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


EFD representations of an MNIST [2] digit. Shows progressive improvement of approximation by order of Fourier series.


$ 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.

General usage examples

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)

OpenCV example

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(

# 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
        numpy.squeeze(cnt), order=10))

Using EFD as features

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

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:

$ python test

or with Pytest:

$ py.test

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


A Python implementation of the method described in [3] and [4] for calculating Fourier coefficients for characterizing closed contours.


[3](1, 2, 3) F. P. Kuhl and C. R. Giardina, “Elliptic Fourier Features of a Closed Contour,” Computer Vision, Graphics and Image Processing, Vol. 18, pp. 236-258, 1982.
[4](1, 2, 3) Oivind Due Trier, Anil K. Jain and Torfinn Taxt, “Feature Extraction Methods for Character Recognition - A Survey”, Pattern Recognition Vol. 29, No.4, pp. 641-662, 1996

Created by hbldh <> on 2016-01-30.


Calculate the \(A_0\) and \(C_0\) coefficients of the elliptic Fourier series.

Parameters:contour (numpy.ndarray) – A contour array of size [M x 2].
Returns:The \(A_0\) and \(C_0\) coefficients.
Return type:tuple
pyefd.elliptic_fourier_descriptors(contour, order=10, normalize=False)[source]

Calculate elliptical Fourier descriptors for a contour.

  • contour (numpy.ndarray) – A contour array of size [M x 2].
  • order (int) – The order of Fourier coefficients to calculate.
  • normalize (bool) – If the coefficients should be normalized; see references for details.

A [order x 4] array of Fourier coefficients.

Return type:


pyefd.normalize_efd(coeffs, size_invariant=True)[source]

Normalizes an array of Fourier coefficients.

See [3] and [4] for details.

  • coeffs (numpy.ndarray) – A [n x 4] Fourier coefficient array.
  • size_invariant (bool) – If size invariance normalizing should be done as well. Default is True.

The normalized [n x 4] Fourier coefficient array.

Return type:


pyefd.plot_efd(coeffs, locus=(0.0, 0.0), image=None, contour=None, n=300)[source]

Plot a [2 x (N / 2)] grid of successive truncations of the series.


Requires matplotlib!

  • coeffs (numpy.ndarray) – [N x 4] Fourier coefficient array.
  • tuple or numpy.ndarray locus (list,) – The \(A_0\) and \(C_0\) elliptic locus in [3] and [4].
  • n (int) – Number of points to use for plotting of Fourier series.

Indices and tables