Surface data has been recently used for volumetric computation in various fields of science, such as geographical data analysis, engineering, molecular modeling and simulation, and scientific visualization. This paper contributes to the extension of scalar functions from surfaces to volumes, by iterative interpolation-approximation procedures based on topological characteristics of surfaces: critical points (maxima, minima), saddle points, the Euler invariant, and so on.

The authors describe a two-step approach. First, they define a topology-driven approximation of the scalar function, based on the relevant critical points, in order to capture the global properties. Then, an error-driven term is added to the global model, using information about the local behavior of the function, in order to obtain the target approximation accuracy. The authors also define a least-squares and a constrained approximation scheme to study the flexibility of the proposed approach.

The presentation is organized in seven sections, with 26 well-selected figures. The first section introduces the reader to the universe of volumetric science and outlines the structure of the paper. After presenting the theoretical background and reviewing the relevant literature in the second section, the authors develop the topology-driven approximation scheme in Section 3. An implicit interpolation scheme using radial basis functions is used to develop a bounded number of iterations for obtaining the global component of the extension. Some properties of the iterative scheme follow:

(1) Initially, the number of critical points of each approximation increases until a maximum is reached; then, it decreases until convergence is obtained.

(2) An increasing chain of sets containing points (critical, 1-star type) is generated toward a good approximation.

In Section 4, the authors obtain the improvement of the global component, using locally and compactly supported basis functions. They address details of both computing the least-squares function approximation, based on the pseudoinverse operator, and a constrained optimization problem for the function approximation with least-squares constraints on the set of critical points with error estimation considerations.

The properties of the interpolation-approximation model obtained are described in Section 5, where the following aspects are detailed: the gradient field of the volume-based model, volume-based harmonic approximation as special choice, upper bounds for the volume-based model energy, upper bounds for the surface-based model (both on surface and triangles), and analysis of critical points of the obtained models. The next section is dedicated to applications: enhanced visualization, simplified critical points for persistence assurance, scalar function approximation with weak constraints, and the treatment of the degenerate cases. The last section gives some remarks on future developments.

The reader will find links to adequate references, both old and new, that are well selected for the topic addressed. With the exception of the fifth reference, which is not completely specified, all references are used in the text.

The results presented in the paper are important for researchers, graduates, and those who deal with geometric modeling (both theory and applications). As the authors state in the introduction, the novelty of their approach “resides in the use of the critical points ... to drive the approximation process.”