computed tomography - pusan national...

Computed Tomography

Ho Kyung [email protected]

Pusan National University

Computed Tomography

Introduction

Etymology

2

X‐ray computed tomography (CT)

• An imaging modality that produces "cross‐sectional" images representing the x‐ray “attenuation properties” of the body

• Tomo + graphy = (slice) + (to write)

Radiograph, 𝑝 𝑢, 𝑣 Tomograph, 𝑓 𝑥, 𝑦 𝜇 𝑥, 𝑦

Contrast

Taken from W. A. Kalender's Text Material (2000)3

63 3563 35 2⁄

100% 57%1738 1734

1738 1734 2⁄100% 0.23%

Classic tomography

4

Linear tomography Axial transverse tomography

Only line P1‐P2 stays in focuswhereas all others appear blurred

In principle, it simulates the backprojectionprocedure used in current times

Digital tomosynthesis

5

A compromise b/w radiography and CT

• Requiring a low number (typically tens) of projection images to compute 3D images with limited depth information

• Reconstructed by simple backprojection (or shift‐and‐add), FBP, or iterative method

• Available for chest imaging (commercially in 2004) and mammography (research prototype in 1999)

DBT prototype at the Univ. of Michigan

Park et al., Radiographics (2007)Dobbins & McAdams, EJR (2009) 6

Generations

7

1st‐generationrotate‐translate type5 min/slice

2nd‐generationmultiple detector elements20 s/slice

3rd‐generationfan‐beam geometry0.5 s/slicemost popular

4th‐generationexpensivedifficult to scatt'd radiation

Sinogram

Collection of projection data as a function of 𝜃

8

0 180

Detecto

r nu

mb

er

𝜃 𝜃

𝜃

𝑢

𝑢

𝑠

𝐼

𝐼 𝑢; 𝜃 𝐼 𝑒 𝐱

𝑝 𝑢; 𝜃 ln𝐼𝐼

𝜇 𝐱 d𝑠

Projection= ray sum= line integral= Radon transform¶

= x‐ray transform

¶ Radon transform is an integral over a plane in 3D while the x-ray transform is described as a line integral for any dimension

𝑢 𝑢

𝑢 𝑢

𝜃 𝜃

Overview

9

X

Y

Z

X‐ray source

Detector array

Isocenter

Image reconstruction

Sinogram

Reconstructed images

System

10

Collimator

• limiting the transmitted x rays to the selected slice

• preventing useless irradiation to the patient

Post‐patient collimator or antiscatter grid

• limiting the detected scattered radiation

• usually, solid‐state detectors employs external in‐plane septa

Slip rings

• to transmit power thru a brush slip ring

• to transmit data via an RF or optical slip ring

• sliding contacts that eliminate the mechanical problems

Gantry

• containing the rotating parts

• can be tilted over a limited angle for imaging oblique slices

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Detectors

12

Energy integrating detectors

• Scintillation crystal with photomultiplier tube (PMT)

‒ Scintillation crystals: NaI, CaF2, BGO …

• Converting x rays into visible light (scintillations)

‒ PMT

• Converting light into an electric current

‒ Pros: high quantum efficiency, fast response time

‒ Cons: low packing density

Scintillator

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• Gas ionization chambers

‒ Consisting of a pressurized (10 – 30 bars) gas chamber (Xenon) with electrodes

‒ Gas ionization drifts electron‐ion pairs along field lines induction of electric currents

‒ Pros: high packing density

‒ Cons: low quantum efficiency (~ 60%), slow response time (~ 700 s)

14

• Scintillation crystals with photodiode (most recent commercial CT detectors)

‒ Scintillation crystals: CdWO4, Y2O3, CsI, Gd2O2S

• Individual scintillator pieces are assembled into a reflector matrix in order to define the detector cells

• Few mm thickness to have a very high absorption efficiency (96%)

» Considering the finite thickness of the septa in the antiscatter grid, the absorption efficiency is limited by the area fill fraction (~80%)

• Good transfer of light to the photodiode

• Very fast response time (~s)‒ Solid‐state or semiconductor detectors (photodiodes)

‒ Multichannel readout electronics or data acquisition system (DAS)

• Integrating the photocurrent from the diode and converting the electric charge signal to voltage using a transimpedance amplifier

