Ultrasound Computed Tomography 20020612 Introduction Conventional Xray image

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Ultrasound Computed Tomography 何祚明 陳彥甫 2002/06/12

Ultrasound Computed Tomography 何祚明 陳彥甫 2002/06/12

Introduction § Conventional X-ray image is the superposition of all the planes normal to

Introduction § Conventional X-ray image is the superposition of all the planes normal to the direction of propagation. § The tomography image is effectively an image of a slice taken through a 3 -D volume. § An estimated 10 -15% of breast cancers evade detection by mammography. § Poor differentiation of malignant tumors from highly common cysts (while ultrasound can do so with accuracies of 90 -100%). § UCT can provide not only structural/density information, but also tissue compressibility and speed of sound maps

Tomography § § Time of flight or intensity attenuation Array transducer : 1 -D

Tomography § § Time of flight or intensity attenuation Array transducer : 1 -D data (only tf) Rotation : 2 -D data (tf on range and angle q) Scan : 3 -D data (y position, tf and angle q) A B

Tomography reconstruct A R q

Tomography reconstruct A R q

Reconstruction Method § Attenuation method § Iterative method – Algebraic Reconstruction Technique (ART) §

Reconstruction Method § Attenuation method § Iterative method – Algebraic Reconstruction Technique (ART) § Direct reconstruction – Fourier transform § Alternative direct reconstruction – Back projection – Filtered back projection

Central Section Theorem y y’ x f (x, y) g 90(y’)

Central Section Theorem y y’ x f (x, y) g 90(y’)

Ambiguity Angle q R q r y q x f(x, y) xcosq + ysinq

Ambiguity Angle q R q r y q x f(x, y) xcosq + ysinq = R

Equations • 1 D FT of projection function 2 D FT

Equations • 1 D FT of projection function 2 D FT

Cont’d § 2 D Fourier transform § (u, v) in polar coordinates is (rcosq,

Cont’d § 2 D Fourier transform § (u, v) in polar coordinates is (rcosq, rsinq) § 2 D inverse Fourier Transform

Figures gq(R) R y v Gq(r) q x f(x, y) r q u

Figures gq(R) R y v Gq(r) q x f(x, y) r q u

Algorithm solve coordinate problems (polar to rectangular coordinates) 1. 1 D FT each of

Algorithm solve coordinate problems (polar to rectangular coordinates) 1. 1 D FT each of the projections gq(R) Gq(r) 2. Integrate Gq(r) in q and r -- avoid coordinate transformation problems gq(R) Gq(r) =F(u, v) f (x, y) 1 D FT

Back Projection Reconstruction § Back projection (at one angle) § Integration of all ambiguity

Back Projection Reconstruction § Back projection (at one angle) § Integration of all ambiguity angle q

Cont’d § Drawback of back projection method § So, we integrate this function after

Cont’d § Drawback of back projection method § So, we integrate this function after added 1/r

Disadvantage § FT{1/r} 1/r : blurring function § f (x, y) convolves 1/r §

Disadvantage § FT{1/r} 1/r : blurring function § f (x, y) convolves 1/r § Additional filter is needed object 1/r y x Backprojected object y x y x

Filter Design § Design filter at spatial frequency domain 1/r § Cone filter v

Filter Design § Design filter at spatial frequency domain 1/r § Cone filter v § Spatial filter u -r 0

Signal Processing Flow § Backprojection all angles, gq(R) Fb(q) § Forward 2 D FT

Signal Processing Flow § Backprojection all angles, gq(R) Fb(q) § Forward 2 D FT to get: § Apply filter on to get F (u, v) § Inverse 2 D FT to f (x, y) gq(R) Fb(q) 2 D FT F(u, v) f (x, y) Filtering 2 D IFT

Algebraic Reconstruction Technique (ART) § The iterative process starts with all values set to

Algebraic Reconstruction Technique (ART) § The iterative process starts with all values set to a constant, such as the mean or 0. § At each iteration, the difference between the measured projection data for a given projection and the sum of all reconstructed elements along the line defining the projection is computed. § This difference is then evenly divided among the N reconstruction elements.

