The adaptive algorithm for track finding

Radiation Measurements, Voi. 25, Nos 1-4, pp. 761-764, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 1350-4487/95 $9.50 + .00

Pergamon

1350.,4487(95)00244-8 THE ADAPTIVE ALGORITHM FOR TRACK FINDING

N. D. DIKOUSSAR Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia

ABSTRACT A new fast and stable with respect to errors and backgrounds recurrent algorithm for track finding is suggested. The algorithm realizes a family of digital adaptive projective filters (APF) with known nonlinear weight functions: projective invariants. APF can be used in adequate control systems for processing and compression of data, including tracing for contours and track segments. KEYWORDS Track finding; image processing; pattern recognition; digital adaptive filters; projective invariants; least square fitting. CROSS-RATIO FUNCTIONS AND "4-POINTS" TRANSFORMATIONS Track finding (TF) is the part of the classic problem of pattern recognition and digital image processing (Duda and Hart, 1973), wich consist in detecting curves (contours) with gaps and backgrounds. The mathematical formalism of APF (adaptive projective filters) has been described in (Dikoussar, 1991, 1994). In this paper the new approach to solving of the TF problem is suggested. The TF process is considered as a model of linear system (Ljung, 1987) with known nonlinear weight functions: cross-ratio (CR) functions or projective invariants. CR-functions are defined by the special algorithm (Dikoussar, 1994) of the cross-ratio of four collinear points. For four points on the real axis 0X with coordinates X0, )(1,)(2 and X we get the pair of triplets CR-functions pi(A) and di(/k), where (A) - (A,L,T) = ( X 1 - X o , X 2 - X o , X - X o ) ,

i = 1,2,3.

Formulae of these functions are given in the Table 1. Table 1. Formulas of CR-functions CR-functions pi(A, L, T)

di(A,L, T)

25:114-YY

i=l L~

(f-A)(L-A)

-~ff-L) A(L-A)

761

i=2

i=3

-a~

~L

~ff-A)

ff-D(~-L) AL

(r-L)(L-A) (r-A)(r-L) L(L-A)

762

N.D. DIKOUSSAR

The system of functions pi(/x), di(A), (i = 1,2, 3) has significant properties. These properties were used for definition of discrete projective (or "4-points') transformations (direct and inverse): 3

Yo = (P,~') = ~_,p~(A,L, r)Yi, and vice versa

3

Y(r) = (D, Z) = ~ di( A,L, r)Zi,

(1)

(2)

i=1

where/6 = (pl,/~,p3)T, /~ = (dl,d2, da)T, ~. = (y1,Y2,y)T, g = (y1,Y2,Y0)T. For X0 = 0 the curve Y = aX 2 + bX + c, (a, b, c - r e a l numbers) is transformed by (1) into Const (Y = c) and eq. (2) yields the parametric notation of the quadratic equation

Y(X) = -~X(X

- L) + - ~ X ( X

- A) + ~°L (X - A)(X - L),

(3)

where M = L - A. The actual form of this curve depends upon the relative position of three points (X0, Y0), (A, Y~), (L, Y2) on the Euclidean plane. All these coordinates are direct measurement parameters of the straight line or quadratic parabola. ALGORITHM OF A P F In most cases, actual data of a track image are sampled to form an image space consisting of I ( , ~ × Jm~ pixels or of discrete coordinates (X, l7") : /4: {(Xk, l/kj)[ k = 1, 2, ..., gm,~; j = 0,1, 2, ..., Nk}, which have a structure similar to TV signals, where Kin,, - a number of rows and Nk numbers of points on k - t h row. We assume the model of the track segment (TS) is defined by the linear-quadratic functions (LQ-model). Eqs.(2) and (3) are based on the LQ-model of TS and have the sructure of the linear regression. Such structures are typical in the theory of systems (Ljung, 1987). In our case eq. (1) is equation of the linear system 3

h(n) = ~ , p i ( n , A , L ) Y ( n -

sl),

(4)

i=1

where Y(n) and h(n) are input and output signals and 8i = n - A, n - L, 0, i = 1,2, 3. The pulse response AL P3= (n-A)(n-L)' nCA, nCL, is the reaction of the system (4) on the input unit signal for discrete values n = 0, 1,2... and fixed A, L. (Functions pl and p2 are created from p3 by means of formal substitutions: A ~ n and L ~ n).(see Table 1). If Y(n) - Y(n0) = AY3 = AY3 + e3 have measurement errors e3 = e(n) with zero means and normal distribution, then following eq. (4), we get the structure of the model of the error equation (Ljung,1987). In this case the signal of the white noise is transformed dynamically through the denominator of the system with the amplitude response p3(A, L, n).

