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Singular system algorithm for particle size analysis
C. Dimofte, R. Mondescu, L. Mihut
To cite this version:
C. Dimofte, R. Mondescu, L. Mihut. Singular system algorithm for particle size analysis. Journal de Physique III, EDP Sciences, 1993, 3 (12), pp.2271-2285. �10.1051/jp3:1993274�. �jpa-00249082�
Classification Physics Abstracts
42.20 02.60 07.60
Singular system algorithm for particle size analysis
C. Dimofte, R. P. Mondescu and L. Mihut
Institute of Atomic Physics, I-F-T-M-, Lab. 130, P-O- Box MG-6, R-76900 Magurele-Bucharest, Romania
(Received 27 April 1993, reiis~d 20 July 1993, accepted16 September1993)
Abstract. This paper presents the problem of light scattering in particle sizing in a totally different approach than that developed in a previous work [Il. Basic mathematics is briefly exposed as well as the advantages and the drawbacks of this new algorithm in comparison with the preceding one. The results of the tests we carried out including simulations and real data tests
emphasize the good resolution and numericbl stability of this method. The algorithm substantially reduces the broadening effect of narrow dimensional distributions and small intensity maxima in
more complex distributions are clearly distinguished. The important role of signal-to-noise ratio (SIN ) in data acquisition is extensively discussed and argued.
1. Introduction.
The problem of light scattering by small particles is a widely discussed problem during the last decades and reference monographic works are often mentioned [2]. Almost all existing papers
are dealing with the scattering of light by spherical particles [3]. When the panicles are large enough larger than several wavelengths the Fraunhofer diffraction theory is the generally accepted model to describe the phenomenon.
Within the limits of this model the energy diffracted by a volumetric size distribution V(a) in a ring of radii r~~ and r~~~ is :
amjx v (~ j
~(~>ni> ~exi) " C
i m>n [J~(X>ni + J((X>ni) J~(Xext) J~ (Xeu)1~ da (I)
~
with
x~~~ ~~j =
"~
sin arctan ~'~~'~~~ (2)
lo f
where f
=
the focal length of the lens which forms the diffraction image in the detection plane ;
a =
the particle diameter A~ =
the wavelength of light a~,~,
~~~ =
the extreme diameters of
the distribution; Jo(x), Jj(x)= the Bessel function of order 0 and I, respectively;
Cj
= a positive constant.
2. The mathematical basis.
We have followed here the excellent general presentation of inverse problems made in [4, 5].
Our problem is the inversion of a first kind Fredholm integral equation :
I(o)
= C2 ~~~ Jj "~~ )/ "~~ aV(a) da (3)
A
~
with I(o)
= the diffracted radiation intensity at an angle « made with the optical axis
C~
= a positive constant.
The symmetrical kernel of the integral equation is :
K(a6)
= (Jj I )/( I ~ (4)
A A
This expression leads to a more practical one : amJx
~(~,nt,extl ~ m>n K(~, ~int, ~ext V (~ d~ (5)
where the kernel of the integral equation is :
5~(a, rjnt, ~eM)
(6) K(a, r,nt> ~ext)
" Cl
a
3~(a, r,~~, r~~~) being the expression in square brackets from equation (I). Similar equations
may be written for each one of the N ring-shaped cells of the detector.
~ ~ ~n(~,nt~. ~ext,,), n = i, N (7)
is the discrete experimentally measured data vector.
Since the functions K~ (a, r,~t~, r~,t~) are square integrable on the interval [a~,~, a~~ ], we are
looking for a solution V(a) in the same class of functions X
= L~[a~~~, a~~~], with the data
vector e belonging to a N-dimensional space, E~, whose scalar product is defined as
N
~~' ~
~EN I ~'n ~n ~'J ~~~
n
where the weights w~ > 0 must be appropriately chosen.
