Introduction to PET Image Reconstruction. Tomographic Imaging. Projection Imaging. PET Image Reconstruction 11/6/07

Transcription

1 Introduction to PET Image Recontruction Adam Aleio Nuclear Medicine Lecture Imaging Reearch Laboratory Diviion of Nuclear Medicine Univerity of Wahington Fall Baic Stage of PET I. Radionuclide Production Make radio-iotope II. Radiochemitry Make radiopharmaceutical - Label a tracer III. Imaging 1. Adminiter radiotracer 2. Poitron decay - annihilation 3. Anti-parallel photon travel through patient (ome interact) 4. Photon enter detector (mot interact) 5. Detected Photon paired into coincident event 6. Store event in inogram format (data) IV. Data Analyi 1. Correct data for phyical effect 2. Recontruction into image and interpret 2 Projection Imaging overlay of all information (non quantitative) Tomographic Imaging Tomo + graphy = Greek: lice + picture ource y x z detector x p(x,z) = dy z orbiting ource + detector data for all angle true cro-ectional image 3 4 Adam Aleio, aaleio@u.wahington.edu 1

2 Type of imaging ytem From photon detection to data in form of Sinogram The number of event detected along an (LOR) i proportional to the integral of activity (i.e. FDG concentration) along that line. Projection: collection of parallel LOR (a ingle view) ource point ource Tranmiion (TX) Emiion (EM) but ame mathematic of tomography 511 kev photon detection Patient FDG ditribution 180 o 0 o Sinogram (all view) ingle projection ine wave traced out by point ource 5 6 Sinogram Example IV. Data Analyi Scanner B Source Object D C A A D Sinogram P(, ) C B Order of correction (common application): Start with Raw Data: Prompt Event = True + Random + Scatter Delayed Event = Approximation of Random 1. Random correction (Yr = Prompt-Delayed) 2. Detection efficiency normalization (Yn = Yr * Norm) 3. Deadtime (Yd = Yn * Dead) 4. Scatter (Y = Yd - Scat) The inogram i p(,) organized a a 2D hitogram - Radon Tranform of the object S 5. Attenuation ( Ya = Y * ACF) attenuation correction factor 6. Image Recontruction 7 8 Adam Aleio, aaleio@u.wahington.edu 2

3 IV. Data Analyi Sinogram recontructed into image Analytical Method: Data Formation? y Projection domain x Image domain Recontruction Problem All modalitie that collect line-integral data (inogram) can employ the ame method for recontruction: 1. Analytical Method - mot popular: FBP- Filtered Backprojection 2. Iterative Method - mot popular: OSEM - ordered ubet expectation maximization Point Source - Forward Projection to 3 projection Can get etimate of point ource with Back-projection 9 10 Analytical Method: imple back-projection of point ource Image & Sinogram Space Angle Intead of getting a point ource, end up with: 1/r = Ditance Backprojected Image No filtering ==> Reult = true 1/r Need to do Filtered Back-Projection FBP To undo thi, filter projection with ramp filter before backprojecting Adam Aleio, aaleio@u.wahington.edu 3

4 Analytical Method: Filtered back-projection Ramp filter accentuate high frequency - Not good for noie High Frequency 0 frequency High Frequency Either clip ramp filter or often ue filter to clip ramp to reduce noie: ex: Hanning filter IDL Demo FBP Characteritic PROS: Analytic method ( invere Radon tranform ) FBP i exact IF: No noie No attenuation Complete, continuouly ampled data Uniform patial reolution Eay to implement Computationally Fat Linear, other propertie well undertood (2x uptake = 2x intenity in image) Can Adjut filter window to trade off bia v. variance CONS: cannot model noie in data, cannot model non-idealitie of ytem, (reolution recovery method) doe not eaily work with unuual geometrie, cannot include knowledge about the image (like non-negative activity) Iterative Recontruction Characteritic Pro Reduce variance (noie) for a given level of accuracy (bia, reolution, etc.) Reduce or eliminate treak Incorporate (model) phyical effect Counting tatitic (noie) Confidence weighting Ditance dependent reolution Scatter, attenuation, detector efficiency, deadtime, random a priori information (non-negativity, anatomical information, etc.) Con Slow (computationally intenive) Non-linear -- hard to analyze A lot of knob to adjut: moothing parameter, number iteration, etc. Streak replaced with different noie character (e.g. blob ) The Recontruction Problem: An Invere Problem Oberved PET data ytem matrix x i N x 1 image vector (typically N ~ 128 x 128) y i M x 1 data vector (typically M ~ 280 x 336) Unknown image Error in obervation P i M x N ytem matrix (provide probability entry j from x will be placed in entry i of y) Adam Aleio, aaleio@u.wahington.edu 4

