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BRAGG-PEAK LOCATION EMPLOYING A MAXIMUM-ENTROPY FORMALISM
M. Lehmann, T. Robinson, S. Wilkins
To cite this version:
M. Lehmann, T. Robinson, S. Wilkins. BRAGG-PEAK LOCATION EMPLOYING A MAXIMUM- ENTROPY FORMALISM. Journal de Physique Colloques, 1986, 47 (C5), pp.C5-55-C5-62.
�10.1051/jphyscol:1986507�. �jpa-00225824�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment au n o 8, Tome 47, aoOt 1986
BRAGG-PEAK LOCATION EMPLOYING A MAXIMUM-ENTROPY FORMALISM
M.S. LEHMANN, T.E. ROBINSON and S.W. WILKINS"
Institut Laue-Langevin, 156X, F-38042 Grenoble Cedex, France
"CSIRO, Clayton, 3168 Victoria, Australia
R e s W - La M t h o d e d e l ' e n t r o p i e maximale a ete a p p l i q u e aux domes d g u n m u l t i d e t e c t e u r 8 deux dimensions. Des c o n t r a i n t e s m a t h h a t i q u e s o n t Bte i n t r o d u i t e s pour imposer que les s p e c t r e s ne contiennent qu'une s e u l e raie d e Bragg et que l e profil d e raie f i n a l e ait une forme raisomablement lisse. La d t h o d e a ete teste s u r un large nambre de s p e c t r e s s y n t h e t i q u e s a v e c I / a ( I )
c 9. Lee re2sultats lnontrent que les e r r e u r s s y s t h a t i q u e s s o n t tr&s p e t i t e s , e t que l e temps d e c a l c u l permet un t r a i t e m e n t e n temps -1. La vale- e n I / o ( I ) au-dessus d e l a q u e l l e p l u s que 50% d e raies d e Bragg s o n t abeerpees correctement, est 4.
Abstract - The miaximum entropy method has been tried on simulated d a t a from a mall 2-dimensional p o s i t i o n - s e n s i t i v e detectpr. C o n s t r a i n t s were introduced t o account for smoothness and the fact that o n l y one peak w a s found w i t h i n the frame o f the recording. Analysis of a large number of weak Bragg peaks w i t h I / U ( I ) c 9 and of d i f f e r e n t s i z e and background showed the method t o g i v e v i r t u a l l y bias-free r e s u l t s . The computing time is s u f f i c i e n t l y lcw to a l l o w real time use on measurements o f a s i n g l e Bragg spot w i t h a small d e t e c t o r . me r e l i a b i l i t y of the method, as measured by the v a l u e o f I / a ( I ) where the method b e g i n s t o fail i n around 50% o f the test cases, w a s found t o be about 4.
In t h e r e c o r d i n g o f a 2-diarensional s c a t t e r i n g p a t t e r n w i n g p o s i t i o n - s e n s i t i v e d e t e c t i o n t e c h n i q u e s t h e d e t e c t o r is moet roDrmonly d i v i d e d ( p h y s i c a l l y or l o g i c a l l y ) i n t o N p i x e l s of e q u a l size, and the r e s u l t o f the measurement is a set of numbers f P - 3 , where P . is t h e accumulated count i n p i x e l j . I n t h e case of d i f f r a c t i o n d a t a f& a s i n g l e b y ~ t a l the object is t h e n t o i n t e g r a t e t h e Bragg pealcs p r e s e n t i n the recording, b u t t o d o t h i s t h e i r l o c a t i o n s have t o be i d e n t i f i e d , o f t e n from the n o i s y d a t a . The first g o a l is t h e r e f o r e t o find the most l i k e l y p o s i t i o n o f the Bragg peak. F o r a total number o f accumulated c o u n t s P i n the d e t e c t o r a l a r g e 'number o f p o s s i b l e sets o f {P.) exist. and t o f i n d the one most l i k e l y t o o c c u r we c a n u s e an approaCh first devdloped by Boltemann /I/.
Imagine that we are randomly p u t t i n g P particles i n t o the N p i x e l s , one after another, u n t i l a given set of {P.) c o u n t s is obtained. T h i s c a n be done i n W ways, where W is given by the m u l t i n d a l c o e f f i c i e n t
The m o s t l i k e l y set is t h e n t a k e n t o be t h e one that c a n be o b t a i n e d i n most ways,
i.e. t h e one t h a t maximizes W, s u b j e c t t o whatever information, i n the f o m of mathematical c o n s t r a i n t s , c a n be impoeed o n the P T h i s information is t h e
j'
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986507
C5 -5 6 JOURNAL DE PHYSIQUE
recording o f t h e map, t o g e t h e r w i t h information about peak shapes, d e t e c t o r e f f i c i e n c y , p o s i t i o n o f powder l i n e s , e t c .
