Using fractional derivatives as degree of symmetry to characterize two dimensional natural features

We present an approach to the common problem in data analysis namely to detect interesting phenomena without a priori knowledge. In an earlier work [1,2] we put forth method of decomposition in one dimension based on a notion we called degree of symmetry. In this work we present a decomposition model in two dimensions based as well on degree of symmetry. We associate degree of symmetry with fractional order of derivatives and represent each feature along each axis as a linear combination of fractionally differentiated symmetric Schwartz functions. The fractional derivatives for each component complement each other such that the sum of their orders is an odd integer. The parameters of this model, namely the fractional degree of the derivatives, the widths of the Schwartz functions and the ratio of the weights in the linear combination of each component induce a multidimensional representation space to detect, classify and characterize shapes. We present a numerical method using wavelet transform coefficients for determining the parameters of the representation model given a two dimensional shape.