2506:
A nice property of the TPS is that it can always be decomposed into a global affine and a local non-affine component. Consequently, the TPS smoothness term is solely dependent on the non-affine components. This is a desirable property, especially when compared to other splines, since the global pose
837:
1672:
473:
442:
2515:
TPS has been widely used as the non-rigid transformation model in image alignment and shape matching. An additional application is the analysis and comparisons of archaeological findings in 3D and was implemented for
1123:
2257:
62:
or thin sheet of metal. Just as the metal has rigidity, the TPS fit resists bending also, implying a penalty involving the smoothness of the fitted surface. In the physical setting, the deflection is in the
1992:
2483:
represents the TPS kernel. Loosely speaking, the TPS kernel contains the information about the point-set's internal structural relationships. When it is combined with the warping coefficients
2681:
J. Duchon, 1976, Splines minimizing rotation invariant semi-norms in
Sobolev spaces. pp 85–100, In: Constructive Theory of Functions of Several Variables, Oberwolfach 1976, W. Schempp and
1491:
973:
1242:
2461:
1154:
1725:
1365:
1483:
83:
direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the
1821:
1783:
2406:
2091:
2058:
2025:
1876:
1191:
326:
293:
1850:
465:
2147:
1405:
176:
2481:
2374:
2354:
2327:
1307:
1022:
260:
2748:
2726:
1276:
199:
2544:
However, note that splines already in one dimension can cause severe "overshoots". In 2D such effects can be much more critical, because TPS are not objective.
2501:
2300:
2280:
2114:
1896:
1745:
1445:
1425:
1385:
993:
896:
860:
231:
141:
121:
101:
81:
832:{\displaystyle E_{\mathrm {tps} ,\mathrm {smooth} }(f)=\sum _{i=1}^{K}\|y_{i}-f(x_{i})\|^{2}+\lambda \iint \left{\textrm {d}}x_{1}\,{\textrm {d}}x_{2}}
878:. In simple words, "the first term is defined as the error measurement term and the second regularisation term is a penalty on the smoothness of
334:
1030:
2615:
213:
The TPS arises from consideration of the integral of the square of the second derivative—this forms its smoothness measure. In the case where
2774:
2628:
201:
coefficients for correspondences of the control points. These parameters are computed by solving a linear system, in other words, TPS has a
2155:
467:
to control the rigidity of the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimizing:
2927:
875:
1901:
2922:
2521:
912:
The thin plate spline has a natural representation in terms of radial basis functions. Given a set of control points
1667:{\displaystyle f_{tps}(z,\alpha )=f_{tps}(z,d,c)=z\cdot d+\phi (z)\cdot c=z\cdot d+\sum _{i=1}^{K}\phi _{i}(z)c_{i}}
2563:
2850:
2817:
2686:
2825:
915:
1196:
2411:
2789:
Bookstein, F. L. (June 1989). "Principal warps: thin plate splines and the decomposition of deformations".
1131:
2752:
2707:
2574:
907:
2650:
1680:
1312:
2584:
1450:
202:
27:
2877:
2757:
2712:
2589:
2568:
1788:
1750:
43:
2682:
2379:
1157:
38:. "A spline is a function defined by polynomials in a piecewise manner." They were introduced to
46:. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm.
2063:
2030:
1997:
1855:
1163:
298:
265:
2770:
2720:
2624:
2557:
1826:
450:
59:
2830:
Proc. of the 14th
International Conference on Computer Vision Theory and Application (VISAPP)
2119:
1390:
2902:
2798:
2762:
2662:
2579:
146:
39:
2527:
The thin plate spline has a number of properties which have contributed to its popularity:
2466:
2359:
2332:
2305:
1285:
2463:. Each row of each newly formed matrix comes from one of the original vectors. The matrix
998:
236:
1255:
1823:
warping coefficient matrix representing the non-affine deformation. The kernel function
181:
2486:
2285:
2265:
2099:
1881:
1730:
1430:
1410:
1370:
978:
881:
863:
845:
216:
126:
106:
86:
66:
2916:
2651:"Energy-Efficient Post-Failure Reconfiguration of Swarms of Unmanned Aerial Vehicles"
31:
2882:
2907:
2517:
842:
For this variational problem, it can be shown that there exists a unique minimizer
1193:
is a set of mapping coefficients. The TPS corresponds to the radial basis kernel
2821:
2740:
2667:
2553:
871:
867:
2887:
2897:
2766:
975:, a radial basis function defines a spatial mapping which maps any location
35:
437:{\displaystyle E_{\mathrm {tps} }(f)=\sum _{i=1}^{K}\|y_{i}-f(x_{i})\|^{2}}
1118:{\displaystyle f(x)=\sum _{i=1}^{K}w_{i}\varphi (\left\|x-c_{i}\right\|)}
2649:
Tahir, Anam; Haghbayan, Hashem; Böling, Jari M.; Plosila, Juha (2023).
