Program Preview
23
pages
English
Documents
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
23
pages
English
Documents
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
- cours - matière potentielle : guide
- cours - matière potentielle : masters
- leçon - matière potentielle : challenge
- leçon - matière potentielle : from pompeii
- leçon - matière potentielle : learners
- expression écrite
- later hominid groups advantages over earlier groups
- tci
- daily life
- ancient world
- social studies
- technology
- language
- students
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.
Z
Y
1MBOFXBWF
0CKFDUPYZ
Y
E
V
*MMVNJOBUJPO
%FUFDUPSQMBOF
Q
Y
Z
%FUFDUPSQMBOF
6OEJ$ŽSBDUFEpFME
TMJDFUIFPSFN
V
%J$ŽSBDUFEpFME
Y
0CKFDUPYZ
'PVSJFSQSPKFDUJPO
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
Z
u (x) = o(r)dyp
X
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
Z
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
Z Z
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
Z
0 0 0 0 g(r r )o(r )u (r )dr0L
YM
E
V
ZM
PYZ
Z
^
'\V
E
5SBOTNJTTJWFHFPNFUSZ
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 ) +
