Image Reconstructionfrom Projections

J. Anthony Parker, MD PhD

Beth Israel Deaconess Medical Center

Boston, Massachusetts

Caveat Lector

Tony_Parker@BIDMC.Harvard.edu

ProjectionSingle Slice

Axial

Single Axial Slice: 3600

collimator

Ignoring attenuation, SPECT data are projections

Attenuation: 180o = 360o

keV

150

100

80

60

50

x

Tc-99m

htl(140 keV) ≈ 4 cm

Cardiac Perfusion Data Collection Special Case - 180o

AxialCoronal / Sagittal

Multiple simultaneous axial slices

Dual-Head General-Purpose Gamma Camera: 900 “Cardiac” Position

2 heads: 900 rotation = 1800 data

1

2

Inconsistent projections“motion corrected”

Original data

0

0

0

00

0

0

0

Single Axial Slice: 3600

Sinogram: ProjectionsSingle Axial Slice

0 60

060

pro

ject

ion

an

gle

x

x

x

Uniformity & Motion on Sinogram

1 h

ea

d2

4 m

in

2 h

ea

ds

12

min

12

min

Reconstruction by Backprojection

Backprojection tails

Backprojection2 projections2 objects

projection tailsmerge resulting

in blurring

Projection -> Backprojection of a Point

(1/r)

backprojectionlines add atthe point

tails spread point out

Projection -> Backprojection

Projection->Backprojection Smooths

Smooths or “blurs” the image

(Low pass filter)

((Convolution with 1/r))

Nuclear Medicine physics

Square law detector adds pixels

-> always blurs

Different from MRI (phase)

(Projection-Slice Theorem)“k-space (k,)”

detail

lowfrequency

spatial frequency domainspatial domain2D Fouriertransform

Spatial Frequency Basis Functionsf(u,v) ≠ 0, single u,0f(u,v) ≠ 0, single 0,v

f(u,v) ≠ 0, single u = v

Projection -> Backprojection: k-space

1/k

(Density ofslices is 1/k)

(Fourier Transform of 1/r <-> 1/k)

one projectionmultiple projections

Image Reconstruction: Ramp Filter

Projection -> Backprojection

blurs with 1/r in object space

k-space 1/k ( 1/r<-> 1/k)

Ramp filter

sharpen with k

(windowed at Nyquist frequency)k

k

Low Pass Times Ramp Filter

Low pass,Butterworth– noise

Ramp –reconstruct

What’s Good about FPB

Ramp filter exactly reconstructs projection

Efficient

(Linear shift invariant)

(FFT is order of n log(n)

n = number of pixels)

“Easily” understood

New Cardiac Cameras

Solid state - CZT: $$$, energy resolution

scatter rejection, dual isotope

Pixelated detector: count rate &

potential high resolution

poorer uniformity

Non-uniform sampling: sensitivity

potential for artifacts

Special purpose design

closer to patient: system resolution

upright: ameliorates diaphragmatic attenuation

Collimator Resolution*

Single photon imaging (i.e. not PET)

Collimators: image formation

Sensitivity / resolution trade-off

Resolution recovery hype

“Low resolution, high sensitivity ->

image processing = high resolution”

Reality - ameliorates low resolution

Steve Moore: “Resolution: data = target object”

Can do quick, low resolution image

* not resolution from reduced distance due to design

Dual Head: Non-Uniform Sampling

Activity Measurement: Attenuation

keV

150

100

80

60

50htl(140 keV) ≈ 4 cm

Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%)

Gd-153 rods T1/2 240 d e.c. 100% 97 keV 29% 103 keV 21%

2 heads: 900 rotation = 1800 data

Semi-erect: Ameliorates Attenuation

Leaning Forward, < 500 Pounds

Digirad: Patient RotatesX-ray Attenuation Correction

CT: Polychromatic Beam -> Dose

keV

150

100

80

60

50

X-ray Tube Spectra

bremsstrahlung

characteristic X-rays

e- interaction:- ionization- deflection

X-ray tube: electrons on Tungsten or Molybdenum

Digirad X-ray Source: X-rays on Lead

74W

82Pb

X-rays interaction- ionization- no 10 bremsstrahlung

Digirad X-ray Spectrum

New Cardiac Cameras

D-SPECT CardiArc Digirad GE

Detector CZT* NaI(Tl) CsI(Tl) CZT*

Electronics SS* PMT PD*? SS*

Pixelated Y N Y Y

Collimation holes slits*? holes pinholes

Non-uniform Y* Y* ~N Y*

Limited angle Y Y N ~N

Closer to pt Y Y Y ~N

AC N CT? CT* CT

Position ~semi semi erect supine

Soft Tissue Attenuation: Supine

breast

lung

Soft Tissue Attenuation: Prone

breast

Soft Tissue Attenuation: Digirad Erect

breast

post

Sequential Tidal-Breathing Emission and Average-Transmission Alignment

Sensitivity / Resolution Trade-Off

Non-uniform sampling -> sensitivity

Special purpose design -> resolution

Advantages

Throughput at same noise

Patient motion - Hx: 1 head -> 2 head

Cost

Non-uniform sampling -> artifacts

History: 7-pinhole - failed

180o sampling - success

Sequential emission transmission

What’s Wrong with FilteredBackprojection, FBP, for SPECT

Can’t model:

Attenuation

Scatter

Depth dependant resolution

New imaging geometries

(Linear shift invariant model)

Solution

Iterative reconstruction

Uses:

Simultaneous linear equations

Matrix algebra

Can model image physics

(Linear model)

Projections as Simultaneous Equations(Linear Model)

But, exact solution for a largenumber of equations isn’t practical

Iterative Backprojection Reconstruction

Af

n

p

fn-1^ pn-1

^ en-1^

fn^

+

- x

+

f0^

r

H

H

A

object data

projection backprojection

estimate

model

error

estimate

estimateddata

estimate +backprojected

error

Reconstruction, H, can be Approximate

Af

n

p

fn-1^ pn-1

^ en-1^

fn^

+

- x

+

f0^

r

H

H

A

Accuracy of Model, A, is Key

Af

n

p

fn-1^ pn-1

^ en-1^

fn^

+

- x

+

f0^

r

H

H

A

^

Model, A, is Well-known PhysicsProblem: Model of the Body

^

Tc-99m half-tissue layer: 4 cm

Attenuation Map Gd-153 Transmission

Map adds noise to reconstructionand can introduce artifacts

Iterative ReconstructionNoise is “Blobby”

What’s Good About Iterative Reconstruction

Able to model:

Data collection, including new geometries

Attenuation

Scatter

Depth dependant resolution

Fairly efficient given current computers

(Iterative solution, e.g. EM, reasonable)

(OSEM is even better)

((OSEM has about 1/nsubsets of EM iterations))

What’s Wrong with Iterative Reconstruction

(Complicated by ill conditioned model)

((Estimating projections not object))

Noise character bad for oncology

To model attenuation & scatter

- need to measure attenuation

- adds noise

Conclusions

Filtered backprojection, FBP

Efficient

(Models noise)

“Easy” to understand

Iterative reconstruction, OSEM

Moderately efficient

Models noise, attenuation, scatter,

depth dependant resolution,

and new cameras

Applause