Computed Tomography (Part 2)eeweb.poly.edu/~yao/EL6813/Lect5_CT_ch6_part2_F18_JM.pdf · 2018-09-28 · Computed Tomography (Part 2) Jonathan Mamou and Yao Wang. Polytechnic School

Computed Tomography(Part 2)

Jonathan Mamou and Yao WangPolytechnic School of Engineering

New York University, Brooklyn, NY 11201

Based on Prince and Links, Medical Imaging Signals and Systems and Lecture Notes by Prince. Figures are from the book.

EL-GY 6813 / BE-GY 6203 / G16.4426Medical Imaging

F16 Yao Wang @ NYU 2

Last Lecture• Instrumentation

– CT Generations– X-ray source and collimation– CT detectors

• Image Formation– Line integrals– Parallel Ray Reconstruction

• Radon transform• Back projection• Filtered backprojection• Convolution backprojection• Implementation issues


This Lecture• Review of Parallel Ray Projection and Reconstruction• Practical implementation with samples• Fan Beam Reconstruction• Signal to Noise in CT


Review: Projection Slice Theorem• Projection Slice theorem

– The 1D Fourier Transform of a projection at angle θ is a line in the 2D Fourier transform of the image at the same angle.


Reconstruction Algorithm for Parallel Projections

• Backprojection Sum:– Backprojection of each projection– Sum

• Filtered backprojection:– 1D FT of each projection– Filter each projection in frequency domain– Inverse 1D FT– Backprojection– Sum

• Convolution backprojection– Convolve each projection with the ramp filter– Backprojection– Sum

• Which one gives more accurate results? Which one takes less computation?


Practical Implementation• Projections g(l, θ) are only measured at finite intervals

– l=nτ; – τ chosen based on maximum frequency in G(ρ,θ), W

• 1/τ >=2W or τ <=1/2W (Nyquist Sampling Theorem)• W can be estimated by the number of cycles/cm in the projection direction in the most detailed

area in the slice to be scanned• For filtered backprojection:

– Fourier transform G(ρ,θ) is obtained via FFT using samples g(nτ, θ)– If N sample are taken, ideally 2N point FFT should be taken by zero padding

g(nτ, θ)• Recall convolving two signals of length N leads to a single signal of length 2N-1

• For convolution backprojection– The ramp-filter is sampled at l=nτ– Sampled Ram-Lak Filter - limiting the rho-filter to within (-1/2τ , -1/2τ )

( )

==−

==

evennoddnn

nnc

;0;/1

0;4/1)( 2

2

πττ

See paper at http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3708767/

Practical Implementation of Filtered Backprojection


From paper at http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3708767/pdf/1475-925X-12-50.pdfNote this implementation perform L point DFT assuming the projection data has L points. L is odd. The reconstructed values near the boundary may have ringing artifacts.

Note that backprojection is implemented within the evaluation of inverse FT.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3708767/pdf/1475-925X-12-50.pdf

Practical Implementation of Convolution Backprojection


Here again: Back projection is implemented within convolution!


The Ram-Lak Filter (from [Kak&Slaney])


1st Generation CT: Parallel Projections


3G: Fan Beam

Much faster than 2G


Fan Beam: Equiangular Ray

We will focus on the equiangular detector setting on the right in this lecture.


Equiangular Ray Projection

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Derivation not required for this class. Detail can be found at [Kak&Slaney].Note typos in [Prince&Links], 1st ed.