In operator form equation (5) becomes LV
= e (9)
The solution of this equation may not exist if the functions K~ are not linearly independent
more than that, if the solution exists it is unique only if the data vector belongs to a
N'-dimensional subspace, Im L, the image space of the operator L, where N'WN is the
number of linearly independent K~ functions. However, we can still search for a least-squares solution of minimal norm, V+, usually called the generalized solution, which always exists and is unique. It can be expressed in terms of the singular system of the operator L, defined by the equations
Lu;
= A~ v~, k
= I,
,
N' (10a)
L* v, = A~ u~, k
=
I,
,
N' (lob)
where L * is the adjoint of operator L. The numbers A~ are the singular values and are ordered in a decreasing row, A
j m A
~ m m A~. The singular functions u~ form an orthonormal basis in X, while the singular vectors v, form an orthonormal basis in E~. These vectors are the
eigenvectors of the selfadjoint operator LL* E~
- E~
LL*v~=Ajv,, k= I,. ,N' (II)
The action of this operator on a data vector is expressed by (LL * e =
~~~'
K~ (a) (L* e (a) da
=
(
w~ G~~ e~ (12)
a~,~ m
where
G~~ = am»K~ (a K~,(a da (13
am,n
is the Gram matrix of the K~ functions, n
= I,
,
N.
Thus finding the singular values Al and the singular vectors v~ implies computing the eigenvalues and eigenvectors of a real and symmetric matrix M, whose elements are given by
Mnm = W'/ Wl Gnm, n, m
= I,
,
N (14)
The singular functions u~ may then be easily found as (Eq. lob))
Hi(a
=
) L
M',<(vi)n Kn (a (15
n
~
and the generalized solution becomes
V * (a
= ~j M'nl~j 2
(e, Vj )E~ (Vj)n Kn (a ) ( 6)
»
~
~l
When the condition number C
= Aj/A~ [5] is large, we are facing an ill-conditioned problem, and the generalized solution is unstable. The usual way to surpass this obstacle is by applying a regularization procedure [6], which consists in approximating the generalized
inverse operator L+ E~ -X, defined by the equation
V+ (a)
=
L+ e (17)
with a regularization operator, R~ E~ - X, dependent on a regularization parameter p. The
regularization operator must satisfy the following conditions : (.) for any p, R~ is contained in X (..) for any p, ((R~ ( w ((L+ ( = Ill
~ (18)
(.. lim R~ = L+
»-o
Using equations (16) and (17) the action of the regularization operator R~ on a data vector is
R~ e = V
~
(a)
= ~j Wn I W~_
k
2 (e, Vk)E~ Vj.)nlK,,(a)
~l i~ (19)
~k
where V~ (a) denotes the regularized solution. Here W~,, are the window coefficients, their role being to diminish the effect of the small singular values in the expression (16). The
properties of the window coefficients are derived easily from the conditions (18) imposed on
the regularization operator
(.) for any p and for any k
= I,
,
N
,
0 w W
~ , w I ;
(..) for any k
= I,
,
N, lim W~ , = I
p o
The crucial problem of the regularization procedure is the adequate choice of the
regularization parameter. This demand may be satisfied by considering two functions the
norm of the regularized solution
v (v )
=
I v
~ I
,
(20) and the discrepancy function
p (p )
=
( LV
~ e (
~
(2 Ii
Many authors [7, 8] have stressed the importance of some prior knowledge about the
conditions fulfilled by the regularized solution and/or by the data vector. Suppose we are able to estimate an upper limit, e, of the norm in E~ of the data vector. Among the regularized
solutions V~ which are consistent with the data vector in the limit e considered, we choose the solution of minimal norm, I-e- we use the conditions
ILV~ e I
~ w e
,
(V~ (
=
minimum (22)
v
The next step is to use only those sets of window coefficients W~ , which allow the function
N
~~~/~ ~~H ~
"~~ i ~~P.
I )~ (~> Vl)Ev ~ (23)
1=
to be a strictly increasing function of p, and
N j§'2
v~(lL)
~
I v»III
= L j (e, vi
)E~1~ (241
1= 1
a strictly decreasing function of p, respectively.
In these conditions it becomes obvious that only one value of p exists for which
p (p
= e, and the regularized solution corresponding to this value of p has minimal norm.
We conclude this section with two related remarks
the constant e is naturally connected to the signal-to-noise ratio (SIN in determining the
data vector
A
~ SIN ~~~
~
EN @ i '~1 ~~ (25)
unlike other algorithms (that presented in II included) where the input data errors may be or may be not considered, the singular system method requires the SIN ratio as an essential
parameter ; this fact must be seen as an advantage of this method since it represents a more
realistic approach to the problem.
