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НАУЧНО-ТЕХНИЧЕСКИИ ВЕСТНИК ИНФОРМАЦИОННЫХ ТЕХНОЛОГИИ, МЕХАНИКИ И ОПТИКИ ноябрь-декабрь 2015 Том 15 № 6 ISSN 2226-1494 http://ntv.i1mo.ru/
SCIENTIFIC AND TECHNICAL JOURNAL OF INFORMATION TECHNOLOGIES, MECHANICS AND OPTICS November-December 2015 Vol. 15 No 6 ISSN 2226-1494 http://ntv.ifmo.ru/en
METHOD OF TRAINING EXAMPLES IN SOLVING INVERSE ILL-POSED PROBLEMS OF SPECTROSCOPY V.S. Sizikova, A.V. Stepanovb
a ITMO University, Saint Petersburg, 197101, Russian Federation b Vernadsky Crimean Federal University, Simferopol, 95342, Russian Federation Corresponding author: [email protected]
Article info
Received 20.09.15, accepted 12.10.15 doi:10.17586/2226-1494-2015-15-6-1147-1154 Article in English
For citation: Sizikov V.S., Stepanov A.V. Method of training examples in solving inverse ill-posed problems of spectroscopy. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 6, pp. 1147-1154.
Abstract
Subject of Study. The paper deals with further development of the method of computational experiments for solving ill-posed problems, e.g., the inverse spectroscopy problem. This method produces an effective (nonoverstated) estimate for solution error of the first-kind equation. Method of Research. An equation is solved by the Tikhonov regularization method. We have obtained nonoverstated estimate for solution error and a new principle for choosing the regularization parameter on the basis of the truncating singular number spectrum of an operator. It is proposed to estimate the truncation magnitude by results of solving model (training, learning) examples close to an initial example (problem). This method takes into account an additional information about the solution. Main Results. We have derived a new, more accurate estimate for regularized solution error using the truncation parameter g. Ways for determining g according to the results of solving model examples are proposed. The method of modeling or training is applied to solving the inverse spectroscopy problem (restoration of a fine spectrum structure by solving integral equation on the basis of an experimental spectrum and the spread function of a spectral device). The method makes it possible to resolve close lines and select weak lines. Practical Relevance. The proposed method can be used to restore smoothed and noisy spectra, in other words, to enhance the resolution of spectral devices by mathematical and computer processing of experimental spectra. Keywords
ill-posed problems, Tikhonov regularization, solution error, method of training examples, inverse problem of spectroscopy, integral equation, spread function of spectral device, measured spectrum, training spectra, restored spectrum. Acknowledgements
This work was supported by the Russian Foundation for Basic Research (RFBR), grant №13-08-00442.
СПОСОБ ОБУЧАЮЩИХ ПРИМЕРОВ В РЕШЕНИИ ОБРАТНЫХ НЕКОРРЕКТНЫХ ЗАДАЧ СПЕКТРОСКОПИИ
В. С. Сизиков% А.В. Степановb
a Университет ИТМО, Санкт-Петербург, 197101, Российская Федерация
b Крымский федеральный университет им. В.И. Вернадского, Симферополь, 95342, Российская Федерация Адрес для переписки: [email protected]
Информация о статье
Поступила в редакцию 20.09.15, принята к печати 12.10.15 doi:10.17586/2226-1494-2015-15-6-1147-1154 Язык статьи - английский
Ссылка для цитирования: Сизиков В.С., Степанов А.В. Способ обучающих примеров в решении обратных некорректных задач спектроскопии // Научно-технический вестник информационных технологий, механики и оптики. 2015. Т. 15. № 6. С. 1147-1154.
