ROI Reconstruction in CT
|
|
|
- Norman Long
- 6 years ago
- Views:
Transcription
1 Reconstruction in CT From Tomo reconstruction in the 21st centery, IEEE Sig. Proc. Magazine (R.Clackdoyle M.Defrise) L. Desbat TIMC-IMAG September 10, 2013 L. Desbat Reconstruction in CT
2 Outline 1 CT Radiology 2 Radon Transform 3 reconstriction Motivations parallel Hilbert Projection Equality DBP-H Inversion L. Desbat Reconstruction in CT
3 CT Radiology CT scanner: principle L. Desbat Reconstruction in CT
4 X-ray attenuation CT Radiology I D I S Source I D Detector L. Desbat Reconstruction in CT l
5 X-ray attenuation CT Radiology I D I S2 µ 1 I S Source I D Detector e L. Desbat Reconstruction in CT e l
6 X-ray attenuation CT Radiology I D I S2 µ 1 I S Source I D Detector I S 2 2 e e 2e L. Desbat Reconstruction in CT l
7 X-ray attenuation CT Radiology I D I S2 µ 1 I S Source I D Detector I S 2 2 e I S 2 3 e 2e L. Desbat Reconstruction in CT 3e l
8 X-ray attenuation CT Radiology I D I S2 µ 1 I S Source I D Detector I S 2 2 e I S 2 3 I S 2 4 e 2e L. Desbat Reconstruction in CT 3e 4e l
9 X-ray attenuation CT Radiology I D I S Source I D Detector L. Desbat Reconstruction in CT l
10 X-ray attenuation CT Radiology I D I S2 µ 2 I S Source I D Detector e e L. Desbat Reconstruction in CT l
11 X-ray attenuation CT Radiology I D I S2 µ 2 I S Source I D Detector I S 2 2 e e 2e L. Desbat Reconstruction in CT l
12 X-ray attenuation CT Radiology I D I S2 µ 2 I S Source I D Detector I S 2 2 e I S 2 3 e 2e 3e L. Desbat Reconstruction in CT l
13 CT Radiology X-ray attenuation I D I S2 µ 2 I S Source I D Detector I S 2 2 e I S 2 3 I S 2 4 e 2e 3e 4e L. Desbat Reconstruction in CT l
14 X-ray attenuation CT Radiology I S I D Source Detector L. Desbat Reconstruction in CT
15 X-ray attenuation CT Radiology Lambert-Beer s Law I D (x) = I S e µl I S : intensité initiale µ : coefficient d atténuation I S µ I D l : épaisseur du tissus Source Detector l L. Desbat Reconstruction in CT
16 X-ray attenuation CT Radiology Lambert-Beer s Law I D = I S e (µ 1l 1 +µ 2 l 2 +µ 3 l 3 ) µ 1 µ 2 µ 3 I D = I S e i µ i dl i I S Source I D Detector I D = I S e D S µ(l)dl l1 l2 l3 L. Desbat Reconstruction in CT
17 x-ray image formation CT Radiology Metal Bone Fat tissues Water and soft tissues Air L. Desbat Reconstruction in CT
18 From Radiology to Tomography Radon Transform I D = I S e D S µ(l)dl ( ) D ID ln = µ(l)dl With a CT we measure the integral of µ over lines I S S L. Desbat Reconstruction in CT
19 Radon Transform parametrization e2 Ω O e1 L. Desbat Reconstruction in CT
20 Radon Transform parametrization x D Detector e2 Ω O e1 Source x S L. Desbat Reconstruction in CT
21 Radon Transform parametrization ζ(ϕ) x D Detector e2 Ω θ(ϕ) φ [0, 2π[, s R ϕ p(φ, s) = Rµ(φ, s) = R µ (sθ(φ) + lζ(φ)) dl s O e1 Source x S L. Desbat Reconstruction in CT
22 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
23 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
24 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
25 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
26 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
27 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
28 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
29 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
30 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
31 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
32 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
33 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
34 Radon Transform ζ(ϕ) e2 Ω θ(ϕ) ϕ O e1 L. Desbat Reconstruction in CT
35 Radon Transform e2 θ(ϕ) Ω ζ(ϕ) ϕ O e1 L. Desbat Reconstruction in CT
36 Radon Transform θ(ϕ) e2 Ω ζ(ϕ) O ϕ e1 L. Desbat Reconstruction in CT
37 Radon Transform θ(ϕ) e2 Ω ϕ ζ(ϕ) O e1 L. Desbat Reconstruction in CT
38 Radon Transform θ(ϕ) e2 Ω ϕ O e1 ζ(ϕ) L. Desbat Reconstruction in CT
39 Radon Transform e2 Ω θ(ϕ) ϕ O e1 ζ(ϕ) L. Desbat Reconstruction in CT
40 ζ(ϕ) L. Desbat Reconstruction in CT Radon Transform e2 Ω θ(ϕ) ϕ O e1
