Pozitron-Emission TomographyReconstruction

(A computer graphics view)

Szirmay-Kalos László

Scalar fields3D tér pontjaiban egy skalár érték

• Skalár: hőmérséklet, sűrűség nyomás, potenciál, …• Származás: Euler-i szimuláció, Rekonstrukció (tomográfia)

tárolás: 3D textúravagy „voxel tömb”

x

y

z

v(x,y,z)

v(x,y,z)

Visualization of scalar fields• Megjelenítés fényszóró anyag (participating

media) analógiáját felhasználva (belsejébe belelátunk)

• Adott szintfelület kiemelése (külsőt lehámozzuk)

Transzferfüggvény

Sűrűség + deriváltak

v

grad v

Optikai paraméterek

Képszintézis

Kép

Fényszóró közeg

• Albedo a: az elnyelődés valószínűsége feltéve, hogy az ütközés bekövetkezett

• Fekete test: albedo = 0

ds A=

1

Hatáskeresztmetszet,alias kioltási tényező

·ds = P(ütközés)

Sugársűrűség változása

ds

L(s)

s

L(s+ds)

L(s+ds)= L(s) – L(s)·(s)·ds + Le(s)·ds + (s)·a(s)·ds·f(‘,)Li(‘)d‘

// Kiszóródás+abszorbció// Emisszió // Beszóródás

dL(s)/ds = –L(s)·(s) +Le(s)+Linscatter(s)

Megoldás fényelnyelő közegre (emisszió és beszóródás nincs):L(s)= L(0)·exp(–

s (s)ds)

Tomography

L(s)= L(0)·exp(–(s)ds)(s)ds = – log L(s)/L(0)

L(s)= Le(s)ds

Absorption Emission

PNP

NPN

Ne- e+

Mediso NanoPETTM/CT

Reconstruction

FBP = Filtered backprojection

Measurement + back projection

w(x,y)=w(r) 1/ri(x,y)=(x,y)

w(x,y) dxdy=2Rw(r) rdrd=

=2 Rw(r)r dr R

Circle of radius R:

Impulse response:

Correction in frequency domain

o(x,y)=i(x,y) w(x,y) FxFy o[x,y] = FxFy i[x,y]FxFyw[x,y]

1/| |FxFy i[x,y] = FxFy o[x,y] | |

dr

rd

r

Ramp filtering

Algebric back projection

v1 v1

v3 v4

d1 = A11v1 +A13v3

d2

d3

d4

Lin egyenlet (V<D): d = A v, d = [d1,..., dD]

v = [v1,..., vV]

Moore féle pseudo-inverz:

AT d = ATA v

v = (ATA)-1AT d = A+ d

(D) (D×V) (V)

(V×D) (D)(V×V)

Iteratív séma

Skalármező

Skalármező korrekciója

Projekció: vonalintegrálok

Összevetés a mért értékekkel

Light photons

• Relativistic mass is small: E = mc2 = hf• Photon’s energy (wavelength) does not

change upon elastic collision• Absorption probability is energy dependent• Wavelengts can be handled

independently

e- e-

Gamma photons

• Relativistic mass is significant• Photon’s energy (wavelength) changes

upon collision• Absorption probability is energy dependent• Wavelengts cannot be handled

independently

e- e-

Compton scattering and the Klein-Nishina phase function

cos11

1'

2

cmEE

E

e

incident photon

scatteredphoton

z

x

y

φ

collisionE

E’

2

23

sin'''

)(E

E

E

E

E

ECP

Klein-Nishina differential cross section(extinction coefficient):

Compton formula:

P(E, ) = … Rejection sampling

cos

u

Table driven sampling

Electron density

E/mec2

Cross section

E/mec2

1

C(1+cos2)

Gamma photon transport

Compton scattering +Klein-Nishina formula = New

– Direction– Energy (freq)

Detectorgrid

Why is it important?

Measureddetectorresponse

Source distribution

???

Density distribution from CT

Iterative reconstruction

Source estimation

Measureddetectorresponse

Source correctionSource

distribution???

Compute detector response

Estimateddetector response

Density distribution from CT

PET

e-

e+

LOR: y

Intensity: x

Finite-element approximation

x (v) NxV bV (v)

1b1

b2

b3

constant

1 b1

b2

b3

Tri-linear

box tent

Higher order filter• more accurate• more sensitive to noise

Non-CC grid?

BCC and FCC grids

CC

BCC FCC

System matrix

108 LOR responses: 107 voxel intensities:

Probabilities that a photon pairemitted in voxel V is detected by LOR L

• Problems:– Huge matrix: 108×107

– A matrix element is an infinite-dimensional integral– Matrix element depends on the measured object (CT)

• Compute matrix elements on-the-fly (like in radiosity)

voxel

Iterative solver

CorrectionBack-projection

Measures LORs

Computed LORs~ ~ ~ ~

Forward-projection

Current voxelintensities

Back-projectionMaximum likelihood estimation

What is x based on y?

