813 
Tri'-, -r- v r- « 
wî®^3 
METHOD OF BOUNDARY EXTRACTION BASED ON SCHRODINGER EQUATION 
Liantang Lou a,b ’ *, Xin Zhan b , Zhongliang Fu a , Mingyue Ding c 
a School of Remote Sensing and Information Engineering, Wuhan University, Wuhan, China, 430072 - 
louliantang@ 163.com, fuzhl@263.net 
b School of Science, Wuhan Institute of Technology, Wuhan, China, 430073 - hmcge@163.com 
c Institute for Pattern Recognition and Artificial Intelligence, Key Laboratory of Education Ministry for Image 
Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan, China, 430074 - 
mding@imaging.robarts.ca 
PS-44, WgS V/6 
KEY WORDS: Boundary Extraction, Schrodinger Equation, Schrodinger Transformation of Image 
ABSTRACT: 
To overcome the main drawbacks of global minimal for active contour models (L. D. Cohen and Ron Kimmel) that the contour is 
only extracted partially for low SNR images, Method of boundary extraction based on Schrodinger Equation is proposed. Our 
Method is based on computing the numerical solutions of initial value problem for second order nonlinear Schrodinger equation by 
using discrete Fourier Transformation. Schrodinger transformation of image is first given. We compute the probability P(b,a) that a 
particle moves from a point a to another point b according to I-Type Schrodinger transformation of image and obtain boundary of 
object by using quantum contour model.. 
1. INTRODUCTION 
Deformable models based on particle motion in classical 
mechanics, also called snakes, or active contour models, were 
first proposed by Kass and Terzopoulos in 1987. Since then, a 
variety of improvements have been made such as balloon 
models (L.D.Cohen,1991), geometric model (V.Caseles, 
F.Catte,T.Coll, and F.Dibos, 1993), as well as the topology 
adaptive deformable model (T.McInemey and D.Terzopoulos, 
1999). In 1997,Cohen and Kimmel described a method for 
integrating object boundaries by searching the path of a 
minimal active deformable model’s energy between two points. 
Lou and Ding used point tracking by estimating the probability 
of a particle moving from one point to another in quantum 
mechanics, and did not impose any smoothness constraints to 
ensure the extraction of the details of a concave contour 
(Liantang Lou and Mingyue Ding, 2007). Feynman and Hibbs 
had used path integration method to count the kernel of the 
particle In this article, the probability of a particle moving from 
one point to another is directly computed according to the 
relation between the kernel K(b,a) and image gradient G(x) 
(see Figure 1) by using discrete Fourier Transformation. 
Figure 1. The relation between the probability P(b,a) and image gradient G(x) 
2. RELATION BETWEEN THE PROBABILITY AND 
IMAGE GRADIENT 
The active contour model or Snake model had their profound 
physical background. If the parameter s in the deformable 
contour curve x(s) = (x(s), y(s)) could be understood as time t, 
object contour curve x(t) could be considered as the path of the 
particle in plane motion. 
Suppose a particle moves from the position a at the time t a to 
the position at the time t h ,e.g., a = x{t a ), b = x(t b ). According 
to the theory of quantum mechanics, the probability of a 
particle moving from the position a to b at t b , denoted by 
P(b,a), is dependent on the kernel K{b,a), which is the sum of 
all paths contribution between x fl and x A , i.e., 
K(b,a)= £<*(*(')), (1) 
R(a,b) 
Corresponding author, louliantang@163.com.