41 ζ(ϕ) L. Desbat Reconstruction in CT Radon Transform e2 Ω θ(ϕ) ϕ O e1
42 Radon Transform e2 Ω θ(ϕ) O ϕ e1 ζ(ϕ) L. Desbat Reconstruction in CT
43 Center Slice Theorem Definition R θ µf (s) def = Rµf (θ, s) (1) Theorem Let µ L 1 (R 2 ) then R θ µ(σ) = ˆµ(σθ) L. Desbat Reconstruction in CT
44 Proof of Center Slice Theorem Proof. R θ µ(σ) = R θ µ(s)e 2iπσs ds R = µ (sθ + lζ) dle 2iπσs ds R R = µ(x)e 2iπσθ x dx R 2 = ˆµ(σθ) where we made the change of variables (s, l) to (x 1, x 2 ) i.e. x = sθ + lζ. This is a rotation thus dlds = dx 1 dx 2 L. Desbat Reconstruction in CT
45 Filtered Back Projection Theorem Let µ L 1 (R 2 ) sufficiently smooth then π µ(x) = R θ µ(σ) σ e 2iπσx θ dσdφ 0 R L. Desbat Reconstruction in CT
46 Filtered Back Projection, proof Proof. µ(x) = = = ˆµ(ξ)e 2iπξ x dξ R 2 ˆµ(σθ)e 2iπσθ x σ dσdφ 0 R R θ µ(σ) σ e 2iπσθ x dσdφ π π 0 R where we have made the polar change of variable ξ = σθ(φ), (φ, σ) [0π[ R, thus dξ 1 dξ 2 = σ dσdφ, and we have used theorem 2. L. Desbat Reconstruction in CT
47 Non Local Filter We define p H the Hilbert transform of the parallel projection p p H (φ, s) = + p(φ, t)h(s t)dt where h(u) = 1 πu and ĥ(σ) = isgn(σ) (distribution). The ramp filtering p R of p is p R (φ, s) = 1 2π s p H (φ, s) The filter p φ (s) p F φ (σ) p filter φ (σ) σ p F 1 φfiltered (s) is non local!!! σ = 1 2π (2iπσ) ( isgn(σ)) is the Hilbert filtering composed by the derivation. L. Desbat Reconstruction in CT
48 geometry α ζ(α) α v(t) Figure: The variables (t, α) L. Desbat Reconstruction in CT
49 geometry We first define the source trajectory along a curve v : C R 2 t v(t) The fan-beam data are then defined by We remark that g (v t, α) = + 0 µ (v t + lζ(α)) dl (2) p(φ, s) = g(v t, φ) + g(v t, φ + π) where s = v t θ(φ) L. Desbat Reconstruction in CT
50 Inversion We consider the circular trajectory, v t = ( R v cos t, R v sin t), Theorem Let µ L 1 (R 2 ) sufficiently smooth then where µ(x) = 1 2 g WF (v t, φ) = 2π 0 t+π/2 t π/2 where r is the ramp filter (ˆr(σ) = σ ). Proof... change of variables 1 x v t 2 g WF (v t, arg(x v t )) dt R v cos(ψ t)g(v t, ψ)r (sin(φ ψ)) dψ L. Desbat Reconstruction in CT
51 Short Scan Figure: Short scan with (Parker) weight is possible... L. Desbat Reconstruction in CT
52 Trajectories, small detectors Motivations parallel Hilbert Projection Equality DBP-H Inversion Figure: Small detector yields truncated data. L. Desbat Reconstruction in CT
53 Hilbert and Motivations parallel Hilbert Projection Equality DBP-H Inversion We define g H the Hilbert transform of the fan beam projection g g H (v t, φ) = 2π 0 g H (v t, ψ)h (sin(φ ψ)) dψ where h(u) = 1 πu L. Desbat Reconstruction in CT
54 Motivations parallel Hilbert Projection Equality DBP-H Inversion The parallel Hilbert Projection Equality Theorem p H (φ, v t θ) = g H (v t, φ) (3) (idea compute p H (φ, s) from g H (v t, φ) with v t θ = s) v(λ) L. Desbat Reconstruction in CT
55 New Reconstruction Conditions Motivations parallel Hilbert Projection Equality DBP-H Inversion Recall with p R (φ, s) = 1 2π s p H (φ, s) p H (φ, s) = g H (v t, φ) Theorem The point x can be reconstructed from FB non truncated projections provided a fan beam vertex can be found on each line passing through x. L. Desbat Reconstruction in CT
56 New Reconstruction Formula Motivations parallel Hilbert Projection Equality DBP-H Inversion µ(x) = 1 2 where C 1 x v t w (v t, arg(x v t )) g F (v t, arg(x v t )) dt g F (v t, φ) = 1 2π where h is the hilbert filter. t+π/2 t π/2 h (sin(ψ φ)) g t (v t, ψ)dψ L. Desbat Reconstruction in CT
57 Very Short Scan Motivations parallel Hilbert Projection Equality DBP-H Inversion Figure: Very Short Scan... L. Desbat Reconstruction in CT
58 Virtual Fean Beam Motivations parallel Hilbert Projection Equality DBP-H Inversion Figure: Data truncation: spine... L. Desbat Reconstruction in CT
