PMC

Smoothed mandible surfaces using three different techniques with the same bandwidths. They are all expected to be the solution of isotropic diffusion. The x, y and z surface coordinates are treated as functional measurements on the original surface and smoothed. The proposed heat kernel smoothing is done with various bandwidths, σ = 0.5, 20, 50, 100. Iterated kernel smoothing performs iterative kernel smoothing with heat kernel approximated linearly with Gaussian kernel (). Diffusion smoothing directly solves the diffusion equation using the same FEM discretization (). Diffusion smoothing and heat kernel smoothing converge to each other as the bandwidth increases.

A typical solution of C-SVDD when a kernel function is used. The C-SVDD finds a more flexible solution in a high-dimensional feature space without mapping data into the feature space using some kernel function; C-SVDD finds a flexible solution directly in the original input space with a kernel function as shown in the left figure.

The crosses show the full solution of . The dashed line is obtained by taking the value of the capacity obtained with full solution at , and multiplying it by the scaling of the kernel variance . Full dots show the value of capacity obtained with the full solution and the contribution of the kernel variance factored out.

Shows the two particle curving solution φ̇_t = υ_t ◦ φ_t, υ_t(·) = Σ_i=1,2 k(·, φ_t(x_i))p_t(i). Left two panels show particles coming together along geodesics they turn “north” together; kernel σ = 3.5. Right two panels show two particles passing each other along geodesics are attracted within the zone of attraction of the kernel width, curving the trajectories; kernel σ = 3.5