β−φ Convolution weighted backprojection


Typos in [Prince&Links, 1st Ed]• P. 207, Eq. (6.38), change to

• Eq. (6.39) change to

• Eq. (6.40),(6.41)

)(sin'

)sin'(2

γγ

γγ cD

Dc

=


Practical Implementation• Projections p(γ,β) are only

measured at finite intervals– γ=nα; – α chosen based on maximum

frequency in γ direction, W• 1/α>=2W or α<=1/2W

• For convolution backprojection– The filter cf(γ) is sampled at γ=nα– Sampled Filter g(nα)– q(nα,β) = p’ (nα,β) *g(nα)

• For backprojection– For given (r,φ), for a given β,

determine (D’,γ)– Use interpolation to determine

q(γ,β) from known values at γ=nα– Modify the projection based on D’

(weighted backprojection)Back projection and sum

( ) ( )( )

( ))sin()cos(),(tan

)cos()sin(),(' 222

φβφβφγ

φβφβφ

−+−=

−+−+=

rDrr

rrDrD

-1/2


Matlab Functions for Fan Beam CT• Relevant functions:

– fanbeam(), ifanbeam()

Fan Beam Reconstruction Through Rebinning [Smith&Webb]


Constrained Optimization Formulation (Algebraic Reconstruction Techniques or ART)



CT Quality Evaluation• Blurring Effect• SNR


Effect of Area Detector• Practical detector integrates the detected photons over

an area• Mathematically, the detector can be characterized by an

indicator function s(l) (aka impulse response)• The measured projection g’(l,θ) is related to the “real”

projection g(l,θ) by– g’(l,θ)= g(l,θ) * s(l)– G’(ρ,θ)= G(ρ,θ) S(ρ)


Windowing Function• Recall that the ideal filter c(ρ) is typically modified by a window

function W(ρ)

• Overall Effect

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thefrom image tedreconstruc theas of thought becan ),(ˆ

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θθρρθρθρ

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=⇔=


Blurred Projection

h(x,y): PSF of the blurring


Circular Symmetry of Blurring


PSF given by Hankel Transform

222 yxr +=


Circularly Symmetric Functions and Hankel Transform

• Circularly symmetric:– f(x,y) = f(r), only depends on the distance to the origin, not angle

• Fourier transform of circularly symmetric function is also circularly symmetric– F(u,v)=F(ρ)

( )

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==−−=

=

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00

00

00

)sincos(1)(;)2()(2)(

Transform Hankel

)()2()(2)()}cos(2exp{),(

)(),( If

)}cos(2exp{),(

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sin,cos;sin,cosLet

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Bessel function of the first kind and 0

thorder


Common Transform pairs• See Table 2.3

• Scaling property

• Duality: If h(r) <-> H(ρ), then H(r)<->h(ρ)

{ } )/(1f(ar)Hankel 2 aqFa

=

{ }{ }

{ }2

)()(

41)(2)(sinHankel

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Derivation of Hankel transform pairs are not required. But you should be able to use given transform pairs, to determine the blur function.


Example • Example 6.5 in [Prince&Links]

– Detector: rectangular detector with width d

• S(l)=rect(l/d)– Rectangular window function

• W(ρ)=rect(ρ/2ρo); ρo>>1/d

– What is the PSF ?

• Solution– S(l)=rect(l/d) <-> S(ρ)=d sinc(dρ)– ρo >>1/d -> – H(ρ)=S(ρ) W(ρ)~= S(ρ) =dsinc(dρ)

(Hankel transform of h(r))

{ }

{ }

{ }

22

2

2

4

)/(2)(

propertyduality Using

)/(1f(ar)Hankel

41

)(2)(sincHankel

)(Hankel inverse)(

rd

drrectrh

aFa

rectr

Hrh

−=

=

−=

=

π

ρ

ρπ

ρρ

Illustrate and explain h(r)


Noise in CT Measurement

akNVarakNE

kekakN

ij

ij

ak

ij

==

==

=== −

}{

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,...1,0;!

}Pr{


What about the measured projection


CBP Approximation


Definitions and Assumptions

gij are independent because Nij are independentDeriving mean and variance of µ(x,y) based on the independence assumptionSee [Prince&Links] for derivation


• Variance increases with ρo (cut-off freq. of filter), and T (detector spacing), decreases with M (number of angles), N bar (or N0) (x-ray intensity)


SNR of the Reconstructed Image

C: fractional change of µ from \bar µ


SNR in a good design


SNR in Fan Beam

N=Nf/Dw=L/D

SNR decreases as D increases. Reason: Convolution of the projection reading with the ramp filter couples the noise between detectors, and effectively increases the noise as the number of detector increasesBut larger D is desired to obtain a good resolution.