Аннотация
Предмет исследования. Дано дальнейшее развитие способа вычислительных экспериментов решения некорректных задач, например, обратной задачи спектроскопии. Этот способ позволяет получить эффективную (незавышенную) оценку погрешности решения уравнения первого рода. Метод. Уравнение решается методом регуляризации Тихонова. Получены незавышенная оценка погрешности решения и новый способ выбора параметра регуляризации на основе использования усечения спектра сингулярных чисел оператора. Величину усечения предлагается оценивать по результатам решения модельных или обучающих примеров, «близких» исходному примеру (задаче). Данный способ учитывает дополнительную информацию о решении. Основной результат. Выведена новая, более точная оценка погрешности регуляризованного решения с использованием параметра усечения g. Предложены способы определения g по результатам решения модельных примеров. Способ моделирования или обучения применен к решению обратной задачи спектроскопии (восстановлению тонкой структуры спектра путем решения
интегрального уравнения на основе экспериментального спектра и аппаратной функции спектрального прибора). Способ позволил разрешить близкие линии и выделить слабые линии. Практическая значимость. Предложенная методика может быть использована для восстановления заглаженных и зашумленных спектров, другими словами, для повышения разрешающей способности спектральных приборов путем математической и компьютерной обработки экспериментальных спектров.
Ключевые слова: некорректные задачи, метод регуляризации Тихонова, погрешность решения, способ обучающих примеров, обратная задача спектроскопии, интегральное уравнение, аппаратная функция спектрального прибора, измеренный спектр, обучающие спектры, восстановленный спектр. Благодарности. Работа выполнена при поддержке РФФИ (грант № 13-08-00442).
Introduction
As is well known [1-4], it is practically impossible to obtain an effective (nonoverstated) error estimate for solution of ill-posed problem, e.g., the inverse spectroscopy problem without an additional (a priori) information about the solution. In this paper, we develop an adaptive method of computational experiments for estimating the solution error and choosing the regularization parameter a in solving ill-posed problems by the Tikhonov regularization method. The method is also known as: the technique of model, standard, learning, training examples, the way of the pseudoinverse operator [1-8]. This method takes into account an additional (a priori) information about the desired solution (an estimate of the number of maxima, their abscissas and ordinates, etc.) and, in this respect, resembles the methods such as the Tikhonov a-regularization with constraints on the solution [9], solution on a compact [4, 9], the methods of descriptive regularization [10], also taking into account a priori information on the solution (nonnegativity, monotonicity, convexity, parameters of extrema, etc.). However, the specific implementation of the method of computational experiments differs from these methods.
This method has been earlier developed and applied to signal processing [1-3], image restoration [5, 6] and spectroscopy [5-8]. In this paper, we propose its modification and application to the inverse problem of spectroscopy.
Basic relations
Consider an operator equation of the first kind
Ay = f, y e Y, f e F, (1)
where y is desired, and f is given elements of Hilbert spaces Y and F; A is a linear bounded operator from Y into F. The operator A is not expected to be continuously invertible, i.e. the problem of solving equation (1) is ill-posed. However, for the exact f we assume that equation (1) is solvable.
The problem is to find an element y e Y with minimal norm, which supplies the minimum value for the
discrepancy || Ay - f ||F and which is the pseudosolution, in particular, the normal solution [4, 9, 11].
In the zero-order Tikhonov regularization method [4, 9, 11, 12], giving one of the most effective ways for obtaining pseudosolutions, instead of (1) the equation
(aE + B) ya= A * f, (2)
is solved, where
A = A + AA , f = f + Af , ya= y + Aya, (3)
moreover, A, f andy are the exact operator and elements; A , f and ya are their practical values; AA , Af and Aya are their errors; a > 0 is the regularization parameter; B = A * A ; E is the unit operator.
Estimate of solution error. Consider the question of estimating the error Aya of the regularized solution ya and choosing the regularization parameter a.
