# Fitting an ellipse to point data

September 30, 2012
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(This article was first published on Last Resort Software, and kindly contributed to R-bloggers)

Some time ago I wrote an R function to fit an ellipse to point data, using an algorithm developed by Radim Halíř and Jan Flusser1 in Matlab, and posted it to the r-help list. The implementation was a bit hacky, returning odd results for some data.

A couple of days ago, an email arrived from John Minter asking for a pointer to the original code. I replied with a link and mentioned that I'd be interested to know if John made any improvements to the code. About ten minutes later, John emailed again with a much improved version ! Not only is it more reliable, but also more efficient. So with many thanks to John, here is the improved code:

fit.ellipse <- function (x, y = NULL) {  # from:  # http://r.789695.n4.nabble.com/Fitting-a-half-ellipse-curve-tp2719037p2720560.html  #  # Least squares fitting of an ellipse to point data  # using the algorithm described in:   #   Radim Halir & Jan Flusser. 1998.   #   Numerically stable direct least squares fitting of ellipses.   #   Proceedings of the 6th International Conference in Central Europe   #   on Computer Graphics and Visualization. WSCG '98, p. 125-132   #  # Adapted from the original Matlab code by Michael Bedward (2010)  # [email protected]
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#  # Subsequently improved by John Minter (2012)  #   # Arguments:   # x, y - x and y coordinates of the data points.  #        If a single arg is provided it is assumed to be a  #        two column matrix.  #  # Returns a list with the following elements:   #  # coef - coefficients of the ellipse as described by the general   #        quadratic:  ax^2 + bxy + cy^2 + dx + ey + f = 0   #  # center - center x and y  #  # major - major semi-axis length  #  # minor - minor semi-axis length  #  EPS <- 1.0e-8   dat <- xy.coords(x, y)     D1 <- cbind(dat$x * dat$x, dat$x * dat$y, dat$y * dat$y)   D2 <- cbind(dat$x, dat$y, 1)   S1 <- t(D1) %*% D1   S2 <- t(D1) %*% D2   S3 <- t(D2) %*% D2   T <- -solve(S3) %*% t(S2)   M <- S1 + S2 %*% T   M <- rbind(M[3,] / 2, -M[2,], M[1,] / 2)   evec <- eigen(M)$vec cond <- 4 * evec[1,] * evec[3,] - evec[2,]^2 a1 <- evec[, which(cond > 0)] f <- c(a1, T %*% a1) names(f) <- letters[1:6] # calculate the center and lengths of the semi-axes # # see http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2288654/ # J. R. Minter # for the center, linear algebra to the rescue # center is the solution to the pair of equations # 2ax + by + d = 0 # bx + 2cy + e = 0 # or # | 2a b | |x| |-d| # | b 2c | * |y| = |-e| # or # A x = b # or # x = Ainv b # or # x = solve(A) %*% b A <- matrix(c(2*f[1], f[2], f[2], 2*f[3]), nrow=2, ncol=2, byrow=T ) b <- matrix(c(-f[4], -f[5]), nrow=2, ncol=1, byrow=T) soln <- solve(A) %*% b b2 <- f[2]^2 / 4 center <- c(soln[1], soln[2]) names(center) <- c("x", "y") num <- 2 * (f[1] * f[5]^2 / 4 + f[3] * f[4]^2 / 4 + f[6] * b2 - f[2]*f[4]*f[5]/4 - f[1]*f[3]*f[6]) den1 <- (b2 - f[1]*f[3]) den2 <- sqrt((f[1] - f[3])^2 + 4*b2) den3 <- f[1] + f[3] semi.axes <- sqrt(c( num / (den1 * (den2 - den3)), num / (den1 * (-den2 - den3)) )) # calculate the angle of rotation term <- (f[1] - f[3]) / f[2] angle <- atan(1 / term) / 2 list(coef=f, center = center, major = max(semi.axes), minor = min(semi.axes), angle = unname(angle)) } Next here is a utility function which takes a fitted ellipse and returns a matrix of vertices for plotting: get.ellipse <- function( fit, n=360 ) { # Calculate points on an ellipse described by # the fit argument as returned by fit.ellipse # # n is the number of points to render tt <- seq(0, 2*pi, length=n) sa <- sin(fit$angle)   ca <- cos(fit$angle) ct <- cos(tt) st <- sin(tt) x <- fit$center[1] + fit$maj * ct * ca - fit$min * st * sa   y <- fit$center[2] + fit$maj * ct * sa + fit\$min * st * ca     cbind(x=x, y=y) }
And finally, some demo code from John:
create.test.ellipse <- function(Rx=300,         # X-radius                                Ry=200,         # Y-radius                                Cx=250,         # X-center                                Cy=150,         # Y-center                                Rotation=0.4,   # Radians                                NoiseLevel=0.5) # Gaussian Noise level{  set.seed(42)  t <- seq(0, 100, by=1)  x <- Rx * cos(t)  y <- Ry * sin(t)  nx <- x*cos(Rotation)-y*sin(Rotation) + Cx  nx <- nx + rnorm(length(t))*NoiseLevel   ny <- x*sin(Rotation)+y*cos(Rotation) + Cy  ny  <- ny + rnorm(length(t))*NoiseLevel  cbind(x=nx, y=ny)}X <- create.test.ellipse()efit <- fit.ellipse(X)e <- get.ellipse(efit)plot(X) lines(e, col="red") print(efit)
1Halíř R., Flusser J.: Numerically stable direct least squares fitting of ellipses. In: Proceedings of the 6th International Conference in Central Europe on Computer Graphics and Visualization. WSCG '98. CZ, Plzeň 1998, pp. 125-132. (postscript file)