3. Results and discussion.
3.I APPARATUS. The experimental arrangement is shown in figure I. The beam of a
1.5 mW He-Ne laser (item I), is passed through an expander (item 2) which yields a 20 mm diameter plane wave, in order to assure a representative collection of scattering particles. The
light is diffracted by the sample forced through the measuring cell (item 3) by the pump (item 4). The cell is only I mm thick to prevent occultation of the particles. The presence of the
pump and a good homogenization of the sample minimizes the sedimentation. The scattered
light is projected by a lens corrected for the spherical aberration (item 5) onto the detector (item 6), placed in the focal plane of the lens 5.
The 16-cell detector was specially developed for this application [9] in a hybrid structure
(Fig. 2), consisting in a matrix of 12 concentric ring sectors, centred on the optical axis, with
2 3 5 6
Fig. I. Experimental arrangement.
16
l
~
Fig. 2. Detector geometry with 1-12 concentric ring sectors matrix and 13-16 external cells.
mean radii in a geometrical progression ((he mean radius of the smallest ring is 0.194 mm, the ratio is / and the angular
extent of the rings is 42.5°), and in 4 extemal rectangular cells.
The correlation between the dimensions of the detector, the angular distribution of the diffracted light and the focal length of lens 5 (Fig. I) imposes a certain measuring range. One
can easily change this range by changing the focal length.
The algorithm includes corrections for the different sensitivities of the detecting cells and form factors for the external cells.
3 2 pRELIMINARY TESTS. For all the tests we run we chose N =16, a~,~
=
2 ~Lm, a~~~ =
496 ~Lm, A~ =
6 3281 (He-Ne laser), f
=
200 mm and the weights w,
= r, In /,
with r, the mean radii of the detecting cells.
The program uses precalculated data for the singular values A~ and singular vectors
v~, because they do not depend on the input data vector e. The condition number is
C
= A ill
j~ = 912, so we are solving an ill-conditioned problem. The regularization parameter computation is made on the basis of equations (22) and needs as input data the SIN ratio, the number of linearly independent singular vectors, N', a data vector e, the
precision in p determination, the window coefficients W~~, and the singular values and vectors.
Since the speed of the program heavily depends on the speed in p calculation, we tried to
diminish the number of variables involved. We verified that all the singular vectors
v, are linearly independent and in all the tests the value N'
=
16 was used. For any kind of data
vectors considered, a precision of 0.001 in p computation proved to be more than satisfactory.
Five window coefficients were tested
a) Tikhonov window (Tkj [6]
W~_k ~ l
~~(l / +
J1 (26)
b) Top-hat (rectangular) window (Th) [5]
W~_,
=
I if k
w [I/p
and W~_~ =
0 if k
> I/p
,
~~~~
[1/p being the integer part of I/p ; c) Triangular window (Tr) [5]
W~,~ = l (k I )/[I/p if k w [I/p]
and W~,,
=
0 if k
~ II p ; ~~~~
d) Successive approximations window (SA) [10] this is an iterative method, where the
regularization parameter p is the inverse of the number of iterations n, and
W~ , = I (I vi /)", with 0
~ r ~ 2/A ), (29)
e) Hanning window (H) [8]
W~ , =
I + cos ((k I qr/ IIp )/2 if k w Ill p
and W~
, = 0 if k
> [I/p ~~~~
No a priori statement on the usefulness of a window can be made. Although the SA window seemed to be quite promising, the results of all tests were discouraging, as well as those of the
tests performed with Th window. While using SA window leads to very flattened distributions,
the Th window (also known as numerical filtering) exhibits strong numerical oscillations for any simulated data vector and SIN ratio. The Tr window yields good results for distributions
having the maxima toward large particle diameters, but less satisfactory ones for distributions
having maxima shifted toward small diameters.
Particular attention must be paid to the SIN ratio. For each data vector exists a specific
SIN ratio which gives the best results. For values of SIN ratio smaller than optimum the distribution flattens ; for SIN ratio values greater than optimum numerical instabilities appears.