It is known [13] that it is almost impossible to obtain an effective (nonoverstated) estimate of the error Aya without using additional (a priori) information on the solution. In this paper, we propose to use the results of solving "close" model (learning) examples as additional information. Taking into account the ratio A *Ay = A * f , which follows from (1), as well as the ratios (2) and (3), we obtain
(aE + B) Aya = -(aE + A * AA) y + A * Af
or
Aya = (aE + B)-1 A*(Af - AA y) - a(aE + B)-1 y , from where we find the following estimates in the norm of absolute and relative errors of the regularized solution
|| Aya || < ||(aE + B )-1 A * || ■ (|| Af || + || AA || • || y ||) + a||(aE + B )-1||-||y||,
<|| (aE + B)-1 A' || -f ^ +|| AA || | + a || (aE + B)-1 ||. (4)
II y|| I || y|| )
Taking into account that || A || -|| y ||>|| f || or 1/|| y ||<|| A || /1| f ||, we obtain the estimate (4) for the relative error of the regularized solution in the form
^rei(o) - <|| (aE + B)-1 A * || -|| A || (5rei + ^rei) + a || (aE + B)-1 || , (5)
|| y ||
where
я -Щ[ У - Il AA I
0rel и J- и ' yrel "
||f|| || A ||
are the relative errors of the right-hand side f and operator A. The right-hand side of (5) is the upper envelope of the true relative error arel (a). The first summand in the right-hand side of (5) is due to the errors of data, while
the second summand is determined by regularization. In (5), we have (cf. [14-/7]): || (aE + B)-1 A * || <//(2Va), and norm || (aE + B)-1 || can be expressed through minimum singular number ^ of symmetric positively determined operator aE + B :
|| (aE + B)-1 || = 1/(aE + B) = 1/(a + (B)). We obtain (cf. [14-17]):
0„i(o) .1^J^—(6)
|| y || 2va a + ^min(B)
where n = 5rd + £ rel.
However, in practice, estimate (6) (as well as (5)) may give a significant overstatement for arel (a), since, in case of ill-conditioned and ill-posed problems, ^ min( B) is close or equal to zero and then (when
B) = 0)
- ^ < ^ n +1. (7)
|| y || 2Va
The estimate (7) is not only overstated, but not having the minimum with respect to a.
To obtain more effective estimate of arel(a) we use the concept of the pseudoinverse operator having
enclosed in it, however, a sense somewhat different from the pseudo-inverse Moore-Penrose matrix A+ which gives the solution y = A+ f [4, 12, 18] and from the regularized operator (aE + B)-1 A * which gives the solution
ya = (aE + B )-1 A * f . The point is that A+ corresponds to the case a ^ 0 , (B) ^ 0, while regularization is dealing with a finite value of a > 0 and ^min (B) , 0, which leads to an overstatement of arel (a) in both cases. In order to bring the estimate arel (a) nearer to the true estimate of arel (a), we truncate the spectrum of
the operator (matrix in the discrete case) B from below, namely, instead of ^min (B) we use a value g > ^min (B) and write (6) in the form
^rel(a) - ^^^ <s(a), (8)
|| y ||
where
, * || A || a
E(a) -. (9)
2 V a a + g
It was shown in [1, 2, 4] that the function e(a), according to (9), has a (unique) minimum under the condition
"A"^ ,1.30. (10)
vg 4
From the condition e'(a) = 0, we obtain the equation for a (cf. [2, 4])
a -
'Mll^f3 , „^
I .
As shown in [2], in this case e"(a) > 0 , i.e. (11) corresponds to the minimum of the function e(a).
(a + g)3. (11)
According to relations (8) and (9), a relative error estimate || Aya ||/|| y || of regularized solution ya depends on A and n (more exactly, on the product || A || q). Therefore, if we solve a few examples (e.g., a few
spectra are being processed) with the same A and n (spread function and noise), then their error estimates (9) will be identical and nonoverstated (in function of a). It follows that when solving some original example P (i.e. when processing fp) with unknown solution (spectrum) yP, one can use the results of solving other (model, training) example Q with known (given) exact solution (spectrum) yQ , with the same A and n as in example P. Furthermore, when solving example Q, one can calculate the function arel(a)Q = || AyaQ ||/|| yQ || and, based on
this function, find aopt Q (optimal value of a, at whicharel(a)Q = min). This value aopt Q can be used for
solving the original example (spectrum) P.
Estimate of parameter g. Furthermore, it is necessary to determine the parameter g, which comes into (9). An estimate of g can be obtained graphically, namely, by fitting such value of g, at which envelope e(a) contacts curve (or a set of curves) arel (a)Q. The value of a corresponding to the contact point we denote as a .
