forked from alessandroferrari/elliptic-fourier-descriptors
-
Notifications
You must be signed in to change notification settings - Fork 0
/
elliptic_fourier_descriptors.py
267 lines (178 loc) · 7.9 KB
/
elliptic_fourier_descriptors.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
'''
Created on Jan 7, 2013
@author: Alessandro Ferrari
Fast python/numpy implementation of Elliptic Fourier Descriptors.
Original paper of the implemented algorithm is
"Elliptic Fourier Features of a Closed Contour", Frank P. Kuhl, Charles Giardina, 1981
Repository available at http://www.github.com/alessandroferrari
'''
import cv2
import numpy as np
def sort_contours(contours):
#array of points, at 0th row there will be the upper-left mostly point of the 0th contour of list contours, and so on
points_per_contour = np.zeros((len(contours),2), dtype = int)
#retrieving the upper-left mostly point of the contours
counter = 0
for c in contours:
cres = np.reshape(c,(-1,2))
idx = np.lexsort( (cres[:,0],cres[:,1]) )
sorted_contours = cres[idx,:]
points_per_contour[counter,:] = sorted_contours[0,:]
counter = counter + 1
#sort the contours so that the scan order is raster
sorted_idx = np.lexsort((points_per_contour[:,0], points_per_contour[:,1]))
sorted_contours = []
for idx in sorted_idx:
sorted_contours.append(contours[idx])
#voilla..wonderful raster sorted contours
return sorted_contours
def deltaX_deltaY(bin_im):
_, contours_list, hierarchy= cv2.findContours( bin_im, mode=cv2.RETR_EXTERNAL, method=cv2.CHAIN_APPROX_NONE)
contours_list = sort_contours(contours_list)
deltaXY_list = []
for contour in contours_list:
contour = np.reshape(contour,[-1,2])
dim0, dim1 = contour.shape
difference = np.zeros((dim0, dim1))
difference[0:dim0-1,:] = contour[1:dim0,:] - contour[0:dim0-1,:]
difference[dim0-1,:] = contour[0,:] - contour[dim0-1,:]
deltaX = - difference[:,0]
deltaY = - difference[:,1]
deltaXY_list.append((deltaX, deltaY))
return deltaXY_list
def elliptic_fourier_descriptors(bin_im,N):
deltaXY_list = deltaX_deltaY(bin_im)
num_segments = len(deltaXY_list)
efds= np.zeros((num_segments,N,4))
K_array = np.zeros((num_segments))
T_array = np.zeros((num_segments))
counter = 0
for deltaX, deltaY in deltaXY_list:
efd, K, T = elliptic_fourier_descriptors_segment(deltaX,deltaY,N)
efds[counter,:,:] = efd
K_array[counter] = K
T_array[counter] = T
counter = counter + 1
return efds, K_array, T_array
def elliptic_fourier_descriptors_segment(delta_X, delta_Y, N):
efds = np.zeros((N,4))
delta_t = np.sqrt(np.square(delta_X)+np.square(delta_Y))
K = len(delta_X)
t = np.cumsum(delta_t)
T = np.sum(delta_t)
if T == 0: #just a single pixel segments... particular case
return efds, K, T
# there are N-1 a_n, b_n, c_n, d_n coefficients
n_vector = np.arange(start=1,stop=N,step=1)
n, p = np.meshgrid(n_vector, np.arange(K) )
delta_xp = np.take(delta_X, p)
delta_yp = np.take(delta_Y, p)
delta_tp = np.take(delta_t, p)
tp = np.take(t, p)
tp_current = tp[0:K][:]
tp_prev = np.vstack((tp[K-1][:],tp[0:K-1][:]))
arg_grid1 = 2 * np.pi * n * tp_current / T
arg_grid2 = 2 * np.pi * n * tp_prev / T
cos_grid1 = np.cos(arg_grid1)
cos_grid2 = np.cos(arg_grid2)
sin_grid1 = np.sin(arg_grid1)
sin_grid2 = np.sin(arg_grid2)
factor = (T / (2 * np.pi**2 * np.square(n_vector)))
Xfactor = (delta_xp/delta_tp)
Yfactor = (delta_yp/delta_tp)
prod = Xfactor * ( cos_grid1 - cos_grid2 )
#a_n
efds[1:,0] = factor * np.sum( prod , axis=0)
prod = Xfactor * ( sin_grid1 - sin_grid2 )
#b_n
efds[1:,1] = factor * np.sum( prod , axis=0)
prod = Yfactor * ( cos_grid1 - cos_grid2 )
#c_n
