Area-Preserving Mapping of 3D Carotid Ultrasound Images Using Density-Equalizing Reference Map

Abstract

Carotid atherosclerosis is a focal disease at the bifurcations of the carotid artery. To quantitatively monitor the local changes in the vessel-wall-plus-plaque thickness (VWT) and compare the VWT distributions for different patients or for the same patients at different ultrasound scanning sessions, a mapping technique is required to adjust for the geometric variability of different carotid artery models. In this work, we propose a novel method called density-equalizing reference map (DERM) for mapping 3D carotid surfaces to a standardized 2D carotid template, with an emphasis on preserving the local geometry of the carotid surface by minimizing the local area distortion. The initial map was generated by a previously described arclength scaling (ALS) mapping method, which projects a 3D carotid surface onto a 2D non-convex L-shaped domain. A smooth and area-preserving flattened map was subsequently constructed by deforming the ALS map using the proposed algorithm that combines the density-equalizing map and the reference map techniques. This combination allows, for the first time, one-to-one mapping from a 3D surface to a standardized non-convex planar domain in an area-preserving manner. Evaluations using 20 carotid surface models show that the proposed method reduced the area distortion of the flattening maps by over 80% as compared to the ALS mapping method.

Parallelizable global quasi-conformal parameterization of multiply-connected surfaces via partial welding

“…Therefore, it is natural to ask how we can reduce the area distortion of the quasi-conformal parameterization without altering the Beltrami coefficient µ. Given a multiply-connected mesh S = (V, F) and the global quasi-conformal parameterization Φ : S → R 2 obtained by our PGQCM algorithm, we define the area distortion of Φ on a triangle face T ∈ F as follows [8,10]: Specifically, the numerator of d Area (T ) measures the ratio of the area of T to the surface area of S, the denominator measures the ratio of the area of Φ(T ) to the total area of Φ(S), and d Area (T ) is the logged area ratio. Note that d Area (T ) ≈ 0 indicates that the area distortion is small, while a large d Area (T ) indicates that the area distortion is large.…”

“…Existing area-preserving parameterization methods include the locally authalic map [22], Lie advection [85], optimal mass transport (OMT) [21,26,84], density-equaling map (DEM) [8,15] and stretch energy minimization (SEM) [77]. Although the area structure of the input surface can be well-preserved by these methods, the angle structure is usually significantly distorted.…”

Tooth morphometry using quasi-conformal theory

“…To simplify the mapping problem, we begin with flattening S i and S j onto the plane. While there exists other flattening methods such as area-preserving maps [27,28], conformal parameterizations are preferred in our case as they preserve the Beltrami coefficient and hence the conformal distortion under compositions. Following the approach in [21], we compute two conformal maps g i : S i → R i and g j : S j → R j that flatten S i and S j onto two rectangular domains R i , R j on the plane.…”

Adaptive area-preserving parameterization of open and closed anatomical surfaces

“…Therefore, some recent works have focused on the computation of area-preserving parameterizations for genus-0 closed surfaces [25]- [27] and simply-connected open surfaces [28]- [30]. Furthermore, area-preserving parameterizations have been found useful for biomedical visualization [31]- [33] as particular regions of biomedical structures will less likely to be shrunk under area-preserving mappings. More recently, a few works have considered the parameterization of biomedical surfaces onto other target domains.…”