2537:
It has closed-form solutions for both warping and parameter estimation.
233:
is two dimensional, for interpolation, the TPS fits a mapping function
2802:
2690:
2892:
2507:
parameters included in the affine transformation are not penalized.
2252:{\displaystyle E_{tps}(d,c)=\|Y-Xd-\Phi c\|^{2}+\lambda c^{T}\Phi c}
143:
corresponding control points (knots), the TPS warp is described by
2685:, eds., Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977.
2571:(the thin plate spline is a special case of a polyharmonic spline)
2704:
Non-Rigid Point
Matching: Algorithms, Extensions and Applications
2531:
It produces smooth surfaces, which are infinitely differentiable.
447:
The smoothing variant, correspondingly, uses a tuning parameter
178:
parameters which include 6 global affine motion parameters and
2826:"Recovering and Visualizing Deformation in 3D Aegean Sealings"
2791:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
898:." It is in a general case needed to make the mapping unique.
2027:
are chosen to be the same as the set of points to be warped
1987:{\displaystyle \phi _{i}(z)=\|z-x_{i}\|^{2}\log \|z-x_{i}\|}
866:
discretization of this variational problem, the method of
123:
coordinates within the plane. In 2D cases, given a set of
2556:(a discrete version of the thin plate approximation for
2540:
There is a physical explanation for its energy function.
2302:
are just concatenated versions of the point coordinates
58:
refers to a physical analogy involving the bending of a
2617:
Formation
Control of Swarms of Unmanned Aerial Vehicles
2534:
There are no free parameters that need manual tuning.
2489:
2469:
2414:
2382:
2362:
2335:
2308:
2288:
2268:
2158:
2122:
2102:
2066:
2033:
2000:
1904:
1884:
1858:
1829:
1791:
1753:
1733:
1727:
matrix representing the affine transformation (hence
1683:
1494:
1453:
1433:
1413:
1393:
1373:
1315:
1288:
1258:
1199:
1166:
1134:
1033:
1001:
981:
918:
884:
848:
476:
453:
337:
301:
268:
239:
219:
184:
149:
129:
109:
89:
69:
42:by Duchon. They are an important special case of a
2495:
2475:
2455:
2400:
2368:
2348:
2321:
2294:
2274:
2251:
2141:
2108:
2085:
2052:
2019:
1986:
1890:
1870:
1844:
1815:
1777:
1739:
1719:
1666:
1477:
1439:
1419:
1399:
1379:
1359:
1301:
1270:
1236:
1185:
1148:
1117:
1016:
987:
967:
890:
854:
831:
459:
436:
320:
287:
254:
225:
193:
170:
135:
115:
95:
75:
2851:"Tutorial No. 13: Apply TPS-RPM Transformation"
2592:(emerging alternative to spline-based surfaces)
328:that minimizes the following energy function:
2878:Explanation for a simplified variation problem
2749:Society for Industrial and Applied Mathematics
8:
2447:
2421:
2218:
2193:
2080:
2067:
2047:
2034:
2014:
2001:
1981:
1962:
1947:
1927:
1472:
1460:
1180:
1167:
962:
919:
587:
551:
425:
389:
315:
302:
282:
269:
2644:
2642:
2640:
2725:: CS1 maint: location missing publisher (
16:Method of data interpolation and smoothing
2816:Bogacz, Bartosz; Papadimitriou, Nikolas;
2756:
2711:
2666:
2488:
2468:
2441:
2428:
2413:
2381:
2361:
2340:
2334:
2313:
2307:
2287:
2267:
2237:
2221:
2163:
2157:
2127:
2121:
2101:
2074:
2065:
2041:
2032:
2008:
1999:
1994:. Note that for TPS, the control points
1975:
1950:
1940:
1909:
1903:
1883:
1857:
1828:
1790:
1752:
1732:
1682:
1658:
1639:
1629:
1618:
1533:
1499:
1493:
1452:
1432:
1412:
1392:
1372:
1345:
1329:
1314:
1293:
1287:
1257:
1219:
1198:
1174:
1165:
1133:
1101:
1074:
1064:
1053:
1032:
1000:
980:
968:{\displaystyle \{c_{i},i=1,2,\ldots ,K\}}
926:
917:
883:
847:
823:
813:
812:
811:
805:
795:
794:
783:
770:
765:
747:
740:
726:
713:
700:
682:
675:
658:
645:
640:
622:
615:
590:
577:
558:
545:
534:
496:
482:
481:
475:
452:
428:
415:
396:
383:
372:
343:
342:
336:
309:
300:
276:
267:
238:
218:
183:
148:
128:
108:
88:
68:
2609:
2607:
2605:
1252:Suppose the points are in 2 dimensions (
2706:, Yale University, New Haven, CT, USA,
2601:
1237:{\displaystyle \varphi (r)=r^{2}\log r}
2718:
2456:{\displaystyle \phi (\|x_{i}-x_{j}\|)}
1149:{\displaystyle \left\|\cdot \right\|}
7:
2745:Spline models for observational data
2503:, a non-rigid warping is generated.
