Fuzzy Frankot–Chellappa Algorithm for Surface Normal Integration

• Published in: Algorithms Vol. 18; no. 8; p. 488
• Main Authors: Hajighasemi, Saeide, Breuß, Michael
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.08.2025
• Subjects: Accuracy; Algorithms; Analysis; Boundary conditions; Differential equations; Efficiency; Frankot–Chellappa method; fuzzy derivatives; fuzzy Poisson equation; Fuzzy sets; Missing data; Partial differential equations; surface normal integration
• ISSN: 1999-4893; 1999-4893
• DOI: 10.3390/a18080488

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 the presence of noisy normal data. We also show that it can naturally handle missing data (such as gaps) in the normal field by filling them using neighboring information.