Abstract We present a novel methodology that utilizes four‐dimensional (4D) space deformation to simulate a magnification lens on versatile volume datasets and textured solid models. Compared with other magnification methods (e.g. geometric optics, mesh editing), 4D differential geometry theory and its practices are much more flexible and powerful for preserving shape features (i.e. minimizing angle distortion), and easier to adapt to versatile solid models. The primary advantage of 4D space lies at the following fact: we can now easily magnify the volume of regions of interest (ROIs) from the additional dimension, while keeping the rest region unchanged. To achieve this primary goal, we first embed a 3D volumetric input into 4D space and magnify ROIs in the fourth dimension. Then we flatten the 4D shape back into 3D space to accommodate other typical applications in the real 3D world. In order to enforce distortion minimization, in both steps we devise the high‐dimensional geometry techniques based on rigorous 4D geometry theory for 3D/4D mapping back and forth to amend the distortion. Our system can preserve not only focus region, but also context region and global shape. We demonstrate the effectiveness, robustness and efficacy of our framework with a variety of models ranging from tetrahedral meshes to volume datasets.

This paper develops a novel volumetric parameterization and spline construction framework, which is an effective modeling tool for converting surface meshes to volumetric splines. Our new splines are defined upon a novel parametric domain called generalized polycubes (GPCs). A GPC comprises a set of regular cube domains topologically glued together. Compared with conventional polycubes (CPCs), the GPC is much more powerful and flexible and has improved numerical accuracy and computational efficiency when serving as a parametric domain. We design an automatic algorithm to construct the GPC domain while also permitting the user to improve shape abstraction via interactive intervention. We then parameterize the input model on the GPC domain. Finally, we devise a new volumetric spline scheme based on this seamless volumetric parameterization. With a hierarchical fitting scheme, the proposed splines can fit data accurately using reduced number of superfluous control points. Our volumetric modeling scheme has great potential in shape modeling, engineering analysis, and reverse engineering applications.

This paper presents a volumetric modeling framework to construct a novel spline scheme called restricted trivariate polycube splines (RTP-splines). The RTP-spline aims to generalize both trivariate T-splines and tensor-product B-splines; it uses solid polycube structure as underlying parametric domains and strictly bounds blending functions within such domains. We construct volumetric RTP-splines in a top-down fashion in four steps: 1) Extending the polycube domain to its bounding volume via space filling; 2) building the B-spline volume over the extended domain with restricted boundaries; 3) inserting duplicate knots by adding anchor points and performing local refinement; and 4) removing exterior cells and anchors. Besides local refinement inherited from general T-splines, the RTP-splines have a few attractive properties as follows: 1) They naturally model solid objects with complicated topologies/bifurcations using a one-piece continuous representation without domain trimming/patching/merging. 2) They have guaranteed semistandardness so that the functions and derivatives evaluation is very efficient. 3) Their restricted support regions of blending functions prevent control points from influencing other nearby domain regions that stay opposite to the immediate boundaries. These features are highly desirable for certain applications such as isogeometric analysis. We conduct extensive experiments on converting complicated solid models into RTP-splines, and demonstrate the proposed spline to be a powerful and promising tool for volumetric modeling and other scientific/engineering applications where data sets with multiattributes are prevalent.

This paper develops a new trivariate hierarchical spline scheme for volumetric data representation. Unlike conventional spline formulations and techniques, our new framework is built upon a novel parametric domain called Generalized PolyCube (GPC), comprising a set of regular cubes being glued together. Compared with the conventional PolyCube (PC) that could serve as a "one-piece'' 3-manifold domain, GPC has more powerful and flexible representation ability. We develop an effective framework that parameterizes a solid model onto a topologically equivalent GPC domain, and design a hierarchical fitting scheme based on trivariate T-splines. The entire data-spline-conversion modeling framework provides high-accuracy data fitting and greatly reduce the number of superfluous control points. It is a powerful toolkit with broader application appeal in shape modeling, engineering analysis, and reverse engineering.