/usr/share/pyshared/dipy/reconst/dti.py is in python-dipy 0.7.1-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 | #!/usr/bin/python
""" Classes and functions for fitting tensors """
from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
import scipy.optimize as opt
from dipy.utils.six.moves import range
from dipy.data import get_sphere
from ..core.gradients import gradient_table
from ..core.geometry import vector_norm
from ..core.sphere import Sphere
from .vec_val_sum import vec_val_vect
from ..core.onetime import auto_attr
from .base import ReconstModel, ReconstFit
def _roll_evals(evals, axis=-1):
"""
Helper function to check that the evals provided to functions calculating
tensor statistics have the right shape
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor. shape should be (...,3).
axis : int
The axis of the array which contains the 3 eigenvals. Default: -1
Returns
-------
evals : array-like
Eigenvalues of a diffusion tensor, rolled so that the 3 eigenvals are
the last axis.
"""
if evals.shape[-1] != 3:
msg = "Expecting 3 eigenvalues, got {}".format(evals.shape[-1])
raise ValueError(msg)
evals = np.rollaxis(evals, axis)
return evals
def fractional_anisotropy(evals, axis=-1):
r"""
Fractional anisotropy (FA) of a diffusion tensor.
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
fa : array
Calculated FA. Range is 0 <= FA <= 1.
Notes
--------
FA is calculated using the following equation:
.. math::
FA = \sqrt{\frac{1}{2}\frac{(\lambda_1-\lambda_2)^2+(\lambda_1-
\lambda_3)^2+(\lambda_2-\lambda_3)^2}{\lambda_1^2+
\lambda_2^2+\lambda_3^2}}
"""
evals = _roll_evals(evals, axis)
# Make sure not to get nans
all_zero = (evals == 0).all(axis=0)
ev1, ev2, ev3 = evals
fa = np.sqrt(0.5 * ((ev1 - ev2) ** 2 + (ev2 - ev3) ** 2 + (ev3 - ev1) ** 2)
/ ((evals * evals).sum(0) + all_zero))
return fa
def mean_diffusivity(evals, axis=-1):
r"""
Mean Diffusivity (MD) of a diffusion tensor.
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
md : array
Calculated MD.
Notes
--------
MD is calculated with the following equation:
.. math::
MD = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}
"""
evals = _roll_evals(evals, axis)
return evals.mean(0)
def axial_diffusivity(evals, axis=-1):
r"""
Axial Diffusivity (AD) of a diffusion tensor.
Also called parallel diffusivity.
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor, must be sorted in descending order
along `axis`.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
ad : array
Calculated AD.
Notes
--------
AD is calculated with the following equation:
.. math::
AD = \lambda_1
"""
evals = _roll_evals(evals, axis)
ev1, ev2, ev3 = evals
return ev1
def radial_diffusivity(evals, axis=-1):
r"""
Radial Diffusivity (RD) of a diffusion tensor.
Also called perpendicular diffusivity.
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor, must be sorted in descending order
along `axis`.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
rd : array
Calculated RD.
Notes
--------
RD is calculated with the following equation:
.. math::
RD = \frac{\lambda_2 + \lambda_3}{2}
"""
evals = _roll_evals(evals, axis)
return evals[1:].mean(0)
def trace(evals, axis=-1):
r"""
Trace of a diffusion tensor.
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
trace : array
Calculated trace of the diffusion tensor.
Notes
--------
Trace is calculated with the following equation:
.. math::
Trace = \lambda_1 + \lambda_2 + \lambda_3
"""
evals = _roll_evals(evals, axis)
return evals.sum(0)
def color_fa(fa, evecs):
r""" Color fractional anisotropy of diffusion tensor
Parameters
----------
fa : array-like
Array of the fractional anisotropy (can be 1D, 2D or 3D)
evecs : array-like
eigen vectors from the tensor model
Returns
-------
rgb : Array with 3 channels for each color as the last dimension.
Colormap of the FA with red for the x value, y for the green
value and z for the blue value.
Note
-----
It is computed from the clipped FA between 0 and 1 using the following
formula
.. math::
rgb = abs(max(\vec{e})) \times fa
"""
if (fa.shape != evecs[..., 0, 0].shape) or ((3, 3) != evecs.shape[-2:]):
raise ValueError("Wrong number of dimensions for evecs")
return np.abs(evecs[..., 0]) * np.clip(fa, 0, 1)[..., None]
# The following are used to calculate the tensor mode:
def determinant(q_form):
"""
The determinant of a tensor, given in quadratic form
Parameters
----------
q_form : ndarray
The quadratic form of a tensor, or an array with quadratic forms of
tensors. Should be of shape (x, y, z, 3, 3) or (n, 3, 3) or (3, 3).
Returns
-------
det : array
The determinant of the tensor in each spatial coordinate
"""
# Following the conventions used here:
# http://en.wikipedia.org/wiki/Determinant
aei = q_form[..., 0, 0] * q_form[..., 1, 1] * q_form[..., 2, 2]
bfg = q_form[..., 0, 1] * q_form[..., 1, 2] * q_form[..., 2, 0]
cdh = q_form[..., 0, 2] * q_form[..., 1, 0] * q_form[..., 2, 1]
ceg = q_form[..., 0, 2] * q_form[..., 1, 1] * q_form[..., 2, 0]
bdi = q_form[..., 0, 1] * q_form[..., 1, 0] * q_form[..., 2, 2]
afh = q_form[..., 0, 0] * q_form[..., 1, 2] * q_form[..., 2, 1]
return aei + bfg + cdh - ceg - bdi - afh
def isotropic(q_form):
r"""
Calculate the isotropic part of the tensor [1]_.
Parameters
----------
q_form : ndarray
The quadratic form of a tensor, or an array with quadratic forms of
tensors. Should be of shape (x,y,z,3,3) or (n, 3, 3) or (3,3).
