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scale_and_shift_two_focal.py
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import numpy as np
import madpose
def solve_shift_and_scale_two_focal(x1_, x2_, d1, d2):
# Estimates scale, shift, and two focal lengths
# x1 should be four image points, i.e. x1[0] = [x,y,1]
# with corresponding depth value d1[0]
#
# Returns list of solutions (a1,b1,a2,b2,f1, f2) where
# d1_corrected = a1 * d1 + b1
# d2_corrected = a2 * d2 + b2
# and f1 and f2 is the two focal lengths.
# Note that only the first 3 points satisfy the rigidity constraints.
# Normalize focal length
x1 = x1_.copy()
x2 = x2_.copy()
f1_0 = np.abs(x1[:, 0:2].flatten()).mean()
f2_0 = np.abs(x2[:, 0:2].flatten()).mean()
x1[:, 0:2] /= f1_0
x2[:, 0:2] /= f2_0
# Compute coefficients
coeffs = np.zeros((40,))
coeffs[0] = (
2 * x2[0, 0] * x2[1, 0]
+ 2 * x2[0, 1] * x2[1, 1]
- (x2[0, 0] ** 2)
- (x2[0, 1] ** 2)
- (x2[1, 0] ** 2)
- (x2[1, 1] ** 2)
)
coeffs[1] = (
(x1[0, 0] ** 2)
- 2 * x1[0, 1] * x1[1, 1]
- 2 * x1[0, 0] * x1[1, 0]
+ (x1[0, 1] ** 2)
+ (x1[1, 0] ** 2)
+ (x1[1, 1] ** 2)
)
coeffs[2] = (
2 * d2[0] * x2[0, 0] * x2[1, 0]
- 2 * d2[0] * (x2[0, 1] ** 2)
- 2 * d2[1] * (x2[1, 0] ** 2)
- 2 * d2[1] * (x2[1, 1] ** 2)
- 2 * d2[0] * (x2[0, 0] ** 2)
+ 2 * d2[1] * x2[0, 0] * x2[1, 0]
+ 2 * d2[0] * x2[0, 1] * x2[1, 1]
+ 2 * d2[1] * x2[0, 1] * x2[1, 1]
)
coeffs[3] = (
2 * d1[0] * (x1[0, 0] ** 2)
+ 2 * d1[0] * (x1[0, 1] ** 2)
+ 2 * d1[1] * (x1[1, 0] ** 2)
+ 2 * d1[1] * (x1[1, 1] ** 2)
- 2 * d1[0] * x1[0, 0] * x1[1, 0]
- 2 * d1[1] * x1[0, 0] * x1[1, 0]
- 2 * d1[0] * x1[0, 1] * x1[1, 1]
- 2 * d1[1] * x1[0, 1] * x1[1, 1]
)
coeffs[4] = (
2 * d2[0] * d2[1] * x2[0, 0] * x2[1, 0]
- (d2[0] ** 2) * (x2[0, 1] ** 2)
- (d2[1] ** 2) * (x2[1, 0] ** 2)
- (d2[1] ** 2) * (x2[1, 1] ** 2)
- (d2[0] ** 2) * (x2[0, 0] ** 2)
+ 2 * d2[0] * d2[1] * x2[0, 1] * x2[1, 1]
)
coeffs[5] = 2 * d2[0] * d2[1] - (d2[0] ** 2) - (d2[1] ** 2)
coeffs[6] = (
(d1[0] ** 2) * (x1[0, 0] ** 2)
+ (d1[0] ** 2) * (x1[0, 1] ** 2)
+ (d1[1] ** 2) * (x1[1, 0] ** 2)
+ (d1[1] ** 2) * (x1[1, 1] ** 2)
- 2 * d1[0] * d1[1] * x1[0, 0] * x1[1, 0]
- 2 * d1[0] * d1[1] * x1[0, 1] * x1[1, 1]
)
coeffs[7] = (d1[0] ** 2) - 2 * d1[0] * d1[1] + (d1[1] ** 2)
coeffs[8] = (
2 * x2[0, 0] * x2[2, 0]
+ 2 * x2[0, 1] * x2[2, 1]
- (x2[0, 0] ** 2)
- (x2[0, 1] ** 2)
- (x2[2, 0] ** 2)
- (x2[2, 1] ** 2)
)
coeffs[9] = (
