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TitleThe Inversion of the Radon Transform on the Rotational Group and
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Table of Contents
                            1 Introduction
2 Functions on S2 and SO(3)
	2.1 Parameterization of the Domains S2 and SO(3)
	2.2 Legendre Polynomials
	2.3 Spherical Harmonics
	2.4 Chebyshev Polynomials
	2.5 Wigner Functions
	2.6 The Laplace--Beltrami Operator and Sobolev Spaces
3 The Radon Transform on SO(3)
	3.1 Definition and Basic Properties
	3.2 The Radon Transform in Sobolev--Hilbert Spaces
	3.3 Generalizations of the Radon Transform
	3.4 Radially Symmetric Functions on S2 and SO(3)
4 The PDF--to--ODF Inversion Problem
	4.1 Crystallographic Background
	4.2 The Diffraction Experiment
	4.3 The Ill--Posedness of the PDF--to--ODF Inversion Problem
	4.4 The Reproducibility of the ODF
	4.5 ODF Estimation
5 Implementation of the MLS ODF Estimator
	5.1 Fast Fourier Transforms on S2 and SO(3)
	5.2 Discretisation of the MLS ODF Estimator
	5.3 The MLS ODF Estimation Algorithm
	5.4 Numerical Tests
	5.5 Applications
A PDF and ODF Plots
Bibliography
                        
Document Text Contents
Page 1

The Inversion of the Radon Transform
on the Rotational Group

and Its Application to Texture Analysis

Der Fakultät für Mathematik und Informatik
der Technischen Universität Bergakademie Freiberg

eingereichte

DISSERTATION

zur Erlangung des akademischen Grades
doctor rerum naturalium

Dr. rer. nat.
vorgelegt

von Dipl.-Math. Ralf Hielscher
geboren am 09.06.1977 in Löbau

Freiberg, den 12.12.2006

Page 2

Contents

1 Introduction 1

2 Functions on S2 and SO(3) 7
2.1 Parameterization of the Domains S2 and SO(3) . . . . . . . . . . . . . . 7
2.2 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Wigner Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 The Laplace–Beltrami Operator and Sobolev Spaces . . . . . . . . . . . . 20

3 The Radon Transform on SO(3) 25
3.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The Radon Transform in Sobolev–Hilbert Spaces . . . . . . . . . . . . . 29
3.3 Generalizations of the Radon Transform . . . . . . . . . . . . . . . . . . 35
3.4 Radially Symmetric Functions on S2 and SO(3) . . . . . . . . . . . . . . 38

4 The PDF–to–ODF Inversion Problem 47
4.1 Crystallographic Background . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 The Diffraction Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 The Ill–Posedness of the PDF–to–ODF Inversion Problem . . . . . . . . 54
4.4 The Reproducibility of the ODF . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 ODF Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Implementation of the MLS ODF Estimator 81
5.1 Fast Fourier Transforms on S2 and SO(3) . . . . . . . . . . . . . . . . . . 81
5.2 Discretisation of the MLS ODF Estimator . . . . . . . . . . . . . . . . . 84
5.3 The MLS ODF Estimation Algorithm . . . . . . . . . . . . . . . . . . . . 90
5.4 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A PDF and ODF Plots 109

Bibliography 123

ii

Page 62

4 The PDF–to–ODF Inversion Problem

The Ambiguity Due to the Ill–Posedness of the Radon Transform. The inversion of
the planar Radon transform is a classical example of an ill–posed problem. In Section 3.2
we have characterized the one–dimensional Radon transform on O(3) as an isomorphism
between the Sobolev spaces H0(O(3)) and H 1

2
(S2×S2). Hence, the inversion of the one–

dimensional Radon transform on O(3) is an ill–posed problem of order 1
2

(cf. Louis, 1989,
Sec. 3.2). Since the measured diffraction counts are in general effected by measurement
errors one has to apply regularization techniques to avoid amplification of errors (cf.
Bernier and Miller, 2006; van den Boogaart et al., 2006).

4.4 The Reproducibility of the ODF
We are concerned with the following simplified problem. Let SLaue ⊆ O(3) be a Laue
group and let Pi ∈ L2(S2), i = 1, . . . , N , be a list of pole figures with respect to the
crystal directions hi ∈ S2. We are interested in the range of ODFs f ∈ L2(O(3)/SLaue)
that satisfy

X f(hi, ◦) = Pi, i = 1, . . . , N. (4.9)
In other words, here we focus on the ambiguity of the ODF estimation problem neglecting
the ambiguity due to incomplete or superposed pole figures and unknown normalization
coefficients. This problem was first formulated by Matthies (cf. Matthies, 1982, Sec. 31)
and is central in QTA (Schaeben, 1994). Remember that for f ∈ L2(O(3)/SLaue) the
partial pointwise evaluation X f(h, ◦) ∈ L2(S2) is well defined (cf. Remark 3.12) for any
crystal direction h ∈ S2.

