2517:
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
848:
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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
1134:
2268:
73:
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
2003:
2494:
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
2692:
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
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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
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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.
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843:{\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}}
889:. 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
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1041:
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224:
The TPS arises from consideration of the integral of the square of the second derivative—this forms its smoothness measure. In the case where
2785:
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coefficients for correspondences of the control points. These parameters are computed by solving a linear system, in other words, TPS has a
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to control the rigidity of the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimizing:
2938:
886:
1912:
2933:
2532:
923:
The thin plate spline has a natural representation in terms of radial basis functions. Given a set of control points
1678:{\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}}
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2861:
2828:
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2800:
Bookstein, F. L. (June 1989). "Principal warps: thin plate splines and the decomposition of deformations".
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49:. "A spline is a function defined by polynomials in a piecewise manner." They were introduced to
57:. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm.
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2008:
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Proc. of the 14th
International Conference on Computer Vision Theory and Application (VISAPP)
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50:
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The thin plate spline has a number of properties which have contributed to its popularity:
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2474:. Each row of each newly formed matrix comes from one of the original vectors. The matrix
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warping coefficient matrix representing the non-affine deformation. The kernel function
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2662:"Energy-Efficient Post-Failure Reconfiguration of Swarms of Unmanned Aerial Vehicles"
42:
2893:
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2528:
853:
For this variational problem, it can be shown that there exists a unique minimizer
17:
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is a set of mapping coefficients. The TPS corresponds to the radial basis kernel
2832:
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882:
878:
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2777:
986:, a radial basis function defines a spatial mapping which maps any location
46:
448:{\displaystyle E_{\mathrm {tps} }(f)=\sum _{i=1}^{K}\|y_{i}-f(x_{i})\|^{2}}
1129:{\displaystyle f(x)=\sum _{i=1}^{K}w_{i}\varphi (\left\|x-c_{i}\right\|)}
2660:
Tahir, Anam; Haghbayan, Hashem; Böling, Jari M.; Plosila, Juha (2023).
2548:
It has closed-form solutions for both warping and parameter estimation.
244:
is two dimensional, for interpolation, the TPS fits a mapping function
2813:
2701:
2903:
2518:
parameters included in the affine transformation are not penalized.
2263:{\displaystyle E_{tps}(d,c)=\|Y-Xd-\Phi c\|^{2}+\lambda c^{T}\Phi c}
154:
corresponding control points (knots), the TPS warp is described by
2696:, eds., Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977.
2582:(the thin plate spline is a special case of a polyharmonic spline)
2715:
Non-Rigid Point
Matching: Algorithms, Extensions and Applications
2542:
It produces smooth surfaces, which are infinitely differentiable.
458:
The smoothing variant, correspondingly, uses a tuning parameter
189:
parameters which include 6 global affine motion parameters and
2837:"Recovering and Visualizing Deformation in 3D Aegean Sealings"
2802:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
909:." It is in a general case needed to make the mapping unique.
2038:
are chosen to be the same as the set of points to be warped
1998:{\displaystyle \phi _{i}(z)=\|z-x_{i}\|^{2}\log \|z-x_{i}\|}
877:
discretization of this variational problem, the method of
134:
coordinates within the plane. In 2D cases, given a set of
2567:(a discrete version of the thin plate approximation for
2551:
There is a physical explanation for its energy function.
2313:
are just concatenated versions of the point coordinates
69:
refers to a physical analogy involving the bending of a
2628:
Formation
Control of Swarms of Unmanned Aerial Vehicles
2545:
There are no free parameters that need manual tuning.
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matrix representing the affine transformation (hence
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53:by Duchon. They are an important special case of a
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86:
2862:"Tutorial No. 13: Apply TPS-RPM Transformation"
2603:(emerging alternative to spline-based surfaces)
339:that minimizes the following energy function:
2889:Explanation for a simplified variation problem
2760:Society for Industrial and Applied Mathematics
8:
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2736:: CS1 maint: location missing publisher (
27:Method of data interpolation and smoothing
2827:Bogacz, Bartosz; Papadimitriou, Nikolas;
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2019:
2010:
2005:. Note that for TPS, the control points
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979:{\displaystyle \{c_{i},i=1,2,\ldots ,K\}}
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1263:Suppose the points are in 2 dimensions (
2717:, Yale University, New Haven, CT, USA,
2612:
1248:{\displaystyle \varphi (r)=r^{2}\log r}
2729:
2467:{\displaystyle \phi (\|x_{i}-x_{j}\|)}
1160:{\displaystyle \left\|\cdot \right\|}
7:
2756:Spline models for observational data
2514:, a non-rigid warping is generated.