(x1[0, 0] ** 2)
- 2 * x1[0, 1] * x1[2, 1]
- 2 * x1[0, 0] * x1[2, 0]
+ (x1[0, 1] ** 2)
+ (x1[2, 0] ** 2)
+ (x1[2, 1] ** 2)
)
coeffs[10] = (
2 * d2[0] * x2[0, 0] * x2[2, 0]
- 2 * d2[0] * (x2[0, 1] ** 2)
- 2 * d2[2] * (x2[2, 0] ** 2)
- 2 * d2[2] * (x2[2, 1] ** 2)
- 2 * d2[0] * (x2[0, 0] ** 2)
+ 2 * d2[0] * x2[0, 1] * x2[2, 1]
+ 2 * d2[2] * x2[0, 0] * x2[2, 0]
+ 2 * d2[2] * x2[0, 1] * x2[2, 1]
)
coeffs[11] = (
2 * d1[0] * (x1[0, 0] ** 2)
+ 2 * d1[0] * (x1[0, 1] ** 2)
+ 2 * d1[2] * (x1[2, 0] ** 2)
+ 2 * d1[2] * (x1[2, 1] ** 2)
- 2 * d1[0] * x1[0, 0] * x1[2, 0]
- 2 * d1[0] * x1[0, 1] * x1[2, 1]
- 2 * d1[2] * x1[0, 0] * x1[2, 0]
- 2 * d1[2] * x1[0, 1] * x1[2, 1]
)
coeffs[12] = (
2 * d2[0] * d2[2] * x2[0, 0] * x2[2, 0]
- (d2[0] ** 2) * (x2[0, 1] ** 2)
- (d2[2] ** 2) * (x2[2, 0] ** 2)
- (d2[2] ** 2) * (x2[2, 1] ** 2)
- (d2[0] ** 2) * (x2[0, 0] ** 2)
+ 2 * d2[0] * d2[2] * x2[0, 1] * x2[2, 1]
)
coeffs[13] = 2 * d2[0] * d2[2] - (d2[0] ** 2) - (d2[2] ** 2)
coeffs[14] = (
(d1[0] ** 2) * (x1[0, 0] ** 2)
+ (d1[0] ** 2) * (x1[0, 1] ** 2)
+ (d1[2] ** 2) * (x1[2, 0] ** 2)
+ (d1[2] ** 2) * (x1[2, 1] ** 2)
- 2 * d1[0] * d1[2] * x1[0, 0] * x1[2, 0]
- 2 * d1[0] * d1[2] * x1[0, 1] * x1[2, 1]
)
coeffs[15] = (d1[0] ** 2) - 2 * d1[0] * d1[2] + (d1[2] ** 2)
coeffs[16] = (
2 * x2[1, 0] * x2[2, 0]
+ 2 * x2[1, 1] * x2[2, 1]
- (x2[1, 0] ** 2)
- (x2[1, 1] ** 2)
- (x2[2, 0] ** 2)
- (x2[2, 1] ** 2)
)
coeffs[17] = (
(x1[1, 0] ** 2)
- 2 * x1[1, 1] * x1[2, 1]
- 2 * x1[1, 0] * x1[2, 0]
+ (x1[1, 1] ** 2)
+ (x1[2, 0] ** 2)
+ (x1[2, 1] ** 2)
)
coeffs[18] = (
2 * d2[1] * x2[1, 0] * x2[2, 0]
- 2 * d2[1] * (x2[1, 1] ** 2)
- 2 * d2[2] * (x2[2, 0] ** 2)
- 2 * d2[2] * (x2[2, 1] ** 2)
- 2 * d2[1] * (x2[1, 0] ** 2)
+ 2 * d2[2] * x2[1, 0] * x2[2, 0]
+ 2 * d2[1] * x2[1, 1] * x2[2, 1]
+ 2 * d2[2] * x2[1, 1] * x2[2, 1]
)
coeffs[19] = (
2 * d1[1] * (x1[1, 0] ** 2)
+ 2 * d1[1] * (x1[1, 1] ** 2)
+ 2 * d1[2] * (x1[2, 0] ** 2)
+ 2 * d1[2] * (x1[2, 1] ** 2)
- 2 * d1[1] * x1[1, 0] * x1[2, 0]
- 2 * d1[2] * x1[1, 0] * x1[2, 0]
- 2 * d1[1] * x1[1, 1] * x1[2, 1]
- 2 * d1[2] * x1[1, 1] * x1[2, 1]
)
coeffs[20] = (
2 * d2[1] * d2[2] * x2[1, 0] * x2[2, 0]
- (d2[1] ** 2) * (x2[1, 1] ** 2)
- (d2[2] ** 2) * (x2[2, 0] ** 2)
- (d2[2] ** 2) * (x2[2, 1] ** 2)
- (d2[1] ** 2) * (x2[1, 0] ** 2)
+ 2 * d2[1] * d2[2] * x2[1, 1] * x2[2, 1]
)
coeffs[21] = 2 * d2[1] * d2[2] - (d2[1] ** 2) - (d2[2] ** 2)
