Table 1.

Level set distribution model (LSDM) and point distribution model (PDM)

	LSDM	PDM
Shape representation	{Ψ1, Ψ2, …, Ψn}, Ψi : Nd elements	{X1, X2, …, Xn}, Xi : dL elements
Mean shape	$\stackrel{-}{\Psi }=(1∕n){\Sigma }_{i=1}^n{\Psi }_i$	$\stackrel{-}{X}=(1∕n){\Sigma }_{i=1}^n{X}_i$
Mean offset matrix	QΨ	QX
SVD	$Q_{\Psi }=U_{\Psi }{\Sigma }_{\Psi }{V}_{\Psi }^{\mathrm{T}}$	$Q_X=U_X{\Sigma }_X{V}_X^{\mathrm{T}}$
Parametric model	Ψ̃ = Ψ̄ + UΨ,kαΨ	X̃ = X̄ + UX,kαX
Probability of shape	$p\left({\alpha }_{\Psi }\right)=\frac{1}{\sqrt{{\left(2\pi \right)}^k\left|{\Sigma }_{\Psi ,k}\right|}}\cdot \exp \left[-\frac{1}{2}{\alpha }_{\Psi }^{\mathrm{T}}{\Sigma }_{\Psi ,k}^{-1}{\alpha }_{\Psi }\right]$	$p\left({\alpha }_X\right)=\frac{1}{\sqrt{{\left(2\pi \right)}^k\left|{\Sigma }_{X,k}\right|}}\cdot \exp \left[-\frac{1}{2}{\alpha }_X^{\mathrm{T}}{\Sigma }_{X,k}^{-1}{\alpha }_X\right]$