2107:If one substitutes the solution for
2104:in the place of the control points.
2481:
2374:
2254:
2222:
887:nonlinear dimensionality reduction
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361:
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25:
1731:{\displaystyle (D+1)\times (D+1)}
1371:{\displaystyle (1,y_{ix},y_{iy})}
273:between corresponding point-sets
2634:. Finland: University of Turku.
1293:for the point-set where a point
1489:{\displaystyle \alpha =\{d,c\}}
1418:which consists of two matrices
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176:
164:
1:
2909:TPS interactive morphing demo
1827:{\displaystyle K\times (D+1)}
1789:{\displaystyle 1\times (D+1)}
2866:GigaMesh Software Framework
2679:10.1109/ACCESS.2022.3181244
2533:GigaMesh Software Framework
2412:{\displaystyle (K\times K)}
1320:is represented as a vector
1006:in space to a new location
2955:
2939:Multivariate interpolation
2575:Inverse distance weighting
916:
41:-based technique for data
2758:, Philadelphia, PA, USA:
2097:{\displaystyle \{x_{i}\}}
2064:{\displaystyle \{x_{i}\}}
2031:{\displaystyle \{c_{i}\}}
1882:{\displaystyle 1\times K}
1197:{\displaystyle \{w_{i}\}}
332:{\displaystyle \{x_{i}\}}
299:{\displaystyle \{y_{i}\}}
2843:, Prague, Czech Republic
1856:{\displaystyle \phi (z)}
471:{\displaystyle \lambda }
2778:10.1137/1.9781611970128
2419:matrix formed from the
2153:{\displaystyle E_{tps}}
1411:{\displaystyle \alpha }
1378:. The unique minimizer
1291:homogeneous coordinates
2508:
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2334:
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1999:
1903:
1889:vector for each point
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182:{\displaystyle 2(K+3)}
148:
128:
108:
88:
2934:Splines (mathematics)
2586:Radial basis function
2509:
2489:
2487:{\displaystyle \Phi }
2469:
2414:
2382:
2380:{\displaystyle \Phi }
2362:
2360:{\displaystyle x_{i}}
2335:
2333:{\displaystyle y_{i}}
2308:
2288:
2265:
2155:
2122:
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1829:
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1313:{\displaystyle y_{i}}
1284:
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981:
919:Radial basis function
913:Radial basis function
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473:
450:
379:
334:
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184:
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129:
109:
89:
2904:TPS in templated C++
2713:Chui, Haili (2001),
2625:Tahir, Anam (2023).
2498:
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2423:
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2111:
2075:
2071:, so we already use
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2009:
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1893:
1867:
1838:
1800:
1762:
1742:
1692:
1503:
1462:
1442:
1422:
1402:
1398:is parameterized by
1382:
1324:
1297:
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1042:
1028:{\displaystyle f(x)}
1010:
990:
927:
893:
857:
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462:
346:
310:
277:
266:{\displaystyle f(x)}
248:
228:
214:closed-form solution
193:
158:
138:
118:
98:
78:
2601:Subdivision surface
2580:Polyharmonic spline
1909:, where each entry
1796:vector) and c is a
1282:{\displaystyle D=2}
786:
661:
55:polyharmonic spline
2702:10.1007/BFb0086566
2504:
2484:
2464:
2409:
2377:
2357:
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2283:
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2117:
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2028:
1995:
1899:
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1853:
1824:
1786:
1748:
1728:
1675:
1486:
1448:
1428:
1408:
1388:
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1310:
1279:
1245:
1194:
1167:denotes the usual
1157:
1126:
1025:
996:
976:
899:
863:
840:
772:
647:
468:
445:
329:
296:
263:
234:
220:Smoothness measure
205:{\displaystyle 2K}
202:
179:
144:
124:
104:
84:
31:Thin plate splines
18:Thin plate splines
2835:, Hubert (2019),
2787:978-0-89871-244-5
2641:978-951-29-9411-3
2569:manifold learning
2529:triangular meshes
2507:{\displaystyle c}
2306:{\displaystyle X}
2286:{\displaystyle Y}
2120:{\displaystyle f}
1902:{\displaystyle z}
1751:{\displaystyle z}
1451:{\displaystyle c}
1431:{\displaystyle d}
1391:{\displaystyle f}
1035:, represented by
999:{\displaystyle x}
902:{\displaystyle f}
866:{\displaystyle f}
827:
809:
788:
731:
663:
237:{\displaystyle x}
147:{\displaystyle K}
127:{\displaystyle y}
107:{\displaystyle x}
87:{\displaystyle z}
67:thin plate spline
16:(Redirected from
2946:
2894:TPS at MathWorld
2877:
2876:
2874:
2872:
2858:
2852:
2851:
2850:
2848:
2824:
2818:
2817:
2814:10.1109/34.24792
2797:
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2754:, Grace (1990),
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2591:Smoothing spline
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61:Physical analogy
51:geometric design
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2868:
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2859:
2855:
2846:
2844:
2829:Panagiotopoulos
2826:
2825:
2821:
2799:
2798:
2794:
2788:
2769:10.1.1.470.5213
2750:
2749:
2745:
2728:
2724:10.1.1.109.6855
2712:
2711:
2707:
2691:
2687:
2672:: 24768–24779.
