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Aug 12, 1995 - 2.4. Related Work. The ideas of pattern theory were proposed by Grenander in 1976 at a scientific meeting in Loutraki, Greece. This was only a ...
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PlUG~l?-1995

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FROM

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P., 01 form Approved

REPORT DOCUMENTATION PAGE

OMB No. 0704-0188

1

PVOH TOO«" « Burden 'or Ihn g »no f«ve*inq thPäoll«iiOn 0' tMOf-««*w >nd 0} samples from the posterior density Trt(xt(M)) defined over the full parameter space Xt as follows. To simplify the following analysis we introduce some additional notation. Define a deletion operator p for deleting elements from the present configuration such that pf' deletes the jth track while pg removes the last track segment of the jth track, i.e., p{j]

:

(X(3)xK3x^)i2-m=1

xKM-> (M(Z) x$3xi) KLM"

pf

:

(M(3) X$3XA) ^m=1

' x N^-1 ,

' x Nw (.M(3)xSR3x^)^m=1

}

xHM

Also, ©j stands for the addition of elements in the present configuration at jth track, i.e. xt(M) ®j yt(l) represents an M + 1 track configuration formed by adding yt(l) to xt(M) at the jth location. Similarly xt(M) ®j y^ signifies addition of a segment to the jth track of xt{M). 4.2.1

Jump Process

The jump process deals with the assignment of tracks and the choice of model order in two ways. First, the individual tracks are developed by probabilistically placing the track-segments sequentially in the associated track configuration. Secondly, the jump process moves among the subspaces of variable numbers of tracks via the addition and deletion of tracks. For hypothesizing the existence of new tracks and the disappearance of faulty track hypotheses as well as growing and shrinking tracks, we use classical ideas associated with birth-death processes, where a birth corresponds to newly hypothesized track/track-segment in the scene and a death the removal of one. These births/deaths are two of a family of simple jump moves, others corresponding to splitting and fusion of tracks, with the simple moves transforming one model to another. The jump transformations are applied discontinuously and drawn probabilis28

tically from a rich family of transformations. The jumps take one model into another satisfying the condition that given any two models of the dimensions M, M', it should be possible to find a finite chain of transitions leading from one to the other. We allow only those jump moves which result in the following types of transformations through parameter space Addition of track : xt(M)

-*

xt(M) ©j j/i(l),

Addition of track-segment : xt(M)

-*

xt(M) ©j y^ ,

Deletion of track : xt(M)

-»•

p^xt(M) ,

Deletion of track-segment : xt(M)

-+

p%'xt(M) .

It should be noted that the addition of only unit length tracks is allowed. Let Tx(xt(M)) be the set of configuration types that can be reached from xt(M) in one jump move, i.e. T\xt(M)) = {xt(M) ©j yt(l),£t(M) ®j y^,p{^xt(M),pfxt(M)}, Xt(Tl(xt{M))) being the space containing the configurations of these types. The discrete jump moves are performed on the basis of following jump parameters defined for a, b € A* as: • q(a, db) : the transition measure from the configuration a to an infinitesimal neighborhood of b. • q(a) : the intensity of jumping out of configuration a, q(a) = JXt^rl^q(a,db). • Q(a1db) : the transition probability from a to db. Q(a,db) = j—

' iadb))'

The transition measures for the feasible jump moves are given by

q(xt(M),dyt(M

+ l))

=

£ ?r(^t(M),y(M + l))SMM){d(p^y(M + l)))dy(l) , i=i M

q(xt(M),dyt(M)) = £q&t(M),MM))6gtm(d(p{£yt(MJ))dy 3=1 M

+

^qj(St(M),MM))6pu)St{M)(dyt(M)) , i=i M

q{xt{M\dyt{M-l)) = J24^t(M),yt(M-l))Spu)MM)(dyt(M-l)) i=i

29

(15)

and the intensity of jumping out of xt(M) is given by Af+l

q(xt(M))

=

J2 [

qbT(xt(M),xt(M)®Jm))d(m))

M

M

.

qbs(xt(M),xt(M)®yM)dy

W M

+ J24{UM),P^St(M)).

(16)

The birth/death intensities 9x,?y, 95,95 can be derived from the posterior measures of the present and the candidate configurations in two ways. One is analogous to a Gibbs sampling type algorithm while the other is analogous to a Metropolis type acceptance/rejection algorithm. The intensities obtained from the first method are given by, • Gibb's sampling: qbT(xt(M), xt(M) @j yt(l)) =

c-^A2(*) ^ e^A2(fc)]r

(

j =

^)

where X\(k) = ircos( 0} .

(21)

m=0

Then the infinitesimal generator for diffusion Ad becomes Adf(x(M)) = -\ (V2£(£(M)) o V2/(£(M))) + \ £ y^ffiff , V/ € V(A)

where TLM — J2m=ii

tQ

tal number of track-segments in the parameter set x(M), o stands for

the vector dot-product and the gradients V2E(x(M)), V2/(f(M)) are w.r.t the position vector p(M). Substituting this expression in the integral condition we get (Adf(x(M))n{x(M))dx(M)

= 0-E(x{M))

- J -(VE(x(M)) o V/(£(M)) +

[l(3^d>f(x(M))\e-^M)

38

dx(M)

Integration by parts of the second term, with the fact that the function / vanishes at the boundary, results in a terra which is negative of the first term. Therefore the given posterior 7r(s(M)) is the stationary density of the diffusion process. The generator AJ of a jump process is given by the expression, AJf(x(M)) = q(x(M)) f

Q(x(M),dy)(f(y) - f(x(M))) .

When substituted in the stationarity condition, it provides q(x(M))n(x(M))dx(M) = /

q(y,dx(M))Tr(y)dy ,

JX(T-i(x(M)))

which is often called as the detailed balance condition. Therefore, the jump parameters should satisfy this equation for the density ir(x(M)) to be the stationary density of the jump process. We will prove this condition assuming only birth/death of tracks, the treatment for birth/deaths of track segments being similar. Substituting for the transition measures in the detailed balance condition and simplifying we obtain,

Af+i

.

ir(x(M))dx(M)[ T / M+l

dx(M)[J2

M

qU^M),£(M)®jy(l))dy[l) + Y/q^(x(M),p^x(M))}

,

qUS(M)®jy(l),x(M))n(x(M)®jy(l))dy\l)

M

+ J2qbT(p?S(M),x(M)Hp^x(M)) }.

(22)

Substituting the values for q^q^ from 17, and analyzing only the jth terms from sums on both sides, R.H.S.

^r(l)

JXi 4(M + 1) + J_c-[i(*(^))-i(P^(W))J+c-i'(^(l))ff(pÜ)f(M))l

1 dx(M)L_ r / e-mM))e-p(m)®jm)dv(D 1 yy } i)) 4(M +1) JQ> (Z)(ZT(

1 f e-L(S(M)®:y(l)) e-PWM)®jm)d$(l) 4(M i + 1)JQ
= X(l) f]{y(l): L(x(M)) > L(x(M) 0; y(l))} fi< = *(1) f){y(l) • £(W) < I(f(M) @j y(l))} L.H.S. d