• Performing the analog to digital conversion with typical sampling rates (~kHz)

‒ Susceptible to electronic noise introduced by the transimpedance amplifier

• Dominant at low signal levels, leading to noise streaks in the images

15

16

Photon counting detectors

• Based on direct conversion

‒ Detector materials: cadmium telluride (CdTe) or cadmium zinc telluride (CdZnTe or CZT)

‒ Converting x‐ray photon into a certain electronic charge proportional to its energy

• x10 larger than that produced by the scintillator/photodiode combination

• The electronic noise no longer dominates the signal from individual x‐rays

‒ Electronic circuits detect charge packages and count the number of photons instead of integrating their energy

• Improving the CNR by 10 to 20%

‒ Remaining challenges include stability and the count rate limits

Image courtesy: E. Roessl et al., Philips (2009)

17M. J. Willemink et al., Radiology (2018)



20C. H. McCollough et al., Radiology (2015)

21C. H. McCollough et al., Radiology (2015)


Image quality

23Taken from W. A. Kalender's Text Material (2000)

80 80 pixels (w/ slice 13 mm)

1024 1024 pixels

CT number

24

CT measures and computes the spatial distribution of the linear attenuation coefficient 𝜇 𝑥, 𝑦• Note that 𝜇 𝑥, 𝑦 ~𝑓 𝐸, 𝑍• Impossible in direct comparison of images obtained CT systems with different voltages and

filtration

Instead, use CT value as a so called “CT number” in Hounsfield unit (HU)

• Compute attenuation coefficient relative to the attenuation of water

• Hounsfield units (HU) in the range of ‐1000 to 1000

‒ CT number in HU 1000

• Air = ‐1000

• Water = 0

• Bone = the positive side scale but no unique CT number ( 𝜇 ~ composition, structure, 𝐸)

• Immune to different spectra

HU

-1000

waterair 0 T

CT#


CT number differences due to effective atomic numbers decrease for higher energies

Contrast at high energies is dominated by density differences

Negative contrast of fat is caused both by the low effective atomic number and by the low density


27

Typically the CT image consists of 512 512 pixels representing the CT numbers

Due to the large dynamic range, window/leveling must be used to view a CT image

• Choose the center and width (C/W) of the window

Taken from W. A. Kalender's Text Material (2000)

Original

Bone window Lung window

(Bimodal histogram)

Image reconstruction

Inverse problem

• Usually inconsistent, ill‐posed problem

‒ Noise corruption

‒ Huge size of 𝐴‒ 𝐴 does not exist

‒ Solution is not unique

‒ Solution is unstable

28

[System characteristic]{Image} = {Measurement}

Solving methods

• Iterative solvers (discretization)‒ Series expansion methods (e.g., ART)

‒ Statistical methods

‒ Optimization (regularization)

• [MLEM, LS, ART] + [priori/penalty]

• Analytic (kernel) solver

‒ 𝜇 𝐱 d𝐱 𝑘 𝐱 𝑝 𝐱 𝐱

Projection and Radon transform

29

Consider the 2D parallel‐beam geometry.

• 𝜇 𝑥, 𝑦 = the distribution of the linear attenuation coefficient in the 𝑥𝑦‐plane with a diameter FOV

• 𝐼 = the unattenuated intensity of the x‐ray beams

• 𝑟, 𝑠 = a new coordinate system defined by rotating 𝑥, 𝑦 over the angle 𝜃

Transformations

•𝑟𝑠

cos 𝜃 sin 𝜃sin 𝜃 cos 𝜃

𝑥𝑦

•𝑥𝑦

cos 𝜃 sin 𝜃sin 𝜃 cos 𝜃

𝑟𝑠

• with the Jacobian

‒ 𝐽 cos 𝜃 sin 𝜃sin 𝜃 cos 𝜃

1

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For a fixed angle 𝜃, the measured intensity profile as a function of 𝑟 is given by:

• 𝐼 𝑟 𝐼 𝑒,

, 𝐼 𝑒,

,

‒ 𝐿 , = the line that makes an angle 𝜃 with the 𝑦‐axis at distance 𝑟 from the origin

Considering the polyenergetic x‐ray spectrum:

