-
Notifications
You must be signed in to change notification settings - Fork 15
/
pyefd.py
270 lines (217 loc) · 9.42 KB
/
pyefd.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
A Python implementation of the method described in [#a]_ and [#b]_ for
calculating Fourier coefficients for characterizing
closed contours.
References
----------
.. [#a] 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.
.. [#b] 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 <[email protected]> on 2016-01-30.
"""
from __future__ import division
from __future__ import print_function
from __future__ import unicode_literals
from __future__ import absolute_import
import numpy as np
try:
_range = xrange
except NameError:
_range = range
def elliptic_fourier_descriptors(
contour, order=10, normalize=False, return_transformation=False
):
"""Calculate elliptical Fourier descriptors for a contour.
:param numpy.ndarray contour: A contour array of size ``[M x 2]``.
:param int order: The order of Fourier coefficients to calculate.
:param bool normalize: If the coefficients should be normalized;
see references for details.
:param bool return_transformation: If the normalization parametres should be returned.
Default is ``False``.
:return: A ``[order x 4]`` array of Fourier coefficients and optionally the
transformation parametres ``scale``, ``psi_1`` (rotation) and ``theta_1`` (phase)
:rtype: ::py:class:`numpy.ndarray` or (:py:class:`numpy.ndarray`, (float, float, float))
"""
dxy = np.diff(contour, axis=0)
dt = np.sqrt((dxy ** 2).sum(axis=1))
t = np.concatenate([([0.0]), np.cumsum(dt)])
T = t[-1]
phi = (2 * np.pi * t) / T
orders = np.arange(1, order + 1)
consts = T / (2 * orders * orders * np.pi * np.pi)
phi = phi * orders.reshape((order, -1))
d_cos_phi = np.cos(phi[:, 1:]) - np.cos(phi[:, :-1])
d_sin_phi = np.sin(phi[:, 1:]) - np.sin(phi[:, :-1])
a = consts * np.sum((dxy[:, 0] / dt) * d_cos_phi, axis=1)
b = consts * np.sum((dxy[:, 0] / dt) * d_sin_phi, axis=1)
c = consts * np.sum((dxy[:, 1] / dt) * d_cos_phi, axis=1)
d = consts * np.sum((dxy[:, 1] / dt) * d_sin_phi, axis=1)
coeffs = np.concatenate(
[
a.reshape((order, 1)),
b.reshape((order, 1)),
c.reshape((order, 1)),
d.reshape((order, 1)),
],
axis=1,
)
if normalize:
coeffs = normalize_efd(coeffs, return_transformation=return_transformation)
return coeffs
def normalize_efd(coeffs, size_invariant=True, return_transformation=False):
"""Normalizes an array of Fourier coefficients.
See [#a]_ and [#b]_ for details.
:param numpy.ndarray coeffs: A ``[n x 4]`` Fourier coefficient array.
:param bool size_invariant: If size invariance normalizing should be done as well.
Default is ``True``.
:param bool return_transformation: If the normalization parametres should be returned.
Default is ``False``.
:return: The normalized ``[n x 4]`` Fourier coefficient array and optionally the
transformation parametres ``scale``, :math:`psi_1` (rotation) and :math:`theta_1` (phase)
:rtype: :py:class:`numpy.ndarray` or (:py:class:`numpy.ndarray`, (float, float, float))
"""
# Make the coefficients have a zero phase shift from
# the first major axis. Theta_1 is that shift angle.
theta_1 = 0.5 * np.arctan2(
2 * ((coeffs[0, 0] * coeffs[0, 1]) + (coeffs[0, 2] * coeffs[0, 3])),
(
(coeffs[0, 0] ** 2)
- (coeffs[0, 1] ** 2)
+ (coeffs[0, 2] ** 2)
- (coeffs[0, 3] ** 2)
),
)
# Rotate all coefficients by theta_1.
for n in _range(1, coeffs.shape[0] + 1):
coeffs[n - 1, :] = np.dot(
np.array(
[
[coeffs[n - 1, 0], coeffs[n - 1, 1]],
[coeffs[n - 1, 2], coeffs[n - 1, 3]],
]
),
np.array(
[
[np.cos(n * theta_1), -np.sin(n * theta_1)],
[np.sin(n * theta_1), np.cos(n * theta_1)],
]
),
).flatten()
# Make the coefficients rotation invariant by rotating so that
# the semi-major axis is parallel to the x-axis.
psi_1 = np.arctan2(coeffs[0, 2], coeffs[0, 0])
psi_rotation_matrix = np.array(
[[np.cos(psi_1), np.sin(psi_1)], [-np.sin(psi_1), np.cos(psi_1)]]
)
# Rotate all coefficients by -psi_1.
for n in _range(1, coeffs.shape[0] + 1):
coeffs[n - 1, :] = psi_rotation_matrix.dot(
np.array(
[
[coeffs[n - 1, 0], coeffs[n - 1, 1]],
[coeffs[n - 1, 2], coeffs[n - 1, 3]],
]
)
).flatten()
size = coeffs[0, 0]
if size_invariant:
# Obtain size-invariance by normalizing.
coeffs /= np.abs(size)
if return_transformation:
return coeffs, (size, psi_1, theta_1)
else:
return coeffs
def calculate_dc_coefficients(contour):
"""Calculate the :math:`A_0` and :math:`C_0` coefficients of the elliptic Fourier series.
