Vlxx7zz 7xz) + 2Ix7z Vlyz7xy IyyIxz)
Ix VIyy Izz7 2IyIzIyz
+Iy2VIxx + Izz )-2IxIzIxz
where Ix indicates the partial derivative of the image data with respect to x, and Ixz indicates the mixed partial derivative with respect to x and z, etc. and h = I2 + I2 + I2 .
The maximum (Kmax) and minimum (Kmin) principal curvatures of a surface at a point P are given by and
They can be thought of as the reciprocal of the radii of the smallest and largest circles, respectively, that can be placed tangent to P. The greater the curvature, the smaller the osculating circle; the less the curvature, the flatter the surface and the larger the circle.
Equations (1) and (2) require first- and second-order partial derivatives, which are known to be susceptible to noise. To ameliorate this problem, filters can be used to smooth the data and reduce the undesirable effects of noise. Monga and Benayoun used the 3D Deriche filters to both smooth and compute derivatives . The functions jo,/1; j2 are used to smooth and compute first- and second-order partial derivatives, respectively:
These functions are applied to the 3D image data as convolution filters. For efficiency, the functions are applied only to points lying on the desired isosurface within the image. The required partial derivatives are computed using, for example,
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