问题
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
回答1:
Following the comments thread, here are some graphs. I use Momocs since I'm familiar with it and it will shorten examples.
I brief, your problem is that there are two outlines in your outline.
I also include original use of spline by Julien Claude, and two additional examples with bezier curves and elliptic Fourier transforms. All 4 can be used to describe an outline (and reconstruct it) and it's probably worth gathering them here.
The picture gathers the original shape and these 4 methods
Now the code. It's not particularly long but quite repetitive.
# 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))
来源：https://stackoverflow.com/questions/36473986/approximate-a-shape-outline-using-constrained-b-splines