Returns
-------
A_hat: ndarray
The isotropic part of the tensor in each spatial coordinate
Notes
-----
The isotropic part of a tensor is defined as (equations 3-5 of [1]_):
.. math ::
\bar{A} = \frac{1}{2} tr(A) I
.. [1] Daniel B. Ennis and G. Kindlmann, "Orthogonal Tensor
Invariants and the Analysis of Diffusion Tensor Magnetic Resonance
Images", Magnetic Resonance in Medicine, vol. 55, no. 1, pp. 136-146,
2006.
"""
tr_A = q_form[..., 0, 0] + q_form[..., 1, 1] + q_form[..., 2, 2]
n_dims = len(q_form.shape)
add_dims = n_dims - 2 # These are the last two (the 3,3):
my_I = np.eye(3)
tr_AI = (tr_A.reshape(tr_A.shape + (1, 1)) * my_I)
return (1 / 3.0) * tr_AI
def deviatoric(q_form):
r"""
Calculate the deviatoric (anisotropic) part of the tensor [1]_.
Parameters
----------
q_form : ndarray
The quadratic form of a tensor, or an array with quadratic forms of
tensors. Should be of shape (x,y,z,3,3) or (n, 3, 3) or (3,3).
Returns
-------
A_squiggle : ndarray
The deviatoric part of the tensor in each spatial coordinate.
Notes
-----
The deviatoric part of the tensor is defined as (equations 3-5 in [1]_):
.. math ::
\widetilde{A} = A - \bar{A}
Where $A$ is the tensor quadratic form and $\bar{A}$ is the anisotropic
part of the tensor.
.. [1] Daniel B. Ennis and G. Kindlmann, "Orthogonal Tensor
Invariants and the Analysis of Diffusion Tensor Magnetic Resonance
Images", Magnetic Resonance in Medicine, vol. 55, no. 1, pp. 136-146,
2006.
"""
A_squiggle = q_form - isotropic(q_form)
return A_squiggle
def norm(q_form):
r"""
Calculate the Frobenius norm of a tensor quadratic form
Parameters
----------
q_form: ndarray
The quadratic form of a tensor, or an array with quadratic forms of
tensors. Should be of shape (x,y,z,3,3) or (n, 3, 3) or (3,3).
Returns
-------
norm : ndarray
The Frobenius norm of the 3,3 tensor q_form in each spatial
coordinate.
Notes
-----
The Frobenius norm is defined as:
:math:
||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}
See also
--------
np.linalg.norm
"""
return np.sqrt(np.sum(np.sum(np.abs(q_form ** 2), -1), -1))
def mode(q_form):
r"""
Mode (MO) of a diffusion tensor [1]_.
Parameters
----------
q_form : ndarray
The quadratic form of a tensor, or an array with quadratic forms of
tensors. Should be of shape (x, y, z, 3, 3) or (n, 3, 3) or (3, 3).
Returns
-------
mode : array
Calculated tensor mode in each spatial coordinate.
Notes
-----
Mode ranges between -1 (linear anisotropy) and +1 (planar anisotropy)
with 0 representing orthotropy. Mode is calculated with the
following equation (equation 9 in [1]_):
.. math::
Mode = 3*\sqrt{6}*det(\widetilde{A}/norm(\widetilde{A}))
Where $\widetilde{A}$ is the deviatoric part of the tensor quadratic form.
References
----------
.. [1] Daniel B. Ennis and G. Kindlmann, "Orthogonal Tensor
Invariants and the Analysis of Diffusion Tensor Magnetic Resonance
Images", Magnetic Resonance in Medicine, vol. 55, no. 1, pp. 136-146,
2006.
"""
A_squiggle = deviatoric(q_form)
A_s_norm = norm(A_squiggle)
# Add two dims for the (3,3), so that it can broadcast on A_squiggle:
A_s_norm = A_s_norm.reshape(A_s_norm.shape + (1, 1))
return 3 * np.sqrt(6) * determinant((A_squiggle / A_s_norm))
def linearity(evals, axis=-1):
r"""
The linearity of the tensor [1]_
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
linearity : array
Calculated linearity of the diffusion tensor.
Notes
--------
Linearity is calculated with the following equation:
.. math::
Linearity = \frac{\lambda_1-\lambda_2}{\lambda_1+\lambda_2+\lambda_3}
Notes
-----
[1] Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz F.,
"Geometrical diffusion measures for MRI from tensor basis analysis" in
Proc. 5th Annual ISMRM, 1997.
"""
evals = _roll_evals(evals, axis)
ev1, ev2, ev3 = evals
return (ev1 - ev2) / evals.sum(0)
def planarity(evals, axis=-1):
r"""
The planarity of the tensor [1]_
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
linearity : array
Calculated linearity of the diffusion tensor.
Notes
--------
Linearity is calculated with the following equation:
.. math::
Planarity = \frac{2 (\lambda_2-\lambda_3)}{\lambda_1+\lambda_2+\lambda_3}
Notes
-----
[1] Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz F.,
"Geometrical diffusion measures for MRI from tensor basis analysis" in
Proc. 5th Annual ISMRM, 1997.
"""
evals = _roll_evals(evals, axis)
ev1, ev2, ev3 = evals
return (2 * (ev2 - ev3) / evals.sum(0))
def sphericity(evals, axis=-1):
r"""
The sphericity of the tensor [1]_
Parameters
----------
evals : array-like
Eigenvalues of a diffusion tensor.
axis : int
Axis of `evals` which contains 3 eigenvalues.
Returns
-------
sphericity : array
Calculated sphericity of the diffusion tensor.
