Reconstruction of image from projections

173 Pages·2003·1.608 MB·English

by  Palamodov V.

Reconstructive Integral
Geometry
V.P.Palamodov
Tel Aviv University
Preface
One hundred years ago Hermann Minkowski (1904) has started the problem:
2
toreconstructanevenfunctionf onthesphereS fromknowledgeoftheintegrals
Mf (C) = fds
Z
C
over big circles C. Paul Funk (1916) has found an explicit reconstruction for-
mula for f from data of big circle integrals. Johann Radon studied the similar
problem for the Euclidean plane and space. The interest to reconstruction prob-
lems like Minkowski-Funk’s and Radon’s ones grew tremendously in the last four
decades, stimulated by the spectrum of new problems and methods of image
reconstruction. These are X-ray, MRI, gamma and positron radiography, ultra-
sound, thermoacoustic, seismic tomography, electron microscopy, synthetic radar
imaging and others. Analytic methods of reconstruction in two and three dimen-
sions from plane, ray or spherical averages are now in the focus of studies, being
motivated by applications.
The objective of the Chapters 2-5 and 7 of this book is to represent the scope
1
of recent results and new methods in the reconstructive integral geometry in a
uniform way. Keeping in mind the applications to real problems, the problems
with incomplete data are studied in Chapter 6. The phase space analysis is
applied to show the limits of stable reconstruction. We do not touch here the
problems arising in adaptation of analytic methods to numerical reconstruction
algorithms. We refer to the books [63],[64] which are focused on these problems.
Various aspects of relations between integral geometry and difierential equa-
tions are discussed in Chapter 8. The results presented here are partially new.
Necessary information from the harmonic analysis and the distribution theory is
collected in Chapter 1.
The book is an extended version of the lecture course which was read for
students of Tel Aviv University. Not much of additional knowledge is necessary
for reading Chapters 1-4.
1Blaschke’sterm"Integralgeometrie"lookssomehowredundant;theword’geometry’means
itself, since the old Greek time, calculation of lengths, areas, volumes, i.e. some integrals.
Integral geometry in Blaschke’s sense is rather contemplative point of view on the subject,
whereas reconstructive integral geometry is the application motivated part of the geometry.
2
Notations
R;C - the fleld of real, respectively, of complex numbers
: :
j = 2…{; { = p 1
¡ ⁄
V - a vector space over R of flnite dimension V - the dual space
dx or dV - a volume form in V
^
F (f) = f - the Fourier transform of a function f or of the density fdx deflned
in V
^
f (») = exp( j»x)f (x)dx
⁄ V ¡
F (f) =R exp(j»x)f (»)d» - the adjoint Fourier transform
V⁄
D (V) - thRe space of smooth functions of compact support (test functions) in the
vector space V
K (V) - the space of smooth densities of compact support (test densities) in V
S(V) - the Schwartz space of smooth fast decreasing functions in V
¡(‚) - Euler Gamma-function; basic formulae:
( 1)k
¡(1) = 1;¡(1=2) = p…;¡(‚+1) = ‚¡(‚); res¡(‚)d‚ = ¡ ;k = 0;1;2;:::
¡k k!