5 Iterative Recontruction: Baic Component 1. Decription of the form of the image ( pixel, voxel, blob ) 2. Sytem model relating unknown image to each detector meaurement : relate image to data (Can include detector repone, correction for attenuation, efficiencie, etc ) 3. Statitical Model decribing how each meaurement behave around it mean (Poion, Gauian, ) 4. Objective Function defining the bet image etimate (Log-likelihood, WLS,MAP ) 5. Method for maximizing the objective function (EM) Main Point: Lot of option, Not all EM algorithm the ame Overview of common choice for component 5. Method for maximizing the objective function EM (or ome minor variation) i mot common approach Expectation Maximization i a general method for olving all kind of tatitical etimation problem, in tomography reult in eay maximization tep EM Back project Forward project Tomography EM: Method for maximizing the Poion Likelihood Function (ML-EM) Iterative Recon Algorithm y=px + noie Sinogram Data: y Initialize Activity & Attenuation 1. Image Etimate: x n 2. Model Sytem: P 3,4.Objective Function Iterate Common Method 1.EM (ML-EM - maximum likelihood method) 2.OSEM (Ordered Subet Expectation Maximization) Subet A Subet B Variant of EM (till Maximum Likelihood) - Pro: Fat Con: Doe not converge Ue ubet of the data to compute each etimate Subet A then B then then repeat 1.Decription of the form of the image 2.Sytem model relating unknown image to each detector meaurement : relate image to data 3.Statitical Model decribing how each meaurement behave around it mean 4.Objective Function defining the bet image etimate 5.Method for maximizing the objective function 5. Locate Downhill 5. Minimizer: x n+1 19 Then 20 Adam Aleio, aaleio@u.wahington.edu 5

6 OSEM Example Don t Know Thi True Image Data (noiele) Know Thi 1t Gue Firt Image Firt Subet Second Subet 3. Regularized Method Common Method MAP(maximum a poteriori), PWLS(penalized weighted leat quare), GEM (generalized EM) All conit of variation in objective function Third Subet End of Iteration 1 Aume have ome knowledge about the image before we even get the data (a priori knowledge) PROS: Can enforce noie/reolution propertie in final image (don t need to pot-mooth), Can include anatomical information (PET/CT?), Enforce non-negativity, Method converge to final olution, More accurate model of data CONS: Uually take longer, ha more variable to et and undertand (more knob ), can impoe odd noie tructure Slide courtey of Tom Lewellen Example of EM v. FBP: Simulation Simulation with Poion noie baed on an average of 100 count per detector channel FBP v. EM v. MAP (not exhautive comparion) Adam Aleio, aaleio@u.wahington.edu 6

7 Some More Example: Example of different pot filter OSEM, S28, I2, LF4.3, PF6, Z0. OSEM, S28, I2, LF4.3, PF8, Z0. OSEM, S28, I2, LF4.3, PF10, Z1. FBP, 12 mm Hann, no axial moothing OSEM, 10 mm Gauian, no axial moothing Imaging - Fully 3D v. 2D Fully 3D Recontruction Fully 3D PET data increae enitivity of canner (~ 8x) Drawback: Increaed catter!! Significant torage and recontruction computation demand 2D - Direct And Cro Plane Only Fully 3D - Oblique Plane Alo Direct Analytic Approach 3DRP: 3D reprojection (Kinahan and Roger 1988) Iterative Approach Simple conceptual extenion: Jut need ytem model that relate voxel to fully 3D data (a oppoed to a pixel to 2D data) Sytem model become x larger (big computational challenge. (2D:1.5 Billion entrie to 3D: 1000 Billion entrie) Rebinning Approach Reduce Fully 3D data to decoupled et of 2D data, then do normal 2D recontruction FORE (Fourier Rebinning) mot common form Adam Aleio, aaleio@u.wahington.edu 7