Problem Formulation

We are interested in exploiting the relationships between points on the shape boundary in the domain of a Fourier descriptor. Affine homographies have been studied in the Fourier domain  [4], in which the boundary points were represented as complex numbers. We need a richer representation to linearize the affine homography relation and so use a vector of complex numbers as our descriptor for points on the boundary of a shape. Let $P[i] = (u[i], v[i], w[i])$ be the homogeneous coordinates of points on the closed boundary of a planar shape. The shape is represented by a sequence of vectors of complex numbers as shown below.

$${\bf x}[i] = \left[ \begin{array}{c} u[i] + j\, 0 \\ v[i] + j\, 0 \\ w[i] + j\, 0 \end{array} \right]$$

Let the image-to-image transformation of these points from view 0 to view $l$ be given by a $3\times 3$ matrix M. We have,

$${\bf x}^l[i] = {\bf M} {\bf x}^0[i]$$ (1)

Taking the Fourier transform on both sides we get,

$${\bf\bar{X}}^l[k] = {\bf M} {\bf\bar{ X}}^0[k]$$ (2)

where ${\bf\bar{X}}^0$ and ${\bf\bar{X}}^l$ are the Fourier transforms of ${\bf x}^0$ and ${\bf x}^l$, respectively. The sequences ${\bf\bar{X}}^l[k]$ are periodic and conjugate symmetric.