使用受约束的 B 样条近似形状轮廓
Approximate a shape outline using constrained B-splines
我正在寻找生成约束样条以近似形状（在我的例子中是足迹轮廓）的可能性。作为原始数据，我有一个包含数百个 xy 坐标对的 table，它们是从足迹的边界收集的。样条应该只逼近数据点（样条不需要通过数据点）。我希望能够在一定程度上平滑样条曲线。另外，我需要能够约束样条曲线：定义样条曲线必须通过的几个关键数据点。
R 包 "cobs"（约束 B 样条曲线，https://cran.r-project.org/web/packages/cobs/index.html）非常接近于提供解决方案，提供参数以根据需要约束样条曲线。但是，此包不会通过有序的数据点序列进行样条化，这在您希望样条线遵循形状边界时当然是至关重要的。我尝试分别样条化 x 和 y 坐标，但在将它们重新组合后，图中出现了两个不同的形状，所以这似乎不起作用（或者我弄错了什么？）。有人知道解决方案吗？
更新：工作示例（恐龙足迹大纲）
data.txt:
structure(list(V1 = c(124.9, 86.44, 97.22, 81.34, 49.09, 57.18,
-77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72,
-600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18,
-1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59,
-859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09,
-394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97,
344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14,
1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93,
2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03,
1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96,
1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94,
1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32,
697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25,
718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96,
823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72,
735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16,
45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72,
-1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69,
144.87, 142.26, 146.34, 125.24, 124.9, 86.44, 97.22, 81.34, 49.09,
57.18, -77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72,
-600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18,
-1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59,
-859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09,
-394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97,
344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14,
1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93,
2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03,
1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96,
1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94,
1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32,
697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25,
718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96,
823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72,
735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16,
45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72,
-1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69,
144.87, 142.26, 146.34, 125.24), V2 = c(-446.8, -415.83, -394.43,
-259.19, -104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23,
-176.89, -233.68, -321.28, -416.57, -457.97, -458.93, -429.09,
-422.35, -450.27, -431.98, -379.03, -260.63, -123.94, -2.65,
269.76, 455.55, 548.92, 616.3, 691.38, 756.84, 888.72, 1016.97,
1157.18, 1198.02, 1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47,
733.81, 784.18, 835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13,
732.45, 586.79, 394.84, 212.42, 28.64, -111.58, -337.56, -490.03,
-526.07, -528.82, -547.2, -551.97, -552.3, -585.51, -551.34,
-543.16, -526.1, -494.11, -466.88, -355.93, -274.94, -215.04,
-114.3, -194.21, -103.73, -3.62, 104.2, 230.8, 154.25, 380.55,
416.62, 260.07, 295.75, 295.75, 251.47, 220.67, 225.96, 180.72,
121.52, 4.14, -127.23, -176.24, -332.11, -408.35, -494.11, -573.75,
-582.62, -678.88, -730.38, -788.62, -831.94, -846.38, -895.95,
-934.46, -968.15, -1033.12, -1097.62, -1150.08, -1157.3, -1254.04,
-1340.2, -1441.75, -1500.47, -1550.52, -1605.39, -1681.44, -1709.84,
-1715.22, -1672.34, -1607, -1522.59, -1440.57, -1421.18, -1345.62,
-1247.95, -1190.77, -1181.58, -1071.65, -1037.62, -1010.39, -998.82,
-986.57, -937.9, -887.29, -842.05, -831.46, -774.66, -703.91,
-573.75, -533.59, -448.16, -446.8, -415.83, -394.43, -259.19,
-104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23, -176.89,
-233.68, -321.28, -416.57, -457.97, -458.93, -429.09, -422.35,