2659:
2658:
2649:
2642:
2631:
2624:
2623:
2614:
2610:
2561:
2524:
2496:
2495:
2476:
2475:
2448:
2435:
2421:
2420:
2389:
2388:
2369:
2368:
2347:
2342:
2341:
2320:
2315:
2314:
2295:
2294:
2275:
2274:
2244:
2228:
2170:
2165:
2164:
2134:
2129:
2128:
2109:
2108:
2081:
2073:
2072:
2048:
2040:
2039:
2015:
2007:
2006:
1982:
1957:
1947:
1916:
1911:
1910:
1891:
1890:
1865:
1864:
1836:
1835:
1798:
1797:
1760:
1759:
1740:
1739:
1690:
1689:
1665:
1646:
1540:
1506:
1501:
1500:
1460:
1459:
1440:
1439:
1420:
1419:
1400:
1399:
1380:
1379:
1352:
1336:
1322:
1321:
1300:
1295:
1294:
1289:). One can use
1265:
1264:
1261:
1226:
1206:
1205:
1181:
1173:
1172:
1146:
1141:
1140:
1108:
1101:
1097:
1081:
1040:
1039:
1008:
1007:
988:
987:
933:
925:
924:
921:
915:
891:
890:
855:
854:
830:
812:
768:
754:
753:
747:
746:
720:
707:
703:
689:
688:
682:
681:
643:
629:
628:
622:
621:
620:
616:
597:
584:
565:
488:
483:
482:
460:
459:
435:
422:
403:
349:
344:
343:
316:
308:
307:
283:
275:
274:
246:
245:
226:
225:
222:
191:
190:
156:
155:
136:
135:
116:
115:
96:
95:
76:
75:
63:
28:
23:
22:
15:
12:
11:
5:
2952:
2950:
2942:
2941:
2936:
2926:
2925:
2922:
2921:
2916:
2911:
2906:
2901:
2896:
2891:
2884:
2883:External links
2881:
2879:
2878:
2853:
2819:
2808:(6): 567–585.