This behaviour is common to all the window coefficients used and has been already reported II II ; we believe that it is due to the highly oscillating character of the singular functions (in Fig. 3 u4, us and uj~ are represented). When SIN ratio is high, p is small and the windowing
effect is low ; consequently, oscillations of u~ functions are found in the distribution function.
If SIN ratio is low, p is high, the windowing effect is strong, some information contained in the
data vector is lost and the distribution broadens. Yet, the value of SIN ratio is not critical
variations of + / 30 fli do not lead to significant changes in the distribution function.
,
q
~ al )
~ ' _ j I j
Ci '( 1'
~ j /j '
j I ' j / [
~ i
"~ ~
f (i ( / (
£ (I I I
~ :
0
;
I j ' L ~
i ~
; "
i I I (
i ; i
' ,
/ l
~
/ /
;. I,~
n' io2 1o3
Fbrtide diameter a(pm) Fig. 3. Three singular functions u~ (continuous line), u~ (dashed line) and uj~ (dotted line).
This effect of the SIN ritio, which is a conspicuous advantage for low SIN ratio devices,
raises an interesting problem for an ideal device, with a very high SIN ratio the rich
information obtained with such a device cannot be fully extracted by the algorithm, and a
compromise must be made by choosing the optimum value for SIN ratio, although it is lower than the value yielded by the apparatus. This is clearly a drawback of the method, and is the
price to be paid for its subsequent advantages.
These preliminary tests, performed on simulated 3-Dirac functions, led us to several decisions
we were working further only with Tk and H windows ;
the SIN ratio used was lo for Tk window and 7 for H window (fortunately these are very realistic values in granulometric data acquisition process)
a special routine was developed to deal with the non-physical negative values of the
computed distribution function.
Despite the theoretical importance of the distribution function, we were mainly concemed with the histogram the percentage distribution on 16 granulometric classes due to its
wider use in practical applications.
3.3 MAXIMA POSITIONING PRECISION TESTS. We carried out these tests using-16 simulated
b-Dirac data vectors, centred at the middle of each granulometric class. The relative deviations of computed maxima for both Tk and H windows are plotted in figure 4. The input data errors for Tk and H windows are lo fli and 14 fli respectively, corresponding to the SIN ratios already
mentioned. The results emphasize one of the major advantages of this algorithm as compared
to that presented in Ill ; the large relative deviations that can be observed in figure 4 for
particle diameters below lo ~Lm are actually absolute deviations less than ~Lm. For particle
diameters above lo ~Lm the relative deviations of maxima are
~ 5 fli and
~ 7 fli for Tk and H windows respectively.
Another important advantage revealed by these tests is that the broadening effect generated by this algorithm is far less pronounced than that of the II algorithm. In figure 5 we plotted a
97 ~Lm-centred 3-Dirac simulated distribution and the distribution functions as computed with the two algorithms.
3,4 MAXIMA VALUES TESTS. These tests consisted in computing the value of the histogram
for a certain class, for several b-Dirac function data vectors located in that class. Figure 6 presents the results obtained for both window coefficients and for a granulometric class in the
ii Ii
/t i
j I
§ I l
e
~ '
~ l
~ l
W w
> I
fl
z
~ l
I
,'
/ ,---,
/ I' ,
,
io° io' io~ ~o~
Fbrticle dbmeter alpm ) Fig. 4. Relative deviations of maxima (computed values) for Tk window (continuous line) and H
window (dashed line) vs. particle diameters.
b
$
~C
~ ;(
it i
)( i
(I
il j
ii it('
i j
j j
/ j
.,
o ioo 2~v 300 loo soo
Fbrticle diameter a (pm
Fig. 5. 97 ~Lm-centred &Dirac function : simulated (continuous line) and computed with Tk window (dashed line) and with algorithm Ii (dotted line) respectively.
' ',
$c
# i~
Q
30 32 3£ 36 38 CO £2 ii
Farticle ddmeter a (pm) Fig. 6. Percentage values in a granulometric class for Tk window (continuous line) and H window
(dashed line) vs. &Dirac data vector location in the same class.
