Image Reconstruction from Projections
description
Transcript of Image Reconstruction from Projections
Image Reconstructionfrom Projections
J. Anthony Parker, MD PhD
Beth Israel Deaconess Medical Center
Boston, Massachusetts
Caveat Lector
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