I f P is reasonably large*, S t i r l i n g ' s approximation f o r f a c t o r i a l s can be used t o g i v e
I n W I P.S = -P Z P./P ln(P./P)
3 3
An approach t o the i n t e g r a t i o n o f t h e Bragg peak would t h u s be t o use t h e most l i k e l y map as obtained b y maximizing S on t h e observed d a t a , and when t h e peak boundary is found from t h i s map, t o use the raw d a t a t o g e t t h e i n t e n s i t y .
Following thermodynamic language S i s c a l l e d t h e entropy, and t h e method o f d e t e r - mining t h e most l i k e l y count f o r each p i x e l b y maximizing S i s tenned t h e Maximum Entropy Method (MEM). O r i g i n a l l y , t h e use o f MEM was developed i n statistical mechanics /2/. More r e c e n t l y , it has been employed i n image r e c o n s t r u c t i o n and c r y s t a l l o g r a p h y /3/ and numerical techniques are under c o n s t a n t development. I n this paper we report a s t u d y o f t h e a p p l i c a t i o n o f WM f o r t h e peak l o c a t i o n on a s m a l l l-dimensional area d e t e c t o r . Whereas most p o s i t i o n s e n s i t i v e d e t e c t o r s have been designed f o r t h e r e c o r d i n g o f many Bragg r e f l e c t i o n s , c o n s t r u c t i o n is under way at t h e ILL o f s m a l l area d e t e c t o r s (16 x 1 6 o r 32 x 32 d e t e c t o r elements) aimed a t t h e s t u d y of one o r a few Bragg r e f l e c t i o n s a t a given t i m e . These d e t e c t o r s would mainly be used for samples w i t h v e r y s m a l l u n i t c e l l s , and should l e a d both t o a faster d a t a c o l l e c t i o n r a t e , and a more precise knowledge o f t h e Bragg p e a k ( s ) .
11 - MANIPULATION OF TfE MAP
I n o r d e r t o maximize t h e S, s u b j e c t t o a s e r i e s o f c o n s t r a i n t s ( ) and t h e con- d i t i o n that t h e t o t a l n-r o f c o u n t s is c o n s t a n t , we search f o h h e maximum o f t h e
f u n c t i o n
Q = - Z p . . l n p . . - A,? p . . - C A
i j 1 3 1 7 k kfk
with respect t o ( p . . ) , where pi. is t h e MEM count i n p i x e l ( i j ) i n a P-dimensional PSD w i t h N p i x e l s . l3 ( kk} are ~ a & r a n # e undetermined m u l t i p l i e r s , and t h e first o f t h e s e , Ao, is used t o ensure t h a t t h e number o f counts is k e p t c o n s t a n t , i , e .
where 0 . . is t h e recorded count.
1 3
The c o n s t r a i n t s imposed depend on t h e n a t u r e o f t h e problem. I n t h e p r e s e n t list t h e r e are t h r e e c o n s t r a i n t s , namely:
C o n s t r a i n t 1: f l -- x2 = 1 - Z w.. ( p . - o . . ) ~ 1 i j 1 3 ~j 1 7
T h i s is t h e most e s s e n t i a l c o n s t r a i n t , and e n s u r e s that { p - . } a g r e e w i t h ( 0 - - 1 w i t h i n the counting statistical limits. For law c o u n t s it12an be d i f f i c u l t %b
estimate the weight w.. f o r i n d i v i d u a l p i x e l s , so we approximate w.. by t h e mean
over t h e d e t e c t o r , i.;? 13
If f is l a r g e r t h a n 1 t h e n we have not t a k e n f u l l advantage o f the information c o n t a m e d i n { o . . } , w h i l e 1. f less t h a n 1 means t h a t t h e n o i s e has been allowed t o impose structur&'on t h e f i n & map.
* 32 c o u n t s d i s t r i b u t e d on 16 p i x e l s , e.g., would be enough!