2792:
2786:
2743:
2705:
2685:
2647:
2640:
2611:
2609:
2606:
2605:
2604:
2598:
2593:
2588:
2583:
2577:
2572:
2560:
2557:
2553:
2552:
2549:
2546:
2543:
2523:
2520:
2503:
2483:
2463:
2460:
2455:
2451:
2447:
2442:
2438:
2434:
2431:
2428:
2408:
2405:
2402:
2399:
2396:
2376:
2354:
2350:
2327:
2323:
2302:
2282:
2271:
2270:
2259:
2256:
2251:
2247:
2243:
2240:
2235:
2231:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2183:
2180:
2177:
2173:
2147:
2144:
2141:
2137:
2116:
2093:
2088:
2084:
2080:
2060:
2055:
2051:
2047:
2027:
2022:
2018:
2014:
1994:
1989:
1985:
1981:
1978:
1975:
1972:
1969:
1964:
1960:
1954:
1950:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1923:
1919:
1898:
1878:
1875:
1872:
1852:
1849:
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1823:
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1536:
1533:
1530:
1527:
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1516:
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1509:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1447:
1427:
1407:
1387:
1367:
1362:
1359:
1355:
1351:
1346:
1343:
1339:
1335:
1332:
1329:
1307:
1303:
1278:
1275:
1272:
1260:
1257:
1244:
1241:
1238:
1233:
1229:
1225:
1222:
1219:
1216:
1213:
1193:
1188:
1184:
1180:
1169:Euclidean norm
1155:
1152:
1149:
1137:
1136:
1125:
1121:
1115:
1111:
1107:
1104:
1100:
1096:
1093:
1088:
1084:
1078:
1073:
1070:
1067:
1063:
1059:
1056:
1053:
1050:
1047:
1024:
1021:
1018:
1015:
995:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
940:
936:
932:
917:Main article:
914:
911:
898:
881:, is used for
875:finite element
862:
851:
850:
837:
833:
819:
815:
803:
797:
792:
784:
779:
775:
771:
766:
761:
757:
750:
745:
740:
735:
727:
723:
719:
714:
710:
706:
701:
696:
692:
685:
680:
677:
672:
667:
659:
654:
650:
646:
641:
636:
632:
625:
619:
615:
612:
609:
604:
600:
596:
591:
587:
583:
580:
577:
572:
568:
564:
559:
554:
551:
548:
544:
540:
537:
534:
531:
525:
522:
519:
516:
513:
510:
506:
502:
499:
496:
491:
467:
456:
455:
442:
438:
434:
429:
425:
421:
418:
415:
410:
406:
402:
397:
392:
389:
386:
382:
378:
375:
372:
369:
363:
360:
357:
352:
328:
323:
319:
315:
295:
290:
286:
282:
262:
259:
256:
253:
233:
221:
218:
201:
198:
178:
175:
172:
169:
166:
163:
143:
123:
103:
83:
62:
59:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2951:
2940:
2937:
2935:
2932:
2931:
2929:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2886:
2882:
2867:
2863:
2857:
2854:
2842:
2838:
2834:
2831:, Diamantis;
2830:
2823:
2820:
2815:
2811:
2807:
2803:
2796:
2793:
2789:
2783:
2779:
2775:
2770:
2765:
2761:
2757:
2753:
2747:
2744:
2739:
2733:
2725:
2720:
2716:
2709:
2706:
2703:
2699:
2695:
2689:
2686:
2680:
2675:
2671:
2667:
2663:
2656:
2654:
2652:
2648:
2643:
2637:
2630:
2629:
2621:
2619:
2617:
2613:
2607:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2581:
2578:
2576:
2573:
2570:
2566:
2563:
2562:
2558:
2556:
2550:
2547:
2544:
2541:
2540:
2539:
2536:
2534:
2530:
2521:
2519:
2515:
2501:
2453:
2449:
2445:
2440:
2436:
2426:
2403:
2400:
2397:
2352:
2348:
2325:
2321:
2300:
2280:
2257:
2249:
2245:
2241:
2238:
2233:
2225:
2219:
2216:
2213:
2210:
2207:
2201:
2195:
2192:
2189:
2181:
2178:
2175:
2171:
2163:
2162:
2161:
2145:
2142:
2139:
2135:
2114:
2105:
2086:
2082:
2053:
2049:
2020:
2016:
1987:
1983:
1979:
1976:
1970:
1967:
1962:
1952:
1948:
1944:
1941:
1935:
1929:
1921:
1917:
1896:
1876:
1873:
1870:
1847:
1841:
1818:
1815:
1812:
1806:
1803:
1780:
1777:
1774:
1768:
1765:
1745:
1722:
1719:
1716:
1710:
1704:
1701:
1698:
1688:where d is a
1670:
1666:
1659:
1651:
1647:
1641:
1636:
1633:
1630:
1626:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1598:
1592:
1589:
1586:
