Applying Semi-discrete Operators to Calculus

Ever since the early 1980’s, computer scientists have been using an algorithm named "Summed Area Table", also known as "Integral Image". This algorithm was shown to provide a tremendous computational gain, since it fits precisely to the needs of discrete geometry researchers, due to its discrete nature. It was first introduced in 1984 by Crow, and was reintroduced to the computer vision community in 2001 by Viola and Jones. In 2007, Wang and his colleagues suggested a semi-discrete, semicontinuous formulation of an extension to this algorithm (the antiderivative formula), and in this book it is suggested that a decisive parameter at the formulation of the theorem can be naturally defined via a simple pointwise operator. The main operator of this theory is defined by a mixture of the discrete and continuous, to form a semi discrete and efficient operator, given that one aims at classification of monotony. This approach to analyze the monotony of functions is hence suitable for computers (in order to save computation time), and the simplicity of the definition allows further research in other areas of classical analysis. ar X iv :1 01 2. 57 51 v3 [ cs .D M ] 2 5 Ju l 2 01 1 To my mother, Sarit Who always believed in me