Determining g can also be performed analytically. Equating arel(a) and e(a), as well as taking into account the condition e'(a) = 0, we obtain two equations for two unknowns a and g:
Mi!
2л/а ' а + g a = F (a), where, according to (11)
= CTrel(aX
F (a) = x(a + g )4/3, x =
In V/3 A ii q
(12)
(13)
Here, CTrel(a) is the calculated upper curve from a set of curves arel(a)Q = || AyaQ ||/|| yQ ||. The first equation in (12) is the condition of contact of e(a) (according to (9)) and arel(a), whereas the second equation is the minimum condition of function e(a), i.e. s'(a) = 0 at the contact point. The first equation can be resolved relatively to g:
g = а •
CTrel(a) -
i A ii n
2%/a
-1
Then, obtained system of two equations can be solved by iterations:
h = Xi-i (ai-i + gi-i) ,
ao = X,
gt = ai •
X,-1
4gi-i ,
CTrel(ai ) -
i A ii q 2<Ja
-i
i = 1,2,3 ...
This iterative process for a converges to some a = ag, since |F'(a) < i, as follows from (13).
However, since the function arel (a) is given in tabular form, it is more convenient to solve the problem graphically displaying onto a computer monitor the curves arel (a) and e(a) at different g. To enhance the efficiency of this method when working out a model example Q (or several examples) it is necessary to use an additional information about the original example (spectrum) P, namely, an estimate of the number of maxima (spectral lines) in the desired solution (spectrum) yP, ratios of their intensities, values of its abscissa (wavelengths or frequencies), the type of kernel (SF), etc. Such information will be helpful to choose more "successfully" the regularization parameter a and estimate the error of solving the examples (spectra) Q and P.
The modeling method generates a regularizing algorithm (RA), since when q ^ 0, a = o(q2) and finite
|| A || and g, we have for original and model examples according to (8) and (9) (cf. [2, 4]):
CTrel(a) =■
Ay* ii
->0.
|y||
i.e. at zero errors of initial data, the solution ya turns into the exact solution (normal pseudosolution).
а
Remark 1. Although the method of modeling (training, learning) requires a lot of preliminary work on drawing up and solving the model (training) examples, it is very effective in cases when it is required to solve a significant number of "close" examples (to resolve signals for a number of times, to restore several similar spectra in the inverse spectroscopy problem, etc.). Moreover, this method gives the possibility to explore practical potentials of the used method and algorithm applied to a particular problem on a number of training examples (to obtain the real solution error, the possibility of restoring the fine solution structure, etc.).
Remark 2. The objection can arise that, because of the ill-posedness of a problem, even small differences of the model problem (example, spectrum) from the original one can lead to significant differences of the regularization parameter a, the relative solution error || Aya ||/|| y ||, etc. However, firstly, the problem is solved by a stable regularization method and it is the conditionally well-posed (by Tikhonov), and secondly, relations (8) and (9) show that the error estimates for solutions || Aya || /1| y || are the same for the original and
model examples under the condition of identity of || A || q.
An example from spectroscopy
Let us illustrate the foregoing method of modeling (training, learning) by an example from the inverse spectroscopy problem (cf. [8]). The problem is to restore a spectrum via solving the Fredholm integral equation of the first kind (an ill-posed problem)
b
Ay = JK(X,X')y(X')dX' = f(X), c<X<d , (14)
a
where K (X, X') is the spread function of a spectral device, y(X) is the true (desired) spectrum, f (X) is the measured (experimental) spectrum, X is the wavelength, [a, b] are the limits for desired spectrum, [c, d ] are the limits for measured spectrum.
We assume that, instead of exact f and K, we have f and K such that || f - f || <S , || A - A || . Equation (14) is solved by the Tikhonov regularization method according to (2), where A* = AT .
At first, we consider the original example P with known measured noisy spectrum f (X) (Fig. 1) on a uniform grid X = X ■ ,Xmin + h,...,Xm. where Xmin = c = 450 nm, Xmax = d = 650 nm, h = AX = const = 1 nm
' Hull y Illlll y y IIldA y Illlll y IIldA y
is the discretization step, and n = (Xmax - Xmin)/h = 200 is the number of discretization steps in X.
