# Fuzzy Frankot-Chellappa Algorithm for Surface Normal Integration
## 4 Fuzzy Frankot-Chellappa Surface Normal Integration
### Fuzzy Frankot-Chellappa Method  
In the fuzzy domain, the surface reconstruction is governed by a fuzzy Poisson equation with Dirichlet boundary conditions, where the fuzzy gradient field is derived from uncertain normal vectors. The fuzzy Laplacian is discretized using the fuzzy finite difference scheme, and the solution is computed using a fuzzy variant of the discrete sine transform. The main novelty of this approach lies in the fact that it is applying the discrete fuzzy sine transform directly to obtain the solution of a fuzzy Poisson equation. We use the fuzzy inverse discrete sine transformation  
\[\mathcal{F}_{s}\{\tilde{f}_{i,j}\}=\tilde{\tilde{f}}_{k,l}\text{ and }\mathcal{F}_{s}^{-1}\{\tilde{f}_{k,l}\}=\tilde{f}_{i,j}:\]  
\[\tilde{f}_{i,j}=\frac{4}{mn}\sum_{k=1}^{m-1}\sum_{l=1}^{n-1}\tilde{\tilde{f}}_ {k,l}(\sin(\pi\frac{ki}{m})\sin(\pi\frac{lj}{n})) \tag{10}\]  
First we solve it only with a homogeneous Dirichlet condition \(\tilde{z}_{i,j}=0\) for \((i,j)\in\partial\Omega=\{0,m\}\times\{0,n\}\), where \(\Omega=[0,m]\times[0,n]\), in order to later determine the solution for problems with an inhomogeneous Dirichlet boundary conditions using the knowledge gained. We assume Dirichlet boundary conditions (e.g., \(\tilde{z}=0\) on \(\partial\Omega=[0,m]\times[0,n]\)) for the fuzzy Poisson equation, as in the classic case. We seek a fuzzy solution \(\tilde{z}\) satisfying \(\Delta\tilde{z}=\tilde{f}\), where \(\tilde{f}(i,j)\) is the fuzzy divergence of the fuzzy normal field. To solve this numerically, we follow a similar approach to the classical algorithm but operate on the fuzzy quantities. First, we compute the discrete fuzzy Laplace equation analogous to Equation (9) by replacing \(z\) and \(f\) with \(\tilde{z}\) and \(\tilde{f}\) and using \(gH\) difference for subtraction in Equation (9). The resulting Equation (at each \(\alpha\)-cut) has the form  
\[\frac{\tilde{z}_{i+1,j}\oplus\tilde{z}_{i-1,j}\oplus\tilde{z}_{i,j+1}\oplus \tilde{z}_{i,j-1}\odot\alpha\tilde{z}_{i,j}}{h^{2}}\odot_{gH}\tilde{f}_{i,j}=0, \tag{11}\]We again insert the inverse transform from (10) into the discrete Formula (11), leaving \(h\) undetermined:  
\[\frac{4}{mn}\sum_{k=1}^{m-1}\sum_{l=1}^{n-1} \frac{\hat{z}_{k,l}}{h^{2}}\Big{[}\Big{(}\sin\!\left(\frac{\pi k(i+1)} {m}\right)+\sin\!\left(\frac{\pi k(i-1)}{m}\right)\Big{)}\sin\!\left(\frac{\pi lj }{n}\right)\] \[+\sin\!\left(\frac{\pi ki}{m}\right)\!\left(\sin\!\left(\frac{\pi l (j+1)}{n}\right)+\sin\!\left(\frac{\pi l(j-1)}{n}\right)\right)\] \[-4\sin\!\left(\frac{\pi ki}{m}\right)\sin\!\left(\frac{\pi lj}{n} \right)\Big{]}\ominus_{gH}\hat{f}_{k,l}\sin\!\left(\frac{\pi ki}{m}\right)\sin \!\left(\frac{\pi lj}{n}\right)=0.\]  
Since the sine functions form an independent linear family, we can claim that the coefficients in the above equation are zero; in addition, we know \(\sin(x\pm y)=\sin(x)\cos(y)\pm\cos(x)\sin(y)\). All fuzzy operations, including addition, subtraction, multiplication, and division, are performed componentwise on each \(\alpha\)-cut representation of the fuzzy numbers. As a result, the mathematical manipulations reduce to interval arithmetic at each \(\alpha\)-level, avoiding the need for explicit definitions of fuzzy multiplication and division. This \(\alpha\)-cut approach guarantees that all transformations remain consistent with the properties of fuzzy numbers and ensures that the algorithm proceeds in a rigorous and computationally efficient manner.