The density-equalizing map, a technique developed for cartogram creation, has been widely applied to data visualization but only for 2D applications. In this work, we propose a novel method called the volumetric density-equalizing reference map (VDERM) for computing density-equalizing map for volumetric domains. Given a prescribed density distribution in a volumetric domain in $\mathbb{R}^3$, the proposed method continuously deforms the domain, with different volume elements enlarged or shrunk according to the density distribution. With the aid of the proposed method, medical and sociological data can be visualized via deformations of 3D objects. The method can also be applied to adaptive remeshing and shape modeling. Furthermore, by exploiting the time-dependent nature of the proposed method, applications to shape morphing can be easily achieved. Experimental results are presented to demonstrate the effectiveness of the proposed method.

中文翻译：

---

具有应用程序的体积密度均衡参考图

密度均衡地图是一种为创建制图而开发的技术，已广泛应用于数据可视化，但仅适用于 2D 应用程序。在这项工作中，我们提出了一种称为体积密度均衡参考图 (VDERM) 的新方法，用于计算体积域的密度均衡图。给定 $\mathbb{R}^3$ 中体积域中的规定密度分布，所提出的方法不断变形域，根据密度分布放大或缩小不同的体积元素。借助所提出的方法，可以通过 3D 对象的变形来可视化医学和社会学数据。该方法还可以应用于自适应重新网格划分和形状建模。此外，通过利用所提出方法的时间相关性，可以轻松实现形状变形的应用。