59 Idea Motivations parallel Hilbert Projection Equality DBP-H Inversion The idea of the Differentiated Backprojection is to compute the Hilbert transform of µ along a direction α from the back projection of the devivation of the projection. µ is then reconstructed from the inversion of the Hilbert transform. L. Desbat Reconstruction in CT
60 Truncated projections Motivations parallel Hilbert Projection Equality DBP-H Inversion Figure: Data truncation: spine... L. Desbat Reconstruction in CT
Sampling in tomography
in tomography Echantillonnage en tomographie L. Desbat TIMC-IMAG, UMR 5525, UJF-Grenoble 1 15 June 2012 L. Desbat in tomography Outline CT, Radon transform and its inversion 1 CT, Radon transform and its
EE 4372 Tomography. Carlos E. Davila, Dept. of Electrical Engineering Southern Methodist University
EE 4372 Tomography Carlos E. Davila, Dept. of Electrical Engineering Southern Methodist University EE 4372, SMU Department of Electrical Engineering 86 Tomography: Background 1-D Fourier Transform: F(
LECTURES ON MICROLOCAL CHARACTERIZATIONS IN LIMITED-ANGLE
LECTURES ON MICROLOCAL CHARACTERIZATIONS IN LIMITED-ANGLE TOMOGRAPHY Jürgen Frikel 4 LECTURES 1 Today: Introduction to the mathematics of computerized tomography 2 Mathematics of computerized tomography
Two-Material Decomposition From a Single CT Scan Using Statistical Image Reconstruction
/ 5 Two-Material Decomposition From a Single CT Scan Using Statistical Image Reconstruction Yong Long and Jeffrey A. Fessler EECS Department James M. Balter Radiation Oncology Department The University
Feature Reconstruction in Tomography
Feature Reconstruction in Tomography Alfred K. Louis Institut für Angewandte Mathematik Universität des Saarlandes 66041 Saarbrücken http://www.num.uni-sb.de louis@num.uni-sb.de Wien, July 20, 2009 Images
Topics. EM spectrum. X-Rays Computed Tomography Direct Inverse and Iterative Inverse Backprojection Projection Theorem Filtered Backprojection
Bioengineering 28A Principles of Biomedical Imaging Fall Quarter 24 X-Rays/CT Lecture Topics X-Rays Computed Tomography Direct Inverse and Iterative Inverse Backprojection Projection Theorem Filtered Backprojection
Maximal Entropy for Reconstruction of Back Projection Images
Maximal Entropy for Reconstruction of Back Projection Images Tryphon Georgiou Department of Electrical and Computer Engineering University of Minnesota Minneapolis, MN 5545 Peter J Olver Department of
Image Reconstruction from Projection
Image Reconstruction from Projection Reconstruct an image from a series of projections X-ray computed tomography (CT) Computed tomography is a medical imaging method employing tomography where digital
Numerical Methods for geodesic X-ray transforms and applications to open theoretical questions
Numerical Methods for geodesic X-ray transforms and applications to open theoretical questions François Monard Department of Mathematics, University of Washington. Nov. 13, 2014 UW Numerical Analysis Research
Topics. EM spectrum. X-Rays Computed Tomography Direct Inverse and Iterative Inverse Backprojection Projection Theorem Filtered Backprojection
Bioengineering 28A Principles of Biomedical Imaging Fall Quarter 25 X-Rays/CT Lecture Topics X-Rays Computed Tomography Direct Inverse and Iterative Inverse Backprojection Projection Theorem Filtered Backprojection
Efficient Solution Methods for Inverse Problems with Application to Tomography Radon Transform and Friends
Efficient Solution Methods for Inverse Problems with Application to Tomography Radon Transform and Friends Alfred K. Louis Institut für Angewandte Mathematik Universität des Saarlandes 66041 Saarbrücken
Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2005 X-Rays/CT Lecture 1. Topics
Bioengineering 28A Principles of Biomedical Imaging Fall Quarter 25 X-Rays/CT Lecture Topics X-Rays Computed Tomography Direct Inverse and Iterative Inverse Backprojection Projection Theorem Filtered Backprojection
The Radon transform. Chris Stolk. December 18, 2014
The Radon transform Chris Stolk December 18, 2014 1 Introduction In two dimensions the Radon transform is an integral transform that maps a function to its integrals over lines. Let θ S 1 and s R then