-450.27, -431.98, -379.03, -260.63, -123.94, -2.65, 269.76, 455.55,
548.92, 616.3, 691.38, 756.84, 888.72, 1016.97, 1157.18, 1198.02,
1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47, 733.81, 784.18,
835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13, 732.45, 586.79,
394.84, 212.42, 28.64, -111.58, -337.56, -490.03, -526.07, -528.82,
-547.2, -551.97, -552.3, -585.51, -551.34, -543.16, -526.1, -494.11,
-466.88, -355.93, -274.94, -215.04, -114.3, -194.21, -103.73,
-3.62, 104.2, 230.8, 154.25, 380.55, 416.62, 260.07, 295.75,
295.75, 251.47, 220.67, 225.96, 180.72, 121.52, 4.14, -127.23,
-176.24, -332.11, -408.35, -494.11, -573.75, -582.62, -678.88,
-730.38, -788.62, -831.94, -846.38, -895.95, -934.46, -968.15,
-1033.12, -1097.62, -1150.08, -1157.3, -1254.04, -1340.2, -1441.75,
-1500.47, -1550.52, -1605.39, -1681.44, -1709.84, -1715.22, -1672.34,
-1607, -1522.59, -1440.57, -1421.18, -1345.62, -1247.95, -1190.77,
-1181.58, -1071.65, -1037.62, -1010.39, -998.82, -986.57, -937.9,
-887.29, -842.05, -831.46, -774.66, -703.91, -573.75, -533.59,
-448.16)), .Names = c("V1", "V2"), class = "data.frame", row.names = c(NA,
-280L))
require(cobs)
xy <- dget(data.txt)
#Cumchord function (from Claude, 2008): Cumulative chordal distance vector
cumchord<-function(M)
{cumsum(sqrt(apply((M-rbind(M[1,],
M[-(dim(M)[1]),]))^2,1,sum)))}
z <- cumchord(xy)
#Calculating B-spline for x and y values separately
x <- cobs(z,xy[,1],nknots=50)
y <- cobs(z,xy[,2],nknots=50)
#Plot spline
plot(xy)
lines(x$fitted,y$fitted)
结果图的图像
在评论线程之后，这里有一些图表。我使用 Momocs 因为我熟悉它，它会缩短示例。
我介绍一下，你的问题是你的大纲里面有两个大纲。
我还包括 Julien Claude 对 spline 的原始使用，以及另外两个使用贝塞尔曲线和椭圆傅立叶变换的示例。所有 4 个都可以用来描述一个大纲（并重建它），可能值得在这里收集它们。
图合原形和这4种方法
现在是代码。不是特别长，但是很重复。
# devtools::install_github("vbonhomme/Momocs")
library(Momocs) # version 1.0.3
xy <- structure(list(
V1 = c(124.9, 86.44, 97.22, 81.34, 49.09, 57.18, -77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72, -600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18, -1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59, -859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09, -394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97, 344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14, 1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93, 2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03, 1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96, 1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94, 1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32, 697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25, 718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96, 823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72, 735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16, 45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72, -1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69, 144.87, 142.26, 146.34, 125.24, 124.9, 86.44, 97.22, 81.34, 49.09, 57.18, -77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72, -600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18, -1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59, -859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09, -394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97, 344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14, 1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93, 2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03, 1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96, 1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94, 1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32, 697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25, 718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96, 823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72, 735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16, 45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72, -1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69, 144.87, 142.26, 146.34, 125.24),
