Fuzzy transform and least-squares approximation: Analogies, differences, and generalizations

Investigating the relations between the least-squares approximation techniques and the Fuzzy Transform, in this paper we show that the Discrete Fuzzy Transform is invariant with respect to the interpolating and least-squares approximation. Additionally, the Fuzzy Transform is evaluated at any point by simply resampling the continuous approximation underlying the input data. Using numerical linear algebra, we also derive new properties (e.g., stability to noise, additivity with respect to the input data) and characterizations (e.g., radial and dual membership maps) of the Discrete Fuzzy Transform. Finally, we define the geometry- and confidence-driven Discrete Fuzzy Transforms, which take into account the intrinsic geometry and the confidence weights associated to the data.