MB-JASS D Reconstruction. Hannes Hofmann March 2006
MB-JASS 2006 2-D Reconstruction Hannes Hofmann 19 29 March 2006 Outline Projections Radon Transform Fourier-Slice-Theorem Filtered Backprojection Ramp Filter 19 29 Mar 2006 Hannes Hofmann 2 Parallel Projections
The mathematics behind Computertomography
Radon transforms The mathematics behind Computertomography PD Dr. Swanhild Bernstein, Institute of Applied Analysis, Freiberg University of Mining and Technology, International Summer academic course 2008,
1D Convolution. Convolution g[m] = g[0]δ[m] + g[1]δ[m 1] + g[2]δ[m 2] h[m',k] = L[δ[m k]] = h[ m $ k]
![1D Convolution. Convolution g[m] = g[0]δ[m] + g[1]δ[m 1] + g[2]δ[m 2] h[m',k] = L[δ[m k]] = h[ m $ k]](/thumbs/90/104468011.jpg "1D Convolution. Convolution g[m] = g[0]δ[m] + g[1]δ[m 1] + g[2]δ[m 2] h[m',k] = L[δ[m k]] = h[ m $ k]")
cm 5 cm Bioengineering 28A Principles of Biomeical Imaging Assume =/2 Fall Quarter 24 X-Rays Lecture 2 cm 5 cm cm cm Assume =/2 Assume =/2 Convolution g[m] = g[]δ[m] + g[]δ[m ] + g[2]δ[m 2] h[m,k] = L[δ[m
System Modeling for Gamma Ray Imaging Systems
System Modeling for Gamma Ray Imaging Systems Daniel J. Lingenfelter and Jeffrey A. Fessler COMMUNICATIONS & SIGNAL PROCESSING LABORATORY Department of Electrical Engineering and Computer Science The University
Microlocal Methods in X-ray Tomography
Microlocal Methods in X-ray Tomography Plamen Stefanov Purdue University Lecture I: Euclidean X-ray tomography Mini Course, Fields Institute, 2012 Plamen Stefanov (Purdue University ) Microlocal Methods
Lecture 9 February 2, 2016
MATH 262/CME 372: Applied Fourier Analysis and Winter 26 Elements of Modern Signal Processing Lecture 9 February 2, 26 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long, Edited by E. Bates Outline Agenda:
Restoration of Missing Data in Limited Angle Tomography Based on Helgason- Ludwig Consistency Conditions
Restoration of Missing Data in Limited Angle Tomography Based on Helgason- Ludwig Consistency Conditions Yixing Huang, Oliver Taubmann, Xiaolin Huang, Guenter Lauritsch, Andreas Maier 26.01. 2017 Pattern
Recent progress on the explicit inversion of geodesic X-ray transforms
Recent progress on the explicit inversion of geodesic X-ray transforms François Monard Department of Mathematics, University of Washington. Geometric Analysis and PDE seminar University of Cambridge, May
A Brief Introduction to Medical Imaging. Outline
A Brief Introduction to Medical Imaging Outline General Goals Linear Imaging Systems An Example, The Pin Hole Camera Radiations and Their Interactions with Matter Coherent vs. Incoherent Imaging Length
Inversions of ray transforms on simple surfaces
Inversions of ray transforms on simple surfaces François Monard Department of Mathematics, University of Washington. June 09, 2015 Institut Henri Poincaré - Program on Inverse problems 1 / 42 Outline 1
The Radon Transform and the Mathematics of Medical Imaging
Colby College Digital Commons @ Colby Honors Theses Student Research 2012 The Radon Transform and the Mathematics of Medical Imaging Jen Beatty Colby College Follow this and additional works at: http://digitalcommons.colby.edu/honorstheses
Quadrature Formula for Computed Tomography
Quadrature Formula for Computed Tomography orislav ojanov, Guergana Petrova August 13, 009 Abstract We give a bivariate analog of the Micchelli-Rivlin quadrature for computing the integral of a function
ON A CLASS OF GENERALIZED RADON TRANSFORMS AND ITS APPLICATION IN IMAGING SCIENCE
ON A CLASS OF GENERALIZED RADON TRANSFORMS AND ITS APPLICATION IN IMAGING SCIENCE T.T. TRUONG 1 AND M.K. NGUYEN 2 1 University of Cergy-Pontoise, LPTM CNRS UMR 889, F-9532, France e-mail: truong@u-cergy.fr
Midterm Review. Yao Wang Polytechnic University, Brooklyn, NY 11201
Midterm Review Yao Wang Polytechnic University, Brooklyn, NY 11201 Based on J. L. Prince and J. M. Links, Medical maging Signals and Systems, and lecture notes by Prince. Figures are from the textbook.