1583:
1580:
1577:
1571:
1568:
1565:
1562:
1559:
1551:
1548:
1545:
1541:
1537:
1531:
1528:
1525:
1517:
1514:
1511:
1507:
1499:
1498:
1497:
1480:
1477:
1474:
1468:
1465:
1445:
1425:
1405:
1385:
1360:
1357:
1353:
1349:
1344:
1341:
1337:
1333:
1330:
1305:
1301:
1292:
1276:
1273:
1270:
1258:
1256:
1242:
1239:
1236:
1231:
1227:
1223:
1217:
1211:
1186:
1182:
1170:
1150:
1113:
1109:
1105:
1102:
1091:
1086:
1082:
1076:
1071:
1068:
1065:
1061:
1057:
1051:
1045:
1038:
1037:
1036:
1019:
1013:
993:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
938:
934:
920:
912:
910:
896:
888:
884:
880:
876:
860:
835:
831:
817:
813:
801:
795:
790:
782:
777:
773:
764:
759:
748:
743:
738:
733:
725:
721:
712:
708:
699:
694:
683:
678:
675:
670:
665:
657:
652:
648:
639:
634:
623:
617:
613:
610:
607:
602:
589:
585:
578:
575:
570:
566:
557:
552:
549:
546:
542:
538:
532:
504:
489:
481:
480:
479:
465:
440:
427:
423:
416:
413:
408:
404:
395:
390:
387:
384:
380:
376:
370:
350:
342:
341:
340:
321:
317:
288:
284:
257:
251:
231:
219:
217:
215:
199:
196:
173:
170:
167:
161:
141:
121:
101:
81:
72:
68:
60:
58:
56:
52:
48:
44:
43:interpolation
40:
36:
32:
19:
2869:. Retrieved
2865:
2856:
2845:, retrieved
2840:
2822:
2805:
2801:
2795:
2755:
2746:
2714:
2708:
2688:
2669:
2665:
2627:
2554:
2537:
2525:
2522:Applications
2516:
2272:
2106:
1687:
1290:
1262:
1138:
922:
879:elastic maps
852:
457:
223:
66:
64:
34:
30:
29:
2666:IEEE Access
2565:Elastic map
883:data mining
2928:Categories
2899:TPS in C++
2608:References
2919:TPS in JS
2764:CiteSeerX
2719:CiteSeerX
2694:K. Zeller
2482:Φ
2459:‖
2446:−
2433:‖
2427:ϕ
2401:×
2375:Φ
2255:Φ
2242:λ
2230:‖
2223:Φ
2220:−
2211:−
2205:‖
2160:becomes:
1993:‖
1980:−
1974:‖
1971:
1959:‖
1945:−
1939:‖
1918:ϕ
1874:×
1842:ϕ
1807:×
1769:×
1711:×
1648:ϕ
1627:∑
1617:⋅
1605:⋅
1593:ϕ
1584:⋅
1532:α
1466:α
1406:α
1240:
1212:φ
1151:⋅
1106:−
1092:φ
1062:∑
965:…
770:∂
756:∂
718:∂
705:∂
691:∂
645:∂
631:∂
614:∬
611:λ
599:‖
576:−
563:‖
543:∑
466:λ
437:‖
414:−
401:‖
381:∑
65:The name
47:smoothing
2914:TPS in R
2847:28 March
2762:(SIAM),
2732:citation
2559:See also
1154:‖
1148:‖
1120:‖
1099:‖
37:) are a
2871:3 March
2531:in the
2784:
2766:
2721:
2638:
2596:Spline
2367:, and
2273:where
1259:Spline
1139:where
873:. The
39:spline
2752:Wahba
2632:(PDF)
2387:is a
1863:is a
1758:is a
71:plate
2873:2019
2849:2019
2833:Mara
2782:ISBN
2738:link
2636:ISBN
2340:and
2293:and
1438:and
1171:and
885:and
306:and
45:and
2810:doi
2774:doi
2698:doi
2674:doi
1968:log
1496:).
1237:log
114:or
35:TPS
2930::
2864:.
2839:,
2806:11
2804:.
2780:,
2772:,
2734:}}
2730:{{
2670:11
2668:.
2664:.
2650:^
2615:^
2535:.
2127:,
1255:.
216:.
2875:.
2816:.
2812::
2776::
2740:)
2700::
2682:.
2676::
2644:.
2571:)
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2462:)
2454:j
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2404:K
2398:K
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2326:i
2322:y
2301:X
2281:Y
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2250:T
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2239:+
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2217:d
2214:X
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2202:=
2199:)
2196:c
2193:,
2190:d
2187:(
2182:s
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2079:{
2059:}
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2046:{
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2021:i
2017:c
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1575:)
1572:c
1569:,
1566:d
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1557:(
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1523:(
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.