V2 = c(-446.8, -415.83, -394.43, -259.19, -104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23, -176.89, -233.68, -321.28, -416.57, -457.97, -458.93, -429.09, -422.35, -450.27, -431.98, -379.03, -260.63, -123.94, -2.65, 269.76, 455.55, 548.92, 616.3, 691.38, 756.84, 888.72, 1016.97, 1157.18, 1198.02, 1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47, 733.81, 784.18, 835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13, 732.45, 586.79, 394.84, 212.42, 28.64, -111.58, -337.56, -490.03, -526.07, -528.82, -547.2, -551.97, -552.3, -585.51, -551.34, -543.16, -526.1, -494.11, -466.88, -355.93, -274.94, -215.04, -114.3, -194.21, -103.73, -3.62, 104.2, 230.8, 154.25, 380.55, 416.62, 260.07, 295.75, 295.75, 251.47, 220.67, 225.96, 180.72, 121.52, 4.14, -127.23, -176.24, -332.11, -408.35, -494.11, -573.75, -582.62, -678.88, -730.38, -788.62, -831.94, -846.38, -895.95, -934.46, -968.15, -1033.12, -1097.62, -1150.08, -1157.3, -1254.04, -1340.2, -1441.75, -1500.47, -1550.52, -1605.39, -1681.44, -1709.84, -1715.22, -1672.34, -1607, -1522.59, -1440.57, -1421.18, -1345.62, -1247.95, -1190.77, -1181.58, -1071.65, -1037.62, -1010.39, -998.82, -986.57, -937.9, -887.29, -842.05, -831.46, -774.66, -703.91, -573.75, -533.59, -448.16, -446.8, -415.83, -394.43, -259.19, -104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23, -176.89, -233.68, -321.28, -416.57, -457.97, -458.93, -429.09, -422.35, -450.27, -431.98, -379.03, -260.63, -123.94, -2.65, 269.76, 455.55, 548.92, 616.3, 691.38, 756.84, 888.72, 1016.97, 1157.18, 1198.02, 1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47, 733.81, 784.18, 835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13, 732.45, 586.79, 394.84, 212.42, 28.64, -111.58, -337.56, -490.03, -526.07, -528.82, -547.2, -551.97, -552.3, -585.51, -551.34, -543.16, -526.1, -494.11, -466.88, -355.93, -274.94, -215.04, -114.3, -194.21, -103.73, -3.62, 104.2, 230.8, 154.25, 380.55, 416.62, 260.07, 295.75, 295.75, 251.47, 220.67, 225.96, 180.72, 121.52, 4.14, -127.23, -176.24, -332.11, -408.35, -494.11, -573.75, -582.62, -678.88, -730.38, -788.62, -831.94, -846.38, -895.95, -934.46, -968.15, -1033.12, -1097.62, -1150.08, -1157.3, -1254.04, -1340.2, -1441.75, -1500.47, -1550.52, -1605.39, -1681.44, -1709.84, -1715.22, -1672.34, -1607, -1522.59, -1440.57, -1421.18, -1345.62, -1247.95, -1190.77, -1181.58, -1071.65, -1037.62, -1010.39, -998.82, -986.57, -937.9, -887.29, -842.05, -831.46, -774.66, -703.91, -573.75, -533.59, -448.16)), .Names = c("V1", "V2"), class = "data.frame", row.names = c(NA, -280L))
### First thing first: double outline ---------------------
coo_plot(xy)
ldk_labels(xy) # blurry since superimposed
coo_plot(xy[1:140, ], lwd=3) # first shape
coo_draw(xy[-(1:140), ], border="white") # second shape
# so from here, we will use the first 140th points from xy and name it 'shp' to avoid confusion
# if you're bored with dinos footprints, you can use beer bottles with shp <- bot[9] for a guinness
shp <- xy[1:140, ]
### 1.Natural splines ---------------------
shp_cumchord <- coo_perimcum(shp) # cumchord equivalent
shp_spline <- cbind(spline(shp_cumchord, shp[, 1], method="natural", n=120)$y,
spline(shp_cumchord, shp[, 2], method="natural", n=120)$y)
coo_plot(shp, main="natural spline", zoom=1.2)
coo_draw(shp_spline, border="blue", lwd=2)
### 2. B-splines with cobs ---------------------
library(cobs)
shp_bspline <- cbind(cobs(shp_cumchord, shp[, 1], nknots=50)$fitted,
cobs(shp_cumchord, shp[, 2], nknots=50)$fitted)
coo_plot(shp, main = "bspline", zoom=1.2)
coo_draw(shp_bspline, border="blue")
### 3. Bezier curves ---------------------
# built in function so it's shorter
shp_bezier <- shp %>% bezier() %>% bezier_i()
coo_plot(shp, main = "bezier", zoom=1.2)
coo_draw(shp_bezier, border="blue")
### 4. elliptic Fourier transforms ---------------------
# another built in function
shp_eft <- shp %>% efourier() %>% efourier_i()
coo_plot(shp, main = "bspline", zoom=1.2)
coo_draw(shp_eft, border="blue")
### 5. A panel of original shape and 4 methods ---------
Out(list(original=shp,
nat_spline=shp_spline, bspline=shp_bspline,
bezier=shp_bezier, eft=shp_eft)) %>%
panel(names=TRUE, dim=c(1, 5))