MATH 162. FINAL EXAM ANSWERS December 17, 2006
MATH 6 FINAL EXAM ANSWERS December 7, 6 Part A. ( points) Find the volume of the solid obtained by rotating about the y-axis the region under the curve y x, for / x. Using the shell method, the radius
Tomography and Reconstruction
Tomography and Reconstruction Lecture Overview Applications Background/history of tomography Radon Transform Fourier Slice Theorem Filtered Back Projection Algebraic techniques Measurement of Projection
The relationship between image noise and spatial resolution of CT scanners
The relationship between image noise and spatial resolution of CT scanners Sue Edyvean, Nicholas Keat, Maria Lewis, Julia Barrett, Salem Sassi, David Platten ImPACT*, St George s Hospital, London *An MDA
ANALYZING RECONSTRUCTION ARTIFACTS FROM ARBITRARY INCOMPLETE X-RAY CT DATA
ANALYZING RECONSTRUCTION ARTIFACTS FROM ARBITRARY INCOMPLETE X-RAY CT DATA LEISE BORG 1, JÜRGEN FRIKEL2, JAKOB SAUER JØRGENSEN 3, AND ERIC TODD QUINTO 4 Abstract. This article provides a mathematical analysis
Rich Tomography. Bill Lionheart, School of Mathematics, University of Manchester and DTU Compute. July 2014
Rich Tomography Bill Lionheart, School of Mathematics, University of Manchester and DTU Compute July 2014 What do we mean by Rich Tomography? Conventional tomography reconstructs one scalar image from
z 0 > 0 z = 0 h; (x, 0)
n = (q 1,..., q n ) T (,, t) V (,, t) L(,, t) = T V d dt ( ) L q i L q i = 0, i = 1,..., n. l l 0 l l 0 l > l 0 {x} + = max(0, x) x = k{l l 0 } +ˆ, k > 0 ˆ (x, z) x z (0, z 0 ) (0, z 0 ) z 0 > 0 x z =
Iterative Data Refinement for Soft X-ray Microscopy
Iterative Data Refinement for Soft X-ray Microscopy Presented by Joanna Klukowska August 2, 2013 Outline Iterative Data Refinement Transmission X-ray Microscopy Numerical Tests Joanna Klukowska IDR for
Mathematics 2203, Test 1 - Solutions
Mathematics 220, Test 1 - Solutions F, 2010 Philippe B. Laval Name 1. Determine if each statement below is True or False. If it is true, explain why (cite theorem, rule, property). If it is false, explain
Arc Length. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Arc Length Today 1 / 12
Philippe B. Laval KSU Today Philippe B. Laval (KSU) Arc Length Today 1 / 12 Introduction In this section, we discuss the notion of curve in greater detail and introduce the very important notion of arc
Introduction to the Mathematics of Medical Imaging
Introduction to the Mathematics of Medical Imaging Second Edition Charles L. Epstein University of Pennsylvania Philadelphia, Pennsylvania EiaJTL Society for Industrial and Applied Mathematics Philadelphia
From Electrical Impedance Tomography to Network Tomography
From Electrical Impedance Tomography to Network Tomography Carlos A. Berenstein Institute for Systems Research ISR University of Maryland College Park With the collaboration of Prof. John Baras and Franklin
Interpolation procedure in filtered backprojection algorithm for the limited-angle tomography
Interpolation procedure in filtered backprojection algorithm for the limited-angle tomography Aleksander Denisiuk 1 Abstract A new data completion procedure for the limited-angle tomography is proposed.
Ke Li and Guang-Hong Chen
Ke Li and Guang-Hong Chen Brief introduction of x-ray differential phase contrast (DPC) imaging Intrinsic noise relationship between DPC imaging and absorption imaging Task-based model observer studies
6: Positron Emission Tomography
6: Positron Emission Tomography. What is the principle of PET imaging? Positron annihilation Electronic collimation coincidence detection. What is really measured by the PET camera? True, scatter and random
Deuteron from CLAS/EG1B Data. Spin Structure Functions of the OUTLINE. Nevzat Guler (for the CLAS Collaboration) Old Dominion University
Spin Structure Functions of the Deuteron from CLAS/EGB Data Nevzat Guler (for the CLAS Collaboration) Old Dominion University OULINE Formalism Experimental setup Data analysis Results and Conclusion Motivation
Hamburger Beiträge zur Angewandten Mathematik
Hamburger Beiträge zur Angewandten Mathematik Error Estimates for Filtered Back Projection Matthias Beckmann and Armin Iske Nr. 2015-03 January 2015 Error Estimates for Filtered Back Projection Matthias
1-D Fourier Transform Pairs
1-D Fourier Transform Pairs The concept of the PSF is most easily explained by considering a very small point source being placed in the imaging field-of-view The relationship between the image, I, and
1 Radon Transform and X-Ray CT
Radon Transform and X-Ray CT In this section we look at an important class of imaging problems, the inverse Radon transform. We have seen that X-ray CT (Computerized Tomography) involves the reconstruction
