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Education– Image Reconstruction I
3D Filtered BackprojectionFundamentals, Practicalities, and Applications
Jeff Siewerdsen, PhDDepartment of Biomedical Engineering
Johns Hopkins University
Johns Hopkins UniversitySchools of Medicine and Engineering
I-STAR LaboratoryImaging for Surgery, Therapy, and Radiology
www.jhu.edu/istar
Hopkins CollaboratorsJW Stayman, W Zbijewski (BME)Y Otake, J Lee (Comp. Science)
R Taylor, G Hager (Comp. Science)J Prince (Electrical Engineering)D Reh (Head and Neck Surgery)
G Gallia (Neurosurgery)J Khanna (Spine Surgery)
J Wong (Radiation Oncology)J Carrino (MSK Radiology)
Funding SupportNational Institutes of Health
Carestream Health (Rochester NY)Siemens Healthcare (XP, Erlangen)
DisclosuresMedical Advisory Board, Carestream Health
Medical Advisory Board, Siemens HealthcareElekta Oncology Systems
Acknowledgments
Computed TomographyGenerationsNatural history and new technology
3D Image ReconstructionFiltered backprojectionBasics and practicalitiesOpen-source resources
Image QualityArtfiactsSpatial ResolutionContrast ResolutionNoise
Proliferation and ApplicationsDiagnostic imagingImage-guided interventions
Overview
x
Scan and Rotate:
Linear scan of source and detector
Line integral measured
at each position: p(x)
Rotate source-detector Dq
Repeat linear scan…
Projection data: p(x,q)
x x x x x x x
p(x)
First-Generation CT
WA Kalender, Computed Tomography, 2nd Edition (2005)
1st Generation (1970)
Pencil BeamTranslation / Rotation
2nd Generation (1972)
Fan BeamTranslation / Rotation
CT “Generations”
CT “Generations”
3rd Generation (1976)
Fan BeamContinuous Rotation
4th Generation (1978)
Fan BeamContinuous Tube Rotation
Stationary Detector
WA Kalender, Computed Tomography, 2nd Edition (2005)
WA Kalender, Computed Tomography, 2nd Edition (2005)
Pitch =Table increment / rotation (mm)
Beam collimation width (mm)
Pitch <1 :OverlapHigher z-resolutionHigher dose
Pitch >1:Non-overlapLower z-resolutionLower patient dose
Helical (Spiral CT)
Dual-Source CT
Two complete x-ray and data acquisition systems on one gantry.330 ms rotation time
(effective 83 ms scan time)
Siemens Healthcare– Somatom Definition
zoomUniversity of Zurich
Recent Advances: Cone-Beam CTFully 3-D Volumetric CT
Conventional CT:
Fan-Beam
1-D Detector Rows
Slice Reconstruction
Multiple Rotations
Cone-Beam CT:
Cone-Beam Collimation
Large-Area Detector
3-D Volume Images
Single Rotation
Implementation
Projection at angle q
p(x,q)
Filtered Projection
g(x,q)
Backproject g(x,q).
Add to image m(x,y)
m(x,y)
Loop
ov
er a
ll v
iew
s (a
ll q
)
p(x)
The Sinogram p(x,q)
The Sinogram:
Line integral projection p(x)
… measured at each angle q
p(x;q) “Sinogram”
x
q
p(x;q)
Simple Backprojection:
Trace projection data p(x;q)
through the reconstruction matrix
from the detector (x) to the source
Simple backprojection yields
radial density (1/r)
Therefore, a point-object is
reconstructed as (1/r)
Solution: “Filter” the projection data
by a “ramp filter” |r|
p(x;q)
X-ray source
Filtered Backprojection
p(x;q)
p(x;q)*RampKernel(x)
x
q
The Filtered Sinogram:Convolve with RampKernel(x)
p(x)*RampKernel(x)
Equivalent to Fourier product
M(fx)|fx|
p(x;q)
X-ray source
Filtered Backprojection
Open-Source Resources
MATLABSource code (m)Function call (m)Executable UI (exe)
Intended userEducationResearch
Input data typesjpg, raw, png, etc.
Flexible, transparentFilters, voxel size,Reconstruction filters
OSCaROpen-Source Cone-Beam Reconstructor
www.jhu.edu/istar/downloads
Open-Source ResourcesOpen-Source Resources
C# / C++Source code (C#)Executable UI (exe)
Intended userEducationResearch
Input data typesjpg, raw, png, etc.
