BioE153:Imaging As An Inverse Problem Grant T. Gullberg

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

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

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

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

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

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

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

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

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

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

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

X-ray CT Projections 13

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

15 different attenuation coefficients

Image Matrix 16 pixelized array of attenuation coefficients

Projections example of projections for a particular pixelized array of attenuation coefficients

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

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

20 the system of linear equations 6 equations in 9 unknowns

21 the inclusion of a third projection

solve for the unknown attenuation coefficients from a set of three projections

23 the system of linear equations 11 equations in 9 unknowns

F 24 Matrix Equation

Reconstruction is an Inverse Solution

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

Reconstruction Original 27 solution from two projection measurements

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 6 equations in 9 unknowns the system of linear equations

Reconstruction 30 solution from three projection measurements

31 the system of linear equations 11 equations in 9 unknowns

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