# Analytic Form Fitting in Poor Triangular Meshes
## 3 Methodology  
Fitting of an analytic primitive \(g()\) to the support (triangle) subset \(S\) requires (obviously) the determination of \(S\). Extraction of support subset \(S\) from the input set \(M\) is heavily based on triangle neighborhood interrogations on \(M\). The acceleration of neighborhood queries on \(M\) was implemented by building the triangular boundary representation of the triangle set. This construction was itself accelerated by applying a spatial hashing process on input \(M\) (Spatial Hashing based on techniques from [18; 19]). Spatial hashing and B-Rep construction are reported in Section 3.1 on preprocessing. Section 3.2 discusses the specific problem of inferring the analytic form \(g()\) from a triangle set \(S\). Sections 3.3 and 3.4 address the determination of triangle subset \(S\) for applying Section 3.2. Section 3.4 focuses on cases where the border between support subset \(S\) and \(M-S\) is sharp (\(C^{0}\)-continuity), thus allowing identification of \(S\) by using continuity information alone. Section 3.4 addresses more difficult cases where continuity at the border between \(S\) and \(M-S\) is \(C^{1}\) or better. In these cases, \(S\) is legitimated by a good fitting \(g()\), but at the same time, \(g()\) depends on \(S\). In these cases, fitting (\(g()\)) and segmentation (\(S\)) are codependent and cannot be executed independently from each other.  
**Preprocessing**  
The process of extracting and identifying analytic forms from a manifold triangle set (Figure 1) is preceded by a preprocessing stage encompassing: (1) a spatial hashing; (2) boundary representation (B-Rep) construction.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Approach** & **Refs.** & **Advantages** & **Disadvantages** \\ \hline
**Stochastic Methods.** Randomly apply local regressions of alternative goal primitives and keeping the best fit. & [2; 3; 10; 11; 12] & (1) Robust to outliers; (2) application-specific & (1) Requires dense, isotropic, homogeneous input dataset. \\ \hline
**Mapping to Parameter Space.** Implement a mapping to the parameter space, identify the number of parameters needed and search for clusters of similar parameters. & [4; 5; 13; 14] & (1) Robust to noise and outliers; (2) work with missing data & (1) High computational cost; (2) requires dense datasets \\ \hline
**Mesh Segmentation Fitting Methods.** Create clusters of the dataset using different grouping techniques. Fitting can occur during or after the segmentation. & [6; 7; 8; 9; 15; 16; 17] & (1) Spatial consistency & (1) Sensitive to noise and outliers; (2) require supplementary information in the dataset. \\ \hline
**Our approach.** A primitive fitting in poor triangular meshes. Segmentation driven by dihedral angle is implemented when continuity is strict \(C^{0}\). Segmentation and fitting are codependent for \(C^{1}\) or higher continuity. & (1) Requires the type of analytic form intended to be fitted as an input. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Competitor approaches vs. current approach.  
Figure 1: Overall primitive extraction process.