efds[1:,2] = factor * np.sum( prod , axis=0)
prod = Yfactor * ( sin_grid1 - sin_grid2 )
#d_n
efds[1:,3] = factor * np.sum( prod , axis=0)
#A0, C0
efds[0,0], efds[0,2] = continuous_components(delta_X = delta_X, delta_Y = delta_Y,
delta_t = delta_t, t = t, T = T, K = K )
efds = rotation_and_scale_invariance(efds)
return efds, K, T
def continuous_components(delta_X, delta_Y, delta_t, t, T, K):
p = np.arange(K)
delta_xp = np.take(delta_X, p)
delta_yp = np.take(delta_Y, p)
delta_tp = np.take(delta_t, p)
tp = np.take(t, p)
tp = np.hstack( ( np.array([0]) , tp ) )
first_term_xi = np.cumsum(delta_X[0:K-1])
second_term_xi = (delta_X[1:K]/delta_t[1:K]) * np.cumsum(delta_t[0:K-1])
xi = np.hstack( ( np.array([0]), first_term_xi - second_term_xi ) )
first_term_delta = np.cumsum(delta_Y[0:K-1])
second_term_delta = (delta_Y[1:K]/delta_t[1:K]) * np.cumsum(delta_t[0:K-1])
delta = np.hstack( ( np.array([0]), first_term_delta - second_term_delta ) )
A0 = (1/T)*np.sum( (delta_xp/(2*delta_tp) * (np.square(tp[1:K+1]) - np.square(tp[0:K]))) + \
xi * (tp[1:K+1] - tp[0:K]))
C0 = (1/T)*np.sum( (delta_yp/(2*delta_tp) * (np.square(tp[1:K+1]) - np.square(tp[0:K]))) + \
delta * (tp[1:K+1] - tp[0:K]))
return A0, C0
def reconstruct(efds, T, K):
T=np.ceil(T)
N=len(efds)
reconstructed = np.zeros((T,2))
n = np.arange(start=1,stop=N,step=1)
t = np.arange(T)
n_grid, t_grid = np.meshgrid( n, t )
a_n_grid = np.take(efds[:,0], n_grid)
b_n_grid = np.take(efds[:,1], n_grid)
c_n_grid = np.take(efds[:,2], n_grid)
d_n_grid = np.take(efds[:,3], n_grid)
arg_grid = n_grid * t_grid / T
cos_term = np.cos( 2 * np.pi * arg_grid )
sin_term = np.sin( 2 * np.pi * arg_grid )
reconstructed[:,0] = efds[0,0] + np.sum(a_n_grid * cos_term + b_n_grid * sin_term, axis=1)
reconstructed[:,1] = efds[0,0] + np.sum(c_n_grid * cos_term + d_n_grid * sin_term, axis=1)
return reconstructed
def rotation_and_scale_invariance(efds):
N = len(efds)
a1 = efds[1,0]
b1 = efds[1,1]
c1 = efds[1,2]
d1 = efds[1,3]
theta1 = (1.0/2)*np.arctan( 2 * ( a1 * b1 + c1 * d1 ) / ( a1**2 + c1**2 - b1**2 - d1**2 ) )
x11 = a1*np.cos(theta1) + b1*np.sin(theta1)
y11 = c1*np.cos(theta1) + d1*np.sin(theta1)
E0_1 = np.sqrt( np.square(x11) + np.square(y11) )
theta2 = theta1 + np.pi / 2
x22 = a1*np.cos(theta2) + b1*np.sin(theta2)
y22 = c1*np.cos(theta2) + d1*np.sin(theta2)
E0_2 = np.sqrt( np.square(x22) + np.square(y22) )
if E0_1 >= E0_2:
semimajor_axis = E0_1
theta = theta1
else:
semimajor_axis = E0_2
theta = theta2
thetagrid = theta * np.arange(N)
efds_star = efds.copy()
efds_star[:,0] = np.cos(thetagrid) * efds[:,0] + np.sin(thetagrid) * efds[:,1]
efds_star[:,1] = -np.sin(thetagrid) * efds[:,0] + np.cos(thetagrid) * efds[:,1]
efds_star[:,2] = np.cos(thetagrid) * efds[:,2] + np.sin(thetagrid) * efds[:,3]
efds_star[:,3] = -np.sin(thetagrid) * efds[:,2] + np.cos(thetagrid) * efds[:,3]
if (efds_star[1,2]!=0):
if (efds_star[1,0]>=0):
phi1 = np.arctan(efds_star[1,2] / efds_star[1,0])
else:
phi1 = np.arctan(efds_star[1,2] / efds_star[1,0]) + np.pi
else:
if (efds_star[1,2]>0):
phi1 = np.arctan(efds_star[1,2] / efds_star[1,0])
else:
phi1 = np.arctan(efds_star[1,2] / efds_star[1,0]) + np.pi
alpha = np.cos(phi1)
beta = np.sin(phi1)
efds_2star = efds_star.copy()
efds_2star[:,0] = (alpha * efds_star[:,0] + beta * efds_star[:,2]) / semimajor_axis
efds_2star[:,1] = (alpha * efds_star[:,1] + beta * efds_star[:,3]) / semimajor_axis
efds_2star[:,2] = (-beta * efds_star[:,0] + alpha * efds_star[:,2]) / semimajor_axis
efds_2star[:,3] = (-beta * efds_star[:,1] + alpha * efds_star[:,3]) / semimajor_axis
return efds_2star