Table 1.

Summary of notation and registration parameters.

	Step	Symbol	Definition
	Images	I0	Moving image (e.g., preoperative CT or MR)
		I1	Fixed image (intraoperative CBCT)
		I|transformation	An image I after a given transformation (e.g., GM rigid, GM nonrigid, distance transformation, Demons)
➀Pre-processing	Model structures	Sk	Segmentation of the volume of interest (VOI) i.e. tongue and hyoid bone in Ik
		$I_k^{\mathit{Mask}}$	VOI in Ik
	Point cloud extraction	$P_k^{\prime }=[{\mathit{x}}_{k,1}^{\prime },{\mathit{x}}_{k,2}^{\prime },\dots ,{\mathit{x}}_{k,3}^{\prime }]$	A set of n surface points extracted from components in Sk (i.e., tongue and hyoid bone)
		Pk=[xk,1,xk,2,…,xk,3]	$P_k^{\prime }$ after normalization.
		${\mathit{x}}_k^{\prime }$	Spatial location of a point in $P_k^{\prime }$ in R3
		xk	${\mathit{x}}_k^{\prime }$ after normalization.
		n	Number of points in Pk
Gaussian mixture registration	Gaussian Mixture (GM)	φ(x|Θk)	Gaussian mixture density of Pk with a set of mixture parameters Θk.
	➁ GM Rigid	${\sigma }_{\mathit{GMRigid}}^{(\ell )}$	Width of the isotropic Gaussian components at each registration level ℓ.
		TGMRigid	GM rigid transformation
	➂ GM Nonrigid	${\sigma }_{\mathit{GMNonRigid}}^{(\ell )}$	Width of the isotropic Gaussian components at each registration level ℓ.
		${\lambda }_{\mathit{GMNonRigid}}^{(\ell )}$	Regularization parameter at each registration level ℓ.
		TGMNonRigid	GM nonrigid transformation
		DGMNonRigid	GM nonrigid displacement field
		GM	Concatenation of GM rigid and GM nonrigid registrations (combined transform of steps ➁ and ➂)
➃ Demons registration	Demons	DT	Distance transformation
		σUpdateField	Width of the Gaussian filter to regularize the update field
		σDisplacementField	Width of the Gaussian filter to regularize the displacement field
		MSL	Maximum update step length
		DDemons	Demons displacement field