2096:If one substitutes the solution for
2093:in the place of the control points.
2470:
2363:
2243:
2211:
876:nonlinear dimensionality reduction
758:
744:
706:
693:
679:
633:
619:
512:
509:
506:
503:
500:
497:
489:
486:
483:
350:
347:
344:
14:
1720:{\displaystyle (D+1)\times (D+1)}
1360:{\displaystyle (1,y_{ix},y_{iy})}
262:between corresponding point-sets
2623:. Finland: University of Turku.
1282:for the point-set where a point
1478:{\displaystyle \alpha =\{d,c\}}
1407:which consists of two matrices
2450:
2418:
2395:
2383:
2187:
2175:
1921:
1915:
1839:
1833:
1810:
1798:
1772:
1760:
1714:
1702:
1696:
1684:
1651:
1645:
1590:
1584:
1563:
1545:
1523:
1511:
1354:
1316:
1209:
1203:
1142:
1136:
1112:
1108:
1087:
1083:
1043:
1037:
1011:
1005:
583:
570:
524:
518:
421:
408:
362:
356:
249:
243:
165:
153:
1:
2898:TPS interactive morphing demo
1816:{\displaystyle K\times (D+1)}
1778:{\displaystyle 1\times (D+1)}
2855:GigaMesh Software Framework
2668:10.1109/ACCESS.2022.3181244
2522:GigaMesh Software Framework
2401:{\displaystyle (K\times K)}
1309:is represented as a vector
995:in space to a new location
2944:
2928:Multivariate interpolation
2564:Inverse distance weighting
905:
30:-based technique for data
2747:, Philadelphia, PA, USA:
2086:{\displaystyle \{x_{i}\}}
2053:{\displaystyle \{x_{i}\}}
2020:{\displaystyle \{c_{i}\}}
1871:{\displaystyle 1\times K}
1186:{\displaystyle \{w_{i}\}}
321:{\displaystyle \{x_{i}\}}
288:{\displaystyle \{y_{i}\}}
2832:, Prague, Czech Republic
1845:{\displaystyle \phi (z)}
460:{\displaystyle \lambda }
2767:10.1137/1.9781611970128
2408:matrix formed from the
2142:{\displaystyle E_{tps}}
1400:{\displaystyle \alpha }
1367:. The unique minimizer
1280:homogeneous coordinates
2497:
2477:
2457:
2402:
2370:
2350:
2323:
2296:
2276:
2253:
2143:
2110:
2087:
2054:
2021:
1988:
1892:
1878:vector for each point
1872:
1846:
1817:
1779:
1741:
1721:
1668:
1634:
1479:
1441:
1421:
1401:
1381:
1361:
1303:
1272:
1238:
1187:
1150:
1119:
1069:
1018:
989:
969:
892:
856:
833:
550:
461:
438:
388:
322:
289:
256:
227:
195:
172:
171:{\displaystyle 2(K+3)}
137:
117:
97:
77:
2923:Splines (mathematics)
2575:Radial basis function
2498:
2478:
2476:{\displaystyle \Phi }
2458:
2403:
2371:
2369:{\displaystyle \Phi }
2351:
2349:{\displaystyle x_{i}}
2324:
2322:{\displaystyle y_{i}}
2297:
2277:
2254:
2144:
2111:
2088:
2055:
2022:
1989:
1893:
1873:
1847:
1818:
1780:
1742:
1722:
1669:
1614:
1480:
1442:
1422:
1402:
1382:
1362:
1304:
1302:{\displaystyle y_{i}}
1273:
1239:
1188:
1151:
1120:
1049:
1019:
990:
970:
908:Radial basis function
902:Radial basis function
893:
857:
834:
530:
462:
439:
368:
323:
290:
257:
228:
196:
173:
138:
118:
98:
78:
2893:TPS in templated C++
2702:Chui, Haili (2001),
2614:Tahir, Anam (2023).