In Proposition 4.11 we have shown that the range of such ODFs is in general un-
bounded with respect to the maximum norm and with respect to the L2–norm. How-
ever, there exist ODFs f ∈ C(O(3)/SLaue) such that there is an one to one relation to
the corresponding PDF. A class of such ODFs is described by the next proposition. For
simplicity we restrict ourself to the triclinic case, i.e. to SLaue = Stric = {Id,−Id}. Then
the orientation space simplifies to O(3)/Stric = SO(3).

Proposition 4.12. Let ftrue ∈ C(SO(3)) be a triclinic ODF localized within a ball of
diameter π

2
around a certain orientation g0 ∈ SO(3), i.e. ftrue(g) = 0 for all g ∈ SO(3)

with ](g−10 g) ≥
π
2
. Then ftrue is uniquely determined by the corresponding pole density

function Ptrue = X ftrue.

Proof. First of all we notice that the condition ftrue(g) = 0 for all g ∈ SO(3) with
](g−10 g) ≥

π
2

is equivalent to the condition P (h, r) = 0 for all h, r ∈ S2 with ](g0h, r) =
π
2
. This is due to the identity of the sets{

g ∈ SO(3)
∣∣∣ ](g,g0) ≥ π

2

}
=
{

g ∈ G(h, r)
∣∣∣ h, r ∈ S2,](g0h, r) = π

2

}
and the non–negativity of ftrue. Consequently, the assumptions of the proposition can
be derived from the pole density function Ptrue directly.

60

Page 63

4 The PDF–to–ODF Inversion Problem

Let h, r ∈ S2 such that ](g0h, r) = π2 . By inequality (2.5) any rotation g ∈ SO(3) with
gg0h = r satisfies ](g0,g) ≥ π2 . Hence, the condition ftrue(g) = 0 for all rotations g ∈
SO(3) with ](g0,g) ≥ π2 implies ftrue(g) = 0 for all rotations g ∈ G(h, r). Consequently
Rf(h, r) = 0 and we conclude that the Radon transform of the true ODF ftrue is uniquely
determined by the true PDF Ptrue thanks to

Rftrue(g0h, r) =

{
Ptrue(g0h, r) if ](g0h, r) ≤ π2 ,
0 otherwise,

for any h, r ∈ S. By Theorem 3.10 the Radon transform is injective and hence the ODF
ftrue is uniquely determined by the PDF Ptrue.

Our purpose in this section is to relax the assumptions of Proposition 4.12 such that
it applies to arbitrary ODFs and to single pole figures Pi = P (hi, ◦), i = 1, . . . , N .

General Framework.

Definition 4.13. Let ψ : [0, π]→ R+ be some non–negative, square integrable function
and let S ⊆ S2 be an arbitrary subset. We define the concentration of a non–negative,
square integrable function P : S2 → R+ with respect to the set S and the weighting
function ψ as

σψ(P, S) =
1




S2
ψ(](S, r))P (r) dr.

Here, ](r, S) denoted the angular distance between the vector r and the set S.
Analogously we define the concentration of any non–negative, square integrable func-

tion f : O(3) → R+ in some subset Q ⊆ O(3) with respect to the weighting function ψ
by

σψ(f,Q) =
1

16π2


O(3)

ψ(](Q,g))f(g) dg.

Let P : S2 → R+ and f : O(3)→ R+ be probability density functions. Then there are
two important special cases for the choice of the function ψ which allow for a statistical
interpretation of the concentrations σψ(P, S) and σψ(f,Q). If ψ(t) = 1[0,ε] is the indicator
function then σψ(P, S) and σψ(f,Q) represent the mass located within the distance ε > 0
to the sets S and Q, respectively. If ψ(t) = t2 and S and Q are single elements which
correspond to the mean values of P and f then σψ(P, S) and σψ(f,Q) are the variances
of P and f , respectively. It is emphasized that Definition 4.13 allows for the presence
of crystal symmetries, i.e. for ODFs defined on factor spaces O(3)/SLaue. In this case
the set Q has to be chosen such that Q = QSLaue. Since any Laue group contains the
inversion −Id ∈ O(3) we have Q = −Q in all cases of interest.

Now we are ready to formulate the main theorem of this section relating the concen-
trations of an ODF and its pole figures.

61

Page 124

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