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

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

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

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

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

Flood Field Apparatus Segmented Crystal Continuous Crystal Tungsten Mask Movable Source

Flood Field Image

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

Filtering

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.

Peak Finding Example of missed peak

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!

Exhaustive Solution Example with Four Points Distorted Peaks: 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….

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

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

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

Voronoi Diagram

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

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

Regression Technique 1) Take log of data points 2) Get least squared coefficients of polynomial C1 + C2(X) + C2(Y) + C3(XY) + C4X^2Y ) 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

Basis Function Regression of X- Component of Distortion

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

Final Residual Error of Basis Function Regression

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?????