coeffs[22] = (
(d1[1] ** 2) * (x1[1, 0] ** 2)
+ (d1[1] ** 2) * (x1[1, 1] ** 2)
+ (d1[2] ** 2) * (x1[2, 0] ** 2)
+ (d1[2] ** 2) * (x1[2, 1] ** 2)
- 2 * d1[1] * d1[2] * x1[1, 0] * x1[2, 0]
- 2 * d1[1] * d1[2] * x1[1, 1] * x1[2, 1]
)
coeffs[23] = (d1[1] ** 2) - 2 * d1[1] * d1[2] + (d1[2] ** 2)
coeffs[24] = (
2 * x2[0, 0] * x2[3, 0]
+ 2 * x2[0, 1] * x2[3, 1]
- (x2[0, 0] ** 2)
- (x2[0, 1] ** 2)
- (x2[3, 0] ** 2)
- (x2[3, 1] ** 2)
)
coeffs[25] = (
(x1[0, 0] ** 2)
- 2 * x1[0, 1] * x1[3, 1]
- 2 * x1[0, 0] * x1[3, 0]
+ (x1[0, 1] ** 2)
+ (x1[3, 0] ** 2)
+ (x1[3, 1] ** 2)
)
coeffs[26] = (
2 * d2[0] * x2[0, 0] * x2[3, 0]
- 2 * d2[0] * (x2[0, 1] ** 2)
- 2 * d2[3] * (x2[3, 0] ** 2)
- 2 * d2[3] * (x2[3, 1] ** 2)
- 2 * d2[0] * (x2[0, 0] ** 2)
+ 2 * d2[0] * x2[0, 1] * x2[3, 1]
+ 2 * d2[3] * x2[0, 0] * x2[3, 0]
+ 2 * d2[3] * x2[0, 1] * x2[3, 1]
)
coeffs[27] = (
2 * d1[0] * (x1[0, 0] ** 2)
+ 2 * d1[0] * (x1[0, 1] ** 2)
+ 2 * d1[3] * (x1[3, 0] ** 2)
+ 2 * d1[3] * (x1[3, 1] ** 2)
- 2 * d1[0] * x1[0, 0] * x1[3, 0]
- 2 * d1[0] * x1[0, 1] * x1[3, 1]
- 2 * d1[3] * x1[0, 0] * x1[3, 0]
- 2 * d1[3] * x1[0, 1] * x1[3, 1]
)
coeffs[28] = (
2 * d2[0] * d2[3] * x2[0, 0] * x2[3, 0]
- (d2[0] ** 2) * (x2[0, 1] ** 2)
- (d2[3] ** 2) * (x2[3, 0] ** 2)
- (d2[3] ** 2) * (x2[3, 1] ** 2)
- (d2[0] ** 2) * (x2[0, 0] ** 2)
+ 2 * d2[0] * d2[3] * x2[0, 1] * x2[3, 1]
)
coeffs[29] = 2 * d2[0] * d2[3] - (d2[0] ** 2) - (d2[3] ** 2)
coeffs[30] = (
(d1[0] ** 2) * (x1[0, 0] ** 2)
+ (d1[0] ** 2) * (x1[0, 1] ** 2)
+ (d1[3] ** 2) * (x1[3, 0] ** 2)
+ (d1[3] ** 2) * (x1[3, 1] ** 2)
- 2 * d1[0] * d1[3] * x1[0, 0] * x1[3, 0]
- 2 * d1[0] * d1[3] * x1[0, 1] * x1[3, 1]
)
coeffs[31] = (d1[0] ** 2) - 2 * d1[0] * d1[3] + (d1[3] ** 2)
coeffs[32] = (
2 * x2[1, 0] * x2[3, 0]
+ 2 * x2[1, 1] * x2[3, 1]
- (x2[1, 0] ** 2)
- (x2[1, 1] ** 2)
- (x2[3, 0] ** 2)
- (x2[3, 1] ** 2)
)
coeffs[33] = (
(x1[1, 0] ** 2)
- 2 * x1[1, 1] * x1[3, 1]
- 2 * x1[1, 0] * x1[3, 0]
+ (x1[1, 1] ** 2)
+ (x1[3, 0] ** 2)
+ (x1[3, 1] ** 2)
)
coeffs[34] = (
2 * d2[1] * x2[1, 0] * x2[3, 0]
- 2 * d2[1] * (x2[1, 1] ** 2)
- 2 * d2[3] * (x2[3, 0] ** 2)
- 2 * d2[3] * (x2[3, 1] ** 2)
- 2 * d2[1] * (x2[1, 0] ** 2)
+ 2 * d2[1] * x2[1, 1] * x2[3, 1]
+ 2 * d2[3] * x2[1, 0] * x2[3, 0]
+ 2 * d2[3] * x2[1, 1] * x2[3, 1]
)
coeffs[35] = (
2 * d1[1] * (x1[1, 0] ** 2)
+ 2 * d1[1] * (x1[1, 1] ** 2)
+ 2 * d1[3] * (x1[3, 0] ** 2)
+ 2 * d1[3] * (x1[3, 1] ** 2)
