Refinement of the Coherent Point Drift Registration Results by the Example of Cephalometry Problems

Personalizing the three-dimensional atlas model (template) of an organ under analysis based on the data of patient’s three-dimensional examination is an important task for modern digital medicine. The template should be represented by a set of points Y and marked with the key points for the model (landmarks). The template is personalized by the non-rigid registration of the set Y with the set of points X that represents patient’s tomogram. Presently, the coherent point drift (CPD) is the most popular method for solving the problem of non-rigid alignment. In this paper, we propose and explore an approach that substantially improves the CPD result in the problem of automatic registration of cephalometric points (CPs). The proposed algorithm remains robust in the presence of significant skull deformations. Traditionally, 3D cephalometry uses the geometric descriptor of a CP, which refines the position of the point on the bone surface. However, the result of applying the descriptor depends significantly on the accuracy of its anatomical basis reconstruction. The proposed approach solves this problem by clarifying the basis of geometric descriptors, the supporting elements of which are orbital planes and the Frankfurt (orbital-ear) horizontal. For this purpose, additionally marked points of the orbitals (YO) are included into the template Y. Once Y and X are aligned by the CPD method, the plane positions of the orbitals are refined by solving a linear regression problem on the subsets of YO for the left and right orbitals. Cases of refinement with and without use of Tikhonov regularization (ridge-regression) are analyzed. The effect of the increase in the cardinality of the set X relative to Y on the registration accuracy is investigated. It is found that the condition |X| < |Y| has a negative effect on the accuracy, which increases when the cardinality of X relative to Y decreases. The refinement of CPs by the geometric descriptor is carried out on tomogram data in the region around CPs that is found by the CPD method. The dimensions of this region along three coordinates are determined by the anatomical domain of a particular CP descriptor. The quality of the algorithm is measured by the Euclidean distance between hand-marked and automatically found points. The template Y for the algorithm is built on a trauma-deformed skull. The algorithm is quantitatively verified by registering the CPs of orbitals and cheekbones from the data of four tomograms for the deformed skull. A key feature and source of high accuracy of the approach is the use of linear regression with Tikhonov regularization to refine it. As a result, 81.25% of the points found fall within a radius of 2 mm from the hand-marked points and 100% of the points fall within a radius of 4 mm.