education image reconstruction i 3d filtered backprojection · pdf filec# / c++ source code...

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

http://www.jhu.edu/istar

Evolution and Proliferation of CT

Sir Godfrey Hounsfield

Nobel Prize, 1979

c. 1975 c. 2011

Evolution and Proliferation of CT

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



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

Filtered

Backprojection:

The Basics

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



Projection Data

p(x,q)3D Reconstruction (Axial Slice)

m(x,y,z)

Cone-Beam GeometryDefinitions and Coordinate Systems

3D Filtered Backprojection

Raw Projection Data


Offset-Gain (and Defect) Correction


Log Normalization


Cosine Weighting(Feldkamp Weights)

u

v


Data Redundancy (Parker) Weights

u

q


Ramp Filter


Smoothing / Apodization Filter

Evolution

of the

Projection

Data


Backprojection

# of voxels# of projections

Repeat

Evolution

of the 3D

Recon

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

Artifacts

Artifacts

Rings Shading

Lag

Motion

Metal

Streaks

“Cone-Beam”Truncation

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







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

Minimum resolvable

line-pair group

Spatial Resolution

(line-pairs per mm)

Spatial Resolution

(Modulation Transfer Function)

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


Axial Plane (x,y)

S(fx, fy)

Axial NPS

Sagittal Plane (x,z)

S(fx, fz)


Sagittal NPS

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:

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


J. M. Boone (UC Davis) Iodine-Enhanced Tumor Imaging

Breast Screening / Diagnosis


Iodine

Calcium

Morphology, Function,

and Quantitation Dual-Energy CBCTUpper Extremities

Lower Extremities Weight-Bearing


Skull Base Surgery

Spine / Orthopaedics Thoracic Surgery

CBCT C-Arm


Head and Neck

LungProstate

Sarcoma

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

education image reconstruction i 3d filtered backprojection · pdf filec# / c++ source code...

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