It is assumed that the spread function (SF) K = K(X, X') of spectral device has a variable width, i.e. it is nondifference. As it is known [8, 19], the SF width w(X) at level of 0.5 is proportional to wavelength X. Therefore, we assume w(X) = qX , where q = 0.015. This corresponds to w(c) = w(450 nm) = 6.75 nm, w(485 nm) = 7.275 nm, w(620 nm) = 9.3 nm, and w(d) = w(650 nm) = 9.75 nm.
We use the dispersion SF
K(X, X') =-w(X)/2* , 2 . (15)
(X-X')2 +[w(X)/2]2
It is shown in [8] that SF of this type gives one of the most accurate restorations of a spectrum. To characterize the SF, along with the width w(X) at a level of 0.5, one may also use the integral SF width W (X) (the ratio of the SF area to its height)
W(X) = I K(X, X') dX'/K(X, X).
For a dispersion SF, we have: W(X) = (tc/2) w(X) «1.571 w(X).
Figure 1 shows the SF K(X, X ') (17) at X = 485 and 620 nm.
Analysis of Fig. 1 shows that the true (unknown) spectrum has, most likely, two close lines in the vicinity of X « 525 nm and near X « 620 nm, but they are poorly resolved in the measured spectrum f (X). Moreover, there is an indication that there is one weak line at X « 507 nm, as well as at X « 543 nm in true spectrum y(X). Thus, everything indicates that there are at least nine spectral lines in the spectrum y(X), although the number of lines in the measured spectrum f (X) is fewer (6 or 7).
In connection with this, the second (model, training) example Q "close" to the original P was modeled. The true spectrum of example Q contains 9 as well as 8 and 10 spectral lines in the form of Gaussian (cf. [3, 8]), i.e. several examples Q were modeled.
'3 5
¡3
% 4 о 4
о
& 3
■Н
л ~
о
(D
а 1
0,
2
1
3 т /\ 4
т * / / \ У
450 470 490
510 530 550
X, X', nm
570 590 610 630 650
Fig. 1. Example P. 1 - exact spectrum f (X); 2 - noisy spectrum f (X) and two cross-sections of SF: 3 - 10K(485,X') and 4 - 10K(620,X') (conventional units)
The measured spectra fQ (X) in examples Q were numerically calculated by formula
b
fQ (X) = J K(X, X') yQ (X') d X', c < x < d .
a
Furthermore, a = 460 nm , b = 640 nm.
The measurement errors 5 of the spectrum fP (X) were estimated at about 1%, which corresponds to the standard deviation (SD) « 0.02. Therefore, the values of fQ (X) were noisy by random errors with SD from 0.01 to 0.04, which corresponds to 8rel « 0.5 - 2% (because the value 8rel is known inexactly). The SF in example Q was taken in the form (15), moreover (since the SD is also known inexactly), w(X) was taken to be w(X) = q(1 + C)X, where C e [-0.02, 0.04], which corresponds to |rel « 0 -4%.
Further, the "close" model examples Q were solved by the quadrature method with Tikhonov regularization via solving equation (2) at SF (15) for several values of the regularization parameter a. The dependence of the relative error of regularized solutions ya (X) = ya Q (X) was calculated with respect to exact
solutions y(X) = yQ (X):
„ (a) llya (X) - y(X)||
CTrei (a) =-.
lly(X)||
Figure 2 shows dependences arel (a)Q for series of "close" model examples and for several values of errors Srel = 8/1| f || and |rel = 1| A || (the region between the curves 1 and 2). Note that the curve 3 in the modeling method is supposed to be unknown; it is given for illustration only.
2
H g
g
<D <D
"<3
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 1
\ \ \ \
\ \ 1 X \
\ ard(a) ч \ \ \
\ N \
ч \ g=0
V» xV "" \ \
1 \ s(a) '-//Ж / As, A/
\ ч ч / / / // • /j g
sV.