Notes
--------
Linearity is calculated with the following equation:
.. math::
Sphericity = \frac{3 \lambda_3)}{\lambda_1+\lambda_2+\lambda_3}
Notes
-----
[1] Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz F.,
"Geometrical diffusion measures for MRI from tensor basis analysis" in
Proc. 5th Annual ISMRM, 1997.
"""
evals = _roll_evals(evals, axis)
ev1, ev2, ev3 = evals
return (3 * ev3) / evals.sum(0)
def apparent_diffusion_coef(q_form, sphere):
r"""
Calculate the apparent diffusion coefficient (ADC) in each direction of a
sphere.
Parameters
----------
q_form : ndarray
The quadratic form of a tensor, or an array with quadratic forms of
tensors. Should be of shape (..., 3, 3)
sphere : a Sphere class instance
The ADC will be calculated for each of the vertices in the sphere
Notes
-----
The calculation of ADC, relies on the following relationship:
.. math ::
ADC = \vec{b} Q \vec{b}^T
Where Q is the quadratic form of the tensor.
"""
bvecs = sphere.vertices
bvals = np.ones(bvecs.shape[0])
gtab = gradient_table(bvals, bvecs)
D = design_matrix(gtab)[:, :6]
return -np.dot(lower_triangular(q_form), D.T)
class TensorModel(ReconstModel):
""" Diffusion Tensor
"""
def __init__(self, gtab, fit_method="WLS", *args, **kwargs):
""" A Diffusion Tensor Model [1]_, [2]_.
Parameters
----------
gtab : GradientTable class instance
fit_method : str or callable
str can be one of the following:
'WLS' for weighted least squares
dti.wls_fit_tensor
'LS' or 'OLS' for ordinary least squares
dti.ols_fit_tensor
'NLLS' for non-linear least-squares
dti.nlls_fit_tensor
'RT' or 'restore' or 'RESTORE' for RESTORE robust tensor fitting [3]_
dti.restore_fit_tensor
callable has to have the signature:
fit_method(design_matrix, data, *args, **kwargs)
args, kwargs : arguments and key-word arguments passed to the
fit_method. See dti.wls_fit_tensor, dti.ols_fit_tensor for details
References
----------
.. [1] Basser, P.J., Mattiello, J., LeBihan, D., 1994. Estimation of
the effective self-diffusion tensor from the NMR spin echo. J Magn
Reson B 103, 247-254.
.. [2] Basser, P., Pierpaoli, C., 1996. Microstructural and
physiological features of tissues elucidated by quantitative
diffusion-tensor MRI. Journal of Magnetic Resonance 111, 209-219.
.. [3] Lin-Ching C., Jones D.K., Pierpaoli, C. 2005. RESTORE: Robust
estimation of tensors by outlier rejection. MRM 53: 1088-1095
"""
ReconstModel.__init__(self, gtab)
if not callable(fit_method):
try:
self.fit_method = common_fit_methods[fit_method]
except KeyError:
raise ValueError('"' + str(fit_method) + '" is not a known fit '
'method, the fit method should either be a '
'function or one of the common fit methods')
self.design_matrix = design_matrix(self.gtab)
self.args = args
self.kwargs = kwargs
def fit(self, data, mask=None):
""" Fit method of the DTI model class
Parameters
----------
data : array
The measured signal from one voxel.
mask : array
A boolean array used to mark the coordinates in the data that
should be analyzed that has the shape data.shape[-1]
"""
# If a mask is provided, we will use it to access the data
if mask is not None:
# Make sure it's boolean, so that it can be used to mask
mask = np.array(mask, dtype=bool, copy=False)
data_in_mask = data[mask]
else:
data_in_mask = data
params_in_mask = self.fit_method(self.design_matrix, data_in_mask,
*self.args, **self.kwargs)
dti_params = np.zeros(data.shape[:-1] + (12,))
dti_params[mask, :] = params_in_mask
return TensorFit(self, dti_params)
class TensorFit(object):
def __init__(self, model, model_params):
""" Initialize a TensorFit class instance.
"""
self.model = model
self.model_params = model_params
def __getitem__(self, index):
model_params = self.model_params
N = model_params.ndim
if type(index) is not tuple:
index = (index,)
elif len(index) >= model_params.ndim:
raise IndexError("IndexError: invalid index")
index = index + (slice(None),) * (N - len(index))
return type(self)(self.model, model_params[index])
@property
def shape(self):
return self.model_params.shape[:-1]
@property
def directions(self):
"""
For tracking - return the primary direction in each voxel
"""
return self.evecs[..., None, :, 0]
@property
def evals(self):
"""
Returns the eigenvalues of the tensor as an array
"""
return self.model_params[..., :3]
@property
def evecs(self):
"""
Returns the eigenvectors of the tensor as an array
"""
evecs = self.model_params[..., 3:]
return evecs.reshape(self.shape + (3, 3))
@property
def quadratic_form(self):
"""Calculates the 3x3 diffusion tensor for each voxel"""
# do `evecs * evals * evecs.T` where * is matrix multiply
# einsum does this with:
# np.einsum('...ij,...j,...kj->...ik', evecs, evals, evecs)
return vec_val_vect(self.evecs, self.evals)
def lower_triangular(self, b0=None):
return lower_triangular(self.quadratic_form, b0)
@auto_attr
def fa(self):
"""Fractional anisotropy (FA) calculated from cached eigenvalues."""
return fractional_anisotropy(self.evals)
@auto_attr
def mode(self):
"""
Tensor mode calculated from cached eigenvalues.
"""
return mode(self.quadratic_form)
@auto_attr
def md(self):
r"""
Mean diffusitivity (MD) calculated from cached eigenvalues.
Returns
---------
md : array (V, 1)
Calculated MD.
