Clustering by finding valleys of densities
Graph based implementation
PDE based valley seeking
PDE: Partial Differential Equation
PDE's can be used to minimize energy functionals
- Simple Example
Consider 1-D curves
$ u(x): [0,1] \rightarrow \Re $ with fixed end points $ u(0)=a $, $ u(1)=b $ (3-1)
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Suppose Energy of curve is $ E(u)=\int _0 ^1 F(u, u')dx $ for some function $ F: \Re ^2 \rightarrow \Re $
e.g.) $ F(u,u')=|u'|^2 $ (3-2)
The curve that minimizes (or maximizes) E(u) satisfies Euler Equation
$ \frac{\partial F} {\partial u} -\frac{d}{dx}(\frac{\partial F}{\partial u'})=0 $ (3-3)
sometimes written as $ E'=0 \Rightarrow \frac{\partial E}{\partial u}=0 $ (3-4)
Similarly if $ E=\int _0 ^1 F(u,u',u'')dx $ (3-5)
e.g.) $ F(u,u',u'')=|u''|^2 $ (3-6)
Then Euler equation is $ \frac{\partial F}{\partial u} - \frac{d}{dx}(\frac{\partial F}{\partial u'}) + \frac{d}{dx^2}(\frac{\partial F}{\partial u''})=0 $ (3-7)
Similarly, for surface in $ \Re ^2 $
$ u(x,y): [0,1] \times [0,1] \rightarrow \Re $ (3-8)
Suppose energy is given by
$ E(u)=\int _{surface} F(u,u_x, u_y, u_{xx}, u_{xy},u_{yy})dxdy $ (3-9)
e.g.) $ F={u_x}^2+{u_y}^2 $ (3-10)
Then Euler equation is
