Education– Image Reconstruction I
3D Filtered Backprojection
Fundamentals, Practicalities, and Applications
Jeff Siewerdsen, PhD
Department of Biomedical Engineering
Johns Hopkins University
Johns Hopkins University
Schools of Medicine and Engineering
Acknowledgments
I-STAR Laboratory
Imaging for Surgery, Therapy, and Radiology
www.jhu.edu/istar
Hopkins Collaborators
JW 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 Support
National Institutes of Health
Carestream Health (Rochester NY)
Siemens Healthcare (XP, Erlangen)
Disclosures
Medical Advisory Board, Carestream Health
Medical Advisory Board, Siemens Healthcare
Elekta Oncology Systems
Evolution and Proliferation of CT
Sir Godfrey Hounsfield
Nobel Prize, 1979
Evolution and Proliferation of CT
c. 1975
c. 2011
Overview
Computed Tomography
Generations
Natural history and new technology
3D Image Reconstruction
Filtered backprojection
Basics and practicalities
Open-source resources
Image Quality
Artfiacts
Spatial Resolution
Contrast Resolution
Noise
Proliferation and Applications
Diagnostic imaging
Image-guided interventions
First-Generation CT
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)
p(x)
xxx xxx x x
CT “Generations”
1st Generation (1970)
2nd Generation (1972)
Pencil Beam
Translation / Rotation
Fan Beam
Translation / Rotation
WA Kalender, Computed Tomography, 2nd Edition (2005)
CT “Generations”
3rd Generation (1976)
4th Generation (1978)
Fan Beam
Continuous Rotation
Fan Beam
Continuous Tube Rotation
Stationary Detector
WA Kalender, Computed Tomography, 2nd Edition (2005)
Helical (Spiral CT)
Pitch <1 :
Overlap
Higher z-resolution
Higher dose
Pitch >1:
Non-overlap
Lower z-resolution
Lower patient dose
Pitch =
Table increment / rotation (mm)
Beam collimation width (mm)
WA Kalender, Computed Tomography, 2nd Edition (2005)
Dual-Source CT
University of Zurich
zoom
Two complete x-ray and data acquisition systems on one gantry.
330 ms rotation time
(effective 83 ms scan time)
Siemens Healthcare– Somatom Definition
Recent Advances: Cone-Beam CT
Fully 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
Filtered
Backprojection:
The Basics
Loop over all views (all q)
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)
The Sinogram p(x,q)
p(x)
The Sinogram:
Line integral projection p(x)
… measured at each angle q
 p(x;q) “Sinogram”
q
p(x;q)
x
Filtered Backprojection
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|
X-ray source
Filtered Backprojection
The Filtered Sinogram:
p(x;q)
Convolve with RampKernel(x)
p(x)*RampKernel(x)
Equivalent to Fourier product
M(fx)|fx|
p(x;q)
q
p(x;q)*RampKernel(x)
x
X-ray source
Filtered Backprojection
Projection Data
p(x,q)
3D Reconstruction (Axial Slice)
m(x,y,z)
Cone-Beam Geometry
Definitions and Coordinate Systems
3D Filtered Backprojection
Raw Projection Data
3D Filtered Backprojection
Offset-Gain (and Defect) Correction
3D Filtered Backprojection
Log Normalization
3D Filtered Backprojection
Cosine Weighting
(Feldkamp Weights)
v
u
3D Filtered Backprojection
Data Redundancy (Parker) Weights
q
u
3D Filtered Backprojection
Ramp Filter
3D Filtered Backprojection
Smoothing / Apodization Filter
Evolution
of the
Projection
Data
3D Filtered Backprojection
Backprojection
Repeat 

# of voxels
# of projections
Evolution
of the 3D
Recon
Open-Source Resources
OSCaR
Open-Source Cone-Beam Reconstructor
MATLAB
Source code (m)
Function call (m)
Executable UI (exe)
Intended user
Education
Research
Input data types
jpg, raw, png, etc.
Flexible, transparent
Filters, voxel size,
Reconstruction filters
www.jhu.edu/istar/downloads
Open-Source Resources
PortoRECO
Portable Reconstruction
C# / C++
Source code (C#)
Executable UI (exe)
Intended user
Education
Research
Input data types
jpg, raw, png, etc.
