The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
If the matrix F + a|y| 2 can be diagonalized, that is, 
V + a|y| 2 = P~ l DP , then 
u(y) = P-'Diag{e ud ', e - ud \:.,e- ud -)Pv{y) (13) 
where D = Diag(d i ,d 2 ,---,d mn ) was diagonal matrix. Eq.(13) 
degenerates into Eq.(7) when v(x) = 0 . The discrete Schrodinger 
transformation of image <p(x) based on v(x) can be obtained 
using Eq.(3). 
4. BOUNDARY EXTRACTION BASED ON 
SCHRODINGER TRANSFORMATION OF IMAGE 
Schrodinger transformation of image can be applied to image 
processing and analysis, such as, boundary extraction, edge 
enhancement, image inpainting, image restoration,etc. We 
extract boundary of object according to the approach given in 
( Liantang Lou and Mingyue Ding, 2007 ) . The steps are 
listed as follows: 
(1) Compute discrete Schrodinger transformation of gradient 
image G(x), denoted by u(x,t 0 ), where t 0 is a small positive 
constant. 
(2) Compute the probability P(x) = u(x,t 0 ). 
(3) Estimate the expectation position of the particle at the next 
time t a + At by: 
¿>€S 
6eS 
where the set S are possible positions after the interval time At 
when a particle moves from a starting position a at the time t a . 
From Eq. (14), the next boundary point position is estimated. 
The whole boundary is tracked by repeating the expectation 
position calculation procedure iteratively. 
5. EXPERIMENTAL RESULT 
The following experiments (see Figure 2) show the meaning 
and function of Schrodinger transformation of image, that is, 
Schrodinger transformation of image can be seen as the result 
of original image shrinking inside and spreading outward, like 
as interference wave. The bigger at in Eq.(7) is, the more 
obvious the interference is. If we estimate the probability of a 
particle appearing in some point using Schrodinger 
transformation of gradient image, we can obtain the same 
conclusion given in the article (Liantang Lou and Mingyue 
Ding, 2007, Figure 1). The quantum contour model produces a 
contour around the true object boundary with the jagged 
particle trajectory while the deformable model produces a 
smooth but biased contour. Though quantum contour is zigzag 
and the contour extracted using deformable model was quite 
smooth, quantum contour has smaller system deviation. 
The quantum contours of object are given in Figure 2. Figure 2 
show that the contour with Schrodinger transformations is 
smoother than the contour without using Schrodinger 
transformations. 
(a) (b) (c) 
Figure 2. Schrodinger transformation of image 
(a) The original image, (b), (c) are Schrodinger transformations of image. The constant at is 0.0005, 0.001, respectively. 
(a) (b) (c) 
Figure 3. contours extracted by using the quantum contour based approach. 
(a) The original image, (b) quantum contour without using Schrodinger transformations(Liantang Lou and Mingyue Ding, 2007), (c) 
quantum contour based on Schrodinger transformations with the constant at is 0.000001. 
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