Density-Equalizing Maps for Simply Connected Open Surfaces

Abstract

In this paper, we are concerned with the problem of creating flattening maps of simplyconnected open surfaces in R 3 . Using a natural principle of density diffusion in physics, we propose an effective algorithm for computing density-equalizing flattening maps with any prescribed density distribution. By varying the initial density distribution, a large variety of mappings with different properties can be achieved. For instance, area-preserving parameterizations of simply-connected open surfaces can be easily computed. Experimental results are presented to demonstrate the effectiveness of our proposed method. Applications to data visualization and surface remeshing are explored.

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

“…Figs. 5ce show the results generated at the non-convex corner of the L-shaped domain by the density-equalizing map (DEM) [28], the shape-prescribed density-equalizing map (SPDEM) [27] and the proposed DERM method. For both DEM and SPDEM, the density was set to be the face area for computing area-preserving flattening maps.…”

“…We consider zooming into the non-convex corner of the L-shaped domain as illustrated in a. In b, c, d, e, the mapping results at that region produced by the ALS mapping method [23], the density-equalizing map (DEM) [28], the shape-prescribed density-equalizing map (SPDEM) [27] and the proposed DERM method are respectively shown. It can be observed that our method is advantageous in adjusting for the geometric variability of the carotid surfaces without causing any overlaps.…”

“…Recently, Choi and Rycroft [27] developed a method for computing surface flattening maps by extending a technique for cartogram projection called the density-equalizing maps [28]. Although the shape-prescribed density-equalizing map method [27] is successful in producing an area-preserving map from a 3D surface onto a 2D domain, the bijectivity of the map is not guaranteed if the target 2D domain is non-convex.…”

“…Recently, Choi and Rycroft [27] developed a method for computing surface flattening maps by extending a technique for cartogram projection called the density-equalizing maps [28]. Although the shape-prescribed density-equalizing map method [27] is successful in producing an area-preserving map from a 3D surface onto a 2D domain, the bijectivity of the map is not guaranteed if the target 2D domain is non-convex. In this work, we developed and validated a novel method called density-equalizing reference map (abbreviated as DERM) for computing area-preserving flattening maps of carotid surfaces onto the 2D nonconvex L-shaped domain, using an improved formulation of density-equalizing maps.…”

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.…”