Rule of Thumb


Aliasing Artifacts• Nyquist Sampling theorem:

– If the maximum freq of a signal is fmax, it should be sampled with a freq fs>=2max, or sampling interval T<=1/2fmax

– If sampled at a lower freq. without pre-filtering, aliasing will occur• High freq. content fold over to low freq

– Prefilter to lower fmax, and then sample

• If the number of samples in each projection (D) or the number of projection angles (M) are not sufficiently dense, the reconstructed image will have streak artifacts– Caused by aliasing– Practical detectors are area detectors and perform pre-filtering

implicitly


From [Kak&Slaney] Fig. 5.1


Summary• Parallel projection reconstruction

– Backprojection summation– Fourier method (projection slice theorem)– Filtered backprojection– Convolution backprojection– Practical implementation: using finite samples

• Fan beam projection and reconstruction– Weighted backprojection

• Blurring due to non-ideal filters and detectors– Approximate the overall effect by a filter:

• h(l)=w(l)*s(l); H(ρ)=W(ρ) S(ρ)– Circularly symmetric functions and Hankel transform

• Equivalent spatial domain filter h(r)=inverse Hankel {H(q)}• Noise in measurement and reconstructed image

– Factors influencing the SNR of reconstructed image• Average X-ray intensity, Number of angles (M), number of samples per angle (D), filter

cut-off ρo

• Impact of number of projection angles and samples on reconstruction image quality

– Nyquist sampling theorem– Streak artifacts


Reference• Prince and Links, Medical Imaging Signals and Systems, Chap 6.• A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging.

Originally published by IEEE, 1998. E-copy available at http://www.slaney.org/pct/

– Chap 3 Contain detailed derivation of reconstruction algorithms both for parallel and fan beam projections. Have discussions both in continuous domain and implementation with sampled discrete signals.

– Chap 5 discusses noise in measurement and reconstructed image.– Chap 5 also covers aliasing effect with more mathematical interpretations

• A useful lecture note from Prof. Fessler– http://web.eecs.umich.edu/~fessler/course/516/l/c-tomo.pdf

• Lecture note by Prof. Parra– http://bme.ccny.cuny.edu/faculty/parra/teaching/med-imaging/lecture4.pdf– http://bme.ccny.cuny.edu/faculty/parra/teaching/med-imaging/lecture5.pdf

• For more discussion on aliasing due to under sampling– https://engineering.purdue.edu/~malcolm/pct/CTI_Ch05.pdf

http://www.slaney.org/pct/

http://web.eecs.umich.edu/%7Efessler/course/516/l/c-tomo.pdf

http://bme.ccny.cuny.edu/faculty/parra/teaching/med-imaging/lecture4.pdf

http://bme.ccny.cuny.edu/faculty/parra/teaching/med-imaging/lecture5.pdf


Homework Due: 9:50 am on Friday October 12

• Reading: – Prince and Links, Medical Imaging Signals and Systems, Chap 6,

Sec.6.3.4-6.5• Note down all the corrections for Ch. 6 on your copy of the textbook

based on the provided errata.• Problems for Chap 6 of the text book:

– P.6.9– P.6.10 (part e is not required)– P.6.13.

• Hint: solution for part (a) should be

– P.6.19– P.6.20

( )( )

≤≤−≤≤−+

=otherwise

allalala

lg0

2/02/302/2/3

)60,( µµ


Computer AssignmentDue: 9:50 am on Friday October 12

Will send assignment via NYU Classes today.

Documents

Computed Tomography (Part 2)eeweb.poly.edu/~yao/EL6813/Lect5_CT_ch6_part2_F18_JM.pdf · 2018-09-28 · Computed Tomography (Part 2) Jonathan Mamou and Yao Wang. Polytechnic School