2487:
2467:
2412:
2380:
2360:
2333:
2306:
2286:
2266:
2156:
2120:
2100:
2064:
2060:, so we already use
2031:
1998:
1902:
1882:
1856:
1827:
1789:
1751:
1731:
1681:
1492:
1451:
1431:
1411:
1391:
1387:is parameterized by
1371:
1313:
1286:
1256:
1197:
1164:
1132:
1031:
1017:{\displaystyle f(x)}
999:
979:
916:
882:
846:
474:
451:
335:
299:
266:
255:{\displaystyle f(x)}
237:
217:
203:closed-form solution
182:
147:
127:
107:
87:
67:
2590:Subdivision surface
2569:Polyharmonic spline
1898:, where each entry
1785:vector) and c is a
1271:{\displaystyle D=2}
775:
650:
44:polyharmonic spline
2691:10.1007/BFb0086566
2493:
2473:
2453:
2398:
2366:
2346:
2319:
2292:
2272:
2249:
2139:
2106:
2083:
2050:
2017:
1984:
1888:
1868:
1842:
1813:
1775:
1737:
1717:
1664:
1475:
1437:
1417:
1397:
1377:
1357:
1299:
1268:
1234:
1183:
1156:denotes the usual
1146:
1115:
1014:
985:
965:
888:
852:
829:
761:
636:
457:
434:
318:
285:
252:
223:
209:Smoothness measure
194:{\displaystyle 2K}
191:
168:
133:
113:
93:
73:
20:Thin plate splines
2824:, Hubert (2019),
2776:978-0-89871-244-5
2630:978-951-29-9411-3
2558:manifold learning
2518:triangular meshes
2496:{\displaystyle c}
2295:{\displaystyle X}
2275:{\displaystyle Y}
2109:{\displaystyle f}
1891:{\displaystyle z}
1740:{\displaystyle z}
1440:{\displaystyle c}
1420:{\displaystyle d}
1380:{\displaystyle f}
1024:, represented by
988:{\displaystyle x}
891:{\displaystyle f}
855:{\displaystyle f}
816:
798:
777:
720:
652:
226:{\displaystyle x}
136:{\displaystyle K}
116:{\displaystyle y}
96:{\displaystyle x}
76:{\displaystyle z}
56:thin plate spline
2935:
2883:TPS at MathWorld
2866:
2865:
2863:
2861:
2847:
2841:
2840:
2839:
2837:
2813:
2807:
2806:
2803:10.1109/34.24792
2786:
2780:
2779:
2760:
2743:, Grace (1990),
2737:
2731:
2730:
2724:
2716:
2715:
2699:
2693:
2679:
2673:
2672:
2670:
2646:
2635:
2634:
2622:
2611:
2580:Smoothing spline
2502:
2500:
2499:
2494:
2482:
2480:
2479:
2474:
2462:
2460:
2459:
2454:
2446:
2445:
2433:
2432:
2407:
2405:
2404:
2399:
2375:
2373:
2372:
2367:
2355:
2353:
2352:
2347:
2345:
2344:
2328:
2326:
2325:
2320:
2318:
2317:
2301:
2299:
2298:
2293:
2281:
2279:
2278:
2273:
2258:
2256:
2255:
2250:
2242:
2241:
2226:
2225:
2174:
2173:
2148:
2146:
2145:
2140:
2138:
2137:
2115:
2113:
2112:
2107:
2092:
2090:
2089:
2084:
2079:
2078:
2059:
2057:
2056:
2051:
2046:
2045:
2026:
2024:
2023:
2018:
2013:
2012:
1993:
1991:
1990:
1985:
1980:
1979:
1955:
1954:
1945:
1944:
1914:
1913:
1897:
1895:
1894:
1889:
1877:
1875:
1874:
1869:
1851:
1849:
1848:
1843:
1822:
1820:
1819:
1814:
1784:
1782:
1781:
1776:
1746:
1744:
1743:
1738:
1726:
1724:
1723:
1718:
1673:
1671:
1670:
1665:
1663:
1662:
1644:
1643:
1633:
1628:
1544:
1543:
1510:
1509:
1484:
1482:
