check an equation of curvature in 2D image?
please i have a question about an equation given in this article: Curvature and Bending Energy in Digitized 2D and 3D Images). paragraphe 3. Isophote curvature in 2D
he said :
f(x, y) be a grey-value image and fx and fy respectively the derivatives in the x and y-direction
g = ( fx , fy ) (5)
c = (- fy , fx ) (6)
Ө= arccos(- fy /|| g|| ) = arcsin( fx/|| g|| ) = arctan(- fx/ fy ) (7)
||g|| = sqrt (fx^2 + fy^2)
To differentiate along the curve with respect to the arc length we use the operator
d/ds=cosӨ *d/dx+ sinӨ *d/dy= -fy/||g||d/dx+fx/||g||d/dy (8)
Applying eq. (8) to eq. (7) we get
dӨ/ds=-(fxx fy^2-2 fx fy fxy+fyy fx^2)/(fx^2+fy^2)^3/2 (9)
i want to know did he got equation (9) from equations (8) and (7) (step by step).
thanks an advance.