• 𝐼 𝑟 𝐼 𝐸 𝑒, ,

, d𝐸

• Practically, it is assumed that x rays are monochromatic

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Transforming each intensity profile into an attenuation profile (log transform):

• 𝑝 𝑟 ln 𝜇 𝑟 cos 𝜃 𝑠 sin 𝜃 , 𝑟 sin 𝜃 𝑠 cos 𝜃 d𝑠,

‒ 𝑝 𝑟 = the projection of the function 𝜇 𝑥, 𝑦 along the angle 𝜃‒ 𝑝 𝑟 = zero for |r| FOV/2

‒ Sufficient to measure 𝑝 𝑟 for 𝜃 ranging from 0 to 𝜋 as far as parallel‐beam geometry because of concurrent beams

Sinogram: 2D dataset 𝑝 𝑟, 𝜃 by stacking all these projections 𝑝 𝑟

Radon transform: the transformation of any function 𝑓 𝑥, 𝑦 into its sinogram 𝑝 𝑟, 𝜃

• 𝑝 𝑟, 𝜃 ℛ 𝑓 𝑥, 𝑦 𝑓 𝑟 cos 𝜃 𝑠 sin 𝜃 , 𝑟 sin 𝜃𝑠 cos 𝜃 d𝑠

• Periodic in 𝜃 with period 2𝜋: 𝑝 𝑟, 𝜃 2𝜋• Symmetric in 𝜃 with period 𝜋 : 𝑝 𝑟, 𝜃 𝜋

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Projection and back‐projection of a single dot:

Sampling

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In practice, available are a limited number 𝑀 of projections or views and a limited number 𝑁 of detector samples

• Discrete sinogram 𝑝 𝑛∆𝑟, 𝑚∆𝜃‒ A matrix with 𝑀 rows and 𝑁 columns

‒ ∆𝑟 = detector sampling distance

‒ ∆𝜃= the rotation interval b/w subsequent views

To limit aliasing: ∆ ∆

or ∆𝑟∆

(two samples/beam width

∆𝑠)

What is the min number of views?

• maxdist 𝐿 , , 𝐿 , maxdist 𝐿 , , 𝐿 ,

• maxdist 𝐿 , , 𝐿 ,·

°

• maxdist 𝐿 , , 𝐿 ,

• number of views per 360° 𝜋 · number of detector samp• e.g., if FOV = 50 cm & ∆𝑠 = 1 mm 𝑁 = 1000 & 𝑀 = ~3000

𝑝 𝑟

∆𝑠

sampledprojection

smoothedprojection

samplingat ∆𝑟

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Typically the number of views per 360 is on the order of the number of detector channels

• GE scanners with 888 detector channels acquire 984 views per rotation

• Siemens scanners with 768 detector cells use 1056 views per 360

Means to improve sampling include quarter detector offset and in‐plane focal spot wobbleor deflection

Backprojection

35

• How to reconstruct the distribution 𝜇 𝑥, 𝑦 (or, 𝑓 𝑥, 𝑦 ) for the given sinogram 𝑝 𝑟, 𝜃 ?

– Assign the value 𝑝 𝑟, 𝜃 to all points 𝑥, 𝑦 along a line 𝑟, 𝜃 and repeat this for 𝜃 ranging from 0 to 𝜋 backprojection:

– 𝑏 𝑥, 𝑦 ℬ 𝑝 𝑟, 𝜃 𝑝 𝑥 cos 𝜃 𝑦 sin 𝜃 , θ d𝜃

– Note that the resultant image is blurred a simple backprojection is unsatisfactory

– Discrete version of BP:

– 𝑏 𝑥 , 𝑦 ℬ 𝑝 𝑟 , 𝜃 ∑ 𝑝 𝑥 cos 𝜃 𝑦 sin 𝜃 , 𝜃 ∆𝜃

– Note that the discrete positions rn generally do not coincide with the discrete values 𝑥 cos 𝜃 𝑦 sin 𝜃

– Interpolation is required; the corresponding projection value is calculated by interpolation b/w its neighboring measured values (called “pixel‐driven” or “voxel‐driven” backprojection).