1481:
1476:
1446:
1444:
1443:
1438:
1426:
1424:
1423:
1418:
1406:
1404:
1403:
1398:
1386:
1384:
1383:
1378:
1366:
1364:
1363:
1358:
1353:
1352:
1337:
1336:
1308:
1306:
1305:
1300:
1298:
1297:
1277:
1275:
1274:
1269:
1243:
1241:
1240:
1235:
1224:
1223:
1192:
1190:
1189:
1184:
1179:
1178:
1155:
1153:
1152:
1147:
1145:
1124:
1122:
1121:
1116:
1111:
1107:
1106:
1105:
1079:
1078:
1068:
1063:
1023:
1021:
1020:
1015:
994:
992:
991:
986:
974:
972:
971:
966:
931:
930:
897:
895:
894:
889:
861:
859:
858:
853:
838:
836:
835:
830:
828:
827:
818:
817:
814:
810:
809:
800:
799:
796:
793:
789:
788:
787:
782:
778:
776:
774:
769:
756:
752:
751:
741:
731:
730:
725:
721:
719:
718:
717:
705:
704:
691:
687:
686:
676:
663:
662:
657:
653:
651:
649:
644:
631:
627:
626:
616:
595:
594:
582:
581:
563:
562:
549:
544:
517:
516:
515:
492:
466:
464:
463:
458:
443:
441:
440:
435:
433:
432:
420:
419:
401:
400:
387:
382:
355:
354:
353:
327:
325:
324:
319:
314:
313:
294:
292:
291:
286:
281:
280:
261:
259:
258:
253:
232:
230:
229:
224:
200:
198:
197:
192:
177:
175:
174:
169:
142:
140:
139:
134:
122:
120:
119:
114:
102:
100:
99:
94:
82:
80:
79:
74:
50:Physical analogy
40:geometric design
2943:
2942:
2938:
2937:
2936:
2934:
2933:
2932:
2913:
2912:
2874:
2869:
2859:
2857:
2849:
2848:
2844:
2835:
2833:
2818:Panagiotopoulos
2815:
2814:
2810:
2788:
2787:
2783:
2777:
2758:10.1.1.470.5213
2739:
2738:
2734:
2717:
2713:10.1.1.109.6855
2701:
2700:
2696:
2680:
2676:
2661:: 24768–24779.
2648:
2647:
2638:
2631:
2620:
2613:
2612:
2603:
2599:
2550:
2513:
2485:
2484:
2465:
2464:
2437:
2424:
2410:
2409:
2378:
2377:
2358:
2357:
2336:
2331:
2330:
2309:
2304:
2303:
2284:
2283:
2264:
2263:
2233:
2217:
2159:
2154:
2153:
2123:
2118:
2117:
2098:
2097:
2070:
2062:
2061:
2037:
2029:
2028:
2004:
1996:
1995:
1971:
1946:
1936:
1905:
1900:
1899:
1880:
1879:
1854:
1853:
1825:
1824:
1787:
1786:
1749:
1748:
1729:
1728:
1679:
1678:
1654:
1635:
1529:
1495:
1490:
1489:
1449:
1448:
1429:
1428:
1409:
1408:
1389:
1388:
1369:
1368:
1341:
1325:
1311:
1310:
1289:
1284:
1283:
1278:). One can use
1254:
1253:
1250:
1215:
1195:
1194:
1170:
1162:
1161:
1135:
1130:
1129:
1097:
1090:
1086:
1070:
1029:
1028:
997:
996:
977:
976:
922:
914:
913:
910:
904:
880:
879:
844:
843:
819:
801:
757:
743:
742:
736:
735:
709:
696:
692:
678:
677:
671:
670:
632:
618:
617:
611:
610:
609:
605:
586:
573:
554:
477:
472:
471:
449:
448:
424:
411:
392:
338:
333:
332:
305:
297:
296:
272:
264:
263:
235:
234:
215:
214:
211:
180:
179:
145:
144:
125:
124:
105:
104:
85:
84:
65:
64:
52:
17:
12:
11:
5:
2941:
2939:
2931:
2930:
2925:
2915:
2914:
2911:
2910:
2905:
2900:
2895:
2890:
2885:
2880:
2873:
2872:External links
2870:
2868:
2867:
2842:
2808:
2797:(6): 567–585.