Notes
--------
MD is calculated with the following equation:
.. math::
MD = \frac{\lambda_1+\lambda_2+\lambda_3}{3}
"""
return self.trace / 3.0
@auto_attr
def rd(self):
r"""
Radial diffusitivity (RD) calculated from cached eigenvalues.
Returns
---------
rd : array (V, 1)
Calculated RD.
Notes
--------
RD is calculated with the following equation:
.. math::
RD = \frac{\lambda_2 + \lambda_3}{2}
"""
return radial_diffusivity(self.evals)
@auto_attr
def ad(self):
r"""
Axial diffusivity (AD) calculated from cached eigenvalues.
Returns
---------
ad : array (V, 1)
Calculated AD.
Notes
--------
RD is calculated with the following equation:
.. math::
AD = \lambda_1
"""
return axial_diffusivity(self.evals)
@auto_attr
def trace(self):
r"""
Trace of the tensor calculated from cached eigenvalues.
Returns
---------
trace : array (V, 1)
Calculated trace.
Notes
--------
The trace is calculated with the following equation:
.. math::
trace = \lambda_1 + \lambda_2 + \lambda_3
"""
return trace(self.evals)
@auto_attr
def planarity(self):
r"""
Returns
-------
sphericity : array
Calculated sphericity of the diffusion tensor [1]_.
Notes
--------
Sphericity is calculated with the following equation:
.. math::
Sphericity = \frac{2 (\lambda2 - \lambda_3)}{\lambda_1+\lambda_2+\lambda_3}
Notes
-----
[1] Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz
F., "Geometrical diffusion measures for MRI from tensor basis
analysis" in Proc. 5th Annual ISMRM, 1997.
"""
return planarity(self.evals)
@auto_attr
def linearity(self):
r"""
Returns
-------
linearity : array
Calculated linearity of the diffusion tensor [1]_.
Notes
--------
Linearity is calculated with the following equation:
.. math::
Linearity = \frac{\lambda_1-\lambda_2}{\lambda_1+\lambda_2+\lambda_3}
Notes
-----
[1] Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz
F., "Geometrical diffusion measures for MRI from tensor basis
analysis" in Proc. 5th Annual ISMRM, 1997.
"""
return linearity(self.evals)
@auto_attr
def sphericity(self):
r"""
Returns
-------
sphericity : array
Calculated sphericity of the diffusion tensor [1]_.
Notes
--------
Sphericity is calculated with the following equation:
.. math::
Sphericity = \frac{3 \lambda_3}{\lambda_1+\lambda_2+\lambda_3}
Notes
-----
[1] Westin C.-F., Peled S., Gubjartsson H., Kikinis R., Jolesz
F., "Geometrical diffusion measures for MRI from tensor basis
analysis" in Proc. 5th Annual ISMRM, 1997.
"""
return sphericity(self.evals)
def odf(self, sphere):
"""
The diffusion orientation distribution function (dODF). This is an
estimate of the diffusion distance in each direction
Parameters
----------
sphere : Sphere class instance.
The dODF is calculated in the vertices of this input.
Returns
-------
odf : ndarray
The diffusion distance in every direction of the sphere in every
voxel in the input data.
"""
lower = 4 * np.pi * np.sqrt(np.prod(self.evals, -1))
projection = np.dot(sphere.vertices, self.evecs)
with warnings.catch_warnings():
warnings.simplefilter("ignore")
projection /= np.sqrt(self.evals)
odf = (vector_norm(projection) ** -3) / lower
# Zero evals are non-physical, we replace nans with zeros
any_zero = (self.evals == 0).any(-1)
odf = np.where(any_zero, 0, odf)
# Move odf to be on the last dimension
odf = np.rollaxis(odf, 0, odf.ndim)
return odf
def adc(self, sphere):
r"""
Calculate the apparent diffusion coefficient (ADC) in each direction on
the sphere for each voxel in the data
Parameters
----------
sphere : Sphere class instance
Returns
-------
adc : ndarray
The estimates of the apparent diffusion coefficient in every
direction on the input sphere
Notes
-----
The calculation of ADC, relies on the following relationship:
.. math ::
ADC = \vec{b} Q \vec{b}^T
Where Q is the quadratic form of the tensor.
"""
return apparent_diffusion_coef(self.quadratic_form, sphere)
def predict(self, gtab, S0=1):
r"""
Given a model fit, predict the signal on the vertices of a sphere
Parameters
----------
gtab : a GradientTable class instance
This encodes the directions for which a prediction is made
S0 : float array
The mean non-diffusion weighted signal in each voxel. Default: 1 in
all voxels.
Notes
-----
The predicted signal is given by:
.. math ::
S(\theta, b) = S_0 * e^{-b ADC}
Where:
.. math ::
ADC = \theta Q \theta^T
$\theta$ is a unit vector pointing at any direction on the sphere for
which a signal is to be predicted and $b$ is the b value provided in
the GradientTable input for that direction
"""
# Get a sphere to pass to the object's ADC function. The b0 vectors
# will not be on the unit sphere, but we still want them to be there,
# so that we have a consistent index for these, so that we can fill
# that in later on, so we suppress the warning here:
with warnings.catch_warnings():
warnings.simplefilter("ignore")
sphere = Sphere(xyz=gtab.bvecs)
adc = self.adc(sphere)
# Predict!
if np.iterable(S0):
# If it's an array, we need to give it one more dimension:
S0 = S0[...,None]
pred_sig = S0 * np.exp(-gtab.bvals * adc)
# The above evaluates to nan for the b0 vectors, so we predict the mean
# S0 for those, which is our best guess:
pred_sig[...,gtab.b0s_mask] = S0
return pred_sig
def wls_fit_tensor(design_matrix, data, min_signal=1):
r"""
Computes weighted least squares (WLS) fit to calculate self-diffusion
tensor using a linear regression model [1]_.