Projection theorem

36

More mathematically instead of the previous intuitive answer we need a mathematical expression for the inverse Radon transform: 𝑓 𝑥, 𝑦 ℛ 𝑝 𝑟, 𝜃

Projection theorem, called central (or Fourier) slice theorem:

• 𝐹 𝑘 , 𝑘 𝑓 𝑥, 𝑦 𝑒 d𝑥d𝑦 or 𝑃 𝑘 𝑝 𝑟 𝑒 · d𝑟

• If 𝜃 is variable, 𝑃 𝑘 𝑃 𝑘, 𝜃

• The projection theorem states that 𝑃 𝑘 𝐹 𝑘 , 𝑘 with

𝑘 𝑘 cos 𝜃𝑘 𝑘 sin 𝜃

𝑘 𝑘 𝑘

• “The 1D FT w.r.t. variable 𝑟 of the Radon transform of a 2D function is the 2D FT of that function”

37

𝜃 135°

𝜃 135°

𝜃 135°

38

Proof:

𝐹 𝑘 , 𝑘 𝑓 𝑥, 𝑦 𝑒 d𝑥d𝑦

𝑓 𝑥, 𝑦 𝑒 · · · · d𝑥d𝑦

𝑓 𝑟 · cos 𝜃 𝑠 · sin 𝜃 , 𝑠 · cos 𝜃 𝑟 · sin 𝜃 𝑒 · · · · · · d𝑠d𝑟

𝑓 𝑟 · cos 𝜃 𝑠 · sin 𝜃 , 𝑠 · cos 𝜃 𝑟 · sin 𝜃 𝑒 · d𝑠d𝑟

𝑓 𝑟 · cos 𝜃 𝑠 · sin 𝜃 , 𝑠 · cos 𝜃 𝑟 · sin 𝜃 d𝑠 𝑒 · d𝑟

𝑝 𝑟, 𝜃 𝑒 · d𝑟

𝑃 𝑘, 𝜃

Direct Fourier reconstruction

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1. Calculate the 1D FT of all the projections 𝑝 𝑟, 𝜃 : ℱ 𝑝 𝑟 𝑃 𝑘2. Put all the values of the 1D function P(k) on a polar grid to obtain the 2D function 𝑃 𝑘, 𝜃 ,

which is equal to 𝐹 𝑘 , 𝑘

3. Calculate the 2D IFT of 𝐹 𝑘 , 𝑘 : ℱ 𝐹 𝑘 , 𝑘 𝑓 𝑥, 𝑦

This method requires the interpolation in step 2, and which causes artifacts, making this method less popular than the filtered backprojection method.

Filtered backprojection (FBP)

40

• To avoid interpolation:

• FBP scheme:

① Filter the sinogram 𝑝 𝑟, 𝜃 : ∀𝜃: 𝑝∗ 𝑟 𝑝 𝑟 ∗ 𝑞 𝑟 or 𝑃∗ 𝑘 𝑃 𝑟 · 𝑘

② Backproject the filtered sinogram 𝑝∗ 𝑟, 𝜃 : ∀𝜃: 𝑓 𝑥, 𝑦 𝑝∗ 𝑥 cos 𝜃 𝑦 sin 𝜃 , 𝜃 d𝜃

ramp filter

convolution kernel

𝑓 𝑥, 𝑦 𝑃 𝑘, 𝜃 𝑘 𝑒 d𝑘d𝜃 with 𝑟 𝑥 cos 𝜃 𝑦 sin 𝜃

𝑃∗ 𝑘, 𝜃 𝑒 d𝑘d𝜃

𝑝∗ 𝑟, 𝜃 d𝜃

𝑝∗ 𝑟, 𝜃 𝑝 𝑟 , 𝜃 𝑞 𝑟 𝑟 d𝑟′

𝑞 𝑟 ℱ 𝑘 𝑘 𝑒 d𝑘

with 𝑟 𝑥 cos 𝜃 𝑦 sin 𝜃

with 𝑟 𝑥 cos 𝜃 𝑦 sin 𝜃

𝑃∗ 𝑘, 𝜃 𝑃 𝑘, 𝜃 𝑘

𝑝∗ 𝑟, 𝜃 𝑃∗ 𝑘, 𝜃 𝑒 d𝑘

Filter functions

41

In practice, the continuous filter 𝑘 is not useful because of its divergent nature