¡(‚)¡(1 ‚) = …=sin…‚; ¡(2‚) = …¡1=222‚¡1¡(‚)¡(‚+1=2):
¡
3
4
Contents
1 Fourier transform and distribution theory 9
1.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Fourier-Plancherel transform and inversion . . . . . . . . . . . . . 13
1.3 Distributions and generalized functions . . . . . . . . . . . . . . . 16
1.4 Tempered distributions and Fourier-Schwartz transform . . . . . . 19
1.5 Bandlimited functions and interpolation . . . . . . . . . . . . . . 21
1.6 Distributions of several variables . . . . . . . . . . . . . . . . . . . 24
1.7 Manifolds and difierential forms . . . . . . . . . . . . . . . . . . . 27
1.8 Pull down and pull back . . . . . . . . . . . . . . . . . . . . . . . 29
2 Radon transform 33
2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Inversion formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Alternative formulae . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Range conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 Frequency analysis of the Radon transform . . . . . . . . . . . . . 43
3 The Funk transform 47
3.1 Factorable mappings . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Spaces of constant curvature . . . . . . . . . . . . . . . . . . . . . 51
3.3 Inversion of the Funk transform . . . . . . . . . . . . . . . . . . . 54
3.4 Radon’s inversion via Funk’s inversion . . . . . . . . . . . . . . . 55
3.5 Unifled form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Funk-Radon transform and wave fronts . . . . . . . . . . . . . . . 58
3.7 Transformation of boundary discontinuities . . . . . . . . . . . . . 61
3.8 Neighborhood of an osculant hyperplane . . . . . . . . . . . . . . 66
3.9 Nonlinear artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.10 Appendix: Pizetti formula for arbitrary signature . . . . . . . . . 69
5
4 Reconstruction from line integrals 71
4.1 Line integrals and John equation . . . . . . . . . . . . . . . . . . 71
4.2 Sources at inflnity . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Reconstruction of the Radon transform from ray integrals . . . . . 77
4.4 Rays tangent to a surface . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Sources on a proper curve . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Reconstruction from plane collimated radiation . . . . . . . . . . 83
4.7 The attenuated ray transform . . . . . . . . . . . . . . . . . . . . 84
4.8 Inversion formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.9 Range conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Integral transform in Euclidean space 91
5.1 A–ne integral transform . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Geometry of a–ne subspaces . . . . . . . . . . . . . . . . . . . . . 92
5.3 Odd-dimensional subspaces. . . . . . . . . . . . . . . . . . . . . . 92
5.4 Even dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.5 Range conditions for the a–ne transform . . . . . . . . . . . . . . 98
5.6 Duality in integral geometry . . . . . . . . . . . . . . . . . . . . . 99
5.7 Fourier transform of homogeneous functions . . . . . . . . . . . . 99
5.8 Duality for the Funk transform . . . . . . . . . . . . . . . . . . . 102
5.9 Duality in Euclidean space . . . . . . . . . . . . . . . . . . . . . . 103
5.10 A–ne transform of difierential forms . . . . . . . . . . . . . . . . 105
6 Incomplete data problems 107
6.1 Completeness condition . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Radon transform of Gabor functions . . . . . . . . . . . . . . . . 109
6.3 Reconstruction from limited angle data . . . . . . . . . . . . . . . 110
6.4 Exterior problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 The parametrix method . . . . . . . . . . . . . . . . . . . . . . . 113
7 Spherical transform and inversion 117
7.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2 Reconstruction from arc integrals . . . . . . . . . . . . . . . . . . 118
7.3 Hemispherical integrals . . . . . . . . . . . . . . . . . . . . . . . . 122
7.4 Limited data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.5 Spheres centered on a sphere . . . . . . . . . . . . . . . . . . . . . 126
7.6 Spherical mean transform . . . . . . . . . . . . . . . . . . . . . . 128
7.7 Characteristic Cauchy problem for the Darboux equation . . . . . 131
7.8 Fundamental solution in odd dimensions . . . . . . . . . . . . . . 134
7.9 Even dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6
8 Funk transform on algebraic varieties 141
8.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.3 Multiplicative difierential equations . . . . . . . . . . . . . . . . . 145
8.4 Funk transform of Leray forms . . . . . . . . . . . . . . . . . . . . 147
8.5 Difierential equations for hypersurface integrals . . . . . . . . . . 149
8.6 Howard’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.7 Herglotz-Petrovsky formulae . . . . . . . . . . . . . . . . . . . . . 153
8.8 Range of difierential operators . . . . . . . . . . . . . . . . . . . . 155
8.9 Decreasing solutions of Maxwell’s system . . . . . . . . . . . . . . 156
8.10 Symmetric difierential forms . . . . . . . . . . . . . . . . . . . . . 158
9 Notes and bibliography 163
7
Chapter 1
Fourier transform and
distribution theory
1.1 Fourier transform
Let X be a space supplied with a Lebesgue measure dx: A (measurable) function
f : V ! C is called integrable in (X;dx); if the integral of jfjdx is flnite. We
shall use few facts from the Lebesgue theory:
R
Theorem 1.1 [Dominated convergence theorem] Let F be an integrable function
in V and f ;i = 1;2;::: a sequence of (measurable) functions such that jf j • F
i i
and f ! f almost everywhere in X : Then
i
f dx ! fdx
i
ZV ZV
Theorem 1.2 [Fubini’s theorem] Let X;Y be spaces endowed with the Lebesgue
measures dx;dy; respectively, andf be a function in X£Y integrable with respect
to the measure dxdy in X £Y. Then the function f(¢;y) is integrable in X for
:
almost all y 2 Y; the function g(y) = f (x;y)dx is integrable in Y and
X
R
dy f (x;y)dx = f (x;y)dxdy:
ZY ZX ZX£Y
From Fubini’s theorem we conclude that
f(x;y)dx dy = f(x;y)dy dx
Z (cid:181)Z ¶ Z (cid:181)Z ¶
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.e. we may change the order of integrations for any integrable f in X £Y.
