Data processing: measuring – calibrating – or testing – Measurement system – Measured signal processing
Reexamination Certificate
2000-06-12
2002-04-09
Shah, Kamini (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system
Measured signal processing
C356S327000
Reexamination Certificate
active
06370490
ABSTRACT:
FIELD OF THE INVENTION
The invention relates to a method of determining an intrinsic spectrum f of radiation emitted by an object to be examined, which intrinsic spectrum f is represented by a set of N data points f
1
. . . f
N
and is determined from a measured spectrum h which is represented by a set of M measuring points h
1
. . . h
M
and is measured by means of an analysis apparatus having a given apparatus transfer function G in the form of an M×N matrix, which method includes the following steps:
a) forming an approximated intrinsic spectrum g of N data points g
1
. . . g
N
;
b) determining a measure of misfit &khgr;
2
between the approximated intrinsic spectrum, convoluted with the apparatus transfer function G, and the measured spectrum;
c) determining the value of a predetermined regularizing function S by inserting the approximated intrinsic spectrum in this function;
d) forming a functional F=&khgr;
2
+&agr;S containing a regularizing constant &agr;;
e) solving the regularizing constant &agr; from said functional, the N eigenvalues &lgr;
1
. . . &lgr;
N
of an N×N auxiliary matrix A formed from the apparatus transfer function and the approximated intrinsic spectrum being determined during said solution process;
f) executing a minimizing process on the functional F with the last regularizing constant &agr; found, using the N data points of the intrinsic spectrum as variables, the N data points thus found constituting a next approximated intrinsic spectrum;
g) repeating the steps b) to f), if necessary, until a predetermined convergence criterion has been satisfied;
h) identifying the approximated intrinsic spectrum then valid as the intrinsic spectrum searched.
The invention also relates to a storage medium which can be read by a computer and is provided with a computer program for carrying out said method, and also to a radiation analysis apparatus which is suitable to carry out the method.
DESCRIPTION OF PRIOR ART
An algorithm for carrying out such a method is known from the article “Bayesian Interpolation” by David J. C. MacKay, Neural Computation 4, pp. 415-447. Algorithms of this kind are known as “Maximum Entropy Algorithms”.
The cited article, notably chapter 4 thereof, describes how the intrinsic variation of a quantity can be determined from a set of measuring values of the relevant quantity which suffer from noise and other disturbing effects, i.e. the variation of this quantity if all disturbing effects exerted by, for example, the measuring equipment and/or static processes were removed.
A situation of this kind occurs, for example, during the measurement of an intensity spectrum as is done in X-ray diffraction. An object to be examined (a crystalline sample) is then irradiated by X-rays which are emitted again by the sample in a manner which is characteristic of the relevant material. The intensity of such emitted radiation is dependent on the angle at which the radiation is incident on the lattice faces of the crystalline material to be examined. An intensity spectrum of the emitted radiation is measured as a function of the take-off angle by moving an X-ray detector around the sample during the measurement.
In order to achieve a suitable angular resolution of the measurement, a limiting gap with a gap width of the order of magnitude of from 20 to 200 &mgr;m is arranged in front of the detector; in the case of a circumscribed circle having a radius of 30 cm this means that a measurement of a spectrum of one revolution yields a number of measuring points N of the order of magnitude of from N=10
4
to N=10
5
.
As is known from the practice of measurement by means of such apparatus, the spectrum of measuring points is the convolution of the intrinsic spectrum with the apparatus transfer function, possibly increased by noise and contributions by other disturbing effects. The transfer function takes into account the effect of all optical elements in the radiation path from the radiation source to the detector, notably the finite width of the detector slit; furthermore, it is in general also dependent on the location of the measurement (i.e. the magnitude of the take-off angle) so that, as is known for such apparatus, for numerical processing the transfer function takes the form of an M×N matrix (where N and M are of the same order of magnitude), i.e. a matrix with from M×N=10
8
to M×N=10
10
matrix elements.
The determination of the intrinsic spectrum according to the algorithm disclosed in the cited article by MacKay is based on an approximation of the intrinsic spectrum. This approximation may be based on prior theoretical knowledge of the spectrum to be measured, but the measured spectrum consisting of a set of M measuring points may also be taken as the approximated intrinsic spectrum; in order to obtain a set of N points, interpolation can be performed between the M measuring points (if M<N) or a part of the M measuring points can be ignored (if M>N). Subsequently, a measure of misfit &khgr;
2
is formed between the approximated intrinsic spectrum, convoluted with the apparatus transfer function, and the measured spectrum. Because of the absence of, for example noise and other statistical functions, this measure of misfit does not have the value zero.
In conformity with the Maximum Entropy Algorithm rule there is formed a functional F=&khgr;
2
+&agr;S in which the regularizing function S is dependent on the N data points of the intrinsic spectrum. The appearance of this regularizing function S is dependent on the nature of the process to be measured; in the case of X-ray diffraction, the appearance of this function may be &Sgr;(f
i
)*log(f
i
), in which the quantities f
i
represent the intensities of the measuring points. The regularizing function S includes a factor &agr; which is referred to as the regularizing constant. According to the Maximum Entropy Algorithm, the functional F must be minimized in dependence on the value of the data points of the intrinsic spectrum. The spectrum of data points at which the functional F is minimum then constitutes the intrinsic spectrum searched. For numerical execution of the minimizing process, however, it is first necessary to determine the value of the regularizing constant &agr;; moreover, an assumption must be made in respect of numerical initial values of the intrinsic spectrum, both in the regularizing function S and in the quantity &khgr;
2
. As has already been stated, the measured spectrum can often be chosen for the numerical initial values of the intrinsic spectrum.
The cited article describes a process for solving the regularizing constant &agr; from the functional F. Therein, an N×N auxiliary matrix is first determined from the apparatus transfer function and the approximated intrinsic spectrum. The process of forming the auxiliary matrix A is described in chapter 4.3 of the cited article. The set of eigenvalues of this auxiliary matrix is determined. A relation can then be derived between the regularizing constant &agr; and the set of eigenvalues &lgr;
1
. . . &lgr;
N
, so that the regularizing constant &agr; can be determined from this relation. This process is described notably in chapter 4.4 of the cited article; said relation is found by equating the formulas (4.8) and (4.9) described therein; it then follows that:
2
α
S
=
∑
i
=
1
i
=
N
λ
i
λ
i
+
α
(
1
)
(The quantity E
W
MP
stated in the cited article equals the regularizing function S). The value of the regularizing constant &agr; can then be determined from expression (1) by means of standard solution methods. The value thus found can be inserted in the functional F, after which the minimum value of the functional F is determined in known manner, the N data points of the intrinsic spectrum then being the variables of the functional F. Those values of the data points at which the minim of F occurs constitute a better approximation of the intrinsic spectrum than the values of the init
Janssen Augustus Josephus Elizabeth Maria
Reefman Derk
Rey William Jacques Jean
Shah Kamini
U.S. Philips Corporation
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