Fourier Descriptors

This is a method used to quantify circle-ness

Fourier descriptors are derived from the Fourier transform of the shape’s boundary. If the boundary is represented as a complex signal (z(t) = x(t) + iy(t)), the Fourier descriptors (a_n) are given by:

where:

• is the number of boundary points.
• is the (k)-th boundary point in complex form.
• is the frequency index.

import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import fft
 
def compute_fourier_descriptors(boundary):
    # Convert boundary points to complex numbers
    complex_boundary = boundary[:, 0] + 1j * boundary[:, 1]
 
    # Compute the Fourier Transform
    fourier_descriptors = fft(complex_boundary)
 
    return fourier_descriptors
 
def normalize_fourier_descriptors(fourier_descriptors):
    """
    Normalize Fourier Descriptors to make them invariant to scale, rotation, and translation.
    :param fourier_descriptors: Fourier Descriptors (complex numpy array).
    :return: Normalized Fourier Descriptors.
    """
    # Discard the first descriptor (DC component) to make it translation-invariant
    fourier_descriptors = fourier_descriptors[1:]
 
    # Normalize by the magnitude of the second descriptor to make it scale-invariant
    fourier_descriptors = fourier_descriptors / np.abs(fourier_descriptors[0])
 
    return fourier_descriptors
 
def reconstruct_boundary(fourier_descriptors, num_points):
 
    # Perform the inverse Fourier Transform
    reconstructed_complex = np.ff✅ Yes	t.ifft(fourier_descriptors, n=num_points)
 
    # Convert back to (x, y) coordinates
    reconstructed_boundary = np.column_stack((np.real(reconstructed_complex), np.imag(reconstructed_complex)))
 
    return reconstructed_boundary