2 2 c o n s t r a i n t 2: f 2 = 1/2N T. wij [(pij - pij-l) + ( p . . - pi-lj ) 1 < 6
i j 13
W e do know t h a t t h e r e s o l u t i o n function is slowly varying, and t h e Bragg peak is t h e r e f o r e smooth. Although t h e MEM using only c o n s t r a i n t 1 by its nature w i l l lead t o a smooth set o f { p . . } , t h e r e is no d i r e c t coupling between neighbours (except v i a t h e condition t h a t th$'sum of p . . is c o n s t a n t ) . Reducing f 2 t h e r e f o r e serves t o enforce l o c a l smoothness. The 3 a n t i t y 6 is chosen so t h a t t h e mean square d i f f e r - ence between neighbouring p i x e l s does not exceed a c e r t a i n f r a c t i o n of the variance of p . 11 ..
c o n s t r a i n t 3: f 3 = I = P . - r . - / 2 P - - i j 11 13 i j 13
The sum is over p o i n t s with p . . l a r g e r than t h e mean count, and r . . is t h e d i s t a n c e from t h e c e n t r e o f g r a v i t y o $ b e peak. W e have i n o u r a n a l y s i s ~ s u m e d t h a t a l l t h e Bragg i n t e n s i t y is found on one region of t h e d e t e c t o r surface. Under t h e s e conditions t h e influence o f small, a r t i f i c i a l peaks can be reduced by lowering t h e moment of i n e r t i a I ( t h e variance o f t h e peak d i s t r i b u t i o n f u n c t i o n ) . This c o n s t r a i n t i s o f limited importance, s e r v i n g mainly cosmetic purposes.
The { p . . } a r e determined from t h e N equations 13
which can be w r i t t e n as
A. is r e l a t e d t o {Xk,k=l,3} by C p - - = P.
i j 13 The d e r i v a t i v e s o f f are
k
where 6 is t h e mean over t h e four n e a r e s t neighbours t o p . . . 13
For a given {Xk,k=l,3), t h e equations are solved by an i t e r a t i v e p m & u r e /4/
i n p . . and Xo u n t i l Zp.
-
P is reached.1 3 i j l j
The v a r i a t i o n o f {Xk,k-1,s) is done f o r one Xk at a time. F i r s t X1 is adjusted t o g e t X2 e! 1. This is done i t e r a t i v e l y . Improved estimates o f Xl are obtained as
o l d 2 2 2 2
XI .X old, where X old is t h e value of x f o r XlOld. , x is within g i w n limits o f 1 the process is stopped.
The q u a n t i t y f a is then c a l c u l a t e d f o r fixed XI, and i f it is l a r g e r than 6,
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XP is increased u n t i l this is no longer t h e case while ensuring x 2 s t a y s belon an upper limit. New estimates of X 2 are obtained i n a way similar t o that f o r XI.
Finally A3 is estimated i n a single cycle. It is obvious from t h e expressions f o r p. . and af3/ap. . t h a t pij is most sensitive t o points far from t h e centre of
13 13 2
gravity. The point giving largest r . . p.. is found, and X3 is then determined
13 13
from t h e r a t i o between pij and t h e mean value E pij/N. I t was ij
found t h a t s e t t i n g X j = p. ./( L pij/LU) gave s a t i s f a c t o r y r e s u l t s f o r the
13 ij
reduction i n s i z e of these p...
13
The peak can now be located from t h e wrooth map {p..) . In exceptional cases t h i s is done by v i s u a l inspection of t h e data. In t h e m a j 8 i t y of cases, however, t h i s is done by a crmputer algorithm, which aims at contouring t h e map, s e t t i n g a limit between t h e peak and t h e backgmund. A main requirement of such an algorithm is speed, and i f the c r i t i c a l contour l e v e l is known t h i s can e a s i l y be achieved.
Unfortunately, most often t h i s is not 80, and f o r weak and consequently very flat peaks even m a l l changes i n t h e l e v e l of t h e limiting contour lead t o large changes i n the s i z e of t h e peak. The other approach is t o use knowledge about t h e size of t h e peak, which is e i t h e r obtained from an analysis of t h e d i s t r i b u t i o n of (p..) o r i s based on prior observations. One intereating method is t o study carefullyYhe histogram of {o..} /5/, but again f o r weak peaks good r e s u l t s are d i f f i c u l t t o obtain. In t h e y r e s e n t analysis we have therefore assumed that t h e numbsr of points i n t h e peak, M, is knawn approximately, and report below same t e s t s on how u c h e r r o r can be tolerated i n t h e choice of M.
The procedure ia then caaputationally simple. The { p - - ) are sorted, and t h e M largest values define t h e peak. Same of t h e points l s a t e d w i l l not be connected with t h e main group of points. They a r e therefore renmmd using t h e condition that a point can only belong t o t h e peak i f two o r more of its neighbours are peak points. Secondly, t h e edge of t h e peak can be somerwhat ragged, and t h i s is cured by including i t e r a t i v e l y a l l pointe that have four o r more neighbours belonging t o t h e Peak.
