A Geometric Method for Accelerated Sphere Tracing of Implicit Surfaces

Sphere tracing is a common raytracing technique used for rendering implicit surfaces defined by a signed distance function (SDF). However, these distance functions are often expensive to compute, prohibiting several real-time applications despite recent efforts to accelerate it. This paper presents a method to precompute a slightly augmented distance field that hugely accelerates rendering. This novel method called quadric tracing supports two configurations: (i) accelerating raytracing without losing precision, so the original SDF is still needed; (ii) entirely replacing the SDF and tracing an interpolated surface. Quadric tracing can offer 20% to 100% speedup in rendering static scenes and thereby amortizing the slowdown caused by the complexity of the geometry.

Medical Data and Mathematically Modeled Implicit Surface Real-Rime Visualization in Web Browsers

Raycasting can display volumetric medical data in fine details and reveal crucial inner imaging information, while implicit surface is able to effectively model complex objects with high flexibility, combining these two rendering modalities together will provide comprehensive information of the scene and has wide applications in surgical simulation, image-guided intervention, and medical training. However, medical data rendering is based on texture depth at every sampling point, while mathematically modeled implicit surfaces do not have geometric information in texture space. It is a challenging task to visualize both physical scalar data and virtual implicit surfaces simultaneously. To address this issue, in this paper, we present a new dual-casting ray-based double modality data rendering algorithm and web-based software platform to visualize volumetric medical data and implicit surface in the same browser. The algorithm runs on graphics processing unit and casts two virtual rays from camera to each pixel on the display panel, where one ray travels through the mathematically defined scene for implicit surface rendering and the other one passes the 3D texture space for volumetric data visualization. The proposed algorithm can detect voxel depth information and algebraic surface models along each casting ray and dynamically enhance the visualized dual-modality data with the improved lighting model and transparency adjustment function. Moreover, auxiliary innovative techniques are also presented to enhance the shading and rendering features of interest. Our software platform can seamlessly visualize volumetric medical data and implicit surfaces in the same web browser over Internet.

Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies

In a recent paper by Iglesias, Rumpf and Scherzer (Found. Comput. Math. 18(4), 2018) a variational model for deformations matching a pair of shapes given as level set functions was proposed. Its main feature is the presence of anisotropic energies active only in a narrow band around the hypersurfaces that resemble the behavior of elastic shells. In this work we consider some extensions and further analysis of that model. First, we present a symmetric energy functional such that given two particular shapes, it assigns the same energy to any given deformation as to its inverse when the roles of the shapes are interchanged, and introduce the adequate parameter scaling to recover a surface problem when the width of the narrow band vanishes. Then, we obtain existence of minimizing deformations for the symmetric energy in classes of bi-Sobolev homeomorphisms for small enough widths, and prove a $\Gamma$-convergence result for the corresponding non-symmetric energies as the width tends to zero. Finally, numerical results on realistic shape matching applications demonstrating the effect of the symmetric energy are presented.

Combination Constraints of Multiple Fields for Implicit Modeling of Ore Bodies

In this paper, we introduce combination constraints for modeling ore bodies based on multiple implicit fields interpolation. The basic idea of the method is to define a multi-labeled implicit function that combines different sub-implicit fields by the combination operations, including intersection, union and difference operators. The contribution of this paper resides in the application of combination of more general implicit fields with combination rules for the implicit modeling of ore bodies, such that the geologist can construct constraints honoring geological relationships more flexibly. To improve the efficiency of implicit surface reconstruction, a pruning strategy is used to avoid unnecessary calculations based on the hierarchical bounding box of the operation tree. Different RBF-based methods are utilized to study the implicit modeling cases of ore bodies. The experimental results of several datasets show that the combination constraints are useful to reconstruct implicit surfaces for ore bodies with mineralization rules involving multiple fields.