Parameters
----------
design_matrix : array (g, 7)
Design matrix holding the covariants used to solve for the regression
coefficients.
data : array ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
min_signal : default = 1
All values below min_signal are repalced with min_signal. This is done
in order to avaid taking log(0) durring the tensor fitting.
Returns
-------
eigvals : array (..., 3)
Eigenvalues from eigen decomposition of the tensor.
eigvecs : array (..., 3, 3)
Associated eigenvectors from eigen decomposition of the tensor.
Eigenvectors are columnar (e.g. eigvecs[:,j] is associated with
eigvals[j])
See Also
--------
decompose_tensor
Notes
-----
In Chung, et al. 2006, the regression of the WLS fit needed an unbiased
preliminary estimate of the weights and therefore the ordinary least
squares (OLS) estimates were used. A "two pass" method was implemented:
1. calculate OLS estimates of the data
2. apply the OLS estimates as weights to the WLS fit of the data
This ensured heteroscadasticity could be properly modeled for various
types of bootstrap resampling (namely residual bootstrap).
.. math::
y = \mathrm{data} \\
X = \mathrm{design matrix} \\
\hat{\beta}_\mathrm{WLS} = \mathrm{desired regression coefficients (e.g. tensor)}\\
\\
\hat{\beta}_\mathrm{WLS} = (X^T W X)^{-1} X^T W y \\
\\
W = \mathrm{diag}((X \hat{\beta}_\mathrm{OLS})^2),
\mathrm{where} \hat{\beta}_\mathrm{OLS} = (X^T X)^{-1} X^T y
References
----------
.. [1] Chung, SW., Lu, Y., Henry, R.G., 2006. Comparison of bootstrap
approaches for estimation of uncertainties of DTI parameters.
NeuroImage 33, 531-541.
"""
tol = 1e-6
if min_signal <= 0:
raise ValueError('min_signal must be > 0')
data = np.asarray(data)
data_flat = data.reshape((-1, data.shape[-1]))
dti_params = np.empty((len(data_flat), 4, 3))
#obtain OLS fitting matrix
#U,S,V = np.linalg.svd(design_matrix, False)
#math: beta_ols = inv(X.T*X)*X.T*y
#math: ols_fit = X*beta_ols*inv(y)
#ols_fit = np.dot(U, U.T)
ols_fit = _ols_fit_matrix(design_matrix)
min_diffusivity = tol / -design_matrix.min()
for param, sig in zip(dti_params, data_flat):
param[0], param[1:] = _wls_iter(ols_fit, design_matrix, sig,
min_signal, min_diffusivity)
dti_params.shape = data.shape[:-1] + (12,)
dti_params = dti_params
return dti_params
def _wls_iter(ols_fit, design_matrix, sig, min_signal, min_diffusivity):
''' Helper function used by wls_fit_tensor.
'''
sig = np.maximum(sig, min_signal) # throw out zero signals
log_s = np.log(sig)
w = np.exp(np.dot(ols_fit, log_s))
D = np.dot(np.linalg.pinv(design_matrix * w[:, None]), w * log_s)
# D, _, _, _ = np.linalg.lstsq(design_matrix * w[:, None], log_s)
tensor = from_lower_triangular(D)
return decompose_tensor(tensor, min_diffusivity=min_diffusivity)
def _ols_iter(inv_design, sig, min_signal, min_diffusivity):
''' Helper function used by ols_fit_tensor.
'''
sig = np.maximum(sig, min_signal) # throw out zero signals
log_s = np.log(sig)
D = np.dot(inv_design, log_s)
tensor = from_lower_triangular(D)
return decompose_tensor(tensor, min_diffusivity=min_diffusivity)
def ols_fit_tensor(design_matrix, data, min_signal=1):
r"""
Computes ordinary least squares (OLS) fit to calculate self-diffusion
tensor using a linear regression model [1]_.
Parameters
----------
design_matrix : array (g, 7)
Design matrix holding the covariants used to solve for the regression
coefficients.
data : array ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
min_signal : default = 1
All values below min_signal are repalced with min_signal. This is done
in order to avaid taking log(0) durring the tensor fitting.
Returns
-------
eigvals : array (..., 3)
Eigenvalues from eigen decomposition of the tensor.
eigvecs : array (..., 3, 3)
Associated eigenvectors from eigen decomposition of the tensor.
Eigenvectors are columnar (e.g. eigvecs[:,j] is associated with
eigvals[j])
See Also
--------
WLS_fit_tensor, decompose_tensor, design_matrix
Notes
-----
.. math::
y = \mathrm{data} \\
X = \mathrm{design matrix} \\
\hat{\beta}_\mathrm{OLS} = (X^T X)^{-1} X^T y
References
----------
.. [1] Chung, SW., Lu, Y., Henry, R.G., 2006. Comparison of bootstrap
approaches for estimation of uncertainties of DTI parameters.
NeuroImage 33, 531-541.
"""
tol = 1e-6
data = np.asarray(data)
data_flat = data.reshape((-1, data.shape[-1]))
evals = np.empty((len(data_flat), 3))
evecs = np.empty((len(data_flat), 3, 3))
dti_params = np.empty((len(data_flat), 4, 3))
#obtain OLS fitting matrix
#U,S,V = np.linalg.svd(design_matrix, False)
#math: beta_ols = inv(X.T*X)*X.T*y
#math: ols_fit = X*beta_ols*inv(y)
#ols_fit = np.dot(U, U.T)
min_diffusivity = tol / -design_matrix.min()
inv_design = np.linalg.pinv(design_matrix)
for param, sig in zip(dti_params, data_flat):
param[0], param[1:] = _ols_iter(inv_design, sig,
min_signal, min_diffusivity)
dti_params.shape = data.shape[:-1] + (12,)
dti_params = dti_params
return dti_params
def _ols_fit_matrix(design_matrix):
"""
Helper function to calculate the ordinary least squares (OLS)
fit as a matrix multiplication. Mainly used to calculate WLS weights. Can
be used to calculate regression coefficients in OLS but not recommended.