我正在寻找生成约束样条以近似形状（在我的例子中是足迹轮廓）的可能性。作为原始数据，我有一个包含数百个 xy 坐标对的 table，它们是从足迹的边界收集的。样条应该只逼近数据点（样条不需要通过数据点）。我希望能够在一定程度上平滑样条曲线。另外，我需要能够约束样条曲线：定义样条曲线必须通过的几个关键数据点。
R 包 "cobs"（约束 B 样条曲线，https://cran.r-project.org/web/packages/cobs/index.html）非常接近于提供解决方案，提供参数以根据需要约束样条曲线。但是，此包不会通过有序的数据点序列进行样条化，这在您希望样条线遵循形状边界时当然是至关重要的。我尝试分别样条化 x 和 y 坐标，但在将它们重新组合后，图中出现了两个不同的形状，所以这似乎不起作用（或者我弄错了什么？）。有人知道解决方案吗？
更新：工作示例（恐龙足迹大纲）
data.txt:
structure(list(V1 = c(124.9, 86.44, 97.22, 81.34, 49.09, 57.18,
-77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72,
-600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18,
-1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59,
-859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09,
-394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97,
344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14,
1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93,
2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03,
1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96,
1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94,
1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32,
697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25,
718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96,
823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72,
735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16,
45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72,
-1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69,
144.87, 142.26, 146.34, 125.24, 124.9, 86.44, 97.22, 81.34, 49.09,
57.18, -77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72,
-600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18,
-1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59,
-859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09,
-394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97,
344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14,
1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93,
2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03,
1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96,
1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94,
1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32,
697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25,
718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96,
823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72,
735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16,
45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72,
-1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69,
144.87, 142.26, 146.34, 125.24), V2 = c(-446.8, -415.83, -394.43,
-259.19, -104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23,
-176.89, -233.68, -321.28, -416.57, -457.97, -458.93, -429.09,
-422.35, -450.27, -431.98, -379.03, -260.63, -123.94, -2.65,
269.76, 455.55, 548.92, 616.3, 691.38, 756.84, 888.72, 1016.97,
1157.18, 1198.02, 1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47,
733.81, 784.18, 835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13,
732.45, 586.79, 394.84, 212.42, 28.64, -111.58, -337.56, -490.03,
-526.07, -528.82, -547.2, -551.97, -552.3, -585.51, -551.34,
-543.16, -526.1, -494.11, -466.88, -355.93, -274.94, -215.04,
-114.3, -194.21, -103.73, -3.62, 104.2, 230.8, 154.25, 380.55,
416.62, 260.07, 295.75, 295.75, 251.47, 220.67, 225.96, 180.72,
121.52, 4.14, -127.23, -176.24, -332.11, -408.35, -494.11, -573.75,
-582.62, -678.88, -730.38, -788.62, -831.94, -846.38, -895.95,
-934.46, -968.15, -1033.12, -1097.62, -1150.08, -1157.3, -1254.04,
-1340.2, -1441.75, -1500.47, -1550.52, -1605.39, -1681.44, -1709.84,
-1715.22, -1672.34, -1607, -1522.59, -1440.57, -1421.18, -1345.62,
-1247.95, -1190.77, -1181.58, -1071.65, -1037.62, -1010.39, -998.82,
-986.57, -937.9, -887.29, -842.05, -831.46, -774.66, -703.91,
-573.75, -533.59, -448.16, -446.8, -415.83, -394.43, -259.19,
-104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23, -176.89,
-233.68, -321.28, -416.57, -457.97, -458.93, -429.09, -422.35,
-450.27, -431.98, -379.03, -260.63, -123.94, -2.65, 269.76, 455.55,