It is clear that a simple manipulation of peak and background points as described above is i n i t s e l f a powerful method /5/, and it may be asked whether t h e prior Bmoothing using MEM adds anything t o t h i s . I f we f i r s t consider constraint 1, then i n our f o m l a t i o n it does not radically change t h e r e l a t i v e d i s t r i b u t i o n of values in {p--1. w i l l lead t o scamwhat smaller pi -, and small oi. give somewhat l a r g e P p . . , Z Z A ? e r t h e M w e s t values are o ~ i u I B a f r a t (0. . I 3. {pi. 1 is u c h t h e samef3 Constraint 2, however, gives a penalty t o singular po?dts being %oo Urge, i.e. points outside t h e peak, but increases l o w values i n a high count environment, and v i a constraint 3 any point above the mean lying f a r fram t h e peak w i l l effectively be removed from the group of t h e M largest points. W e therefore have good reason t o believe (confirmed by numerical te&s), that t h e MEM does improve t h e s t a r t i n g values f o r t h e manipulation.
When the peak hae been located, t h e integrated intensity I and t h e standard- deviation O( I ) are then calculated w i n g the r a w data toi. } . By selecting a t B m e
&age the largest values i n ( p . . } as being the peak, t h e h e g r a t e d intensity does harever r i s k having a positive%iae. This biaa is p a r t i a l l y o r t o t a l l y rews3d ly taking away pointe that abviously Qo not belong t o t h e peak, and by adding points t o t h e boundary of the peak ae described above. How succeseful t h i s approach ie can best be determined from simulation calculatione, and we have therefore undertalcen the d a t a reduction of synthetic peaks of d i f f e r e n t intensities and sizes.
Although eventually the s t r e n g t h o f any d a t a reduction method w i l l be judged from its use on *real1 data, t h e s e experiences do not e a s i l y t e l l u s haw c o r r e c t t h e i n t e n s i t y e x t r a c t i o n is. We have t h e r e f o r e produced a r t i f i c i a l d a t a with knawn i n t e n s i t i e s and counting e r r o r s and used t h e s e t o probe t h e method. Throughout t h i s a n a l y s i s a g r i d o f 16 x 16 p i x e l s was used, and t h e Bragg pealcs were constructed as a sum of a constant background and a 2-dimensional Gaussian d i s t r i b u t i o n function.
The peak was assumed t o cover t h e region defined by twice the standard deviations o f the d i s t r i b u t i o n . Peak i n t e n s i t i e s w e r e chosed i n the range fram 0 t o 180 counts, i.e. weak peaks, and the average background count i n a p i x e l was from 2 t o 4 i n t h e case o f the estimates o f bias, and from 1 t o 27 f o r the estimate o f s e n s i t i v i t y . After the model peak had been produced t h e d a t a was converted t o "experimental" data using a r o u t i n e by Antoniadis e t al. /6/ which f o r a given Poisson d i s t r i b u t i o n supplies individual samples. For l a t e r camparison the i n t e n s i t y o f t h e
"experimental" map was then calculated using the known peak location. S t a r t i n g values f o r t h e c a l c u l a t i o n were t h e flat map, and the parameters i n t h e d a t a reduction program were:
x 2 was i n i t i a l l y required t o f a l l between 0.95 and 1.05, and t o s t a y b e l o w 1 . 2 when adjusting X 2 .
0 was set t o 0.005*Epij/N, i.e. the mean d i f f e r e n c e between neighbouring p o i n t s was approxinately 7% o f t h e standard deviation. 8 could be varied considerably without causing much change i n t h e f i n a l r e s u l t s . It was found, however, t h a t f o r peaks above o r around 300 counts t h i s condition was t o o strict, b u t as at present we are mainly considering weak peaks, t h e study o f t h i s was postponed.
To i l l u s t r a t e the procedure an example is shown i n Pig. 1. The first p l o t ( a ) shows the model peak with background 1, i n t e n s i t y 150 and 69 p o i n t s i n the peak. The second p l o t ( b ) shows t h e experimental map. The t h i r d p l o t ( c ) Shows {pi 1. To obtain e better i d e a of {p. .}, it was rescaled, and t h i s is ahawn i n the %our& p l a t ( d ) . The r e s u l t s are encoaaging.
