Computed Tomography (Part 2)

Transcription

1 EL-GY 6813 / BE-GY 603 / G Medical Imaging Computed Tomography (Part ) Yao Wang Polytechnic School of Engineering New York University, Brooklyn, NY 1101 Based on Prince and Links, Medical Imaging Signals and Systems and Lecture Notes by Prince. Figures are from the book.

2 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 Last Lecture F16 Yao NYU

3 This Lecture Review of Parallel Ray Projection and Reconstruction Practical implementation with samples Fan Beam Reconstruction Signal to Noise in CT F16 Yao NYU 3

4 Review: Projection Slice Theorem Projection Slice theorem The Fourier Transform of a projection at angle θ is a line in the Fourier transform of the image at the same angle. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. F16 Yao NYU 4

5 Reconstruction Algorithm for Parallel Projections Backprojection Sum: Backprojection of each projection Sum Filtered backprojection: FT of each projection Filtering each projection in frequency domain Inverse FT Backprojection Sum Convolution backprojection Convolve each projection with the ramp filter Backprojection Sum Which one gives more accurate result? Which one takes less computation? F16 Yao NYU 5

6 Practical Implementation Projections g(l, q) are only measured at finite intervals l=nt; t chosen based on maximum frequency in G(ρ, θ), W 1/t >=W or t <=1/W (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(nt, q) If N sample are taken, ideally N point FFT should be taken by zero padding g(nt, q) Recall convolving two signals of length N leads to a single signal of length N-1 For convolution backprojection The ramp-filter is sampled at l=nt Sampled Ram-Lak Filter (limiting the rho-filter to within (-1/t, -1/t ) c( n) = 1/ 4τ ; 1/ ( nπτ ) 0; ; n = 0 n = odd n = even See paper at F16 Yao NYU 6

7 Practical Implementation of Filtered P θ j (l):!projection!data!along!θ j ; R(m,n)= π K! Backprojection s θ j (k):!dft!of!p θ j (l); B θ j (m,n):!backprojected!image!from!filtered!projection R(m,n):Sum!of!backprojections. L:!number!of!samples!along!each!projection,!assuming!L!is!odd K :!number!of!projection!angles. s θ j (k)= L 1 P θ (l)exp iπ lk j L, k = L 1,...,L 1,L 1 = floor L l= L 1 B θ j (m,n)= L 1 k= L 1 K j=1 s θ (k) k j L exp iπ mcosθ + nθ j j k L B θ j (m,n) From paper at Note 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. F16 Yao NYU 7

8 Practical Implementation of Convolution Backprojection c(l):!predesigned!filter B θ j (m,n)= L 1 l= L 1 ( ) P θ (l) c mcosθ j j + nsinθ j l R(m,n)= π! K K j=1 B θ j (m,n) Here again: Back projection is implemented within convolution! F16 Yao NYU 8

9 The Ram-Lak Filter (from [Kak&Slaney]) F16 Yao NYU 9

10 1 st Generation CT: Parallel Projections F16 Yao NYU 10

11 3G: Fan Beam Much faster than G F16 Yao NYU 11

12 Fan Beam: Equiangular Ray We will focus on the equiangular detector setting on the right in this lecture. F16 Yao NYU 1

13 Equiangular Ray Projection Source location is described by ( β, D) D is typically fixed, β varies to provide a largeview angle. To provide complete view, β (0,π ) a For a given source with angle β, γ specifies the detector position or the projection line. For each β, γ varies over a range( γ, γ m m ) (D, β, γ ) completely specifies the line of projection : θ = β + γ, l = D sin( γ ) Instead of g(l, θ ), we can use p( γ, β ) to represent a projection β + α + γ = π / θ + α = π / β + γ = θ F16 Yao NYU 13

14 Equiangular Ray Reconstruction g D r φ β φ Reconstructed image is represented in the polar coordinate using (r,ϕ). The relative position of a pixel at (r,ϕ) to the source at (D, β) is specified by (D',γ) : ( ) + ( r cos(β ϕ) ) ( r cos(β ϕ) ) ( D + rsin(β ϕ) ) D' (r,ϕ) = D + rsin(β ϕ) tanγ(r,ϕ) = Reconstruction formular: f (r,ϕ) = π 0 1 q(γ, β)dβ ( D' ) q(γ, β) = p'(γ, β)*c f (γ) p'(γ, β) = p(γ, β)cos(γ) c f (γ) = 1 D # γ & % ( $ sinγ ' Convolution weighted backprojection c(γ) c(γ) is the ramp filter used in parallel projection. c f (γ) is weighted based on the angle γ. Derivation not required for this class. Detail can be found at [Kak&Slaney]. Note typos in [Prince&Links], 1 st ed. F16 Yao NYU 14