548.92, 616.3, 691.38, 756.84, 888.72, 1016.97, 1157.18, 1198.02,
1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47, 733.81, 784.18,
835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13, 732.45, 586.79,
394.84, 212.42, 28.64, -111.58, -337.56, -490.03, -526.07, -528.82,
-547.2, -551.97, -552.3, -585.51, -551.34, -543.16, -526.1, -494.11,
-466.88, -355.93, -274.94, -215.04, -114.3, -194.21, -103.73,
-3.62, 104.2, 230.8, 154.25, 380.55, 416.62, 260.07, 295.75,
295.75, 251.47, 220.67, 225.96, 180.72, 121.52, 4.14, -127.23,
-176.24, -332.11, -408.35, -494.11, -573.75, -582.62, -678.88,
-730.38, -788.62, -831.94, -846.38, -895.95, -934.46, -968.15,
-1033.12, -1097.62, -1150.08, -1157.3, -1254.04, -1340.2, -1441.75,
-1500.47, -1550.52, -1605.39, -1681.44, -1709.84, -1715.22, -1672.34,
-1607, -1522.59, -1440.57, -1421.18, -1345.62, -1247.95, -1190.77,
-1181.58, -1071.65, -1037.62, -1010.39, -998.82, -986.57, -937.9,
-887.29, -842.05, -831.46, -774.66, -703.91, -573.75, -533.59,
-448.16)), .Names = c("V1", "V2"), class = "data.frame", row.names = c(NA,
-280L))
require(cobs)
xy <- dget(data.txt)
#Cumchord function (from Claude, 2008): Cumulative chordal distance vector
cumchord<-function(M)
{cumsum(sqrt(apply((M-rbind(M[1,],
M[-(dim(M)[1]),]))^2,1,sum)))}
z <- cumchord(xy)
#Calculating B-spline for x and y values separately
x <- cobs(z,xy[,1],nknots=50)
y <- cobs(z,xy[,2],nknots=50)
#Plot spline
plot(xy)
lines(x$fitted,y$fitted)
结果图的图像
在评论线程之后，这里有一些图表。我使用 Momocs 因为我熟悉它，它会缩短示例。
我介绍一下，你的问题是你的大纲里面有两个大纲。
我还包括 Julien Claude 对 spline 的原始使用，以及另外两个使用贝塞尔曲线和椭圆傅立叶变换的示例。所有 4 个都可以用来描述一个大纲（并重建它），可能值得在这里收集它们。
图合原形和这4种方法
现在是代码。不是特别长，但是很重复。
# devtools::install_github("vbonhomme/Momocs")
library(Momocs) # version 1.0.3
xy <- structure(list(
V1 = c(124.9, 86.44, 97.22, 81.34, 49.09, 57.18, -77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72, -600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18, -1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59, -859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09, -394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97, 344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14, 1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93, 2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03, 1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96, 1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94, 1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32, 697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25, 718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96, 823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72, 735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16, 45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72, -1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69, 144.87, 142.26, 146.34, 125.24, 124.9, 86.44, 97.22, 81.34, 49.09, 57.18, -77.6, -191.95, -284.67, -383.18, -379.27, -492.85, -547.72, -600.67, -713.29, -814.36, -868.27, -926.99, -958.76, -1025.18, -1077.16, -1105.07, -1126.25, -1112.77, -1087.74, -989.56, -911.59, -859.61, -745.06, -656.5, -682.01, -637.25, -601.71, -539.09, -394.79, -219.17, -170.17, -201.48, -122.52, -43.56, 127.97, 344.42, 539.09, 686.11, 987.63, 1253.31, 1283.15, 1536.32, 1741.14, 1832.35, 1700.3, 1787.43, 1911.31, 2017.49, 2097.81, 2135.93, 2093.73, 2066.96, 2063.78, 2022.94, 1978.69, 1919.44, 1904.03, 1895.37, 1854.22, 1810.23, 1771.09, 1741.48, 1642.45, 1553.96, 1472.96, 1396.04, 1141.65, 1085.82, 1055.02, 1358.24, 1325.94, 1031.91, 1287.14, 1265.36, 931.15, 872.12, 811.48, 755.65, 738.32, 697.41, 682.49, 647.35, 628.25, 620.09, 629.62, 675.22, 709.25, 718.78, 717.42, 551.09, 535.21, 540.98, 534.73, 546.76, 811.96, 823.03, 822.07, 607.4, 626.18, 637.73, 659.87, 756.13, 753.72, 735.91, 720.99, 676.71, 576.6, 508.26, 339.8, 179.53, 121.16, 45.6, 12.93, -9.87, -12.59, 16, 27.91, 37.78, 49.35, 8.51, 2.72, -1.02, 59.22, 58.2, 51.73, 54.45, 0.96, 10.59, 138.62, 149.69, 144.87, 142.26, 146.34, 125.24),