Fig. 1. Example o f Bragg peak
a. Model peak
b. Observecl peak
JOURNAL DE PHYSIQUE
d. MEM rescaled
F i r s t a test o f the s e n s i t i v i t y was done. The procedures used were e i t h e r t o keep the peak count fixed and increase t h e background, o r t o keep t h e background fixed and vary the peak. For each case 50 frames w e r e treated, the r e s u l t was estimated by v i s u a l inspection and t h e percentages of success w e r e determined. f i e peak determination w a s considered t o fail i f a l l peak p o i n t s were not connected, i f t h e peak l o c a t i o n was wrong, o r i f t h e peak s i z e was t o o large o r t o o small. The r e s u l t s are p l o t t e d i n Fig. 2 as a function o f I / a ( I ) . Open d o t s are f o r peaks with 45 points, dark p o i n t s are f o r peaks with 69 p o i n t s . It is obviously e a s i e r t o determine sharp peaks ( i .e. 45 p o i n t s ) than broad peaks, and the range o f
s e n s i t i v i t y (measured as t h e I / a ( I ) corresponding t o 50% success) lies i n t h e range from 3.5 t o 4.5.
7' SUCCESS
Pig. 2. Percentage of success i n d a t a reduction. Squares are f o r conetant i n t e n s i t y (150 counts), and circles a M triangles are f o r constant background (1 and 3 counts/pixel, respectively).
The b i a s was determined f o r peaks with s i z e s of 45 p o i n t s and 69 p o i n t s , and t h e r e s u l t s a r e shown i n Pig. 3. Here t h e i n t e n s i t y obtained i n t h e d a t a reduction is p l o t t e d v e r s u s t h e i n t e n s i t y c a l c u l a t e d with t h e c o r r e c t boundary, and each point is t h e average of 100 s y n t h e t i c Bragg-peaks. There is indication o f a small p o s i t i v e b i a s f o r t h e l a r g e r peaks. It should be noted t h a t I/o( I ) f o r t h e p o i n t s p l o t t e d a r e i n t h e range from 0 t o 9, s o we are d e a l i n g with low i n t e n s i t y peaks. I n t h e worst c a s e t h e b i a s is about 20% of t h e standard d e v i a t i o n o ( 1 ) .
Pig. 3. Bias i n i n t e n s i t y determination.
f ,, is model i n t e n s i t y , f OUT is derived i n t e n s i t y . Squares are f o r Gaussian peak with 45 p o i n t s and circles are f o r Gaussian peak with 69 p o i n t s . Open symbols are f o r background of 2 counts/pixel, while
f i l l e d symbols a r e f o r 4 counts/- p i x e l .
As t h e number of peak p o i n t s , H, is assumed t o be known, t h e s e n s i t i v i t y of I with respect t o t h i s number was then t e s t e d by doing s i m i l a r c a l c u l a t i o n s varying M from 80% t o 120% of its o r i g i n a l value. The r e s u l t s a r e shown i n Pig. 4. I n t h i s case we p l o t t h e least squares l i n e going through t h e p o i n t s f o r t h e d i f f e r e n t cases. As seen t h e s e n s i t i v i t y is -11, s o w e have a large margin f o r choice o f M. This value would probably be estimated by v i s u a l i n s p e c t i o n o f s t r o n g e r peaks.
150
Fig. 4 C o r r e l a t i o n between choice of M and b i a s . Axis as i n Fig. 3. The l i n e s are t h e l e a s t squares l i n e s through
p o i n t s f o r given value of M. 100
YI - CONCLUSION
The lpaximvm entropy method is a f l e x i b l e t o o l f o r obtaining from a noisy measurement of a d i f f r a c t i o n p a t t e r n a smooth map revealing t h e location o f any d i f f r a c t i o n peak present i n t h e recording. C o n s t r a i n t s based on knowledge of t h e nature of t h e Bragg peak can e a s i l y be introduced, b u t t h e computing might be heavy. In t h i s paper we have s t u d i e d t h e c o n s t r a i n t of ~moothness i n t h e c a s e where o n l y one peak is present, and f o r s i m p l i c i t y we have o n l y taken t h e case of 2 - d ~ n s i o n a l d a t a . We f i n d t h a t t h e bias is very limited, and f o r t h e non-optimised program a v a i l a b l e at present t h e computing time on a VAX 750 is less than 2 s e c . f o r a frame of 16 x 16
C5-62 JOURNAL DE PHYSIQUE
d a t a p o i n t s . It should t h u s be p o s s i b l e t o u s e t h e method i n real time o n d a t a from s m a l l p o s i t i o n s e n s i t i v e d e t e c t o r s . These d e t e c t o r s a r e under c o n s t r u c t i o n and test, so w i t h i n t h e n e x t y e a r we w i l l be a b l e t o try t h e method i n a r o u t i n e f a s h i o n o n real d a t a .
REFERENCES
/1/ Jaynes, E.T. i n The Maximum Entropy Formalism, Ed. R.D. Levine and M. Tribus, M.T. P r e s s (1978) 15.
/2/ Jaynes, E.T., Phys. Rev. 106 (1957) 620.
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