Flexible, transparent
Filters, voxel size,
Reconstruction filters
www.jhu.edu/istar/downloads
Artifacts
Artifacts
Rings
Lag
Shading
Metal
Streaks
Motion
Truncation
“Cone-Beam”
Artifacts: X-ray Scatter
1) Reduced contrast
Reduction of DCT#
2) Image artifacts
Cupping and streaks
3) Increased image noise
Reduced DQE
Contrast (mm-1)
0.0012
Reduced soft-tissue detectability
BR-SR1 Breast in Water
0.001
0.0008
0.0006
0.0004
1  1  SPR e C0d
C '  Co 
ln
d  1  SPR
0.0002



0
0
20
40
60
80 100 120 140
SPR (%)
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 contrast
Reduction of DCT#
2) Image artifacts
Cupping and streaks
3) Increased image noise
Reduced 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
100
2) Image artifacts
Cupping and streaks
3) Increased image noise
Reduced DQE
Abdomen
Chest
SPR (%)
1) Reduced contrast
Reduction of DCT#
Pelvis
Extremity
10
Air
Reduced soft-tissue detectability
1
0
5
10
15
20
Field of View (z) (cm)
A big problem for cone-beam CT:
SPR is very large for large cone angles (i.e., large FOV)
25
Image Quality
Key Image Quality Metrics
Image Uniformity
CT # Accuracy
Spatial resolution
Contrast resolution
Noise (and NPS)
CNR
NEQ
Uniformity / Stationarity
• Signal Uniformity
(3.8 ± 4.2)
Stationarity of the mean
Shading artifacts
Beam-hardening
Truncation
Stationarity of the noise
WSS of second-order statistics
Physical effects:
Quantum noise
Bowtie filter
Sampling effects:
Intrinsic to FBP
Number of projections
View aliasing
(5.6 ± 2.4)
(-1.3 ± 6.2)
(4.6 ± 3.2)
= 3.3 HU
(4.4 ± 4.2)
Mean Signal (/mm)
• Noise Uniformity
DHU = (4.6-1.3) HU
0.20
SPR ~0
SPR ~100%
0.00
-10
0
Distance (mm)
+10
Uniformity / Stationarity
Variance Maps
• Signal Uniformity
s
2(x,y)
Stationarity of the mean
Shading artifacts
Beam-hardening
Truncation
• Noise Uniformity
Cylinder + Bowtie
Water Cylinder
Cylinder + Bowtie
Variance
Stationarity of the noise
WSS of second-order statistics
Physical effects:
Quantum noise
Bowtie filter
Sampling effects:
Intrinsic to FBP
Number of projections
View aliasing
Water Cylinder
Air
Air
-10
0
Distance (mm)
+10
s2
(/mm)2
Spatial Resolution
(line-pairs per mm)
Minimum resolvable
line-pair group
Spatial Resolution
(Modulation Transfer Function)
Spatial Resolution
(Modulation Transfer Function)
127 mm Wire in H2O
1.0
J
J
JJ
J
J
J
Steel Wire
Signal (mm-1)
J
J
0.8
J
J
J
J
J
0.6
J
J
System MTF
J
J
J
J
0.4
J
J
J
J
J
J
J
Measured
0.2
J
J
JJ
JJ
J
JJ
JJ
JJ
JJ
JJJ
0.0
0.0
0.5
1.0
1.5
-1
Spatial Frequency (mm )
MTF  f x , f y   FT LSF x, y 
JJJ
JJJJ
JJ
2.0
Image Noise
• CT image noise depends on
– Dose
– Detector efficiency
– Voxel size
• Axial, axy
• Slice thickness, az
– Reconstruction filter
k E 1 K xy
s 
3
Do  axy az
2
Barrett, Gordon, and Hershel (1976)
Image Noise
Dose
Reconstruction Filter
60
s ~ a
X
40
20
10
0
0
0.5 1.0 1.5 2.0 2.5 3.0
Dose (mGy)
Sharp
30
Smooth
Noise (CT#)
50
b
Noise-Power Spectrum
• The NPS describes
Frequency content (correlation)
ax a y az
2
NPS  fx , f y , f z  
DFT DI x, y, z 
Lx Ly Lz
Magnitude of fluctuations
Noise-Power Spectrum
Axial Plane (x,y)
Axial NPS
S(fx, fy)
Noise-Power Spectrum
Sagittal Plane (x,z)
Sagittal NPS
S(fx, fz)
Noise-Power Spectrum
NPS(fx, fy, fz)
•Axial domain:
“Filtered-ramp”
Mid-Pass
•Longitudinal domain:
“Band-limited”
Low-Pass
Contrast
A “large-area transfer characteristic”
Defined:
 As an absolute difference in mean pixel values:
For example:
C = |0.18 cm-1 – 0.20 cm-1|
= 0.02 cm-2
or
C = |-100 HU – 0 HU|
= 100 HU
 As a relative difference in mean pixel values:
For example:
C = |0.18 cm-1 – 0.20 cm-1|
0.19 cm-1
~ 10%
ROI #1
ROI #2
Contrast-to-Noise Ratio
3.5
103 HU
100 kVp
3.0
CNR
23.3 mGy
Soft-Tissue-Simulating Spheres
2.5
88 HU
2.0
66 HU
1.5
9.6 mGy
45 HU
1.0
25 HU
0.5
22 HU
0.0
2.9 mGy
11 HU
0
5
10
15
20
Dose to Isocenter (mGy)
25
0.6 mGy
Proliferation of CT
Technologies and Applications
Diagnostic Imaging
Specialty Applications
and
Image-Guided Procedures
Trade names removed.
Proliferation of CT
Technologies and Applications
Breast Screening / Diagnosis
J. M. Boone (UC Davis)
Iodine-Enhanced Tumor Imaging
Proliferation of CT
Technologies and Applications
Morphology, Function,
and Quantitation
Upper Extremities
Dual-Energy CBCT
Iodine
Calcium
Lower Extremities
Weight-Bearing
Proliferation of CT
Technologies and Applications
Skull Base Surgery
CBCT C-Arm
Spine / Orthopaedics
Thoracic Surgery
Proliferation of CT
Technologies and Applications
Head and Neck
Lung
Sarcoma
Prostate
Summary and Conclusions
Proliferation of CT
Increased utilization (and related challenges)
New technology (e.g., cone-beam CT)
Specialty applications
Open Source
OSCaR, PortoRECO
Others? (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)
Information and Handouts:
www.jhu.edu/istar