The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
^„,( v i)=( Q: /| v i ~ v h| 2 +$| v m ~ 2v i +v w | 2 )/2 (4) 
As external forces, Ei mage and E con are determined by image in 
formation and user definition, typical external force is: 
E m =-| V/M 2 
E exl =-\v(Gfx,y)*l(x,y)f (5) 
Esnake can be acquired after Snake initialization. Potential en 
ergy is transformed to kinetic energy, and consumed by next 
energy. Snake energy is minimized and drew to more stable 
status. That is to say, the key problem is to calculate the mini 
mum energy. 
2D curve on images can be represented in parametric form as 
i=0 
/=0 
where X, and Y t are the coefficients of the B-spline curve in x 
and y direction repectively. Nfs) is the normalized 3th B- 
spline between knots 5, and S i+4 (Hongwei zhagn,2004). 
3. GENERALIZED POINT PHOTOGRAMMETRY 
Traditional photogrammetry is based on feature points, and 
points in photogrammetry means only physical or visible points, 
such as dots, crosses and comers (Zuxun Zhang, Jianqing 
Zhang, 2005). According to camera model, the collinearity 
equations are: 
.. .. f ci i {X~X s )^(Y-Y s ) + c x (Z-Z s ) 
° a.iX-X^+byY-Y^+cyZ-Z,) 
^(X-X^+byY-Y^jZ-Zy 
70 a,(jr-X s )+6,(r-r s )+c,(Z-Z s ) 
where x and y are the observations, 
X, Y and Z are the coordinates of ground point, 
, a,-, b h Cj (i=1,2,3) are the orientation matrix composed of 
rotation angles q>, oo and k, where Y-axis is taken as the 
primary axis. 
X s , Y s , Z s , (p, co, k, f x 0 , y 0 are the exterior and interior 
parameters. 
In fact, the equations need to be linearized during calculating. 
Usually, a space curve can be parameterized as 
x = /(,) 
• Y = g(t) (a<t<b) (9) 
Z = h(t) 
where f(t), g(t) and h(t) are the locus of points on the space 
curve as a function of curve parameter t, ranging from a to b. 
Incorporating equation (9) into equation (7) and (8) leads to: 
.. ■■ ^q(/t0--y,)+A(g(0-y i )+q(^)-Z t ) nm 
** \f(p-Xj,+bigg)-Yj)+cjm-Z s ) 
f ch(f{t)-X s Vhl^)-Y s Vc 2 {m-Z s ) 
Figurel. curve reconstruction with generalized point 
photogrammetry 
Suppose the tangential vector of a point on the observed image 
curvéis a (as shown in figúrela), equation (10) is used for 
exterior orientation and 3D reconstruction if 45° Si a ^ 135° 
or 225° a ^ 315°, otherwise equation (11) is used. The 
above model can be used for reconstruction of space curves. 
For a space curve, its disparity between the observed image 
feature and the projected space feature is shown in figure la. 
The disparity becomes smaller and smaller during iterations, 
and usually converges within several iterations (as shown in 
figure lb). A space curve is photographed in at least two stereo 
images, the image parameters and the model of the curve may