• The useful Fourier content is limited to frequencies smaller than 𝑘 1 ∆𝑠⁄ 1 2∆𝑟⁄

• Ram‐Lak filter (Ramachandran & Lakshiminarayanan): difference of a block & a triangle

• In space domain: 𝑞 𝑟

• Usually, frequencies slightly below 𝑘 are unreliable because of aliasing & noise– Application of a smoothing window (Hanning (𝛼 = 0.5), Hamming (𝛼 = 0.54), Shepp-Logan,

Butterworth) suppresses the highest spatial frequencies & reduces these artifacts

– 𝐻 𝑘𝛼 1 𝛼 cos for 𝑘 𝑘

0 for 𝑘 𝑘

Hamming window

Hanning window

Fan‐beam FBP

42

Changed coordinates from 𝑟, 𝜃 to 𝛾, 𝛽 :

• 𝛾 = the angle b/w a particular ray & the center line of the corresponding fan

• 𝛽 = the angle b/w the source and the 𝑦‐axis• Fan‐angle = the angle formed by the fan

• Required a range from 0 to (𝜋+ fan‐angle) to include all line measurements

43

Rebinning: reordering the data into parallel data with interpolation

Adaptive FBP algorithm:

• Recall the FBP method: 𝑓 𝑥, 𝑦 𝑝 𝑟 , 𝜃 𝑞 𝑥 cos 𝜃 𝑦 sin 𝜃 𝑟′ d𝑟′d𝜃/

/

‒ 1/2 for compensating the modification of the integration limits from 0 to 2‒ Coordinate transforms: 𝜃 𝛾 𝛽 and 𝑟 𝑅 sin 𝛾

• 𝑓 𝑥, 𝑦 𝑝 𝑟 , 𝛽 𝑞 𝑥 cos 𝛾 𝛽 𝑦 sin 𝛾 𝛽 𝑅 sin 𝛾 ′ 𝑅 sin 𝛾 ′d𝛾′d𝛽

• After a few calculations

‒ 𝑓 𝑥, 𝑦 𝑅 cos 𝛾′ · 𝑝 𝑟 , 𝛽 𝑞 𝛾 𝛾′ d𝛾′d𝛽

• Modified FBP weighted with 1/𝐿• Inner integral is a convolution of 𝑝 𝑟, 𝛽 , weighted with 𝑅 cos 𝛾, with a modified filter kernel

𝑞 𝛾

44

For the detectors lying on a straight line perpendicular to the center line of the fan;

• Changed coordinates from 𝑟, 𝜃 to 𝑡, 𝛽 :‒ 𝑡 = the distance from the origin to the ray through 𝑥, 𝑦 measured

parallel to the detector array

• The weighted FBP:

‒ 𝑓 𝑥, 𝑦/

· 𝑝 𝑡 , 𝛽 · 𝑞 𝑡 𝑡′ d𝑡′d𝛽

• 𝑈 = the projection of the source‐to‐point distance onto the central ray of the fan

Entire volume in one single scan

Required a 2D array of detector elements

Various 3D CBCT algorithms

• Exact direct solutions (Grangeat): special constraints on the orbit of the source

• Iterative reconstructions: computationally very expensive

• Approximate reconstructions (FDK algorithm)

‒ Derived from the standard fan‐beam formula

‒ Obtain the density in a particular voxel as the sum of the contributions from all projections thru that voxel

‒ Correctly consider the angle b/w two views and the distance from the source to the reconstructed voxel

‒ Unavoidable cone‐beam artifacts

Cone‐beam reconstructions

45

𝑓 𝑥, 𝑦1

𝑈/𝑅𝑅

𝑅 𝑡′ 𝜁· 𝑝 𝑡 , 𝜁, 𝛽 ·

12

𝑞 𝑡 𝑡′ d𝑡′d𝛽

Dedicated scanners

46

Circular cone‐beam scanning with flat‐panel detectors

Limited FOV and not critical scan time

Oral and maxillofacial CT

47

Interventional CT

• C‐arm systems

• O‐arm systems

Brest CT

Robotic trajectory

48Kalender & Kyriakou, EJR (2007)

A courtesy of Siemens Medical Systems

computed tomography - pusan national...

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