See Also:
---------
wls_fit_tensor, ols_fit_tensor
Example:
--------
ols_fit = _ols_fit_matrix(design_mat)
ols_data = np.dot(ols_fit, data)
"""
U, S, V = np.linalg.svd(design_matrix, False)
return np.dot(U, U.T)
def _nlls_err_func(tensor, design_matrix, data, weighting=None,
sigma=None):
"""
Error function for the non-linear least-squares fit of the tensor.
Parameters
----------
tensor : array (3,3)
The 3-by-3 tensor matrix
design_matrix : array
The design matrix
data : array
The voxel signal in all gradient directions
weighting : str (optional).
Whether to use the Geman McClure weighting criterion (see [1]_
for details)
sigma : float or float array (optional)
If 'sigma' weighting is used, we will weight the error function
according to the background noise estimated either in aggregate over
all directions (when a float is provided), or to an estimate of the
noise in each diffusion-weighting direction (if an array is
provided). If 'gmm', the Geman-Mclure M-estimator is used for
weighting (see Notes.
Notes
-----
The GemanMcClure M-estimator is described as follows [1]_ (page 1089): "The
scale factor C affects the shape of the GMM [Geman-McClure M-estimator]
weighting function and represents the expected spread of the residuals
(i.e., the SD of the residuals) due to Gaussian distributed noise. The
scale factor C can be estimated by many robust scale estimators. We used
the median absolute deviation (MAD) estimator because it is very robust to
outliers having a 50% breakdown point (6,7). The explicit formula for C
using the MAD estimator is:
.. math ::
C = 1.4826 x MAD = 1.4826 x median{|r1-\hat{r}|,... |r_n-\hat{r}|}
where $\hat{r} = median{r_1, r_2, ..., r_3}$ and n is the number of data
points. The multiplicative constant 1.4826 makes this an approximately
unbiased estimate of scale when the error model is Gaussian."
References
----------
[1] Chang, L-C, Jones, DK and Pierpaoli, C (2005). RESTORE: robust estimation
of tensors by outlier rejection. MRM, 53: 1088-95.
"""
# This is the predicted signal given the params:
y = np.exp(np.dot(design_matrix, tensor))
# Compute the residuals
residuals = data - y
# If we don't want to weight the residuals, we are basically done:
if weighting is None:
# And we return the SSE:
return residuals
se = residuals ** 2
# If the user provided a sigma (e.g 1.5267 * std(background_noise), as
# suggested by Chang et al.) we will use it:
if weighting == 'sigma':
if sigma is None:
e_s = "Must provide sigma value as input to use this weighting"
e_s += " method"
raise ValueError(e_s)
w = 1/(sigma**2)
elif weighting == 'gmm':
# We use the Geman McClure M-estimator to compute the weights on the
# residuals:
C = 1.4826 * np.median(np.abs(residuals - np.median(residuals)))
with warnings.catch_warnings():
warnings.simplefilter("ignore")
w = 1/(se + C**2)
# The weights are normalized to the mean weight (see p. 1089):
w = w/np.mean(w)
# Return the weighted residuals:
with warnings.catch_warnings():
warnings.simplefilter("ignore")
return np.sqrt(w * se)
def _nlls_jacobian_func(tensor, design_matrix, data, *arg, **kwargs):
"""The Jacobian is the first derivative of the error function [1]_.
Notes
-----
This is an implementation of equation 14 in [1]_.
References
----------
[1] Koay, CG, Chang, L-C, Carew, JD, Pierpaoli, C, Basser PJ (2006).
A unifying theoretical and algorithmic framework for least squares
methods of estimation in diffusion tensor imaging. MRM 182, 115-25.
"""
pred = np.exp(np.dot(design_matrix, tensor))
return -pred[:, None] * design_matrix
def nlls_fit_tensor(design_matrix, data, min_signal=1, weighting=None,
sigma=None, jac=True):
"""
Fit the tensor params using non-linear least-squares.
Parameters
----------
design_matrix : array (g, 7)
Design matrix holding the covariants used to solve for the regression
coefficients.
data : array ([X, Y, Z, ...], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
min_signal : float, optional
All values below min_signal are repalced with min_signal. This is done
in order to avaid taking log(0) durring the tensor fitting. Default = 1
weighting: str
the weighting scheme to use in considering the
squared-error. Default behavior is to use uniform weighting. Other
options: 'sigma' 'gmm'
sigma: float
If the 'sigma' weighting scheme is used, a value of sigma needs to be
provided here. According to [Chang2005]_, a good value to use is
1.5267 * std(background_noise), where background_noise is estimated
from some part of the image known to contain no signal (only noise).
jac : bool
Use the Jacobian? Default: True
Returns
-------
nlls_params: the eigen-values and eigen-vectors of the tensor in each voxel.
