Object-space multiphase implicit functions

Abstract

Implicit functions have a wide range of applications in entertainment, engineering and medical imaging. A standard two-phase implicit function only represents the interior and exterior of a single object. To facilitate solid modeling of heterogeneous objects with multiple internal regions, object-space multiphase implicit functions are much desired. Multiphase implicit functions have much potential in modeling natural organisms, heterogeneous mechanical parts and anatomical atlases. In this paper, we introduce a novel class of object-space multiphase implicit functions that are capable of accurately and compactly representing objects with multiple internal regions. Our proposed multiphase implicit functions facilitate true object-space geometric modeling of heterogeneous objects with non-manifold features. We present multiple methods to create object-space multiphase implicit functions from existing data, including meshes and segmented medical images. Our algorithms are inspired by machine learning algorithms for training multicategory max-margin classifiers. Comparisons demonstrate that our method achieves an error rate one order of magnitude smaller than alternative techniques.

Repairing Inconsistent Curve Networks on Non‐parallel Cross‐sections

“…Implicit representations are particularly attractive for interactive editing due to the ease in manipulating the functions [BW90, SWG05, SWSJ07]. Other popular ways for manipulating the functions include interpolating between given functions (as in shape metamorphosis [Hug92, COSL98, BW01, TO05]), associating the function with a particle system [DG95], evolving the function with a user‐specified speed function (as in the powerful level‐sets method [Set99, TO*03]), and interpolating functional or positional constraints using a variational framework [TO02, KHR02, HKHP11, YYW12].…”

“…We are not aware of any deformation method that is designed for consistency objectives like those in our problem. Our method falls into the class of variational implicit methods [TO02, KHR02, HKHP11, YYW12]. Unlike existing methods that formulate deterministic constraints as simple equalities or inequalities, our method uses additional integer variables to encode the uncertainty of labelling along intersection lines, which results in a more challenging optimization problem.…”

“…Curves that partition the plane into two labels can be implicitly represented as the level set of a scalar function. To represent a curve network that partitions the plane into n > 2 labels, we consider a commonly used implicit representation that utilizes a vector function [LSSF06, FJW10, YYW12, HZCJ17]. We first briefly review such representation, and then discuss our choice of the initial vector function and how it is discretized on input cross‐sections.…”

“…While one could define the vector function in more sophisticated ways (e.g., using higher degree polynomials [YYW12]), we chose signed distance functions not only due to its simplicity but also because of the convenience it offers for formulating the deformation energy (Section 3.2). A key observation is that a Euclidean distance function has a constant gradient magnitude, hence a small change to the function values leads to a bounded spatial displacement of its level set.…”

Topology-controlled reconstruction of multi-labelled domains from cross-sections

“…Typical representations of multi-labeled domains include (regional or global) implicit functions [Feng et al 2010;Kim 2010;Losasso et al 2006;Mitchell et al 2015;Saye 2015;Yuan et al 2012;Zhao et al 1996;Zheng et al 2006] and volume fractions [Ahn and Shashkov 2007;Anderson et al 2008Anderson et al , 2010Bonnell et al 2003]. While implicit function representations are often based on level sets, current works typically utilize a single level as the underlying function evolves (e.g., during a simulation), which creates a univariate family of domains.…”