# Fuzzy Frankot-Chellappa Algorithm for Surface Normal Integration
## 2 Fuzzy Concepts  
In our framework, uncertainty in a value is represented using fuzzy numbers. A fuzzy number is typically described by its membership function \(\mu(x)\), which maps each real number \(x\) to a value in \([0,1]\) indicating its degree of membership. We now recall a list of notations and several theoretical concepts useful in this paper. More information on related fuzzy concepts can be found in [20] and the references therein. **Definition 1** (Fuzzy set and \(\alpha\)-cut). : _A fuzzy set \(\tilde{A}\) in \(\mathbb{R}\) is characterized by a membership function \(\mu_{\tilde{A}}\colon\mathbb{R}\to[0,1]\). For \(\alpha\in(0,1]\), the \(\alpha\)-cut of \(\tilde{A}\) is defined as_  
\[[\tilde{A}]^{\alpha}=\{x\in\mathbb{R}\ |\ \mu_{\tilde{A}}(x)\geq\alpha\}=[\tilde{A }^{-}(\alpha),\tilde{A}^{+}(\alpha)].\]  
**Definition 2** (Triangular fuzzy number). : _A fuzzy number \(\tilde{a}\) is defined as a fuzzy subset of the real numbers \(\mathbb{R}\) with a normalized, convex, and upper semi-continuous membership function \(\mu_{\tilde{a}}\colon\mathbb{R}\to[0,1]\) such that the support of \(\mu_{\tilde{a}}\) is bounded and there exists exactly one point with membership degree 1._  
_In this work, we use triangular fuzzy numbers (TFNs). A triangular fuzzy number \(\tilde{A}\) is defined by a triplet \((a_{1},a_{2},a_{3})\), with \(a_{1}<a_{2}<a_{3}\), where \(a_{2}\) is the peak (core) and \([a_{1},a_{3}]\) is the support._  
\[\mu_{\tilde{A}}(x)= \begin{cases}\frac{x-a_{1}}{a_{2}-a_{1}},&a_{1}\leq x\leq \ a_{2}\\ \frac{a_{3}-x}{a_{3}-a_{2}},&a_{2}<x\leq\ a_{3}\\ 0,&\text{otherwise}\end{cases} \]  
**Definition 3** (gH difference). : _Let \(\tilde{A},\tilde{B}\) be fuzzy numbers. Then the generalized Hukuhara difference \(\tilde{A}\odot_{\tilde{g}H}\tilde{B}\) is a fuzzy number \(\tilde{E}\) such that one of the following holds:_  
\[\tilde{A}=\tilde{B}\oplus\tilde{E}\ \ \text{or}\ \ \tilde{B}=\tilde{A}\oplus(- \tilde{E})\]  
_where \((-\tilde{E})\) is defined as the symmetric of \(\tilde{E}\) with respect to 0, i.e., if \(\tilde{E}=(e_{1},e_{2},e_{3})\) is a triangular fuzzy number (TFN), then \((-\tilde{E})=(-e_{3},-e_{2},-e_{1})\)._  
**Definition 4** (Fuzzy-valued function). : _Let \(\tilde{f}:D\to\mathcal{F}(\mathbb{R})\) be a fuzzy-valued function defined on a compact rectangular domain \(D\subseteq\mathbb{R}^{2}\), where \(\mathcal{F}(\mathbb{R})\) is the set of fuzzy numbers on \(\mathbb{R}\), \(D=[a,b]\times[c,d]\) and \((u,v)\in D\) denote the spatial coordinates. For each \((u,v)\in D\) and \(\alpha\in[0,1]\), the function can be expressed in terms of \(\alpha\)-cuts:_  
\[[\tilde{f}(u,v;\alpha)]^{\alpha}=\left[f^{-}(u,v;\alpha),f^{+}(u,v;\alpha)\right]\]  
_At each point \((u,v)\in D\), the output \(\tilde{f}(u,v)\) is a triangular fuzzy number (TFN) represented by the triple \((f_{1}(u,v),f_{2}(u,v),f_{3}(u,v))\), where \(f_{2}\) is the peak (core) and \([f_{1},f_{3}]\) is the support._  
_Hence, the membership function \(\mu_{f(u,v)}(z)\) of the output fuzzy number \(\tilde{f}(u,v)\) is defined as_  
\[\mu_{f(u,v)}(z)= \begin{cases}\frac{z-f_{1}(u,v)}{f_{2}(u,v)-f_{1}(u,v)},&\text {if }z\in[f_{1}(x,y),f_{2}(u,v)]\\ \frac{f_{3}(x,y)-z}{f_{3}(u,v)-f_{2}(u,v)},&\text{if }z\in[f_{2}(u,v),f_{3}(u,v)]\\ 0,&\text{otherwise}\end{cases} \]  
**Definition 5** (Fuzzy partial derivative using gH difference). : _Let \(\tilde{f}:D\subset\mathbb{R}^{2}\to\mathcal{F}\) be a fuzzy-valued function defined over domain \(D\), where \(\mathcal{F}\) denotes the set of fuzzy numbers. The partial derivative of \(f\) with respect to \(u\) at the point \((u_{0},v_{0})\) is defined using the generalized Hukuhara (gH) difference as follows:_\[\frac{\partial\tilde{f}}{\partial u}(u_{0},v_{0}):=\lim_{h\to 0}\frac{\tilde{f}(u_{0}+h,v_{0}) \ominus_{gH}\tilde{f}(u_{0},v_{0})}{h},\]  
_provided that this limit exists.