:param numpy.ndarray contour: A contour array of size ``[M x 2]``.
:return: The :math:`A_0` and :math:`C_0` coefficients.
:rtype: tuple
"""
dxy = np.diff(contour, axis=0)
dt = np.sqrt((dxy ** 2).sum(axis=1))
t = np.concatenate([([0.0]), np.cumsum(dt)])
T = t[-1]
xi = np.cumsum(dxy[:, 0]) - (dxy[:, 0] / dt) * t[1:]
A0 = (1 / T) * np.sum(((dxy[:, 0] / (2 * dt)) * np.diff(t ** 2)) + xi * dt)
delta = np.cumsum(dxy[:, 1]) - (dxy[:, 1] / dt) * t[1:]
C0 = (1 / T) * np.sum(((dxy[:, 1] / (2 * dt)) * np.diff(t ** 2)) + delta * dt)
# A0 and CO relate to the first point of the contour array as origin.
# Adding those values to the coefficients to make them relate to true origin.
return contour[0, 0] + A0, contour[0, 1] + C0
def reconstruct_contour(coeffs, locus=(0, 0), num_points=300):
"""Returns the contour specified by the coefficients.
:param coeffs: A ``[n x 4]`` Fourier coefficient array.
:type coeffs: numpy.ndarray
:param locus: The :math:`A_0` and :math:`C_0` elliptic locus in [#a]_ and [#b]_.
:type locus: list, tuple or numpy.ndarray
:param num_points: The number of sample points used for reconstructing the contour from the EFD.
:type num_points: int
:return: A list of x,y coordinates for the reconstructed contour.
:rtype: numpy.ndarray
"""
t = np.linspace(0, 1.0, num_points)
# Append extra dimension to enable element-wise broadcasted multiplication
coeffs = coeffs.reshape(coeffs.shape[0], coeffs.shape[1], 1)
orders = coeffs.shape[0]
orders = np.arange(1, orders + 1).reshape(-1, 1)
order_phases = 2 * orders * np.pi * t.reshape(1, -1)
xt_all = coeffs[:, 0] * np.cos(order_phases) + coeffs[:, 1] * np.sin(order_phases)
yt_all = coeffs[:, 2] * np.cos(order_phases) + coeffs[:, 3] * np.sin(order_phases)
xt_all = xt_all.sum(axis=0)
yt_all = yt_all.sum(axis=0)
xt_all = xt_all + np.ones((num_points,)) * locus[0]
yt_all = yt_all + np.ones((num_points,)) * locus[1]
reconstruction = np.stack([xt_all, yt_all], axis=1)
return reconstruction
def plot_efd(coeffs, locus=(0.0, 0.0), image=None, contour=None, n=300):
"""Plot a ``[2 x (N / 2)]`` grid of successive truncations of the series.
.. note::
Requires `matplotlib <http://matplotlib.org/>`_!
:param numpy.ndarray coeffs: ``[N x 4]`` Fourier coefficient array.
:param list, tuple or numpy.ndarray locus:
The :math:`A_0` and :math:`C_0` elliptic locus in [#a]_ and [#b]_.
:param int n: Number of points to use for plotting of Fourier series.
"""
try:
import matplotlib.pyplot as plt
except ImportError:
print("Cannot plot: matplotlib was not installed.")
return
N = coeffs.shape[0]
N_half = int(np.ceil(N / 2))
n_rows = 2
t = np.linspace(0, 1.0, n)
xt = np.ones((n,)) * locus[0]
yt = np.ones((n,)) * locus[1]
for n in _range(coeffs.shape[0]):
xt += (coeffs[n, 0] * np.cos(2 * (n + 1) * np.pi * t)) + (
coeffs[n, 1] * np.sin(2 * (n + 1) * np.pi * t)
)
yt += (coeffs[n, 2] * np.cos(2 * (n + 1) * np.pi * t)) + (
coeffs[n, 3] * np.sin(2 * (n + 1) * np.pi * t)
)
ax = plt.subplot2grid((n_rows, N_half), (n // N_half, n % N_half))
ax.set_title(str(n + 1))
if image is not None:
# A background image of shape [rows, cols] gets transposed
# by imshow so that the first dimension is vertical
# and the second dimension is horizontal.
# This implies swapping the x and y axes when plotting a curve.
if contour is not None:
ax.plot(contour[:, 1], contour[:, 0], "c--", linewidth=2)
ax.plot(yt, xt, "r", linewidth=2)
ax.imshow(image, plt.cm.gray)
else:
# Without a background image, no transpose is implied.
# This case is useful when (x,y) point clouds
# without relation to an image are to be handled.
if contour is not None:
ax.plot(contour[:, 0], contour[:, 1], "c--", linewidth=2)
ax.plot(xt, yt, "r", linewidth=2)
ax.axis("equal")
plt.show()