The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
where R(a,b) is the set of all paths between x a and x b . ^(x(t)) 
is the contribution of a path x(t) with a phase proportional to its 
energy E(x(t)), i.e., 
</>(x(t)) = Ce l2Milhmm) , (2) 
where h is the Planck’s Constant, C is a constant, and 
E = £^y|x'(0| 2 -V(x,t)jdt is the energy functional of path. The 
probability from point x a at time t a to point x b at time t b is the 
square of absolute value of kernel K(b,a) from x o to x b , that 
is, 
P(b,a)=\K(b,a)\ 2 . (3) 
3. SCHRODINGER TRANSFORMATION OF IMAGE 
We must face problem counting the kernel K(b,a) to introduce 
law of particle motion in quantum mechanics into image 
processing and analysis. For a system with a simple Lagrangian 
function, K(b,a) can be calculated directly from the path 
integral (see 0) while for a system with a complex Lagrangian 
function, it is difficult and time-consuming to estimate the value 
of P(b,a) from K(b,a). In order to avoid such difficulty, Lou 
and Ding estimate the probability of a particle moving from 
point 'a to point b directly from specific particle models. In the 
paper, We’ll compute the probability P(b,a) that a particle 
moves from a point a to another point b using Schrodinger 
transformation of image. 
Replacing the kernel K(b,a) with the wave function u(x,t) in 
the position x at the time t, then u(x,t) satisfied the following 
Schrodinger equation: 
where the mark ’ * ’ denotes convolution of two functions, ‘ A ’ 
denotes Fourier transformation of function. When v(x) = 0, both 
u(x,t) and u(y,t) have the following analytic solutions (L. C. 
Evans, 1998): 
u = e~ a " M <p(y) (7) 
u(x,t) = —i—\\ e *^ <P(y)dy, xe/? 2 ,(>0 (8) 
4xait JJ , 
When v(x)*0 , u(x,t) and u(y,t) also have analytic solutions 
(L. C. Evans, 1998), but they are quiet complex so that they 
could not be used to compute their numerical solutions. We 
give the following definition of Schrodinger transformation of 
image because of Eq. (5): 
Schrodinger transformation of image cp(x) based on v(x) was 
defined as the solution of Eq. (5). And the transformation is 
called I-Type Schrodinger transformation when v(x) = 0 , 
otherwise the transformation is called II-Type Schrodinger 
transformation. Supposed both<p(x) and v(x) are mxn images, 
then two-dimensional discrete Schrodinger transformation of 
image (p{x) based on v(x) is expressed with the following 
differential equation which its Fourier transformation satisfies: 
\i-u,=(V + a |yf)« (9) 
1 «|,-o = 9 
where u, is mn -dimensional column vector formed by 
concatenating all the rows of mxn matrix u,. mn*mn matrix 
|y| was diagonal matrix whose diagonal elements express 
distance, mnxmn matrix V is a block cyclic matrix, i.e., 
h 2 f d 2 u dV) 
2m (Sx 2 dy 1 ) 
V(x,t)u(x,t), 
(4) 
K 
K-2 - KJ 
where h =/j/2;t = 1.054x 10“ 27 erg s . 
We could rewrite Eq.(4) as the initial-value problem: 
where V, is a cyclic matrix, 
iu, + aAu = v(x)u 
«L =P(*) 
(5) 
By applying Fourier transform to equation (5) and making use 
of the properties of Fourier transform, we have 
' v(/,0) v(i,n — 1) ••• v(,',l) N 
v(/,l) v(/,0) - v(i, 2) 
K v(i,n-1) v{i,n- 2) ••• v(/,0). 
[/'•«,-a|y| 2 M = v(y)*w 
| «|,=o=0(y) 
(6) 
Obviously, the solution of Eq.(9) is 
«(y) = e~ i,<y+alyl V(y) 
(10) 
(11) 
(12)