2781:
2775:
2732:
2694:
2674:
2636:
2629:
2600:
2598:
2595:
2594:
2593:
2587:
2582:
2577:
2572:
2566:
2561:
2549:
2546:
2542:
2541:
2538:
2535:
2532:
2512:
2509:
2492:
2472:
2452:
2449:
2444:
2440:
2436:
2431:
2427:
2423:
2420:
2417:
2397:
2394:
2391:
2388:
2385:
2365:
2343:
2339:
2316:
2312:
2291:
2271:
2260:
2259:
2248:
2245:
2240:
2236:
2232:
2229:
2224:
2220:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2172:
2169:
2166:
2162:
2136:
2133:
2130:
2126:
2105:
2082:
2077:
2073:
2069:
2049:
2044:
2040:
2036:
2016:
2011:
2007:
2003:
1983:
1978:
1974:
1970:
1967:
1964:
1961:
1958:
1953:
1949:
1943:
1939:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1912:
1908:
1887:
1867:
1864:
1861:
1841:
1838:
1835:
1832:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1736:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1675:
1674:
1661:
1657:
1653:
1650:
1647:
1642:
1638:
1632:
1627:
1624:
1621:
1617:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1542:
1539:
1536:
1532:
1528:
1525:
1522:
1519:
1516:
1513:
1508:
1505:
1502:
1498:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1436:
1416:
1396:
1376:
1356:
1351:
1348:
1344:
1340:
1335:
1332:
1328:
1324:
1321:
1318:
1296:
1292:
1267:
1264:
1261:
1249:
1246:
1233:
1230:
1227:
1222:
1218:
1214:
1211:
1208:
1205:
1202:
1182:
1177:
1173:
1169:
1158:Euclidean norm
1144:
1141:
1138:
1126:
1125:
1114:
1110:
1104:
1100:
1096:
1093:
1089:
1085:
1082:
1077:
1073:
1067:
1062:
1059:
1056:
1052:
1048:
1045:
1042:
1039:
1036:
1013:
1010:
1007:
1004:
984:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
929:
925:
921:
906:Main article:
903:
900:
887:
870:, is used for
864:finite element
851:
840:
839:
826:
822:
808:
804:
792:
786:
781:
773:
768:
764:
760:
755:
750:
746:
739:
734:
729:
724:
716:
712:
708:
703:
699:
695:
690:
685:
681:
674:
669:
666:
661:
656:
648:
643:
639:
635:
630:
625:
621:
614:
608:
604:
601:
598:
593:
589:
585:
580:
576:
572:
569:
566:
561:
557:
553:
548:
543:
540:
537:
533:
529:
526:
523:
520:
514:
511:
508:
505:
502:
499:
495:
491:
488:
485:
480:
456:
445:
444:
431:
427:
423:
418:
414:
410:
407:
404:
399:
395:
391:
386:
381:
378:
375:
371:
367:
364:
361:
358:
352:
349:
346:
341:
317:
312:
308:
304:
284:
279:
275:
271:
251:
248:
245:
242:
222:
210:
207:
190:
187:
167:
164:
161:
158:
155:
152:
132:
112:
92:
72:
51:
48:
15:
13:
10:
9:
6:
4:
3:
2:
2940:
2929:
2926:
2924:
2921:
2920:
2918:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2875:
2871:
2856:
2852:
2846:
2843:
2831:
2827:
2823:
2820:, Diamantis;
2819:
2812:
2809:
2804:
2800:
2796:
2792:
2785:
2782:
2778:
2772:
2768:
2764:
2759:
2754:
2750:
2746:
2742:
2736:
2733:
2728:
2722:
2714:
2709:
2705:
2698:
2695:
2692:
2688:
2684:
2678:
2675:
2669:
2664:
2660:
2656:
2652:
2645:
2643:
2641:
2637:
2632:
2626:
2619:
2618:
2610:
2608:
2606:
2602:
2596:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2570:
2567:
2565:
2562:
2559:
2555:
2552:
2551:
2547:
2545:
2539:
2536:
2533:
2530:
2529:
2528:
2525:
2523:
2519:
2510:
2508:
2504:
2490:
2442:
2438:
2434:
2429:
2425:
2415:
2392:
2389:
2386:
2341:
2337:
