Calibration of Photomultiplier Arrays for Medical Imaging Applications

1. Calibration of Photomultiplier Arrays for Medical Imaging Applications Eric Kvam Engineering Physics Undergraduate

2. • Instrumentation -Beam Position Monitor -Low Cost Telemetry• Data Acquisition Software• Kernel Level Drivers• Autonomous Systems -Software Architecture -Unmanned Aerial Vehicle -Awareness Device The LAIR The Laboratory for Advanced Instrumentation Research

3. Researchers Dr. Hong Liu Dept. of Mathematics Prof. Jack McKisson Dept. of Physical Sciences Brian Maisler Eric Kvam Yishi Li Undergraduate Researchers

4. Physical Processes • Gamma Rays • Scintillator • Depth of Interaction • Segmented Crystal • Efficiency • Anode Array • Charge Collection • A/D Network

5. Strategy for Spatial Correction • Obtain Distorted Measured Locations • Determine Known Corresponding Reference Locations • Generate Correction Map

6. Flood Field Apparatus • Segmented Crystal • Continuous Crystal • Tungsten Mask • Movable Source

7. Flood Field Image

8. Filtering • Peaks become extremely positive • Valleys become extremely negative • Flat regions become negative • Positive-definite restriction significantly reduces noise

9. Filtering

10. Peak Finding • Places peaks at local maxima above a preset threshold. • Problems: • Multiple Hits - Single peak with more than one local maxima. • Missed Peak - Peak below cutoff point. • Doublet - unresolved due to close spacing. • User reviews the image and corrects any inconsistencies in peak identification.

11. Peak Finding Example of missed peak

12. Peak Matching • Gridlike Distribution • Scattered Distribution • Traveling Salesman Problem • Select Candidate Pool • Candidate Cost • Solution Branch Cost • Split Branches Recursively • Avoid Attractive Globally Incompatible Local Minima • Exhaustive Solution Requires Too Much Computation!

13. Exhaustive Solution Example with Four Points Distorted Peaks: 1 2 3 4 Reference Points: A B C D Initially There Are Four Branches: [1A] [1B] [1C] [1D] Each Branch Splits Three Ways: [1A 2B] [1B 2A] [1C 2A] [1D 2A] [1A 2C] [1B 2C] [1C 2B] [1D 2B] [1A 2D] [1B 2D] [1C 2D [1D 2C] Each Branch Splits Two Ways….

14. Too Much Computation! • Every permutation is a possible solution • Number of permutations is the factorial of the number of data points • Cost must be evaluated for each possible solution • Exhaustive solution is NP Complete • Typical data contains over 1600 data points

15. Peak Matching Solution • User provides an initial match seed • Peaks are ordered by distance from seed • Candidate pool limited to closest peaks • Anticipated location determined from previous matches • Candidate cost is distance from anticipated location • Branch cost is sum of candidate costs • Limited branch splitting • Limited number of branches maintained

17. Correction Map • Correction vector known at certain peak locations • What is correction vector throughout the domain of the detector?

18. Voronoi Diagram

19. Voronoi Technique • Entire domain of detector is divided into cells associated with particular peaks • Gamma rays assume the correction vector from the peak of the cell that they fall within

20. Regression Technique • Seek two continuous functions for x and y distortion components • Each function has bi-variate dependence x’ = f(x,y) y’ = f(x,y) • Least square solution • Exponentials of polynomials used for basis function

21. Regression Technique 1) Take log of data points 2) Get least squared coefficients of polynomial C1 + C2(X) + C2(Y) + C3(XY) + C4X^2Y + ... 3) Subtract function from data to get residual error 4) Repeat and fit function to remaining residual error • The final distortion function is the sum of all the basis functions • After about 80 iterations the matrix becomes rank defficient and residual error cannot be further reduced

22. Basis Function Regression of X-Component of Distortion

23. Matlab's “Griddata” Interpolation of X-Component of Distortion

24. Final Residual Error of Basis Function Regression

25. Future Work • Reduce required user interaction • Test these solutions on continuous crystals with tungsten mask • Produce tools that technicians can use • Improve charge cloud centroid determination • Include “Z” correction Any undergrads out there interested?????