15 Typos in [Prince&Links, 1 st Ed] P. 07, Eq. (6.38), change to c( D'sin γ ) = γ D'sin γ c( γ ) Eq. (6.39) change to c f (γ) = 1 D! γ $ # & " sinγ % c(γ) Eq. (6.40),(6.41) p(γ, β) p'(γ, β) p'(γ, β) = p(γ, β)cos(γ) q(γ, β) = p'(γ, β)*c f (γ) F16 Yao NYU 15

16 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>=W or α<=1/w For convolution backprojection The filter c f (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 D' -1/ ( r, φ) = tanγ ( r, φ) = ( D + r sin( β φ) ) + ( r cos( β φ) ) ( r cos( β φ) ) ( D + r sin( β φ )) f (r,ϕ) = π 0 1 q(γ, β)dβ ( D' ) q(γ, β) = p'(γ, β)*c f (γ) p'(γ, β) = p(γ, β)cos(γ) F16 Yao NYU 16

17 Matlab Functions for Fan Beam CT Relevant functions: fanbeam(), ifanbeam() F16 Yao NYU 17

18 Fan Beam Reconstruction Through Rebinning [Smith&Webb] F16 Yao NYU 18

19 Constrained Optimization Formulation (Algebraic Reconstruction Techniques or ART) The!measurement!g(l,θ)!in!general!can!be!related!to!the!unknown!f (m,n)!as g(l,θ) = m,n f (m,n)a(m,n; l,θ) 1 (m,n)!!is!on!the!line!(l,θ) A(m,n;l,θ) = 0 otherwise More!generally,!we!can!order!all!possible!(l,θ)!pairs!into!1D!indices,!and!represent g(l,θ)!!as!a!vector!g,!and!order!all!pixel!positions!(m,n)also!into!!1d!indices,!so!that! all!possible!f (m,n)!can!be!put!into!a!vector!f,!and!all!possible!a(m,n;!l,θ)!can!be!put! into!a!matrix!a.!!then!the!above!equations!can!generally!be!written!as! Af = g So!we!can!reconstruct!the!image!f!by!solving!the!above!matrix!equation. If!there!are!more!measurements!than!the!number!of!pixels,!then!the!problem!can!be!solved using!the!least!squares!formulation,!i.e., min!! Af g,!with!a!unique!solution!f = A T A ( ) '1 A T g When!there!are!fewer!measurements!than!the!number!of!pixels,!we!have!an!under!determined! problem,!and!we!need!to!put!additional!constraints!on!f.!!for!example,!we!may!require!that!the!image!f is!sparse!in!some!transform!domain,!i.e.!!tf!has!only!few!nonizero!coefficients,!or!l0!norm!of!tf!is!small,! and!solve!the!following!problem min!! Af g + λ Tf 0 This!is!the!classifical!compressive!sensing!problem.! Instead!of!using!the!L0!norm,!we!can!relax!to!the!L1!norm,!and!solve!the!following!convex!optimization!probem min!! Af g + λ Tf 1! There!are!many!fast!algorithms!to!solve!the!above!problem. F16 Yao NYU 19

20 Blurring Effect SNR CT Quality Evaluation F16 Yao NYU 0

21 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,q) is related to real projection g(l,q) by g (l,q)= g(l,q) * s(l) G (r,q)= G(r,q) S(r) F16 Yao NYU 1

22 Windowing Function Recall that the ideal filter c(r) is typically modified by a window function W(r) Overall Effect fˆ( x, y) can be thought of as the reconstructedimagefromthe projection gˆ( l, θ), whose Fourier transform is Gˆ( ρ, θ) = G( ρ, θ) S( ρ) W ( ρ) gˆ( l, θ) = g( l, θ)* s( l)* w( l) F16 Yao NYU

