An Python/NumPy implementation of a method for approximating a contour with a Fourier series, as described in [1].

pip install pyefd

Given a closed contour of a shape, generated by e.g. scikit-image or OpenCV, this package can fit a Fourier series approximating the shape of the contour.

This section describes the general usage patterns of pyefd.

from pyefd import elliptic_fourier_descriptors
coeffs = elliptic_fourier_descriptors(contour, order=10)

The coefficients returned are the a_n, b_n, c_n and d_n of the following Fourier series representation of the shape.

The coefficients returned are by default normalized so that they are rotation and size-invariant. This can be overridden by calling:

from pyefd import elliptic_fourier_descriptors
coeffs = elliptic_fourier_descriptors(contour, order=10, normalize=False)

Normalization can also be done afterwards:

from pyefd import normalize_efd
coeffs = normalize_efd(coeffs)

If you are using OpenCV to generate contours, this example shows how to connect it to pyefd.

import cv2 
import numpy
from pyefd import elliptic_fourier_descriptors
# Find the contours of a binary image using OpenCV.
contours, hierarchy = cv2.findContours(
    im, cv2.RETR_TREE, cv2.CHAIN_APPROX_SIMPLE)
# Iterate through all contours found and store each contour's 
# elliptical Fourier descriptor's coefficients.
coeffs = []
for cnt in contours:
    # Find the coefficients of all contours
    coeffs.append(elliptic_fourier_descriptors(
        numpy.squeeze(cnt), order=10))

To use these as features, one can write a small wrapper function:

from pyefd import elliptic_fourier_descriptors
def efd_feature(contour):
    coeffs = elliptic_fourier_descriptors(contour, order=10, normalize=True)
    return coeffs.flatten()[3:]

If the coefficients are normalized, then coeffs[0, 0] = 1.0, coeffs[0, 1] = 0.0 and coeffs[0, 2] = 0.0, so they can be disregarded when using the elliptic Fourier descriptors as features.

See [1] for more technical details.

Run tests with with Pytest:

py.test tests.py

The tests include a single image from the MNIST dataset of handwritten digits ([2]) as a contour to use for testing.

See ReadTheDocs.

[1]: Frank P Kuhl, Charles R Giardina, Elliptic Fourier features of a closed contour, Computer Graphics and Image Processing, Volume 18, Issue 3, 1982, Pages 236-258, ISSN 0146-664X, https://dx.doi.org/10.1016/0146-664X(82)90034-X.

[2]: LeCun et al. (1999): The MNIST Dataset Of Handwritten Digits