- 2 * d1[1] * x1[1, 0] * x1[3, 0]
- 2 * d1[1] * x1[1, 1] * x1[3, 1]
- 2 * d1[3] * x1[1, 0] * x1[3, 0]
- 2 * d1[3] * x1[1, 1] * x1[3, 1]
)
coeffs[36] = (
2 * d2[1] * d2[3] * x2[1, 0] * x2[3, 0]
- (d2[1] ** 2) * (x2[1, 1] ** 2)
- (d2[3] ** 2) * (x2[3, 0] ** 2)
- (d2[3] ** 2) * (x2[3, 1] ** 2)
- (d2[1] ** 2) * (x2[1, 0] ** 2)
+ 2 * d2[1] * d2[3] * x2[1, 1] * x2[3, 1]
)
coeffs[37] = 2 * d2[1] * d2[3] - (d2[1] ** 2) - (d2[3] ** 2)
coeffs[38] = (
(d1[1] ** 2) * (x1[1, 0] ** 2)
+ (d1[1] ** 2) * (x1[1, 1] ** 2)
+ (d1[3] ** 2) * (x1[3, 0] ** 2)
+ (d1[3] ** 2) * (x1[3, 1] ** 2)
- 2 * d1[1] * d1[3] * x1[1, 0] * x1[3, 0]
- 2 * d1[1] * d1[3] * x1[1, 1] * x1[3, 1]
)
coeffs[39] = (d1[1] ** 2) - 2 * d1[1] * d1[3] + (d1[3] ** 2)
# fmt: off
# Setup expanded equation system
coeff_ind0 = [0, 8, 16, 24, 32, 0, 8, 16, 24, 32, 1, 9, 17, 25, 33, 2, 10, 0, 8, 16, 18, 24, 26, 32, 34, 0, 8, 16, 24, 32, 1, 9, 17, 25, 33, 2, 10, 8, 0, 16, 18, 24, 26, 32, 34, 8, 0, 16, 24, 32, 1, 9, 17, 25, 33, 3, 11, 1, 9, 17, 19, 25, 27, 33, 35, 4, 12, 2, 10, 18, 20, 26, 28, 34, 36, 2, 10, 8, 16, 18, 0, 26, 24, 32, 34, 9, 1, 17, 25, 33, 3, 11, 9, 1, 19, 17, 27, 25, 33, 35, 5, 4, 13, 12, 10, 2, 18, 21, 20, 26, 29, 28, 34, 36, 37, 10, 2, 8, 0, 18, 24, 26, 32, 34, 16, 3, 11, 19, 9, 17, 1, 27, 25, 33, 35, 6, 14, 3, 11, 19, 22, 27, 30, 35, 38, 4, 12, 20, 28, 36,
4, 12, 10, 18, 20, 2, 28, 26, 34, 36, 5, 13, 21, 29, 37, 11, 3, 19, 9, 1, 27, 25, 33, 17, 35, 6, 14, 11, 3, 22, 19, 30, 27, 35, 38, 5, 12, 13, 21, 4, 20, 28, 29, 36, 37, 5, 12, 13, 4, 10, 21, 2, 20, 26, 29, 28, 34, 36, 37, 18, 7, 15, 23, 31, 39, 6, 14, 22, 11, 19, 3, 30, 27, 35, 38, 6, 14, 22, 30, 38, 12, 20, 4, 28, 36, 13, 5, 21, 29, 37, 13, 5, 21, 29, 37, 14, 6, 22, 30, 38, 13, 12, 21, 5, 4, 28, 29, 37, 36, 20, 7, 15, 23, 31, 39, 14, 22, 6, 30, 38, 13, 5, 29, 37, 21, 15, 7, 23, 31, 39, 7, 15, 23, 31, 39, 14, 6, 22, 11, 3, 30, 27, 35, 19, 38, 7, 15, 23, 31, 39,]
coeff_ind1 = [15, 7, 31, 39, 23, 15, 7, 23, 31,
39, 14, 6, 30, 38, 22, 15, 23, 7, 31, 39,]
ind0 = [0, 2, 18, 28, 39, 41, 45, 60, 69, 78, 80, 82, 98, 108, 119, 120, 122, 123, 128, 130, 138, 145, 148, 157, 159, 164, 169, 177, 186, 194, 201, 205, 220, 229, 238, 241, 245, 246, 252, 254, 260, 263, 269, 276, 278, 287, 293, 302, 310, 313, 323, 328, 330, 345, 357, 360, 362, 364, 369, 377, 378, 386, 388, 394, 399, 400, 402, 403, 408, 410, 418, 425, 428, 