2314:
2310:
2289:
2269:
2246:
2238:
2234:
2230:
2227:
2222:
2214:
2208:
2205:
2202:
2199:
2196:
2190:
2184:
2181:
2178:
2170:
2167:
2164:
2160:
2152:
2151:
2150:
2134:
2131:
2128:
2124:
2103:
2094:
2075:
2071:
2042:
2038:
2009:
2005:
1976:
1972:
1968:
1965:
1959:
1956:
1951:
1941:
1937:
1933:
1930:
1924:
1918:
1910:
1906:
1885:
1865:
1862:
1859:
1836:
1830:
1807:
1804:
1801:
1795:
1792:
1769:
1766:
1763:
1757:
1754:
1734:
1711:
1708:
1705:
1699:
1693:
1690:
1687:
1677:where d is a
1659:
1655:
1648:
1640:
1636:
1630:
1625:
1622:
1619:
1615:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1587:
1581:
1578:
1575:
1572:
1569:
1566:
1560:
1557:
1554:
1551:
1548:
1540:
1537:
1534:
1530:
1526:
1520:
1517:
1514:
1506:
1503:
1500:
1496:
1488:
1487:
1486:
1469:
1466:
1463:
1457:
1454:
1434:
1414:
1394:
1374:
1349:
1346:
1342:
1338:
1333:
1330:
1326:
1322:
1319:
1294:
1290:
1281:
1265:
1262:
1259:
1247:
1245:
1231:
1228:
1225:
1220:
1216:
1212:
1206:
1200:
1175:
1171:
1159:
1139:
1102:
1098:
1094:
1091:
1080:
1075:
1071:
1065:
1060:
1057:
1054:
1050:
1046:
1040:
1034:
1027:
1026:
1025:
1008:
1002:
982:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
927:
923:
909:
901:
899:
885:
877:
873:
869:
865:
849:
824:
820:
806:
802:
790:
784:
779:
771:
766:
762:
753:
748:
737:
732:
727:
722:
714:
710:
701:
697:
688:
683:
672:
667:
664:
659:
654:
646:
641:
637:
628:
623:
612:
606:
602:
599:
596:
591:
578:
574:
567:
564:
559:
555:
546:
541:
538:
535:
531:
527:
521:
493:
478:
470:
469:
468:
454:
429:
416:
412:
405:
402:
397:
393:
384:
379:
376:
373:
369:
365:
359:
339:
331:
330:
329:
310:
306:
277:
273:
246:
240:
220:
208:
206:
204:
188:
185:
162:
159:
156:
150:
130:
110:
90:
70:
61:
57:
49:
47:
45:
41:
37:
33:
32:interpolation
29:
25:
21:
2858:. Retrieved
2854:
2845:
2834:, retrieved
2829:
2811:
2794:
2790:
2784:
2744:
2735:
2703:
2697:
2677:
2658:
2654:
2616:
2543:
2526:
2514:
2511:Applications
2505:
2261:
2095:
1676:
1279:
1251:
1127:
911:
868:elastic maps
841:
446:
212:
55:
53:
23:
19:
18:
2655:IEEE Access
2554:Elastic map
872:data mining
2917:Categories
2888:TPS in C++
2597:References
2908:TPS in JS
2753:CiteSeerX
2708:CiteSeerX
2683:K. Zeller
2471:Φ
2448:‖
2435:−
2422:‖
2416:ϕ
2390:×
2364:Φ
2244:Φ
2231:λ
2219:‖
2212:Φ
2209:−
2200:−
2194:‖
2149:becomes:
1982:‖
1969:−
1963:‖
1960:
1948:‖
1934:−
1928:‖
1907:ϕ
1863:×
1831:ϕ
1796:×
1758:×
1700:×
1637:ϕ
1616:∑
1606:⋅
1594:⋅
1582:ϕ
1573:⋅
1521:α
1455:α
1395:α
1229:
1201:φ
1140:⋅
1095:−
1081:φ
1051:∑
954:…
759:∂
745:∂
707:∂
694:∂
680:∂
634:∂
620:∂
603:∬
600:λ
588:‖
565:−
552:‖
532:∑
455:λ
426:‖
403:−
390:‖
370:∑
54:The name
36:smoothing
2903:TPS in R
2836:28 March
2751:(SIAM),
2721:citation
2548:See also
1143:‖
1137:‖
1109:‖
1088:‖
26:) are a
2860:3 March
2520:in the
2773:
2755:
2710:
2627:
2585:Spline
2356:, and
2262:where
1248:Spline
1128:where
862:. The
28:spline
2741:Wahba
2621:(PDF)
2376:is a
1852:is a
1747:is a
60:plate
2862:2019
2838:2019
2822:Mara
2771:ISBN
2727:link
2625:ISBN
2329:and
2282:and
1427:and
1160:and
874:and
295:and
34:and
2799:doi
2763:doi
2687:doi
2663:doi
1957:log
1485:).