Research Article Exact Interior Reconstruction with Cone-Beam CT
Biomedical Imaging Volume 2007, Article ID 10693, 5 pages doi:10.1155/2007/10693 Research Article Exact Interior Reconstruction with Cone-Beam CT Yangbo Ye, 1 Hengyong Yu, 2 and Ge Wang 2 1 Department
Brownian Motion and lorentzian manifolds
The case of Jürgen Angst Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg École d été de Probabilités de Saint-Flour juillet 2008, Saint-Flour 1 Construction of the diffusion,
LECTURES ON MICROLOCAL CHARACTERIZATIONS IN LIMITED-ANGLE
LECTURES ON MICROLOCAL CHARACTERIZATIONS IN LIMITED-ANGLE TOMOGRAPHY Jürgen Frikel 4 LECTURES 1 Nov. 11: Introduction to the mathematics of computerized tomography 2 Today: Introduction to the basic concepts
Minimum detection window and inter-helix PI-line with triple-source helical cone-beam scanning
Journal of X-Ray Science and Technology 14 2006) 95 107 95 IOS Press Minimum detection window and inter-helix PI-line with triple-source helical cone-beam scanning Jun Zhao a,, Ming Jiang b, Tiange Zhuang
Reconstruction for Proton Computed Tomography: A Monte Carlo Study
Reconstruction for Proton Computed Tomography: A Monte Carlo Study T Li, Z. Liang, K. Mueller, J. Heimann, L. Johnson, H. Sadrozinski, A. Seiden, D. Williams, L. Zhang, S. Peggs, T. Satogata, V. Bashkirov,
Improving Boundary Conditions in Time- Domain Self-Force Calculations
Improving Boundary Conditions in Time- Domain Self-Force Calculations Carlos F. Sopuerta Institute of Space Sciences National Spanish Research Council Work in Collaboration with Anil Zenginoğlu (Caltech)
1. Which of the following statements is true about Bremsstrahlung and Characteristic Radiation?
BioE 1330 - Review Chapters 4, 5, and 6 (X-ray and CT) 9/27/2018 Instructions: On the Answer Sheet, enter your 2-digit ID number (with a leading 0 if needed) in the boxes of the ID section. Fill in the
The Mathematics of CT-Scans
The Mathematics of CT-Scans Tomography has become one of the most important applications of mathematics to the problems of keeping us alive. Modern medicine relies heavily on imaging methods, beginning
Hyperbolic Geometry on Geometric Surfaces
Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction
Applications of Low KeV Imaging in Abdomen
Applications of Low KeV Imaging in Abdomen Dushyant Sahani, M.D Director of CT Associate Professor of Radiology Massachusetts General Hospital Harvard Medical School Email-dsahani@partners.org Disclosure
The Theory of Diffraction Tomography
The Theory of Diffraction Tomography Paul Müller 1, Mirjam Schürmann, and Jochen Guck Biotechnology Center, Technische Universität Dresden, Dresden, Germany (Dated: October 10, 2016) Abstract arxiv:1507.00466v3
Iterative Correction Methods for the Beam Hardening Artifacts in Computed Tomography. Chelsea Hall Department of Mathematics Oregon State University
Iterative Correction Methods for the Beam Hardening Artifacts in Computed Tomography Chelsea Hall Department of Mathematics Oregon State University December 2013 Abstract In this paper, we discuss three
Medical Imaging. Transmission Measurement
Medical Imaging Image Recontruction from Projection Prof Ed X. Wu Tranmiion Meaurement ( meaurable unknown attenuation, aborption cro-ection hadowgram ) I(ϕ,ξ) µ(x,) I = I o exp A(x, )dl L X-ra ource X-Ra
On stable inversion of the attenuated Radon transform with half data Jan Boman. We shall consider weighted Radon transforms of the form
On stable inversion of the attenuated Radon transform with half data Jan Boman We shall consider weighted Radon transforms of the form R ρ f(l) = f(x)ρ(x, L)ds, L where ρ is a given smooth, positive weight
Accelerators and radiation spectra
X-ray tube Accelerators and radiation spectra Electrons are released from the cathode (negative electrode) by thermionic emission accelerated in an evacuated tube hit the anode (target, positive electrode)
Vector-Valued Functions
Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1
Tomography and Sampling Theory
Tomography and Sampling Theory Adel Faridani Department of Mathematics Oregon State University January 11, 2006 Tomography and Sampling Theory p. 1/34 The Divergent Beam X-Ray Transform D z f(ω) = 0 f(z
An Introduction to X-ray tomography and Radon Transforms
Proceedings of Symposia in Applied Mathematics An Introduction to X-ray tomography and Radon Transforms Eric Todd Quinto Abstract. This article provides an introduction to the mathematics behind X-ray
High Performance Computing for Neutron Tomography Reconstruction
High Performance Computing for Neutron Tomography Reconstruction A Parallel Approach to Filtered Backprojection (FBP) Zongpu Li 1 Cain Gantt 2 1 Department of Physics and Materials Science City University
