Approximate a shape outline using constrained B-splines

Question

I'm looking for a possibility to generate a constrained spline in order to approximate a shape (in my case, a footprint outline). As raw data, I have a table with several hundred xy-coordinate pairs, which have been collected from the boundary of the footprint. The spline should only approximate the data points (the spline does not need to pass the data points). I want to be able to smooth the spline to certain degrees. Also, I need to be able to constrain the spline: Defining several critical data points which the spline has to pass.

The R package "cobs" (COnstrained B-Splines, https://cran.r-project.org/web/packages/cobs/index.html) comes very close to providing a solution, offering parameters to constrain the spline as wanted. However, this package does not spline through an ordered sequence of data points, which of course is crucial when you want the spline to follow the boundary of a shape. I tried to spline x and y coordinates separately, but after recombining them two distinct shapes appear in the plot, so this does not seem to work (Or I got something wrong?). Is anybody aware of a solution?

---

Update: working example (dinosaur footprint outline)

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)

Image of resulting plot

Answer 1

在评论主题之后，这里有一些图表。我使用Momocs，因为我熟悉它，它会缩短示例。

我简要介绍一下，你的问题是你的大纲中有两个大纲。

我还包括Julien Claude对spline的原始用法，以及贝塞尔曲线和椭圆傅立叶变换的另外两个例子。所有4都可用于描述轮廓（并重建它），它可能值得在这里收集它们。

图片收集原始形状和这4种方法

现在的代码。它不是特别长，而是非常重复。

# 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))