Python implementation and interpretation of "Elliptic Fourier Features of a Closed Contour" with enhanced start point and rotation normalization methods.
Python implementation and interpretation of "Elliptic Fourier Features of a Closed Contour" with enhanced start point and rotation normalization methods. In the code, the point having the maimum distance from the centroid is selected as the start point and the rotation is normalized by aligning the start point with horizontal axis of the coordiante system
The EllipticFourier module only depends on numpy, thus any python environment having numpy installed is feasible to run the code. Download the file "EllipticFourier.py" and copy the file to your working direction. OpenCV and matplotlib is required to run all examples.
Given a closed contour of a shape, generated by e.g. OpenCV, this package can fit a Fourier series approximating the shape of the contour.
Using elliptic Fourier descriptor to describe the shape of SpongeBob. Here we used 10 elliptic Fourier coefficients to describe the shape.
import numpy as np
import cv2
from EllipticFourier import EllipticFourier
# load an image
img_src = "SpongeBob.jpg"
img = cv2.imread(img_src)
# extract image contour
gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY) # conver to gray image
_, binary = cv2.threshold(gray, 127, 255, cv2.THRESH_BINARY_INV) # threshold to get binary image. Note that in the example image, the forground is back, thus we use cv2.THRESH_BINARY_INV.
contours, _ = cv2.findContours(binary, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_NONE) # find all possible contours
contour = np.squeeze(contours[0])
# forward elliptic Fourier
efDescriptor = EllipticFourier()
A0, C0, coeffs = efDescriptor.forward(contour=contour, N=10)
print(f"A0: {A0}")
print(f"C0: {C0}")
print("Coefficients:")
print(coeffs)
# backward elliptic Fourier and plot reconstructed contour
reconstructed_contour = efDescriptor.backward(M=4096) # M determines how many points of the reconstructed contour
cv2.drawContours(img, np.around(reconstructed_contour).astype(np.int32).reshape((-1,1,2)), -1, (0, 0, 255), 2, cv2.LINE_AA)
cv2.imwrite("example1.jpg", img)Running the following scripts, you will get
A0: 500.32974835623213
C0: 512.9862975766608
Coefficients:
[[-124.16782725 -331.41636558 -329.99128026 232.88912794]
[ -78.04208755 -60.82582495 -66.03245759 -26.06059374]
[ 0.9755801 -24.00522499 -104.851157 -39.06491599]
[ -3.58647986 -51.69773256 1.07226335 54.5695385 ]
[ -30.71359422 -5.70036968 -1.9184519 3.19809928]
[ -2.10359673 -9.16108099 55.73271607 -33.88256265]
[ 4.60579685 5.11983397 -19.0496939 9.76848141]
[ 1.94547198 -1.32610621 -22.03017032 -14.17918943]
[ 8.34267328 6.48530055 2.94044438 -2.37734919]
[ 9.6104728 6.48107343 7.25323288 -6.96029392]]Reconstruct contour with different numbers of coefficients. The more numbers of coefficients, the more accurate the reconstructed contour. All the reconstructed contour overlay on images were saved in directory named example2.
import os
import numpy as np
import cv2
from EllipticFourier import EllipticFourier
# load an image
img_src = "SpongeBob.jpg"
img = cv2.imread(img_src)
# extract image contour
gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY) # conver to gray image
_, binary = cv2.threshold(gray, 127, 255, cv2.THRESH_BINARY_INV) # threshold to get binary image. Note that in the example image, the forground is back, thus we use cv2.THRESH_BINARY_INV.
contours, _ = cv2.findContours(binary, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_NONE) # find all possible contours
contour = np.squeeze(contours[0])
# forward elliptic Fourier
efDescriptor = EllipticFourier()
efDescriptor.forward(contour=contour, N=100) # only forward elliptic Fourier
# reconstruct contour using different numbers of coefficients and save results
dst = "example2"
if not os.path.exists(dst):
os.mkdir(dst)
for num in range(1, 100+1):
reconstructed_contour = efDescriptor.backward(M=4096, modeStart=1, modeNum=num)
raw_img = img = cv2.imread(img_src)
cv2.drawContours(raw_img, np.around(reconstructed_contour).astype(np.int32).reshape((-1,1,2)), -1, (0, 0, 255), 2, cv2.LINE_AA)
image_with_text = cv2.putText(raw_img, f"Mode numbers: {num}", (5,25), cv2.FONT_HERSHEY_SIMPLEX, 1, (0, 0, 0), 2, cv2.LINE_AA)
cv2.imwrite(os.path.join(dst, f"img_{num:0>4}.jpg"), raw_img)3 same shapes of SpongeBob were parameterized using elliptic Fourier descriptor. The coefficients of them are almost the same due to the existence of margin error.
import numpy as np
import cv2
from EllipticFourier import EllipticFourier
def extract_contour(img_src):
# load an image
img = cv2.imread(img_src)
# extract image contour
gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY) # conver to gray image
_, binary = cv2.threshold(gray, 127, 255, cv2.THRESH_BINARY_INV) # threshold to get binary image. Note that in the example image, the forground is back, thus we use cv2.THRESH_BINARY_INV.
