Derivation of Fourier descriptor

For any above derived 1D signature function u(t), its discrete Fourier transform is given by

This results in a set of Fourier coefficients {an}, which is a representation of the shape. Since shapes generated through rotation, translation and scaling (called similarity transform of a shape) of a same shape are similar shapes, a shape representation should be invariant to these operations.The selection of different start point on the shape boundary to derive u(t) should not affect the representation.From Fourier theory, the general form for the Fourier coefficients of a contour generated by translation, rotation,scaling and change of start point from an original contour is given by [13]:

where   and an are the Fourier coefficients of the original shape and the similarity transformed shape, respectively;  and s are the terms due to change of starting point, rotation and scaling. Except the DC component (a0), all the other coefficients are not affected by translation. Now considering the following expression
 

where bn and    are normalized Fourier coefficients of the derived shape and the original shape, respectively. The normalized coefficient of the derived shape bn and that of the original shape   have only difference of exp[j(n-1)t].If we ignore the phase information and only use magnitude of the coefficients, then

 and are the same. In other words,   is invariant to translation, rotation, scaling and change of start point. The set of magnitudes of the normalized Fourier coefficients of the shape { ,0<n<=N} can now be used as shape descriptors, denoted as {FDn,0<n<=N}. The similarity between a query shape Q and a target shape T is given by the city block distance d
between their FDs.The reason of choosing a0 as the normalization factor is because it is the average energy of the signal. It is normally the largest coefficient, therefore, the normalized FD features is in [0, 1].