Non-radioactive radiation sources. Lesson FYSKJM4710 Eirik Malinen
Non-radioactive radiation sources Lesson FYSKJM4710 Eirik Malinen X-ray tube Electrons are released from the cathode (negative electrode) by thermionic emission accelerated in an evacuated tube hit the
Reconstructing a Function from its V-line Radon Transform in a D
Reconstructing a Function from its V-line Radon Transform in a Disc University of Texas at Arlington Inverse Problems and Application Linköping, Sweden April 5, 2013 Acknowledgements The talk is based
Inverse Problems: From Regularization to Bayesian Inference
1 / 42 Inverse Problems: From Regularization to Bayesian Inference An Overview on Prior Modeling and Bayesian Computation Application to Computed Tomography Ali Mohammad-Djafari Groupe Problèmes Inverses
MATH2321, Calculus III for Science and Engineering, Fall Name (Printed) Print your name, the date, and then sign the exam on the line
MATH2321, Calculus III for Science and Engineering, Fall 218 1 Exam 2 Name (Printed) Date Signature Instructions STOP. above. Print your name, the date, and then sign the exam on the line This exam consists
arxiv: v2 [cs.it] 10 Feb 2016
![arxiv: v2 [cs.it] 10 Feb 2016](/thumbs/74/70148984.jpg "arxiv: v2 [cs.it] 10 Feb 2016")
Sparse Sampling in Helical Cone-Beam CT Perfect Reconstruction Algorithms Tamir Bendory and Arie Feuer Abstract In the current paper we consider the Helical Cone Beam CT. This scanning method exposes the
Electron density and effective atomic number images generated by dual energy imaging with a 320-detector CT system: A feasibility study
Electron density and effective atomic number images generated by dual energy imaging with a 320-detector CT system: A feasibility study Poster No.: C-0403 Congress: ECR 2014 Type: Scientific Exhibit Authors:
Fast variance predictions for 3D cone-beam CT with quadratic regularization
Fast variance predictions for 3D cone-beam CT with quadratic regularization Yingying Zhang-O Connor and Jeffrey A. Fessler a a Department of Electrical Engineering and Computer Science, The University
NIH Public Access Author Manuscript Inverse Probl. Author manuscript; available in PMC 2010 April 20.
NIH Public Access Author Manuscript Published in final edited form as: Inverse Probl. 2010 January 1; 26(3): 350131 3501329. doi:10.1088/0266-5611/26/3/035013. High Order Total Variation Minimization for
Lecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
Chaotic Billiards. Part I Introduction to Dynamical Billiards. 1 Review of the Dierent Billiard Systems. Ben Parker and Alex Riina.
Chaotic Billiards Ben Parker and Alex Riina December 3, 2009 Part I Introduction to Dynamical Billiards 1 Review of the Dierent Billiard Systems In investigating dynamical billiard theory, we focus on
Arc Length and Surface Area in Parametric Equations
Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length
Computed Tomography Lab phyc30021 Laboratory and Computational Physics 3
Computed Tomography Lab phyc30021 Laboratory and Computational Physics 3 By Joshua P. Ellis School of Physics Last compiled 8th August 2017 Contents 1. Introduction 1 1.1. Aim.............................................
A pointwise limit theorem for filtered backprojection in computed tomography
A pointwise it theorem for filtered backprojection in computed tomography Yangbo Ye a) Department of Mathematics University of Iowa Iowa City Iowa 52242 Jiehua Zhu b) Department of Mathematics University
Reproducing formulas associated with symbols
Reproducing formulas associated with symbols Filippo De Mari Ernesto De Vito Università di Genova, Italy Modern Methods of Time-Frequency Analysis II Workshop on Applied Coorbit space theory September
An Inversion Method for the Exponential Radon Transform Basedonthe Harmonic Analysis of the Euclidean Motion Group
An Inversion Method for the Exponential Radon Transform Basedonthe Harmonic Analysis of the Euclidean Motion Group Can Evren Yarman and Birsen Yazıcı Department of Electrical, Computer and System Engineering,
IMPROVED EXACT FBP ALGORITHM FOR SPIRAL CT
IMPROVED EXACT FBP ALGORITHM FOR SPIRAL CT ALEXANDER KATSEVICH Abstract. Proposed is a theoretically exact formula for inversion of data obtained by a spiral CT scan with a 2-D detector array. The detector
BME I5000: Biomedical Imaging
BME I5000: Biomedical Imaging Lecture 9 Magnetic Resonance Imaging (imaging) Lucas C. Parra, parra@ccny.cuny.edu Blackboard: http://cityonline.ccny.cuny.edu/ 1 Schedule 1. Introduction, Spatial Resolution,
An Application of Abel s Method to the Inverse Radon Transform
