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  • cours - matière potentielle : guide
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Contents Letter from Bert Bower, TCI Founder and CEO 2 Benefits of History Alive! The Ancient World 3 TCI Technology 4 Program Contents 6 Program Components 11 How to Use This Chapter 12 Student Edition: Sample Chapter 4: The Rise of Sumerian City-States 14 Lesson Guide 25 Lesson Masters 37 Interactive Student Notebook 41 Visuals 49 Welcome to History Alive! The Ancient World. This document contains everything you need to teach the sample chapter “The Rise of Sumerian City-States.
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Di raction Tomography I: The Fourier Di raction
Theorem
Kerkil Choi
Fitzpatrick Center for Photonics
Duke University
21-Oct-09Outline
Di raction tomography I
The Fourier di raction theorem: Green’s function
decomposition
The Fourier di raction theorem: The Fourier transform
approach
A limit of the Fourier di raction theorem
The Fourier space coverage discussion (synthetic aperture)
We will discussion interpolation methods and ltered backpropagation
methods in the next lecture.
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What we are trying to understand...
Di raction tomography vs. Projection tomographyTomography with di racted or scattered elds
Relationship between the object o(r) and di racted eld: ( r = (x;y))
X-ray projection (u ): undi racted eld (projection)p
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u (x) = o(r)dyp
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EM, acoustic (u ): di racted eldd
u(r) = u (r) +u (r)0 d
2 2 r r r(r +k )u () = o()u(): scalar Helmholtz equationd0
2 2o(r) = k [n (r) 1]: object scattering density0
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0 0 0 0u (r) = g(rjr )o(r )u(r )dr ;d
0exp(jkjr rj)00r rg(j ) = : green’s function
04jr rjThe rst Born approximation
u(r) = u (r) +u (r)0 d
Assumption: u << u : weakly scattering objectd 0
The rst Born approximation
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0 0 0 0 0 0 0 0u (x) = g(r r )o(r )u (r )dr + g(r r )o(r )u (r )drrrd 0 d
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0 0 0 0 g(r r )o(r )u (r )dr0L

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The Fourier di raction theorem
u : incident plane wave0
u : di racted (or scattered) eldd
Theorem: When an object o is illuminated by a plane wave u , the0
Fourier transform of the di racted eld produces the Fourier transform O
of the object along a semicircular arc in 2-D and along the semispherical
surface in 3-D in the spatial frequency domain.The Fourier di raction theorem proof: plane wave
decomposition of Green’s function
Plane wave decomposition of Green’s function
0 0r r r rg(j ) = g( ) =
Z 1 j 1 0 0d exp j[(x x ) +