437, 439, 444, 449, 451, 456, 457, 459, 466, 467, 471, 474, 486, 492, 494, 503, 516, 521, 525, 527, 533, 540, 542, 549, 550, 553, 558, 560, 561, 562, 565, 566, 572, 574, 578, 580, 583, 588, 589, 596, 598, 599, 607, 613, 615, 621, 622, 624, 630, 632, 633, 635, 643, 648, 650, 651, 656, 659, 665, 667, 671, 677, 680, 682, 684, 689, 697, 698, 706, 708, 714, 719, 723, 728, 730, 745, 757, 764, 769, 771, 776, 777, 779, 786, 787, 791, 794,
801, 805, 820, 829, 838, 846, 852, 854, 855, 861, 863, 864, 872, 875, 876, 881, 885, 887, 893, 900, 902, 909, 910, 913, 918, 923, 926, 928, 930, 932, 934, 943, 945, 956, 957, 964, 967, 969, 973, 975, 977, 981, 982, 984, 986, 990, 992, 993, 994, 995, 1000, 1002, 1018, 1028, 1039, 1043, 1048, 1050, 1051, 1056, 1059, 1065, 1067, 1071, 1077, 1084, 1089, 1097, 1106, 1114, 1131, 1136, 1139, 1147, 1151, 1166, 1172, 1174, 1183, 1196, 1207, 1213, 1222, 1230, 1233, 1247, 1253, 1262, 1270, 1273, 1291, 1295, 1296, 1299, 1301, 1304, 1307, 1311, 1312, 1315, 1324, 1329, 1337, 1346, 1354, 1371, 1376, 1379, 1387, 1391, 1415, 1421, 1424, 1432, 1435, 1446, 1452, 1454, 1463, 1476, 1481, 1485, 1500, 1509, 1518, 1526, 1532, 1534, 1535, 1541, 1543, 1544, 1552, 1555, 1556, 1563, 1568, 1570, 1585, 1597,]
ind1 = [15, 21, 24, 32, 35, 47, 53, 62, 70, 73, 95,
101, 104, 112, 115, 131, 136, 139, 147, 151,]
# fmt: on
C0 = np.zeros((40, 40))
C1 = np.zeros((40, 4))
C0[np.unravel_index(ind0, (40, 40), "F")] = coeffs[coeff_ind0]
C1[np.unravel_index(ind1, (40, 4), "F")] = coeffs[coeff_ind1]
# Linear elimination
C2 = np.linalg.solve(C0, C1)
# Setup action matrix
AM = np.array([-C2[36, :], -C2[37, :], -C2[38, :], -C2[39, :]])
# Solve eigenvalue problem and get real solutions
D, V = np.linalg.eig(AM)
sols = np.array(
[
(-C2[35, :] @ V) / V[0, :],
D,
V[1, :] / V[0, :],
V[2, :] / V[0, :],
V[3, :] / V[0, :],
]
).T
sols = sols[np.isreal(D), :]
# Extract solutions
solutions = []
for s in sols:
s = np.real(s)
if s[3] < 0 or s[4] < 0:
continue
solutions.append(
(
1.0,
s[1],
np.sqrt(s[0]),
s[2] * np.sqrt(s[0]),
f1_0 / np.sqrt(s[3]),
f2_0 / np.sqrt(s[4]),
)
)
return solutions
def find_transform(X1, X2):
m1 = np.mean(X1, axis=0)
m2 = np.mean(X2, axis=0)
X1m = X1 - m1
X2m = X2 - m2
u, s, vt = np.linalg.svd(X2m.T @ X1m)
R = u @ np.diag([1.0, 1.0, np.linalg.det(u @ vt)]) @ vt
t = m2 - R @ m1
return R, t
def test_solver():
# Setup instance (with positive depths)
while True:
