# Fuzzy Frankot-Chellappa Algorithm for Surface Normal Integration
## 5 Experimental Results
### Comparison in Terms of Gap Reconstruction  
Another advantage of fuzzy integration is its ability to handle missing data (gaps in the normal field) in a principled way. In classic integration, if some region has no valid normals, one must fill in those normals by interpolation or simply leave that area unreconstructed. In our fuzzy approach, we can interpret missing normals mathematically as extremely uncertain values and let the algorithm naturally fill in the gaps using neighboring information. To simulate a gap, we removed the normals in a square region on each test surface. We then attempted to reconstruct the surface despite this missing data. In the fuzzy method, when computing differences in the discrete Poisson equation, we used the nearest available normals for the missing points. For example, if \((i,j)\) is inside the gap, we replaced \(\tilde{p}_{i,j}\) with \(\tilde{p}_{i-1,j}\) (the closest known neighbor in the \(i\)-direction) and \(\tilde{q}_{i,j}\) with \(\tilde{q}_{i,j-1}\) for the difference formula. In effect, this assumes the gradient in the gap is similar to the gradient just outside the gap, which is a reasonable assumption for a smooth surface. Then \(\tilde{f}_{i,j}\) in Equation (9) needs to be modified for points at the boundary of the gap: if \((i,j)\) is just inside the gap, then one of its neighbors is missing. We handled this by using a one-sided difference: e.g., if \((i+1,j)\) is missing, we use \((i-1,j)\) twice (essentially skipping the missing point, which is equivalent to taking a larger step of \(2h\)). Concretely, we derived  
\[\tilde{f}_{i,j_{(\text{gap})}}=\frac{\tilde{p}_{i,j}\odot_{gH}\tilde{p}_{i-2,j }}{2h}\oplus\frac{\tilde{q}_{i,j}\odot_{gH}\tilde{q}_{i,j-2}}{2h}\, \tag{14}\]  
for points where \((i-1,j)\) or \((i,j-1)\) are missing, so we used \((i-2,j)\), \((i,j-2)\) which are just outside the gap. This essentially uses the nearest known data to approximate the derivative across the gap boundary. Using these adjustments, we solved for \(\bar{z}\) as usual. What this achieves is a continuation of the surface into the gap. In other words, the gap is filled in a membrane-like or smoothest possible way, constrained by the boundary of the gap. This approach is analogous to classic hole-filling techniques (like treating it as a Dirichlet interior problem), but here it arose naturally by extending the fuzzy method. Figure 3 shows the results of gap filling for two example surfaces. In each case, we display the ground-truth surface (with a visible hole where data is missing) and the reconstruction by using Equation (14) in the fuzzy FC method. We see that the fuzzy method successfully reconstructed the shape through the missing region; it yields a smooth fill consistent with surrounding slopes. **Additional clarification on gap filling:** Unlike classical discrete methods such as the original Frankot-Chellappa algorithm, which process input data uniformly without awareness of missing regions, the proposed fuzzy extension inherently handles data gaps. Through the fuzzification stage, regions with missing or highly uncertain data naturally receive very low membership values, effectively reducing their influence in the F-transform \begin{table}
\begin{tabular}{c c c} \hline \hline
**Exact \(s\)** & **RMSE (\(d(\bar{s})\))** & **RMSE (\(d_{new}(\bar{s})\))** \\ \hline \(s_{1}\) & 0.028839 & 0.028714 \\ \(s_{2}\) & 0.343579 & 0.343578 \\ \(s_{3}\) & 0.850193 & 0.731745 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of previous and new fuzzy FC method. computation. A simple thresholding strategy on the fuzzy membership map allows explicit identification of gap regions, enabling selective application of specialized reconstruction formulas in those areas. As a result, during surface reconstruction, gaps are treated adaptively without requiring external boundary condition definitions. Consequently, hole filling emerges from the fuzzy framework, producing smoother and more coherent reconstructions even in the presence of incomplete or noisy input data.