23 Blurred Projection h(x,y): PSF of the blurring F16 Yao NYU 3

24 Circular Symmetry of Blurring F16 Yao NYU 4

25 PSF given by Hankel Transform r = x + y F16 Yao NYU 5

26 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) Let x = r cosφ, y = r sin φ; u = ρ cosθ, v = ρ sinθ F( ρ, θ ) = = f ( r, φ)exp{ jπ ( rρ cosφ cosθ + rρ sin φ sinθ ) f ( r, φ)exp{ jπrρ cos( φ θ )} rdrdφ } rdrdφ If f ( x, y) = F( ρ, θ ) = { exp{ jπrρ cos( φ θ )} dφ} f ( r) f ( r) rdr = π 0 f ( r) J 0 (πρr) rdr = F( ρ) Hankel Transform F( ρ) = π 0 f ( r) J 0 (πρr) rdr; J 0 1 ( r) = π π 0 cos( r sin φ) dφ F16 Yao NYU 6

27 Common Transform pairs See Table.3 Fourier => Hankel π ( x + y ) { e } πr { e } = = e e π ( u πρ + v ) Hankel { sin c( r) } = rect( q) π 1 4q Scaling property 1 Hankel a a { f(ar) } = F( q / ) Duality: If h(r) <-> H(ρ), then H(r)<->h(ρ) Derivation of Hankel transform pairs are not required. But you should be able to use given transform pairs, to determine the blur function. F16 Yao NYU 7

28 Example Example 6.5 in [Prince&Links] Detector: rectangular detector with width d S(l)=rect(l/d) Rectangular window function Solution W(ρ)=rect(ρ/ρ o ); ρ o >>1/d 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)) Illustrate and explain h(r) h( r) = inverse Hankel Hankel h( r) = { sinc( r) } rect ( r / d) 4r { H ( ρ) } rect ( ρ) 1 Hankel{ f(ar) } = F( ρ / a) a Using duality property π d = π 1 4ρ F16 Yao NYU 8

29 Noise in CT Measurement Pr{ N ij = k} = k a k! e a ; k = 0,1,... E{ N ij = k} = a Var{ N ij = k} = a F16 Yao NYU 9

30 What about the measured projection F16 Yao NYU 30

31 CBP Approximation F16 Yao NYU 31

32 Definitions and Assumptions g ij are independent because N ij are independent Deriving mean and variance of µ(x,y) based on the independence assumption See [Prince&Links] for derivation F16 Yao NYU 3

33 Variance increases with ρ 0 (cut-off freq. of filter), and T (detector spacing), decreases with M (number of angles),\bar N (or N 0 ) (x-ray intensity) F16 Yao NYU 33

34 SNR of the Reconstructed Image C: fractional change of m from \bar m F16 Yao NYU 34

35 SNR in a good design F16 Yao NYU 35

36 SNR in Fan Beam N=Nf/D w=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 increases But larger D is desired to obtain a good resolution. F16 Yao NYU 36

37 Rule of Thumb F16 Yao NYU 37

38 Aliasing Artifacts Nyquist Sampling theorem: If the maximum freq of a signal is fmax, it should be sampled with a freq fs>=max, or sampling interval T<=1/fmax 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 F16 Yao NYU 38

39 From [Kak&Slaney] Fig. 5.1 F16 Yao NYU 39

40 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(r)=W(r) S(r) 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 r o Impact of number of projection angles and samples on reconstruction image quality Nyquist sampling theorem Streak artifacts F16 Yao NYU 40

41 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, E-copy available at 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 Lecture note by Prof. Parra For more discussion on aliasing due to under sampling F16 Yao NYU 41

42 Reading: Homework Prince and Links, Medical Imaging Signals and Systems, Chap 6, Sec 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 Hint: solution for part (a) should be P.6.19 P.6.0 3µ g( l,60) = 3µ 0 ( a / + l) ( a / l) a / l 0 0 l a / otherwise F16 Yao NYU 4

43 Computer Assignment 1. Learn how do fanbeam. ifanbeam work; summarize their functionalities. Type demos on the command line, then select toolbox - > image processing -> transform -> reconstructing an image from projection data. Alternatively, you can use help for each particular function.. Write a MATLAB program that 1) generate a phantom image (you can use a standard phantom provided by MATLAB or construct your own), ) produce equiangular fan beam projections; 3) reconstruct the phantom using filtered backprojection algorithm; Your program should allow the user to specify the number of fan beams, and the number of projections per fan beam, the angular spacing between the projections. Run your program with different number of projections for the same view angle, and with different view angles, and compare the quality. Use the same filter and interpolation algorithm for all the comparisons. Compare the reconstructed image quality obtained with different number of view angles and number of projections per view angle. Also, compare the image quality with those obtained with parallel projections for the same phantom image when the same total number of measurements are used (from your last assignment). You can use the fanbeam() and ifanbeam() functions in MATLAB. F16 Yao NYU 43