"""
# Flatten for the iteration over voxels:
flat_data = data.reshape((-1, data.shape[-1]))
# Use the OLS method parameters as the starting point for the optimization:
inv_design = np.linalg.pinv(design_matrix)
sig = np.maximum(flat_data, min_signal)
log_s = np.log(sig)
D = np.dot(inv_design, log_s.T).T
# Flatten for the iteration over voxels:
ols_params = np.reshape(D, (-1, D.shape[-1]))
# 12 parameters per voxel (evals + evecs):
dti_params = np.empty((flat_data.shape[0], 12))
for vox in range(flat_data.shape[0]):
start_params = ols_params[vox]
# Do the optimization in this voxel:
if jac:
this_tensor, status = opt.leastsq(_nlls_err_func, start_params,
args=(design_matrix,
flat_data[vox],
weighting,
sigma),
Dfun=_nlls_jacobian_func)
else:
this_tensor, status = opt.leastsq(_nlls_err_func, start_params,
args=(design_matrix,
flat_data[vox],
weighting,
sigma))
# The parameters are the evals and the evecs:
try:
evals,evecs=decompose_tensor(from_lower_triangular(this_tensor[:6]))
dti_params[vox, :3] = evals
dti_params[vox, 3:] = evecs.ravel()
# If leastsq failed to converge and produced nans, we'll resort to the
# OLS solution in this voxel:
except np.linalg.LinAlgError:
print(vox)
dti_params[vox, :] = start_params
dti_params.shape = data.shape[:-1] + (12,)
return dti_params
def restore_fit_tensor(design_matrix, data, min_signal=1.0, sigma=None,
jac=True):
"""
Use the RESTORE algorithm [Chang2005]_ to calculate a robust tensor fit
Parameters
----------
design_matrix : array of shape (g, 7)
Design matrix holding the covariants used to solve for the regression
coefficients.
data : array of shape ([X, Y, Z, n_directions], g)
Data or response variables holding the data. Note that the last
dimension should contain the data. It makes no copies of data.
min_signal : float, optional
All values below min_signal are repalced with min_signal. This is done
in order to avaid taking log(0) durring the tensor fitting. Default = 1
sigma : float
An estimate of the variance. [Chang2005]_ recommend to use
1.5267 * std(background_noise), where background_noise is estimated
from some part of the image known to contain no signal (only noise).
jac : bool, optional
Whether to use the Jacobian of the tensor to speed the non-linear
optimization procedure used to fit the tensor paramters (see also
:func:`nlls_fit_tensor`). Default: True
Returns
-------
restore_params : an estimate of the tensor parameters in each voxel.
Note
----
Chang, L-C, Jones, DK and Pierpaoli, C (2005). RESTORE: robust estimation
of tensors by outlier rejection. MRM, 53: 1088-95.
"""
# Flatten for the iteration over voxels:
flat_data = data.reshape((-1, data.shape[-1]))
# Use the OLS method parameters as the starting point for the optimization:
inv_design = np.linalg.pinv(design_matrix)
sig = np.maximum(flat_data, min_signal)
log_s = np.log(sig)
D = np.dot(inv_design, log_s.T).T
ols_params = np.reshape(D, (-1, D.shape[-1]))
# 12 parameters per voxel (evals + evecs):
dti_params = np.empty((flat_data.shape[0], 12))
for vox in range(flat_data.shape[0]):
start_params = ols_params[vox]
# Do nlls using sigma weighting in this voxel:
if jac:
this_tensor, status = opt.leastsq(_nlls_err_func, start_params,
args=(design_matrix,
flat_data[vox],
'sigma',
sigma),
Dfun=_nlls_jacobian_func)
else:
this_tensor, status = opt.leastsq(_nlls_err_func, start_params,
args=(design_matrix,
flat_data[vox],
'sigma',
sigma))
# Get the residuals:
pred_sig = np.exp(np.dot(design_matrix, this_tensor))
residuals = flat_data[vox] - pred_sig
# If any of the residuals are outliers (using 3 sigma as a criterion
# following Chang et al., e.g page 1089):
if np.any(np.abs(residuals) > 3 * sigma):
# Do nlls with GMM-weighting:
if jac:
this_tensor, status= opt.leastsq(_nlls_err_func,
start_params,
args=(design_matrix,
flat_data[vox],
'gmm'),
Dfun=_nlls_jacobian_func)
else:
this_tensor, status= opt.leastsq(_nlls_err_func,
start_params,
args=(design_matrix,
flat_data[vox],
'gmm'))
# How are you doin' on those residuals?
pred_sig = np.exp(np.dot(design_matrix, this_tensor))
residuals = flat_data[vox] - pred_sig
if np.any(np.abs(residuals) > 3 * sigma):
# If you still have outliers, refit without those outliers:
non_outlier_idx = np.where(np.abs(residuals) <= 3 * sigma)
clean_design = design_matrix[non_outlier_idx]
clean_sig = flat_data[vox][non_outlier_idx]
if np.iterable(sigma):
this_sigma = sigma[non_outlier_idx]
else:
this_sigma = sigma
if jac:
this_tensor, status= opt.leastsq(_nlls_err_func,
start_params,
args=(clean_design,
clean_sig),
Dfun=_nlls_jacobian_func)
else:
this_tensor, status= opt.leastsq(_nlls_err_func,
start_params,
args=(clean_design,
clean_sig))
# The parameters are the evals and the evecs:
try:
evals,evecs=decompose_tensor(from_lower_triangular(this_tensor[:6]))
dti_params[vox, :3] = evals
dti_params[vox, 3:] = evecs.ravel()
# If leastsq failed to converge and produced nans, we'll resort to the
# OLS solution in this voxel:
except np.linalg.LinAlgError:
print(vox)
dti_params[vox, :] = start_params
dti_params.shape = data.shape[:-1] + (12,)
restore_params = dti_params
return restore_params
_lt_indices = np.array([[0, 1, 3],
[1, 2, 4],
[3, 4, 5]])
def from_lower_triangular(D):
""" Returns a tensor given the six unique tensor elements
Given the six unique tensor elments (in the order: Dxx, Dxy, Dyy, Dxz, Dyz,
Dzz) returns a 3 by 3 tensor. All elements after the sixth are ignored.