V2 = c(-446.8, -415.83, -394.43, -259.19, -104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23, -176.89, -233.68, -321.28, -416.57, -457.97, -458.93, -429.09, -422.35, -450.27, -431.98, -379.03, -260.63, -123.94, -2.65, 269.76, 455.55, 548.92, 616.3, 691.38, 756.84, 888.72, 1016.97, 1157.18, 1198.02, 1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47, 733.81, 784.18, 835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13, 732.45, 586.79, 394.84, 212.42, 28.64, -111.58, -337.56, -490.03, -526.07, -528.82, -547.2, -551.97, -552.3, -585.51, -551.34, -543.16, -526.1, -494.11, -466.88, -355.93, -274.94, -215.04, -114.3, -194.21, -103.73, -3.62, 104.2, 230.8, 154.25, 380.55, 416.62, 260.07, 295.75, 295.75, 251.47, 220.67, 225.96, 180.72, 121.52, 4.14, -127.23, -176.24, -332.11, -408.35, -494.11, -573.75, -582.62, -678.88, -730.38, -788.62, -831.94, -846.38, -895.95, -934.46, -968.15, -1033.12, -1097.62, -1150.08, -1157.3, -1254.04, -1340.2, -1441.75, -1500.47, -1550.52, -1605.39, -1681.44, -1709.84, -1715.22, -1672.34, -1607, -1522.59, -1440.57, -1421.18, -1345.62, -1247.95, -1190.77, -1181.58, -1071.65, -1037.62, -1010.39, -998.82, -986.57, -937.9, -887.29, -842.05, -831.46, -774.66, -703.91, -573.75, -533.59, -448.16, -446.8, -415.83, -394.43, -259.19, -104.69, -4.03, 58.59, -80.26, 52.11, -48.33, -142.23, -176.89, -233.68, -321.28, -416.57, -457.97, -458.93, -429.09, -422.35, -450.27, -431.98, -379.03, -260.63, -123.94, -2.65, 269.76, 455.55, 548.92, 616.3, 691.38, 756.84, 888.72, 1016.97, 1157.18, 1198.02, 1101.37, 1025.14, 929.84, 852.25, 766.48, 717.47, 733.81, 784.18, 835.91, 1225.63, 1198.68, 925.3, 742.4, 814.13, 732.45, 586.79, 394.84, 212.42, 28.64, -111.58, -337.56, -490.03, -526.07, -528.82, -547.2, -551.97, -552.3, -585.51, -551.34, -543.16, -526.1, -494.11, -466.88, -355.93, -274.94, -215.04, -114.3, -194.21, -103.73, -3.62, 104.2, 230.8, 154.25, 380.55, 416.62, 260.07, 295.75, 295.75, 251.47, 220.67, 225.96, 180.72, 121.52, 4.14, -127.23, -176.24, -332.11, -408.35, -494.11, -573.75, -582.62, -678.88, -730.38, -788.62, -831.94, -846.38, -895.95, -934.46, -968.15, -1033.12, -1097.62, -1150.08, -1157.3, -1254.04, -1340.2, -1441.75, -1500.47, -1550.52, -1605.39, -1681.44, -1709.84, -1715.22, -1672.34, -1607, -1522.59, -1440.57, -1421.18, -1345.62, -1247.95, -1190.77, -1181.58, -1071.65, -1037.62, -1010.39, -998.82, -986.57, -937.9, -887.29, -842.05, -831.46, -774.66, -703.91, -573.75, -533.59, -448.16)), .Names = c("V1", "V2"), class = "data.frame", row.names = c(NA, -280L))
### First thing first: double outline ---------------------
coo_plot(xy)
ldk_labels(xy) # blurry since superimposed
coo_plot(xy[1:140, ], lwd=3) # first shape
coo_draw(xy[-(1:140), ], border="white") # second shape
# so from here, we will use the first 140th points from xy and name it 'shp' to avoid confusion
# if you're bored with dinos footprints, you can use beer bottles with shp <- bot[9] for a guinness
shp <- xy[1:140, ]
### 1.Natural splines ---------------------
shp_cumchord <- coo_perimcum(shp) # cumchord equivalent
shp_spline <- cbind(spline(shp_cumchord, shp[, 1], method="natural", n=120)$y,
spline(shp_cumchord, shp[, 2], method="natural", n=120)$y)
coo_plot(shp, main="natural spline", zoom=1.2)
coo_draw(shp_spline, border="blue", lwd=2)
### 2. B-splines with cobs ---------------------
library(cobs)
shp_bspline <- cbind(cobs(shp_cumchord, shp[, 1], nknots=50)$fitted,
cobs(shp_cumchord, shp[, 2], nknots=50)$fitted)
coo_plot(shp, main = "bspline", zoom=1.2)
coo_draw(shp_bspline, border="blue")
### 3. Bezier curves ---------------------
# built in function so it's shorter
shp_bezier <- shp %>% bezier() %>% bezier_i()
coo_plot(shp, main = "bezier", zoom=1.2)
coo_draw(shp_bezier, border="blue")
### 4. elliptic Fourier transforms ---------------------
# another built in function
shp_eft <- shp %>% efourier() %>% efourier_i()
coo_plot(shp, main = "bspline", zoom=1.2)
coo_draw(shp_eft, border="blue")
### 5. A panel of original shape and 4 methods ---------
Out(list(original=shp,
nat_spline=shp_spline, bspline=shp_bspline,
bezier=shp_bezier, eft=shp_eft)) %>%
panel(names=TRUE, dim=c(1, 5))