Fourierdescriptors

Fourier descriptors are a method for representing the boundary of a planar shape by the discrete Fourier transform of its boundary coordinates. They are commonly applied to closed contours in image analysis and pattern recognition, providing a compact, quantitative description that can be used for shape matching, retrieval, and classification.

The typical workflow involves extracting a boundary from an image, resampling the boundary into N evenly spaced points, and forming a complex sequence z(k) = x(k) + i y(k). A discrete Fourier transform is applied to z(k) to obtain a set of coefficients that describe the shape in progressively coarser frequency components. Low-frequency coefficients capture the overall form, while higher-frequency coefficients encode finer boundary details. The shape can be reconstructed by performing an inverse transform using a selected subset of coefficients.

To compare shapes across images, the descriptors are normalized to achieve invariance to translation, rotation, and scaling. Translation invariance is achieved by centering the boundary at its centroid. Rotation invariance is often obtained by aligning phases so that a reference coefficient is real, or by using the magnitudes only. Scaling invariance is achieved by dividing all coefficients by a suitable norm, such as the magnitude of the first nonzero harmonic.

Elliptic Fourier Descriptors (EFD) are a popular variant that expresses the boundary as a sequence of ellipses corresponding to harmonics, offering intuitive interpretation and robust performance for many contour-based tasks. Limitations include sensitivity to noise, contour extraction quality, and the need for consistent sampling and normalization across shapes.