Iterative Rekonstruktion

Transcription

1 Itertive Rekonstruktion Vorlesung FH-Hgenberg SE:MED Principles of Emission Tomogrphy Biologicl processes re visulized using rdioctive trcers Qusi-sttionry ccumultion of trcer is employed for dignostic routines (tumour stging, stroke, hert diseses) sptil detecton of γ-qunt ssocited the loction of the emission to line of response (collimtor- SPECT, coincidence mesurement -PET) Trnsversl slices re reconstructed from proection dt Stcked slices form n imge volume 1

2 Rdon Trnsform - Proection Geometry Proection or dt cquisition. Informtion bout the obect is integrted long the line L nd trnsformed into point-informtion ccording to its co-ordintes l nd Θ. 3D Proection Dt - Sinogrms 2

3 Itertive Reconstruction: Problem Sttement Imge volume nd ll proection dt form the imge vector X nd the mesurement vector Y. System of Equtions Itertive Reconstruction: System Mtri The elements i of the system mtri A give the probbility of the detection-event in bin y i of g- quntum emitted from voel. The following system properties re modelled by mtri A scnner geometry sctter ttenution detector efficicy Y=AX hs no ect solution noise -> inconsitent system of equtions number of equtions number of unknowns Fesible solution by itertive lgorithm 3

4 Itertive Reconstruction: Algorithm 1. Strting imge X 0 2. Pseudo proection Y =AX 0 3. Comprison of mesured dt Y with pseudo proection dt Y 4. Updte of Imge dt X n X n+1 5. X n nd X n+1 meet stopping criterium 1. X n+1 is fesible solution 6. else 1. continue with step 2 Arithmetic Reconstruction Technique (ART) ( n+ 1) Additive ART yi i' l = + ' i' ' i Multiplictive ART ( n+ 1) yi = * ' i' ' 4

5 Convergence of dditive ART Simple emple (2 piel nd 2 proection vlues) for the illustrtion of the convergence of dditive ART. Sttisticl Models: ML-EM Poisson distribution: Epecttion vlue : Mesurement vlue: Log-Likelihood: y ep( λ)* λ pλ ( y) = y! λi = i y i p ( y ) ) = ( yi log( i ) i i Itertion step: ( n+ 1) = * i i i ' yi i i' ' 5

6 ML-EM ( n+ 1) = * i i i ' yi i i' ' Convergence of ML-EM ML-EM is mimum entropy pproch, towrds high itertions noise deteriortes the imge. Itertions: Prior informtion is implemented to provide inherent smoothing during reconstruction. 6

7 Byesin Methods: Penlty Terms MAP-density p( X Y) p( Y X)* p( ) Penlty term ( ) P( X) = e αe X Medin root prior ( X M ) E( X) = 2M 2 One step lte lgorithm OSL ( n + 1 ) l = ( n ) l k kl k kl m y l k km ln P ( X ) ( n ) m = ( n ) l l Accelertion: Ordered Subsets A series of subsets {S} of the mesurement vlues Y is defined ML-EM lgorithm is clculted with ech subset {S} until stopping criterion is fulfilled Convergence is ccelerted ( n+ 1) = * i i { S} ' i yi i i' ' 7

8 Attenution Correction PET SPECT i = ep( i i Lend y ep( µ ( L) dl)* ep( 0 0 µ ( L) dl) i Lend µ ( L) dl) = y i i i y ep( i ep( 0 k L µ ( L) dl) µ ) ik Mthemticl phntom Mthemticl phntom consisting of ellipticl structures for simulting both distribution of rdio-trcer nd non-uniform ttenution: ellipticl bckground (big ellipsis) two ellipticl hot spots uniform ttenution in big ellipsis high ttenution in circulr re between hot spots (colon) Poisson distributed noise in simulted proection vlues 8

9 Mthemticl phntom Distribution of rdio-ctive trcer. Attenution coefficients used in simultion nd non-uniform ttenution Attenution coefficients used for uniform. Mthemticl phntom: results 1 FBP, α=0.8 ML-EM, uniform, 50it ML-EM,non-uniform, 50it 9

10 Mthemticl phntom: medin root prior, 50 itertions No Uniform Non-uniform Jszczk phntom: 2GBq 99mTc, Picker Prism3000, FOV=46cm, Mtri, 3.6mm slice thickness, 3 o nd 60s/step ML-EM, no ttenution ML-EM, non-uniform ttenution MAP-Gibbs prior, α=2, β=0.01, non-uniform ttenution MRP, α=0.1, non-uniform ttenution 10

11 Alderson phntom kidneys 50 MBq 99mTc, ech bldder 150 MBq 99mTc body-bckground 80 MBq 99mTc colon Teflon cylinder 23mm dimeter Picker PRISM3000, 3 o /step, 20s/step, circulr orbit, 46cm FOV,3.6mm slice thickness, mtri Philps Tomoscn SR7000, 120kV, 200mA, 34.8cm FOV, 5mm slice thickness, mtri, plnr-mode 3D rendering nd orthogonl sections of the Alderson phntom. Kidneys, bldder nd colon re segmented. 11

12 Alderson phntom: results1 FBP, α=0.8 ML-EM, no MRP, non-uniform Alderson phntom: ordered subsets (OS) MAP-Gibbs prior, α=2, β=0.01, non-uniform ttenution MRP, α=0.1, non-uniform ttenution MAP-Gibbs prior, 15 subsets, 3 it. / subset MRP, 15 subsets, 3it. / subset 12

13 In-vivo dt from hed nd neck re CT nd SPECT dt from the hed nd neck re: SPECT: Prism3000, 99mTc Sestmibi, , FOV 46cm, 3o, 20s CT: Tomoscn SR7000, 120kV, 400mA, , FOV 185mm Segmenttion of 3 components: bone 0.295/cm soft-tissue: /cm ir:0/cm Overly of CT nd SPECT imge dt Reformtted nd segmented CT imge Results: in-vivo dt MAP-Gibbs prior, α=2, β=0.025, no ttenution MRP, α=0.1, no ttenution MAP-Gibbs prior, α=2, β=0.025, non-uniform ttenution MRP, α=0.1, non-uniform ttenution 13

14 Sctter: problem in 3D Prllel Computing Concepts Single tsk computing systems (MS-DOS) Multitsking OS (Uni, NT) Multitsking OS with multiple CPU s Threds on multitsking systems Sending messges on multitsking OS 14

15 Concepts of Prllel Computing SMP: symmetric multiprocessing multiple processors use shred memory bus bsed communiction distributed shred memory Concepts of Prllel Computing Distributed memory mchines. Ech node hs ist own memory. Communiction vi messge pssing connected by fst Ethernet. Beowulf - Linu Cluster zb. Avlon Cluster t Los Almos 70 Alph Worksttions (533 MHz CPUs) 73 Gflops $ Erth Simultor (NEC) 15