x1 = np.c_[np.random.randn(4, 2), np.ones((4,))]
f1_gt = 1000.0 + 2000.0 * np.random.rand(1)
f2_gt = 1000.0 + 2000.0 * np.random.rand(1)
d1_gt = 1.0 + 5 * np.random.rand(4)
X = x1 * d1_gt[:, None]
R = np.linalg.qr(np.random.randn(3, 3))[0]
R = R * np.linalg.det(R)
t = np.random.randn(3)
X2 = X @ R.T + t
d2_gt = X2[:, 2]
x2 = X2 / d2_gt[:, None]
# Add shared focal length
x1[:, 0:2] *= f1_gt
x2[:, 0:2] *= f2_gt
x2[:, 0:2] += np.random.randn(4, 2) * 1 + 0.5
if np.all(d2_gt > 0):
break
# Shift and scale gt depths
a1_gt = np.random.rand(1)
b1_gt = np.random.randn(1)
a2_gt = np.random.rand(1)
b2_gt = np.random.randn(1)
# d1_gt = a1 * d1 + b1
d1 = (d1_gt - b1_gt) / a1_gt
d2 = (d2_gt - b2_gt) / a2_gt
sols = solve_shift_and_scale_two_focal(x1, x2, d1, d2)
sols_madpose = madpose.solve_scale_and_shift_two_focal(x1.T, x2.T, d1, d2)
# poses = madpose.solve_scale_shift_pose_two_focal(x1.T, x2.T, d1, d2)
print(len(sols), len(sols_madpose))
for k, (a1, b1, a2, b2, f1, f2) in enumerate(sols + sols_madpose):
err = (
np.abs(a2 - a2_gt / a1_gt)
+ np.abs(b1 - b1_gt / a1_gt)
+ np.abs(b2 - b2_gt / a1_gt)
+ np.abs(f1 - f1_gt)
+ np.abs(f2 - f2_gt)
)
focal_err = np.abs(f1 - f1_gt) + np.abs(f2 - f2_gt)
print(f"focal_error={focal_err}")
t_skew = np.array([[0, -t[2], t[1]], [t[2], 0, -t[0]], [-t[1], t[0], 0]])
E = t_skew @ R
K2_ = np.array([[f2, 0, 0], [0, f2, 0], [0, 0, 1]])
K1_ = np.array([[f1, 0, 0], [0, f1, 0], [0, 0, 1]])
F = np.linalg.inv(K2_.T) @ E @ K1_
d1_corr = a1 * d1 + b1
d2_corr = a2 * d2 + b2
x1u = x1.copy()
x1u[:, 0:2] /= f1
x2u = x2.copy()
x2u[:, 0:2] /= f2
X1 = x1u * d1_corr[:, None]
X2 = x2u * d2_corr[:, None]
R_est, t_est = find_transform(X1[0:3, :], X2[0:3, :])
t_est_skew = np.array(
[
[0, -t_est[2], t_est[1]],
[t_est[2], 0, -t_est[0]],
[-t_est[1], t_est[0], 0],
]
)
E_est = t_est_skew @ R_est
K2_est = np.array([[f2, 0, 0], [0, f2, 0], [0, 0, 1]])
K1_est = np.array([[f1, 0, 0], [0, f1, 0], [0, 0, 1]])
F_est = np.linalg.inv(K2_est.T) @ E_est @ K1_est
# normalize F
F = F / np.linalg.norm(F)
F_est = F_est / np.linalg.norm(F_est)
F_error = np.linalg.norm(F - F_est)
print(f"F_error={F_error}")
err_R = np.linalg.norm(R - R_est)
err_t = np.linalg.norm(t / np.linalg.norm(t) - t_est / np.linalg.norm(t_est))
print(f"solution={k}, residual={err}, rotation={err_R}, translation={err_t}")
if __name__ == "__main__":
test_solver()