Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey

This paper provides a tutorial and survey for a specific kind of illustrative visualization technique: feature lines. We examine different feature line methods. For this, we provide the differential geometry behind these concepts and adapt this mathematical field to the discrete differential geometry. All discrete differential geometry terms are explained for triangulated surface meshes. These utilities serve as basis for the feature line methods. We provide the reader with all knowledge to re-implement every feature line method. Furthermore, we summarize the methods and suggest a guideline for which kind of surface which feature line algorithm is best suited. Our work is motivated by, but not restricted to, medical and biological surface models.Comment: 33 page

O zakrivljenostima na trokutnim mrežama

A face-based curvature estimation on triangle meshes is presented in this paper. A flexible disk is laid on the mesh around a given triangle. Such a bent disk is used as a geodesic neighborhood of the face for approximating normal and principal curvatures. The radius of the disk is free input data in the algorithm. Its influence on the curvature values and the stability of estimated principal directions are investigated in the examples.U članku je prikazana procjena zakrivljenosti na trokutnim mrežama, bazirana na stranicama. Gipki disk položen je na mrežu oko danog trokuta. Takav prilagodljiv disk koristi se kao geodetska okolina stranice za aproksimaciju normalnih i glavnih zakrivljenosti. Polumjer diska je nezavisni ulazni podatak u algoritmu. U primjerima se istražuje njegov utjecaj na vrijednosti zakrivljenosti i na stabilnost procijenjenih glavnih smjerova

B-splajn dijelovi koji pristaju na plohe i triangularne mreže

In this paper a technique for the construction of quartic polynomial B-spline patches fitting on analytical surfaces and triangle meshes is presented.The input data are curvature values and principal directions at a given surface point which can be computed directly, if the surface is represented by a vector function. In the case of discrete surface representation, i.e. on a triangle mesh the required input data are computed from a circular neighborhood of a specified triangle face. Such a surface patch may replace a well defined region of the mesh, and can be used e.g. in re-triangulation, mesh-simplification and rendering algorithms.U ovom se radu prikazuje metoda za konstrukciju kvartnog polinoma B-splajn dijela podesnog za analitičke plohe i mreže trokuta. Ulazni podaci su vrijednosti zakrivljenosti i glavni smjerovi u danoj točki plohe, koji se mogu izravno računati za plohu zadanu vektorskom funkcijom. Za slučaj diskretne reprezentacije plohe, tj. za triangularnu mrežu, odgovarajući ulazni podaci računaju se iz kružne okoline određ-enog trokuta mreže. Takvi dijelovi mogu zamijeniti dobro definirano područje mreže, i mogu se upotrijebiti npr. u retriangulaciji, simplifikaciji mreže i renderiranju

Consistent Computation of First- and Second-Order Differential Quantities for Surface Meshes

Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce inconsistent results and require ad hoc fixes. In this paper, we show that the computation of the gradient and Hessian of a height function provides the foundation for consistently computing the differential quantities. We derive simple, explicit formulas for the transformations between the first- and second-order differential quantities (i.e., normal vector and principal curvature tensor) of a smooth surface and the first- and second-order derivatives (i.e., gradient and Hessian) of its corresponding height function. We then investigate a general, flexible numerical framework to estimate the derivatives of the height function based on local polynomial fittings formulated as weighted least squares approximations. We also propose an iterative fitting scheme to improve accuracy. This framework generalizes polynomial fitting and addresses some of its accuracy and stability issues, as demonstrated by our theoretical analysis as well as experimental results.Comment: 12 pages, 12 figures, ACM Solid and Physical Modeling Symposium, June 200