Modeling Uncertain Sparse Data with Fuzzy B-splines

Various interpolation and approximation techniques are employed in order to fit a B-spline surface to a set of sparse data for applications in geographical data analysis, image processing, solid modeling, etc.The sparse data are usually endowed with some sort of uncertainty arising from several sources, e.g. measurement errors, data reduction, modelling errors, etc. An appropriate way of describing data uncertainty is through the concepts of interval/fuzzy arithmetic and applying these methods to the above problem leads to the definition of interval/fuzzy B-splines. An important related issue for applications is that of query or interrogation of the fuzzy B-spline which fits a sparse set of uncertain data points. Such a query may also be phrased in the form of solving fuzzy equations. In this article rigorous algorithms are presented for constructing fuzzy B-splines fitting uncertain sparse data and for their interrogation. An example is also presented related to the description of hazardous areas due to environmental pollution.