1226:log
103:or
24:TPS
2919::
2853:.
2828:,
2795:11
2793:.
2769:,
2761:,
2723:}}
2719:{{
2659:11
2657:.
2653:.
2639:^
2604:^
2524:.
2116:,
1244:.
205:.
2864:.
2805:.
2801::
2765::
2729:)
2689::
2671:.
2665::
2633:.
2560:)
2491:c
2451:)
2443:j
2439:x
2430:i
2426:x
2419:(
2396:)
2393:K
2387:K
2384:(
2342:i
2338:x
2315:i
2311:y
2290:X
2270:Y
2247:c
2239:T
2235:c
2228:+
2223:2
2215:c
2206:d
2203:X
2197:Y
2191:=
2188:)
2185:c
2182:,
2179:d
2176:(
2171:s
2168:p
2165:t
2161:E
2135:s
2132:p
2129:t
2125:E
2104:f
2081:}
2076:i
2072:x
2068:{
2048:}
2043:i
2039:x
2035:{
2015:}
2010:i
2006:c
2002:{
1977:i
1973:x
1966:z
1952:2
1942:i
1938:x
1931:z
1925:=
1922:)
1919:z
1916:(
1911:i
1886:z
1866:K
1860:1
1840:)
1837:z
1834:(
1811:)
1808:1
1805:+
1802:D
1799:(
1793:K
1773:)
1770:1
1767:+
1764:D
1761:(
1755:1
1735:z
1715:)
1712:1
1709:+
1706:D
1703:(
1697:)
1694:1
1691:+
1688:D
1685:(
1660:i
1656:c
1652:)
1649:z
1646:(
1641:i
1631:K
1626:1
1623:=
1620:i
1612:+
1609:d
1603:z
1600:=
1597:c
1591:)
1588:z
1585:(
1579:+
1576:d
1570:z
1567:=
1564:)
1561:c
1558:,
1555:d
1552:,
1549:z
1546:(
1541:s
1538:p
1535:t
1531:f
1527:=
1524:)
1518:,
1515:z
1512:(
1507:s
1504:p
1501:t
1497:f
1473:}
1470:c
1467:,
1464:d
1461:{
1458:=
1447:(
1435:c
1415:d
1375:f
1355:)
1350:y
1347:i
1343:y
1339:,
1334:x
1331:i
1327:y
1323:,
1320:1
1317:(
1295:i
1291:y
1266:2
1263:=
1260:D
1232:r
1221:2
1217:r
1213:=
1210:)
1207:r
1204:(
1181:}
1176:i
1172:w
1168:{
1113:)
1103:i
1099:c
1092:x
1084:(
1076:i
1072:w
1066:K
1061:1
1058:=
1055:i
1047:=
1044:)
1041:x
1038:(
1035:f
1012:)
1009:x
1006:(
1003:f
983:x
963:}
960:K
957:,
951:,
948:2
945:,
942:1
939:=
936:i
933:,
928:i
924:c
920:{
886:f
850:f
825:2
821:x
815:d
807:1
803:x
797:d
791:]
785:2
780:)
772:2
767:2
763:x
754:f
749:2
738:(
733:+
728:2
723:)
715:2
711:x
702:1
698:x
689:f
684:2
673:(
668:2
665:+
660:2
655:)
647:2
642:1
638:x
629:f
624:2
613:(
607:[
597:+
592:2
584:)
579:i
575:x
571:(
568:f
560:i
556:y
547:K
542:1
539:=
536:i
528:=
525:)
522:f
519:(
513:h
510:t
507:o
504:o
501:m
498:s
494:,
490:s
487:p
484:t
479:E
430:2
422:)
417:i
413:x
409:(
406:f
398:i
394:y
385:K
380:1
377:=
374:i
366:=
363:)
360:f
357:(
351:s
348:p
345:t
340:E
316:}
311:i
307:x
303:{
283:}
278:i
274:y
270:{
250:)
247:x
244:(
241:f
221:x
189:K
186:2
166:)
163:3
160:+
157:K
154:(
151:2
131:K
111:y
91:x
71:z
22:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.