contours, _ = cv2.findContours(binary, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_NONE) # find all possible contours
contour = np.squeeze(contours[0])
return contour
contour = extract_contour("SpongeBob.jpg")
contour1 = extract_contour("SpongeBob1.jpg")
contour2 = extract_contour("SpongeBob2.jpg")
efDescriptor = EllipticFourier()
efDescriptor.forward(contour=contour, N=100)
coeffs = efDescriptor.normalize(rotation=True, scale=True)
efDescriptor.forward(contour=contour1, N=100)
coeffs1 = efDescriptor.normalize(rotation=True, scale=True)
efDescriptor.forward(contour=contour2, N=100)
coeffs2 = efDescriptor.normalize(rotation=True, scale=True)
np.savetxt("coeffs.txt", coeffs)
np.savetxt("coeffs1.txt", coeffs1)
np.savetxt("coeffs2.txt", coeffs2)The shape of 3 different SpongeBob were described using elliptic Fourier descriptor. Principal component analysis was performed to reduce and classify shapes.
import numpy as np
import cv2
import matplotlib.pyplot as plt
from EllipticFourier import EllipticFourier
# load an image
img_src = "SpongeBobs.jpg"
img = cv2.imread(img_src)
# extract image contour
contours = []
gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY) # conver to gray image
_, binary = cv2.threshold(gray, 127, 255, cv2.THRESH_BINARY_INV) # threshold to get binary image. Note that in the example image, the forground is back, thus we use cv2.THRESH_BINARY_INV.
contour, _ = cv2.findContours(binary, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_NONE) # find all possible contours
for c in contour:
contours.append(np.squeeze(c))
# get coefficients of all contours
efDescriptor = EllipticFourier()
coefficients = []
for c in contours:
efDescriptor.forward(c, N=100)
coeffs = efDescriptor.normalize(rotation=True, scale=True)
coefficients.append(coeffs.flatten())
coefficients = np.array(coefficients) # [10, 400]
# perform PCA analysis of shapes
coefficients_std = (coefficients - coefficients.mean(axis=0)) / coefficients.std(axis=0)
cov_matrix = np.cov(coefficients, rowvar=False)
eigenvalues, eigenvectors = np.linalg.eigh(cov_matrix)
order_of_importance = np.argsort(eigenvalues)[::-1]
eigenvalues_sorted = eigenvalues[order_of_importance]
eigenvectors_sorted = eigenvectors[:, order_of_importance]
projected_coeffs = np.matmul(coefficients_std, eigenvectors_sorted[:,:2])
explained_variance = np.cumsum(eigenvalues_sorted)/np.sum(eigenvalues_sorted)
explained_variance = np.concatenate([[0], explained_variance])
# plot results
plt.rcParams['font.family'] = 'Times New Roman'
plt.rcParams['font.size'] = 15
plt.rcParams['figure.autolayout'] = True
fig, ax = plt.subplots(1, 2, figsize=(10, 5))
ax[0].scatter(projected_coeffs[:,0], projected_coeffs[:,1])
ax[0].set_xlabel("PC1")
ax[0].set_ylabel("PC2")
ax[0].set_xlim([-6, 6])
ax[0].set_ylim([-6, 6])
ax[1].plot(np.arange(10), explained_variance[:10], marker="o")
ax[1].set_xlabel("Number of principla components")
ax[1].set_ylabel("Total explained vairance")
plt.show()3 same shapes of SpongeBob were parameterized using elliptic Fourier descriptor. The error between reconstructed and original contour were illustrated.
import numpy as np
import cv2
import matplotlib.pyplot as plt
from EllipticFourier import EllipticFourier
def extract_contour(img_src):
# load an image
img = cv2.imread(img_src)
# extract image contour
gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY) # conver to gray image
_, binary = cv2.threshold(gray, 127, 255, cv2.THRESH_BINARY_INV) # threshold to get binary image. Note that in the example image, the forground is back, thus we use cv2.THRESH_BINARY_INV.
contours, _ = cv2.findContours(binary, cv2.RETR_EXTERNAL, cv2.CHAIN_APPROX_NONE) # find all possible contours
contour = np.squeeze(contours[0])
return contour
contour = extract_contour("SpongeBob.jpg")
contour1 = extract_contour("SpongeBob1.jpg")
contour2 = extract_contour("SpongeBob2.jpg")
efDescriptor = EllipticFourier()
error = efDescriptor.error(contour=contour)
error1 = efDescriptor.error(contour=contour1)
error2 = efDescriptor.error(contour=contour2)
# plot results
plt.rcParams['font.family'] = 'Times New Roman'
plt.rcParams['font.size'] = 15
plt.rcParams['figure.autolayout'] = True
fig, ax = plt.subplots(1, 3, figsize=(15, 5))
ax[0].plot(error)
ax[1].plot(error1)
ax[2].plot(error2)
plt.show()- 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. https://doi.org/10.1016/0146-664X(82)90034-X
- Burger, W., Burge, M.J. (2013). Fourier Shape Descriptors. In: Principles of Digital Image Processing. Undergraduate Topics in Computer Science. Springer, London. https://doi.org/10.1007/978-1-84882-919-0_6
- pyefd: Python implementation of "Elliptic Fourier Features of a Closed Contour". https://github.com/hbldh/pyefd
Implementation detials were depicted in the file "EllipticFourierDescriptor.pdf".