Flexible, transparentFilters, voxel size,Reconstruction filters
PortoRECOPortable Reconstruction
www.jhu.edu/istar/downloads
A big problem for cone-beam CT:SPR is very large for large cone angles (i.e., large FOV)
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0 20 40 60 80 100 120 140
Contr
ast
(mm
-1)
SPR (%)
BR-SR1 Breast in Water
SPR1
SPR1ln
1'
0 dC
o
e
dCC
Artifacts: X-ray Scatter
1) Reduced contrastReduction of DCT#
2) Image artifactsCupping and streaks
3) Increased image noiseReduced DQE
Reduced soft-tissue detectability
Artifacts: X-ray Scatter
1) Reduced contrastReduction of DCT#
2) Image artifactsCupping and streaks
3) Increased image noiseReduced DQE
Reduced soft-tissue detectability
A big problem for cone-beam CT:SPR is very large for large cone angles (i.e., large FOV)
Artifacts: X-ray Scatter
1) Reduced contrastReduction of DCT#
2) Image artifactsCupping and streaks
3) Increased image noiseReduced DQE
Reduced soft-tissue detectability
A big problem for cone-beam CT:SPR is very large for large cone angles (i.e., large FOV)
1
10
100
0 5 10 15 20 25S
PR
(%
)Field of View (z) (cm)
Pelvis
Abdomen
Chest
Extremity
Air
Image Quality
Key Image Quality MetricsImage UniformityCT # AccuracySpatial resolutionContrast resolutionNoise (and NPS)CNRNEQ
0.20
0.00
Uniformity / Stationarity
• Signal UniformityStationarity of the mean
Shading artifactsBeam-hardeningTruncation
• Noise UniformityStationarity of the noiseWSS of second-order statistics
Physical effects:Quantum noiseBowtie filter
Sampling effects:Intrinsic to FBPNumber of projectionsView aliasing
(4.6 ± 3.2)(5.6 ± 2.4) (-1.3 ± 6.2)
(3.8 ± 4.2)
(4.4 ± 4.2)
DHU = (4.6-1.3) HU
= 3.3 HU
SPR ~0
Distance (mm)
Mean S
ignal (/
mm
)-10 0 +10
SPR ~100%
Uniformity / Stationarity
• Signal UniformityStationarity of the mean
Shading artifactsBeam-hardeningTruncation
• Noise UniformityStationarity of the noiseWSS of second-order statistics
Physical effects:Quantum noiseBowtie filter
Sampling effects:Intrinsic to FBPNumber of projectionsView aliasing
Air
Water CylinderCylinder + Bowtie
Air
Water Cylinder
Cylinder + Bowtie
Distance (mm)
Variance
-10 0 +10
Variance Maps s2(x,y)s2
(/mm)2
JJJJJJJJJJJJJJJJJJJJ
JJJ
JJ
JJ
JJ
JJ
JJ
JJ
JJ
J
JJ
J
J
JJ
JJ
JJ
JJ
JJJJ
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0
Spatial Frequency (mm-1
)
Steel Wire
Measured
System MTF
,, yxLSFFTffMTF yx
Sig
nal (m
m-1
)
127 mm Wire in H2O
Spatial Resolution
(Modulation Transfer Function)
• CT image noise depends on
– Dose
– Detector efficiency
– Voxel size
• Axial, axy
• Slice thickness, az
– Reconstruction filter
Barrett, Gordon, and Hershel (1976)
zxy
xy
o
E
aa
K
D
k
13
2
s
Image Noise
Image Noise
Sm
ooth
Sharp
X
ba ~s
0
10
20
30
40
50
60
0 0.5 1.0 1.5 2.0 2.5 3.0Dose (mGy)
Nois
e (
CT#
)
Dose Reconstruction Filter
• The NPS describes
Frequency content (correlation)
Magnitude of fluctuations
2
,,,, zyxIDFTL
a
L
a
L
afffNPS
z
z
y
y
x
xzyx D
Noise-Power Spectrum
NPS(fx, fy, fz)
•Axial domain:
“Filtered-ramp”
Mid-Pass
•Longitudinal domain:
“Band-limited”
Low-Pass
Noise-Power Spectrum
Contrast
A “large-area transfer characteristic”
Defined:
As an absolute difference in mean pixel values:
As a relative difference in mean pixel values:
ROI #1
ROI #2For example:C = |0.18 cm-1 – 0.20 cm-1|
= 0.02 cm-2
orC = |-100 HU – 0 HU|
= 100 HU
For example:C = |0.18 cm-1 – 0.20 cm-1|
0.19 cm-1
~ 10%
0 5 10 15 20 25
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
CN
R
Dose to Isocenter (mGy)
100 kVp
11 HU
88 HU
103 HU
66 HU
45 HU
25 HU
22 HU
9.6 mGy
2.9 mGy
0.6 mGy
23.3 mGy
Contrast-to-Noise Ratio
Soft-Tissue-Simulating Spheres
Proliferation of CTTechnologies and Applications
Trade names removed.
Diagnostic Imaging
Specialty Applications
and
Image-Guided Procedures
Proliferation of CTTechnologies and Applications
J. M. Boone (UC Davis) Iodine-Enhanced Tumor Imaging
Breast Screening / Diagnosis
Proliferation of CTTechnologies and Applications
Iodine
Calcium
Morphology, Function,
and Quantitation Dual-Energy CBCTUpper Extremities
Lower Extremities Weight-Bearing
Proliferation of CTTechnologies and Applications
Skull Base Surgery
Spine / Orthopaedics Thoracic Surgery
CBCT C-Arm
Proliferation of CTIncreased utilization (and related challenges)New technology (e.g., cone-beam CT)Specialty applications
Open SourceOSCaR, PortoRECOOthers? (Please contact me.)
More to Come – This Week:1.) Siewerdsen – Filtered Backprojection Fundamentals (MON-A-311)2.) Fessler – Iterative Image Reconstruction (TUE-A-211)3.) Yu – Optimization of image acquisition and recon (WED-E-110)
Summary and Conclusions
Information and Handouts:www.jhu.edu/istar