ALGORITHM FOR TRACK FINDING

763

According to (3), the linear or curve track segment is defined completely by its three points (parameters): the basis and two pivot points. These points give us the crossing point ( ~ ) of TS with row X = X0 and the curvature or the slope of the track segment, which can be defined from (3). As the basis point one can choose an arbitrary point on the basis row, while the Yx and 12L are situeted on the poles X -- Xx and X -- XL accordingly (Fig.l). ~"

l 0

'-~iaa~.~":"-;,~,~,

">--__{-3.-~", ~,| i

) --'~':T~{~i{ E ~ ! I ~

X

X~

......

,at

Xo

Fig. 1 Main parameters of the track segment. Shift the coordinate system to the basis point (Xo, ~ ) we reduce the number parameters by 1, simultaneously changing errors (e. = % - e0) :

L=X

-Xo,

Vo=O,

=

aL =

Let A r is a step of discretization of values r. Then for rn = nAr the weight functions pi and dl are defined in discrete points on rows -4-1, +2, ..., + n , ... (n # $, n # L) in relation to the basis n = 0. Using (2) and (5), the L Q - model of a track segment or the predictor, is written in the form = (&,

+

(6)

where L3n = [dl(A,L,n),d2(A,L,n)l T is the vector of regressors, 1),~ = l)'(n)is prediction of V,, n is the number of the row with respect to the basis row and ~) = [0~, OL]T is considered as a vector of unknown^parameters of TS, c(n) is the additive noise. The behaviour of errors of the prediction V, depends upon the choice of parameters A, L and is estimated as follows:

le,,l
N.D. DIKOUSSAR

754

we construct the recurrent algorithm (APF-algorithm) for finding the fourth coordinate of TS on the nv - th row (v - 1 is the number points, which have been included in TS before). r(n ) =

=

~)n~

(A T a

--

~-IATnV V.n V '

~,ALnpJ~nV]

(8)

n~ = +1,+2,...; v = 1, 2, 3, ...; n~ # $,L, where ~),,0 = [0~, OL]T, the index n~ points out that in estimation of parameters ~) were used the values of regressors on n ~ - t h row and the actual coordinate ~',~, for which the residual r(n~) is less than a given threshold of the prediction I r(n

)I<

At the starting steps of the al(~orithm the small changing of the estimates 0~ and OL is tested by the given threshold To : I 0i,,,~ - 0i.... 1 I< To, i = $, L. Elements of the matrix ATA and of vectors/) axe tabulated as look up tables inside the selected window. To define the new ~),~ the accumulated on the previous steps sums are used. Algorithm (8) has a structure of the adaptive algorithm, in which the prediction on the next step uses the 'information state', accumulated before (information feedback). Examples of runs of this algorithm for TS finding in the case of multitrack events are shown in (Dikoussax, 1994). RESULTS The performance of the APF has been realize as the TP5-program for finding the simulated on the image space (100 x 100 pixels) linear and/or quadratic TS with errors, backgrounds and gaps. The algorithm has confirmed the high stability in following for the TS with respect to the wide variations of measurement errors, gaps and backgrounds. Testing the program for TS finding (Dikoussax, 1994), shows that the APF-algorithm keeps the main positive aspects of known methods in solving the track finding problem and has some advantages with respect to traditional ways: both linear and/or quadratic shapes of TS are processed by the same algorithm, the parameterization of the algorithm, use of local and global predictions, the stability of the algorithm to measurement errors and backgrounds, use of the information feedback in the track following procedure, the effective test for rejection of wrong possible tracks at the initial phase, etc. APF satisfy for many requirements of pattern recognition and provide a wide way of elaborating algorithms for different purposses in a digital image processing. REFERENCES Dikoussar N.D.(1991). Discrete projective transformations on the coordinate plane. Math. Model. 3_, 10, 50-64. (in Russian) Dikoussar N.D.(1994). Adaptive projective filters for track finding. Comput. Phys. Commun. 7_9, 39-51. Duda R. and Hart P.(1973). Pattern Classification and Scene Analysis, Wiley, New York. Ljung L.(1987). System Identification: Theory for User, Prentice-Hall, Inglewood Cliffs.