## 3 Materials and Methods
### Model Architecture  
The composite system itself is based on two directly related models: (1) GS [23], a novel neural RFR technique that predicts 3D statistical, color-based, spatial strokes, or \"splats;\" and (2) SuGaR [25], a mesh extraction technique, itself based on GS. The system reconstructs 3D RFR and textured mesh geometry using 180 sequenced, panoramic church images as inputs, and outputs one 3D mesh model of the said church. These take care of image rasterization, RFR, and mesh extraction (in this order). We first break down the rasterization, RFR, and mesh extraction steps, and then present the implementation diagram. The GS model begins the general process by using a sparse point cloud generated via SfM that places the 3D Gaussians defined by position, covariance matrix, and opacity \(\alpha\). This produces a compact representation of a 3D scene, using spherical harmonics to gauge the directions of the Gaussian splats [23; 24]. Then, the algorithm builds a 3D RFR  
Figure 2: Romanesque-Mudejar style example illustrating the three building states of the church of Nuestra Señora de Arbas in Gordaliza del Pino (León, Spain) as follows: building state 0 (complete building); building state 1 (missing roof and comice); building state 2 (mid-height walls). Figure created by the authors. representation by a series of optimization steps: position, covariance, \(\alpha\)-blend, SH, all interspersed with adaptive control of Gaussian density [23]. As a reminder of the basics of GS and SuGaR processes, we outline a brief overview of their formalizations for training RFR and mesh extraction. In a typical neural point-based rendering approach, each color of a given pixel is computed by blending the N-ordered points overlapping a given pixel, where ref. [23] used as their starting point the following equation:  
\[C=\sum_{i\in N}c_{i}\alpha_{i}\prod_{j=1}^{i-1}(1-\alpha_{j}) \tag{1}\]  
Here, c\({}_{i}\) is the color of each point and \(\alpha_{i}\) is given by evaluating a 2D Gaussian with covariance \(\Sigma\) multiplied by a learned per-point opacity [46]. Plus, they formalized the function for a single Gaussian \(G\), centered at point \(\mu\) (mean), which is then multiplied by \(\alpha\) in the blending process, as follows:  
\[G(\mathbf{x})=e^{-\frac{1}{2}(x)^{T}\Sigma^{-1}(x)} \tag{2}\]  
Furthermore, they defined a scaling matrix \(S\) and a rotation matrix \(R\) to find the corresponding covariance matrix \(\Sigma\) (cf. a world space) that is used to place the 3D Gaussians [47]:  
\[\Sigma=RSS^{T}R^{T}, \tag{3}\]  
where the covariance matrix \(\Sigma\) of a 3D Gaussian is equivalent to describing an ellipsoid. For more details, see [23]. After this step, the SuGaR method takes the Gaussians and, using a loss term, enforces 3D Gaussian and scene-surface alignment during the GS optimization process. This model exploits this alignment for high-detail mesh extraction from Gaussians using signed distance functions. Then, a refinement process that optimizes both mesh and 3D Gaussians on the mesh's surface is set in motion, using GS's rendering operation and producing an editable mesh through Poisson reconstruction (PR) [25; 42] as output. In their original paper, the SuGaR authors defined their \"ideal\" SDF associated with a density function \(d\), in the case that \(d=\overline{d}\); where \(p\) is a sampled point,\(f\) is its SDF, and \(d\) is its density. Their formalization of an ideal function f(p) is as follows:  
\[\mathrm{f}(p)=\pm s_{g*}\sqrt{-2}log(d(p)) \tag{4}\]  
Its approximation, \(\overline{f}(p)\), is as follows:  
\[\overline{f}(p)=\pm s_{g*}\sqrt{-2}log\Big{(}\overline{d}(p)\Big{)} \tag{5}\]  
Using these formulas, they rendered depth maps of the Gaussians, and then sampled points \(p\) in each viewpoint, each according to the distribution of the Gaussians, where \(\overline{f}(p)\) is the 3D distance between \(p\) and the intersection of the line of sight for \(p\) and the given depth map.