Image analysis by two types of Franklin-Fourier moments

In this paper, we first derive two types of transformed Franklin polynomial: substituted and weighted radial Franklin polynomials. Two radial orthogonal moments are proposed based on these two types of polynomials, namely substituted Franklin-Fourier moments and weighted Franklin-Fourier moments &#x0028 SFFMs and WFFMs &#x0029, which are orthogonal in polar coordinates. The radial kernel functions of SFFMs and WFFMs are transformed Franklin functions and Franklin functions are composed of a class of complete orthogonal splines function system of degree one. Therefore, it provides the possibility of avoiding calculating high order polynomials, and thus the accurate values of SFFMs and WFFMs can be obtained directly with little computational cost. Theoretical and experimental results show that Franklin functions are not well suited for constructing higher-order moments of SFFMs and WFFMs, but compared with traditional orthogonal moments &#x0028 e.g., BFMs, OFMs and ZMs &#x0029 in polar coordinates, the proposed two types of Franklin-Fourier Moments have better performance respectively in lower-order moments.