# Fuzzy Frankot-Chellappa Algorithm for Surface Normal Integration
## 5 Experimental Results  
In this section, we investigate our basic method along with the adaptive version using several test surfaces and compare it to the classical method. We consider two main scenarios: (1) uncertain but noise-free normals (to test accuracy), and (2) normals with added noise (to test noise suppression). Additionally, we demonstrate the method's ability to handle missing normal data.  
We use several representative surfaces in our experiments. These surfaces exhibit a variety of smooth and sharp features. Each surface is sampled on a grid over the domain \([-1,1]\times[-1,1]\). The true surface height \(z_{u,v}\) is known analytically for each case, so we can compute the exact normals and hence the ground-truth solution for integration.  
Input description and normal field construction: In our experiments, the input to the reconstruction algorithm is a normal vector field \(\vec{n}(u,v)=(n_{1}(u,v),n_{2}(u,v),n_{3}(u,v))\), derived analytically from synthetic ground-truth surfaces shown in Figure 1.  
To compute the gradients required by the FC method, we extract the partial derivatives:  
\[p(u,v)=-\frac{n_{1}(u,v)}{n_{3}(u,v)},\quad q(u,v)=-\frac{n_{2}(u,v)}{n_{3}(u,v)}.\]  
These gradient fields \((p,q)\) serve as the primary inputs to both the classical and the fuzzy FC reconstruction processes.  
For experiments with noise, Gaussian perturbations are applied directly to \(p\) and \(q\). For fuzzy experiments, these perturbed gradients are encoded as symmetric triangular fuzzy numbers, \(\bar{p}(u,v)\) and \(\bar{q}(u,v)\), to model uncertainty.  
**Error reduction strategy evaluation:** Here, to assess the performance of the proposed algorithm on surface reconstruction tasks, the following steps are undertaken:  
* Ground-truth reconstruction: The original surface is computed.
* Clean reconstructions: Both the classical FC and fuzzified FC methods are used to reconstruct the surface from noise-free data.
* Noisy reconstructions: Reconstructions are performed for selected noise levels.
* RMSE analysis: RMSE values are computed using different noise levels for both classical FC and fuzzified FC methods.  
To further highlight some advantages of the fuzzified approach, controlled noise was introduced into the normal vectors, thereby perturbing the computed gradients. We obtained 3D surface reconstructions using both the classical FC and fuzzy FC methods. In all cases, the fuzzy approach produces smoother reconstructions.  
In this work, the noise level models the degree of uncertainty or perturbation introduced into the normal vector field due to measurement errors or external disturbances. Specifically, we assume the presence of Gaussian noise in the normal gradients, meaning that the observed noisy gradients \(\mathbf{g}^{\mathrm{noisy}}=(p^{\prime},q^{\prime})\) are modeled as  
\[p^{\prime}=p+\eta_{p},\quad q^{\prime}=q+\eta_{q},\]  
where \(\eta_{p}\) and \(\eta_{q}\) are independent zero-mean Gaussian noise components:  
\[\eta_{p}\sim\mathcal{N}(0,\sigma^{2}),\quad\eta_{q}\sim\mathcal{N}(0,\sigma^{ 2}).\]  
The noise level, denoted as \(\sigma_{n}\), is then defined as the ratio of the standard deviation of the noise to the maximum magnitude of the true gradient field:  
\[\sigma_{n}=\frac{\sigma}{\max\sqrt{p^{2}+q^{2}}}.\]  
This relative measure ensures that the noise intensity is expressed proportionally to the inherent scale of the gradients, making it invariant to absolute magnitudes.  
Furthermore, in our fuzzified formulation, the spread of the fuzzy numbers \(\bar{p}\) and \(\bar{q}\) is dynamically adapted based on the estimated noise level.