# Fuzzy Frankot-Chellappa Algorithm for Surface Normal Integration
## 5 Experimental Results
### Comparison in Terms of Gap Reconstruction
**Role of the F-transform:** The F-transform is not a reconstruction method, but it plays a critical preparatory role in the surface reconstruction pipeline. Its main contribution lies in the preprocessing stage and regularizing the input normal field before applying the fuzzy Frankot-Chellappa reconstruction step. * Smoothing and noise reduction real: The F-transform, applied to the fuzzified gradient data, acts as a low-pass filter. This attenuates rapid variations (such as noise) while preserving the underlying shape characteristics. As a result, the gradient field fed into the Poisson solver becomes more regular, which improves the stability and accuracy of the reconstruction. * Implicit gap identification: In addition to denoising, the F-transform naturally identifies unreliable regions. During the fuzzification step, input points with low confidence (due to occlusion, missing data, or inconsistency) receive reduced fuzzy membership. Figure 3: Reconstructing surfaces with areas containing unknown normal vectors. (**a**) Left: Ground-truth surface with a visible gap (missing data region) for \(z_{1}=\sin(uv)+u^{2}-v^{2}\); Right: Reconstruction using the fuzzy FC method, demonstrating smooth and coherent gap-filling based on neighboring slope information for \(z_{1}\); (**b**) Left: Ground-truth surface with a visible gap (missing data region) for \(z_{3}=(1-\frac{u^{2}}{\mu^{2}}-\frac{v^{2}}{\mu^{2}})e^{-\frac{u^{2}-u^{2}}{2 \mu^{2}}}\); Right: Reconstruction using the fuzzy FC method, demonstrating smooth and coherent gap-filling based on neighboring slope information for \(z_{3}\);  
The resulting fuzzy coefficients in these areas become negligible. This behavior allows for implicit detection of gaps in the data without explicit labeling or manual segmentation. * Enabling adaptive Poisson solving: The output of the F-transform is passed to the fuzzy Poisson solver. Regions with confident data retain their structure, while uncertain areas can be handled selectively, either by omission or by applying special gap-filling formulas. Thus, the F-transform prepares the problem domain for adaptive treatment in the reconstruction phase. As a result, the F-transform acts as a smoothing, filtering, and data-awareness mechanism prior to fuzzy Poisson reconstruction. Its integration is essential to ensure robustness in noisy or incomplete settings and enables the subsequent fuzzified Frankot-Chellappa method to operate more reliably, as we can see in Figure 4.  
Figure 4: The **(left column)** shows the approximation results obtained using the classical Frankot-Chellappa method with the discrete sine transform (sT), while the **(right column)** illustrates the corresponding approximations using the proposed fuzzy Frankot–Chellappa method based on the fuzzy discrete sine transform (FsT). In the fuzzy approach, the FST is applied via a fuzzy partition (F-transform) for three representative functions from Table 1, using the \(\alpha\)-cut representation with \(\alpha=0.3\). (**a**) \(\sin(uv)+u^{2}-v^{2}\); (**b**) \((1-\frac{u^{2}}{\sigma^{2}}-\frac{v^{2}}{\sigma^{2}})e^{-\frac{u^{2}v^{2}}{ \sigma^{2}}}\); (**c**) \(-u^{2}-v^{2}\) (\(\frac{3}{\delta}<u^{2}+v^{2}<\frac{4}{\delta}\)).