J. Inverse Ill-Posed Probl.? (????), 1 14 DOI 1.1515/jip-????-??? de Gruyter???? An Application of Abel s Method to the Inverse Radon Transform Shavkat Alimov, Joseph David, Alexander Nolte and Julie Sherman
ELEG 479 Lecture #6. Mark Mirotznik, Ph.D. Associate Professor The University of Delaware
ELEG 479 Lecture #6 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware Summary of Last Lecture X-ray Physics What are X-rays and when are they useful for medical imaging? How are X-rays
Studies on implementation of the Katsevich algorithm for spiral cone-beam CT
Journal of X-Ray Science and Technology 12 (2004) 97 116 97 IOS Press Studies on implementation of the Katsevich algorithm for spiral cone-beam CT Hengyong Yu a and Ge Wang b a College of Communication
Positron Emission Tomography
Positron Emission Tomography Presenter: Difei Wang June,2018 Universität Bonn Contents 2 / 24 1 2 3 4 Positron emission Detected events Detectors and configuration Data acquisition Positron emission Positron
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS OPTION I-2 MEDICAL IMAGING Reading Activity Answers IB Assessment Statements Option I-2, Medical Imaging: X-Rays I.2.1. I.2.2. I.2.3. Define
Non-Relativistic Quantum Mechanics as a Gauge Theory
Non-Relativistic Quantum Mechanics as a Gauge Theory Sungwook Lee Department of Mathematics, University of Southern Mississippi LA/MS Section of MAA Meeting, March 1, 2013 Outline Lifted Quantum Mechanics
BioE Exam 1 10/9/2018 Answer Sheet - Correct answer is A for all questions. 1. The sagittal plane
BioE 1330 - Exam 1 10/9/2018 Answer Sheet - Correct answer is A for all questions 1. The sagittal plane A. is perpendicular to the coronal plane. B. is parallel to the top of the head. C. represents a
Part 2 Introduction to Microlocal Analysis
Part 2 Introduction to Microlocal Analysis Birsen Yazıcı & Venky Krishnan Rensselaer Polytechnic Institute Electrical, Computer and Systems Engineering August 2 nd, 2010 Outline PART II Pseudodifferential
NIH Public Access Author Manuscript Proc Soc Photo Opt Instrum Eng. Author manuscript; available in PMC 2013 December 17.
NIH Public Access Author Manuscript Published in final edited form as: Proc Soc Photo Opt Instrum Eng. 2008 March 18; 6913:. doi:10.1117/12.769604. Tomographic Reconstruction of Band-limited Hermite Expansions
Metal Artifact Reduction and Dose Efficiency Improvement on Photon Counting Detector CT using an Additional Tin Filter
Metal Artifact Reduction and Dose Efficiency Improvement on Photon Counting Detector CT using an Additional Tin Filter Wei Zhou, Dilbar Abdurakhimova, Kishore Rajendran, Cynthia McCollough, Shuai Leng
Initial Studies in Proton Computed Tomography
SCIPP Initial Studies in Proton Computed Tomography L. R. Johnson, B. Keeney, G. Ross, H. F.-W. Sadrozinski, A. Seiden, D.C. Williams, L. Zhang Santa Cruz Institute for Particle Physics, UC Santa Cruz,
Local Tomography in 3-D SPECT
Pubblicazioni del Centro De Giorgi Proceedings of the Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Pisa, Italy, October 2007.
Procesamiento de Imágenes y Bioseñales
Procesamiento de Imágenes y Bioseñales Dr. Víctor Castañeda Agenda Physical basis of X-ray- CT, NMR, Ultrasound, Nuclear Medicine Sensors (cameras, gamma probes, microphone) Computational Tomography (CT)
Nonuniqueness in Anisotropic Traveltime Tomography under the Radon Transform Approximation. Bill Menke, December 2017 and January 2018
Nonuniqueness in Anisotropic Traveltime Tomography under the Radon Transform Approximation Bill Menke, December 2017 and January 2018 (This work follows up upon my 2015 BSSA paper on the same subject).
= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
Empirical Wavelet Transform
Jérôme Gilles Department of Mathematics, UCLA jegilles@math.ucla.edu Adaptive Data Analysis and Sparsity Workshop January 31th, 013 Outline Introduction - EMD 1D Empirical Wavelets Definition Experiments
A GENERAL SCHEME FOR CONSTRUCTING INVERSION ALGORITHMS FOR CONE BEAM CT
IJMMS 23:21, 135 1321 PII. S1611712329315 http://ijmms.hindawi.com Hindawi Publishing Corp. A GENERAL SCHEME FOR CONSTRUCTING INVERSION ALGORITHMS FOR CONE BEAM CT ALEXANDER KATSEVICH Received 3 September
FXA UNIT G485 Module X-Rays. Candidates should be able to : I = I 0 e -μx
1 Candidates should be able to : HISTORY Describe the nature of X-rays. Describe in simple terms how X-rays are produced. X-rays were discovered by Wilhelm Röntgen in 1865, when he found that a fluorescent
Cambridge International Examinations Cambridge Pre-U Certificate
Cambridge International Examinations Cambridge Pre-U Certificate PHYSICS (PRINCIPAL) 9792/02 Paper 2 Written Paper May/June 2018 INSERT *7775658333-I* The question in Section 2 of this paper will relate