expensive  Limitation  Strong sound absorption by bones  Hard in brain studies, or anywhere with bones 基本原理（物理的）：波的折射与反射

5. Computed Tomography  A parallel sheet of X-ray  Mathematical tools needed for successful computed tomography (CT) imaging  Basic principle is discovered more once  1979 Nobel Prize in Physiology and Medicine was awarded to G. Hounsfield & A.M. Cormack jointly for realizing and bringing to medical use

6. Computed Tomography  Not expensive to use  Some drawbacks  Possible tissue damage from ionization  Low contrasts between different type of soft tissues

7. Computed Tomography  There are many applications other than medical tomography:  Astronomy---binary stars, coronal study  Oceangraphy---acoustic probing of ocean conditions  Geophysics---mantle flow, atmospheric studies  Porous Media---

8. Hubble Image

12. Test Object

13. Model problem

14.  Coordinate axes satisfy

15. Radon Transformation  Assume the density function of the test object is given by  The scan data can be written as  Question: how to recover from

16. Least Square Method  To understand the idea behind, consider  3*3 small object  Homogenous blocks  The block densities either 0 or 1  How to achieve the reconstruction?

17. Least Square Method  Density 1:  Density 0:

19. Scan Only 2 Directions

20. More Directions

21. How Matrix Forms

22. Reconstruction with least square

23. Efficiency

24. Computed Tomography 冠状动脉