Find x that maximizes P(y|x)

State: x Measurement result: y

Bayes estimation:P(x|y)=P(y|x)P(x)/P(y)

Back-projection• Measured LOR hits have

Poisson-distribution with means of the current estimate:

• Joint probability:

• Likelihood: logarithm of the joint probability:

• Estimate x maximizes the likelihood:

L

Ly

L

yL

L ey

yyP

~

!

~}hits {

~~

L

Ly

L

yL ey

y ~

!

~log

ek

kPk

!}{

+const

• Gathering not shooting! – Forward: voxels to LORs– Backward: LORs to voxels

Parallel implementation issues

~~

Ratios:Same relative error in all

detector pairsFor each LOR L:

For each voxel V:

Solution alternatives for gamma photon transport

• Physicists’ approach:1. Direct mathematical model of a phenomenon

2. Solution algorithm is the simulation of the nature

3. Porting to the computer

4. Performance tuning

• Computer graphics approach:1. Computer architecture

2. Algorithm that is efficient on this architecture

3. Tuning to the problem

Physicists’ approachDetector grid

Physicists’ approachDetector grid

Physicists’ approachDetector grid

Physicists’ approachDetector grid

Woodcock tracking: Free path length sampling in heterogeneous medium

e-maxx

max

s

max

s

CG approach:x

S

Physicists’ approach• Pros

– Direct simulation of nature– Rejection sampling can mimic the Klein-Nishina phase

function and free path length.– Paths are computed independently, so it scales well

on MIMD (multi-CPU = slow and expensive)

• Cons– Bad on SIMD (rejection sampling is a loop)– Similar absolute error in detectors, the relative error is

1/n where n is the number of hits.– Paths are computed independently, so it cannot

exploit coherence (cannot reuse information)– Random writes

Computer graphics approach

• SIMD-like algorithm (GPU)– Independent threads (gathering)– “No” conditional statements or variable length loops

• Reuses paths– Detector view (gathering)

• Same relative error in all detector pairs– Same number of samples in all detector pairs– Detector view (gathering)

• Solve the adjoint problem – Like path tracing instead of photon tracing– Integral transformation (Jacobi determinant)

CUDA: implementation platform

GPUCPU +Host

program

Kernel programs:

Threads

Thread blocks

Warps, …

Thread block

Fast sharedmemory

SIMD

Quasi-SIMD

Adding two N-element vectors

__global__ void AddVectorGPU( float *C, float *A, float *B, int N ) { int i = blockIdx.x * blockDim.x + threadIdx.x; // thread id if (i < N) C[i] = A[i] + B[i]; }

float C[100000], A[100000], B[100000];

int main ( ) { … int N = 100000; … int blockDim = 256; // number of threads per block int gridDim = (N + blockDim – 1) / blockDim; // number of thread blocks AddVectorGPU<<<gridDim, blockDim>>>(C, A, B, N); …}

Runs on GPU, called from CPU

0 ,…, blockDim.x-10 ,…, gridDim.x-1

Path reuse!

Detector 1 Detector 2

Integral transformation

dz1

dz2

dA: perpendicular area

dv: differentialvolume

dl: chord length

emissionpoint

d2: solid angle in which dz2 is visible from

v

2z

1z

1z

D2

D1 d: solid angle in whichdz1 and dz2 are visible from v

1

θz

2θz

1dz

2dz

21221

21 ddd||

coscosdd zzl

zzv

Forward projection

Detector 1 Detector 2

voxel

Nt

Na

=

=

Back projection

Detector 1 Detector 2

voxel

Nt

Na

~~

It can change in time!• OSEM• sample perturbation

~

Solid angle of the farther detector:

Detector sampling

Rnd Poisson-disk

Detector modeling

Measurements and phantoms

• 18x324 LOR• 1283 … 2563 voxel

Time: NV GeForce GTX 285

GPU utilization: 25%

GPU optimization

• Registers assigned to a thread (32->64)

• Forward and back-projection re-structuring– Forward:

– Back:

Error: 1283 voxels

Reconstruction quality

N_detline=1N_detline=16, Poisson diszk

ref

Single scatter compensation

Sample the location ofthe scattering point

Reusing rays:Evaluate many

integrals in parallel

What to sample?What mimics all

integrands?

Scattering point =+ 3 dimensions

Single scatter compensation

1z

2z

s

accumulatedemission

accumulatedattenuation

accumulatedattenuation

Importance sampling:

General scatter compensation

1s

2s

1. Scattering points

1z

2. Ray marching from detector to scattering points

1s

2s

1z

2z

3. Combination of paths

1s

2s

Scatter compensation

Stochastic iteration

• EM objective function (likelihood):

Expected value has some error!• Deterministic error

• Convergent but to a bad value • Random error

• Not convergent

Stochastic iteration

• New likelihood

• Only in forward projection

Likelihood

Reconstruction error

100 iterations, 1 sample/LOR2 billion photons pair/iteration

FP=4 sec, BP=5 sec

ref n=2/n

n=3/n

n=1/n

Without stochastic iteration

Det

Rand

Scatter compensation

Geometria Single scattering Double scattering