1. BioE153:Imaging As An Inverse Problem Grant T. Gullberg gtgullberg@lbl.gov http://muti.lbl.gov/jonathan/courses/bioe153-2002 510 486-7483 1

2. Introduction 2 Mathematics and Physics of Emerging Biomedical Imaging, National Academy Press, Washington, D.C., 1996

3. Examples X-ray Computed Tomography MRI PET SPECT Ultrasonic Tomography Electrical Source Imaging Electrical Impedance Tomography Magnetic Source Imaging Optical Tomography Photo-Acoustic Imaging 3

4. X-ray CT Inverse Problem x y source detector attenuation distribution 4 projection

5. MRI Inverse Problem x y proton spin density 5 gradient signal z along the bore of the magnet

6. PET Inverse Problem x y isotope concentration attenuation distribution 6 projection detector 2 detector 1

7. SPECT Inverse Problem x y isotope concentration attenuation distribution projection 7 detector

8. Ultrasound Inverse Problem velocity traducer/receiver k b – reference wavenumber G – reference Green’s function   – index of refraction P b – background pressure Pressure traducer receiver Fredholm integral equation ( Lipmann-Schwinger ) 8

9. Electrical Source Inverse Problem potential measurement 9 r v – potential n – surface normal - dipole - dipole - conductivity terms - conductivity terms

10. I g current voltage Electrical Impedance Inverse Problem voltage conductivity sensitivity matrix 10

11. Magnetic Source Inverse Problem potential measurement magnetic field measurement 11 v – potential n – surface normal - dipole - dipole - conductivity terms - conductivity terms b – magnetic vector - free space permeability - free space permeability r

12. A Simple Example of An Imaging Inverse Problem X-ray CT Projections Reconstruction Problem as a Solution to a System of Linear Equations Reconstruction is an Inverse Solution 12

13. X-ray CT Projections 13

14. x source Beer’s Law detector 14 units of length -1 flux of photons

15. 15 different attenuation coefficients

16. Image Matrix 16 pixelized array of attenuation coefficients

17. Projections.01.03.05.15 0 0.09.30.35.33.01 17 example of projections for a particular pixelized array of attenuation coefficients

18. Reconstruction Problem as a Solution to a System of Linear Equations 18

19. Projections.09.30.35.33.01 19 solve for the unknown attenuation coefficients from a set of two projections

20. 20 the system of linear equations 6 equations in 9 unknowns

21. 21 the inclusion of a third projection

22. .09.30.35.33.01.0345.2230.3465.0860 0 22 solve for the unknown attenuation coefficients from a set of three projections

23. 23 the system of linear equations 11 equations in 9 unknowns

24. F 24 Matrix Equation

25. Reconstruction is an Inverse Solution.09.30.35.33.01 25

26. 26 Least Squares Solution to a System of Linear Equations generalized inverse

27. Reconstruction.09.30.35.33.01 -.0433.0633.0266.0700.1400.1333.0266.01.03.05.15 0 0 Original 27 solution from two projection measurements

28. with(linalg): A:=array([[1,1,1,0,0,0,0,0,0],[0,0,0,1,1,1,0,0,0],[0,0,0,0,0,0,1,1,1], [1,0,0,1,0,0,1,0,0],[0,1,0,0,1,0,0,1,],[0,0,1,0,0,1,0,0,1]]); B:=array([.09,.30,.30,.01,.33,.35]);leastsqrs(A,b,’optimize’); Maple Routine 28

29. 29 6 equations in 9 unknowns the system of linear equations

30. .01.03.05.15 0 0.09.30.35.33.01.0345.2230.3465.0860 0 Reconstruction 30 solution from three projection measurements

31. 31 the system of linear equations 11 equations in 9 unknowns

32.  Our examples have been two-dimensional. However, X-ray CT imaging is a three- dimensional inverse problem. Comment: 32