# Fuzzy Frankot-Chellappa Algorithm for Surface Normal Integration
###### Abstract  
In this paper, we propose a fuzzy formulation of the classic Frankot-Chellappa algorithm by which surfaces can be reconstructed using normal vectors. In the fuzzy formulation, the surface normal vectors may be uncertain or ambiguous, yielding a fuzzy Poisson partial differential equation that requires appropriate definitions of fuzzy derivatives. The solution of the resulting fuzzy model is approached by adopting a fuzzy variant of the discrete sine transform, which results in a fast and robust algorithm for surface reconstruction. An adaptive defuzzification strategy is also introduced to improve noise handling in highly uncertain regions. In experiments, we demonstrate that our fuzzy Frankot-Chellappa algorithm achieves accuracy on par with the classic approach for smooth surfaces and offers improved robustness in  
compared to classical Poisson methods. In general, unlike stochastic approaches such as Stochastic Poisson Surface Reconstruction, which rely on Gaussian processes to capture uncertainty, our method introduces fuzziness directly at the level of gradient fields. This enables uncertainty modeling without requiring probabilistic assumptions and integrates naturally with classical Poisson solvers via fuzzy arithmetic. The concept of a fuzzy transform was introduced by [PERSON] et al. in [12; 13] and was applied in image processing by [PERSON] et al. in [14]. The fuzzy transform (\(F\)-transform) provides direct and inverse operations that allow one to approximate a function using fuzzy basis functions. The authors of [13] showed that the inverse \(F\)-transform has good approximation properties and is relatively easy to use, even in the presence of noise. Meanwhile, fuzzy partial differential Equations (PDEs) were studied for problems like the Poisson equation in [15; 16; 17]. In particular, in 2018, [PERSON] et al. discussed a fuzzy Poisson equation with Dirichlet boundary conditions, proving the uniqueness and stability of its solution in [17], and a continuous fuzzy Fourier sine transform was applied to solve a fuzzy wave equation in [18]. We explored for the first time a fuzzified formulation of the discrete sine transform method and used it to solve the Poisson equation with fuzzy boundary conditions in [19]. Fuzzy logic provides a means to represent and process imprecise or uncertain information, which in the context of surface integration can arise from measurement noise or ambiguity in the normal data. By leveraging fuzzy concepts, the integration process can accommodate these uncertainties, potentially yielding more robust results. Here, we extend upon the previous method by providing more extensive experiments. Also, we include developments that further refine that approach by introducing a multiplicative fuzzification of input normals and an adaptive defuzzification strategy that dynamically adjusts reconstruction weights based on local uncertainty. Unlike a fixed defuzzification scheme, this method adjusts how the fuzzy solution is resolved to a crisp surface depending on the spread at each pixel. We showed that this enhanced fuzzy FC method maintains the computational efficiency of the classical algorithm while improving noise resilience. This adaptive algorithm models the uncertainty of each pixel more accurately: in regions with low uncertainty (small fuzzy spread), it trusts the central value, whereas in regions with higher uncertainty, it incorporates more information from the range of possible values. A distinctive contribution of this paper is the introduction of the fuzzy F-transform as a preprocessing mechanism for surface reconstruction with missing data. This is the first application of the F-transform to address the problem of gap reconstruction in surface normal integration.