1 1 lg ag
-8
-7 -6 -5
-4
-3 -2 -1
0
1 lg a
Fig. 2. Relative errors arel(a) for examples Q (1 and 2 - the boundaries above and below, 3 - CTrel(a)P) and envelope curves e(a) for several values of parameter g (non-dimensional)
6
Figure 2 shows also several envelopes e(a) according to (9) at || A || = || A || = 0.843 and q = Srel +|rel = 2 -10"2 for several values g from 0 to 0.1. We choose such value g at which the condition (10) is satisfied and one of the curves e(a) contacts the set of curves arel(a), namely, g = 0.045 . This corresponds to regularization parameter a = a = 10~22. It is seen from Fig. 2 that, despite the scatter of curves arel(a) and e(a), the value of g and, as a consequence, a are estimated reliably.
Figure 3 shows solution (restored spectrum) at a g = 10~22, arel(a g) = 0.073 = 7.3%. We can see that the
spectrum is restored accurately: close lines are resolved and weak lines are separated.
9
I 8
3 7
^ 6
о 6 о
fc 5 m
H 4
л
JS 3
M 1
1
3 2 [л
|| » 1
Л 1 1 Л
/
4 Я> Л ч\ 1 1 1 1 J » if м IT л\
/ vjFsj: If О 'LА i— i \ 4 У I ! I / \ \ч
\ \ 43 \ <Р j i
0
460 480 500 520
540 560
X, X', nm
580 600 620 640
Fig. 3. Example P. 1 - true spectrum yP(X); 2 - measured spectrum fP(X); 3 - restored spectrum yaP(X) at a = a g = 10~22 and two cross-sections of SF: 4 - 10 K (485, X ') and 5 - 10 K (620, X ') (conventional units)
Remark 3. Although we assume in the modeling method that the exact spectrum (solution) y(X) is unknown in original example P, we adduce the exact spectrum yP (X) (Fig. 3) to demonstrate the potential possibilities of a technique. However, the spectrum yP (X) is not used for choosing a .
Conclusion
There are a number of ways for choosing the regularization parameter a and estimating error || Aya || for regularized solution ya. We should note the discrepancy principle [18], the generalized discrepancy principle [9], the modified discrepancy principle (the Raus-Gfrerer rule) [20], the cross-validation method [21], the L-curve criterion [22], the local regularizing algorithm [23], the new criterion of a posteriori choosing [15], the adaptive specialized generalized discrepancy principle [13], the new version of a posteriori choosing [16, 17], etc.
The errors 5 and 4 (as well as sourcewise representability of the solution) are usually used in these methods as the additional information about the solution. As a result, the regularization parameter a at finite 5 and 4 is chosen reliably (but with some overstatement in comparison with aopt). Furthermore, the solution error
|| Aya || is obtained mainly in the form of asymptotic estimates, and the estimate || Aya || at finite 5 and 4 is usually obtained with a large overstatement (see. Fig. 2, the curve g = 0).
We have developed a method of model (training, learning) examples, that gives the possibility to choose a and, most importantly, to obtain nonoverstated error estimate || Aya || (see Fig. 2, the contact of curves arel(a) and e(a) at a = a , as well as Fig. 3, the curve yaP (X)). The method of modeling or training has been applied
to solving the inverse spectroscopy problem (restoration of a fine spectrum structure by solving an integral equation with the use of an experimental spectrum and the spread function of a spectral device). The method has enabled to resolve close lines and to select weak lines. The proposed technique can be used for restoration of smoothed and noisy spectra, i.e. for enhancement of the resolving power of spectral devices by mathematical and computer processing of experimental spectra.
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Valery S. Sizikov Audrey V Stepanov
Сизиков Валерий Сергеевич
Степанов Андрей Валерьевич
D.Sc., Professor, Professor, ITMO University, Saint Petersburg, 197101, Russian Federation, [email protected]
D.Sc., Professor, Head of Chair, Vernadsky Crimean Federal University, Simferopol, 95342, Russian Federation, [email protected]
доктор технических наук, профессор, профессор, Университет ИТМО, Санкт-Петербург, 197101, Российская Федерация, [email protected]
доктор технических наук, профессор, заведующий кафедрой, Крымский федеральный университет им. В.И. Вернадского, Симферополь, 95342, Российская Федерация, [email protected]