For an arbitrary number p ‚ 1 the notation L = L (X);p ‚ 1 stands for the
p p
set of functions in V such that the function jfjp is integrable. This is a C-vector
space, moreover, it is a Banach space with the norm
1=p
kfk = jfjpdx
p
(cid:181)Z ¶
The space L (V) of square-integrable functions is a Hilbert space with the scalar
2
(inner) product
hf;gi = f g„dx
Z
whichsatisflestheinequality: jhf;gij2 • kfk2kgk2:(whichisattributedtoCauchy,
Bunyakovsky and H.Schwarz). We call it triangle inequality since it is equivalent
to kf +gk2 • (kfk+kgk)2 for L -norms.
2
Let V be a flnite dimensional vector space over the fleld R. Fix a coordinate
:
system x = (x ;:::;x ) in V and consider the volume density dx = dx ^:::^dx .
1 n 1 n
This density gives rise to the Lebesgue theory in (V;dx). We need one more fact
from this special theory. Let G ‰ V be a (measurable) set; the indicator of this
set is the function g that is equal to 1 in G and g = 0 otherwise. Fix a system
of coordinates in a flnite dimensional space V; we call a function g in V a step
function if it is equal to a linear combinations of indicators of cubes Q ‰ V. In
the case V = R; an arbitrary flnite interval is a cube.
Theorem 1.3 [Density theorem] For any p ‚ 1 the set of step functions is dense
in the space L (V):
p
The Fourier transform of a function f 2 L1(R) is the integral transformation
1 1
:
^
F(f) = f(») = exp(¡j»x)f(x)dx = exp(¡j»x)f(x)dx
Z¡1 Z¡1
with the parameter » running over the dual line R⁄, i.e. over the space of all
linear functionals on R:
Example 1. For the Gauss distribution function f(x) = …¡1=2(cid:190)¡1exp(¡(cid:190)¡2(x¡
y)2) with the mean value y and dispersion (cid:190) we have
f^(») = exp(¡jy»)exp(¡…2(cid:190)2»2)
It satisfles the inequality
^
jf(»)j • kfk (1.1)
1
10
since
^
jf(»)j • jexp(¡j»x)f(x)jdx = jf(x)jdx = kfk :
1
Z Z
Exercise. Prove the Theorem: for an arbitrary function f 2 L1(R) its Fourier
image f^is a continuous function in R⁄ such that f^(») ! 0 as j»j ! 1 such that
^
f(») ! 0 as j»j ! 1.
Some properties. For a point y 2 R we denote by Ty the translation operator
T f(x) = f(x + y): We have F (T f) = exp(j»y)F (f). Taking derivative with
y y
respect to y we come to
Proposition 1.4 If a function f 2 L1(R) is continuous and possesses almost
everywhere the derivative f0 = df=dx 2 L , then
1
f^0(») = j»f^(»)
If f; xf 2 L1(R), then the Fourier image of f has a continuous derivative and
dF(f)
= ¡jF(xf)
d»
J To check the flrst assertion we note that the condition f0 2 L implies that
1
f ! 0 as jxj ! 1. We integrate partially in the integral t exp(¡j»x)f0(x)dx
¡t
and pass on to the limit as t ! 1. For the second statement we commute the
R
derivative and the Fourier integral. I
Convolution. The integral
(f ⁄g)(x) = f(x¡y)g(y)dy (1.2)
Z
is called convolution of functions f;g 2 L . The mapping (f;g) 7! f⁄g is bilinear
2
commutative and associative operation. It satisfles
jf ⁄gj • kfk kgk (1.3)
2 2
which follows from the triangle inequality.
Proposition 1.5 If f;g 2 L the integral (1.2) converges almost everywhere and
1
kf ⁄gk • kfk kgk
1 1 1
J By Fubini’s theorem and by changing the variables z = x¡y we get
j f(y)g(x¡y)dyjdx • jf(y)jjg(x¡y)jdxdy = jf(y)jdy jg(z)jdz: I
Z Z Z Z Z
11