Algebraic Reconstruction Technique (ART) 1. 2. 3. 4. Projection Step Comparison Step Backprojection Step

Algebraic Reconstruction Technique (ART) 1. 2. 3. 4. Projection Step Comparison Step Backprojection Step Update Step Additive ART : Multiplicative ART :

Algebraic Reconstruction Technique (ART) cross section fij (calculated element) gj (measured projection) N elements

Algebraic Reconstruction Technique (ART) cross section fij (calculated element) gj (measured projection) N elements per line

Simulation Results I. • Using ART to reconstruct : Simulated Data Reconstruction Result 10

Simulation Results I. • Using ART to reconstruct : Simulated Data Reconstruction Result 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100 10 20 30 40 50 60 70 80 90 100

Simulation Results I. • Using direct Fourier transform to reconstruct : Simulated Data Reconstruction

Simulation Results I. • Using direct Fourier transform to reconstruct : Simulated Data Reconstruction Result 10 10 20 20 30 30 40 40 50 60 50 70 60 80 70 90 10 20 30 40 50 60 70 80 90 100 80 10 20 30 40 50 60 70 80

Simulation Results II. • Using ART to reconstruct : Simulated Data Reconstruction Result 10

Simulation Results II. • Using ART to reconstruct : Simulated Data Reconstruction Result 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100 10 20 30 40 50 60 70 80 90 100

Simulation Results II. • Using direct Fourier transform to reconstruct : Simulated Data Reconstruction

Simulation Results II. • Using direct Fourier transform to reconstruct : Simulated Data Reconstruction Result 2 10 4 20 6 30 8 40 10 50 12 60 14 70 16 80 18 90 100 20 10 20 30 40 50 60 70 80 90 100 5 10 15 20

Experiment Architecture x Transmitter 2 mm/step y Transducer Phantom Transducer Receiver Water Tank 2

Experiment Architecture x Transmitter 2 mm/step y Transducer Phantom Transducer Receiver Water Tank 2 degrees/rotate

Experiment Architecture

Experiment Architecture

Breast Phantom

Breast Phantom

Eraser Phantom

Eraser Phantom

Rubber Ball

Rubber Ball

Time-of-flight t 1 t 2 t 3

Time-of-flight t 1 t 2 t 3

Reconstructed Image • Breast phantom ART Direct Fourier Transform 10 10 20 20 30

Reconstructed Image • Breast phantom ART Direct Fourier Transform 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 10 20 30 40 50 60 70 80

Reconstructed Image • Breast phantom, using ART 10 20 30 40 50 60 70

Reconstructed Image • Breast phantom, using ART 10 20 30 40 50 60 70 80

Reconstructed Image • Rubber ball ART Direct Fourier Transform 2 2 4 4 6

Reconstructed Image • Rubber ball ART Direct Fourier Transform 2 2 4 4 6 6 8 8 10 10 12 12 14 14 16 16 18 18 20 20 2 4 6 8 10 12 14 16 18 20 5 10 15 20

Reconstructed Image • Eraser ART Direct Fourier Transform 5 5 10 10 15 15

Reconstructed Image • Eraser ART Direct Fourier Transform 5 5 10 10 15 15 20 20 25 25 30 30 35 35 40 40 5 10 15 20 25 30 35 40

Conclusions § ART reconstruction is much faster than other reconstruction method but must notice

Conclusions § ART reconstruction is much faster than other reconstruction method but must notice the velocity profile normalization. § Direct fourier reconstruction is effective but less efficiency. § Backprojection reconstruction needed additional filter to cancel integration errors.

Future Works § Correct velocity profile estimation § Discuss backprojection method errors and additional

Future Works § Correct velocity profile estimation § Discuss backprojection method errors and additional filter design method § Discuss diffraction errors in ultrasound

References § [1] Albert Macovski, “Medical Imaging Systems” § [2] Matthew O’Donnell, “X-Ray Computed

References § [1] Albert Macovski, “Medical Imaging Systems” § [2] Matthew O’Donnell, “X-Ray Computed Tomography (CT)”, University of Michigan § [3] Douglas C. Noll, “Computed Tomography Notes”, University of Michigan, 2001 § [4] Brian Borchers, “Tomography Lecture Notes”, New Mexico Tech. , March 2000 § [5] James F. Greenleaf et al. , “Clinical Imaging with Transmissive Ultrasonic Computerized Tomography”, IEEE trans. on Biomedical Engineering vol. 28 no. 2 1981