Parameters
-----------
D : array_like, (..., >6)
Unique elements of the tensors
Returns
--------
tensor : ndarray (..., 3, 3)
3 by 3 tensors
"""
return D[..., _lt_indices]
_lt_rows = np.array([0, 1, 1, 2, 2, 2])
_lt_cols = np.array([0, 0, 1, 0, 1, 2])
def lower_triangular(tensor, b0=None):
"""
Returns the six lower triangular values of the tensor and a dummy variable
if b0 is not None
Parameters
----------
tensor : array_like (..., 3, 3)
a collection of 3, 3 diffusion tensors
b0 : float
if b0 is not none log(b0) is returned as the dummy variable
Returns
-------
D : ndarray
If b0 is none, then the shape will be (..., 6) otherwise (..., 7)
"""
if tensor.shape[-2:] != (3, 3):
raise ValueError("Diffusion tensors should be (..., 3, 3)")
if b0 is None:
return tensor[..., _lt_rows, _lt_cols]
else:
D = np.empty(tensor.shape[:-2] + (7,), dtype=tensor.dtype)
D[..., 6] = -np.log(b0)
D[..., :6] = tensor[..., _lt_rows, _lt_cols]
return D
def eig_from_lo_tri(data):
"""Calculates parameters for creating a Tensor instance
Calculates tensor parameters from the six unique tensor elements. This
function can be passed to the Tensor class as a fit_method for creating a
Tensor instance from tensors stored in a nifti file.
Parameters
----------
data : array_like (..., 6)
diffusion tensors elements stored in lower triangular order
Returns
-------
dti_params
Eigen values and vectors, used by the Tensor class to create an
instance
"""
data = np.asarray(data)
data_flat = data.reshape((-1, data.shape[-1]))
dti_params = np.empty((len(data_flat), 4, 3))
for ii in range(len(data_flat)):
tensor = from_lower_triangular(data_flat[ii])
eigvals, eigvecs = decompose_tensor(tensor)
dti_params[ii, 0] = eigvals
dti_params[ii, 1:] = eigvecs
dti_params.shape = data.shape[:-1] + (12,)
return dti_params
def decompose_tensor(tensor, min_diffusivity=0):
""" Returns eigenvalues and eigenvectors given a diffusion tensor
Computes tensor eigen decomposition to calculate eigenvalues and
eigenvectors (Basser et al., 1994a).
Parameters
----------
tensor : array (3, 3)
Hermitian matrix representing a diffusion tensor.
min_diffusivity : float
Because negative eigenvalues are not physical and small eigenvalues,
much smaller than the diffusion weighting, cause quite a lot of noise
in metrics such as fa, diffusivity values smaller than
`min_diffusivity` are replaced with `min_diffusivity`.
Returns
-------
eigvals : array (3,)
Eigenvalues from eigen decomposition of the tensor. Negative
eigenvalues are replaced by zero. Sorted from largest to smallest.
eigvecs : array (3, 3)
Associated eigenvectors from eigen decomposition of the tensor.
Eigenvectors are columnar (e.g. eigvecs[:,j] is associated with
eigvals[j])
"""
#outputs multiplicity as well so need to unique
eigenvals, eigenvecs = np.linalg.eigh(tensor)
#need to sort the eigenvalues and associated eigenvectors
order = eigenvals.argsort()[::-1]
eigenvecs = eigenvecs[:, order]
eigenvals = eigenvals[order]
eigenvals = eigenvals.clip(min=min_diffusivity)
# eigenvecs: each vector is columnar
return eigenvals, eigenvecs
def design_matrix(gtab, dtype=None):
""" Constructs design matrix for DTI weighted least squares or
least squares fitting. (Basser et al., 1994a)
Parameters
----------
gtab : A GradientTable class instance
dtype : string
Parameter to control the dtype of returned designed matrix
Returns
-------
design_matrix : array (g,7)
Design matrix or B matrix assuming Gaussian distributed tensor model
design_matrix[j, :] = (Bxx, Byy, Bzz, Bxy, Bxz, Byz, dummy)
"""
B = np.zeros((gtab.gradients.shape[0], 7))
B[:, 0] = gtab.bvecs[:, 0] * gtab.bvecs[:, 0] * 1. * gtab.bvals # Bxx
B[:, 1] = gtab.bvecs[:, 0] * gtab.bvecs[:, 1] * 2. * gtab.bvals # Bxy
B[:, 2] = gtab.bvecs[:, 1] * gtab.bvecs[:, 1] * 1. * gtab.bvals # Byy
B[:, 3] = gtab.bvecs[:, 0] * gtab.bvecs[:, 2] * 2. * gtab.bvals # Bxz
B[:, 4] = gtab.bvecs[:, 1] * gtab.bvecs[:, 2] * 2. * gtab.bvals # Byz
B[:, 5] = gtab.bvecs[:, 2] * gtab.bvecs[:, 2] * 1. * gtab.bvals # Bzz
B[:, 6] = np.ones(gtab.gradients.shape[0])
return -B
def quantize_evecs(evecs, odf_vertices=None):
""" Find the closest orientation of an evenly distributed sphere
Parameters
----------
evecs : ndarray
odf_vertices : None or ndarray
If None, then set vertices from symmetric362 sphere. Otherwise use
passed ndarray as vertices
Returns
-------
IN : ndarray
"""
max_evecs = evecs[..., :, 0]
if odf_vertices == None:
odf_vertices = get_sphere('symmetric362').vertices
tup = max_evecs.shape[:-1]
mec = max_evecs.reshape(np.prod(np.array(tup)), 3)
IN = np.array([np.argmin(np.dot(odf_vertices, m)) for m in mec])
IN = IN.reshape(tup)
return IN
common_fit_methods = {'WLS': wls_fit_tensor,
'LS': ols_fit_tensor,
'OLS': ols_fit_tensor,
'NLLS': nlls_fit_tensor,
'RT': restore_fit_tensor,
'restore':restore_fit_tensor,
'RESTORE':restore_fit_tensor
}
|