# Fitting an ellipse to point data

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Some time ago I wrote an R function to fit an ellipse to point data, using an algorithm developed by Radim Halíř and Jan Flusser^{1} 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:

<br />fit.ellipse <- function (x, y = NULL) {<br /> # from:<br /> # http://r.789695.n4.nabble.com/Fitting-a-half-ellipse-curve-tp2719037p2720560.html<br /> #<br /> # Least squares fitting of an ellipse to point data<br /> # using the algorithm described in: <br /> # Radim Halir & Jan Flusser. 1998. <br /> # Numerically stable direct least squares fitting of ellipses. <br /> # Proceedings of the 6th International Conference in Central Europe <br /> # on Computer Graphics and Visualization. WSCG '98, p. 125-132 <br /> #<br /> # Adapted from the original Matlab code by Michael Bedward (2010)<br /> # [email protected]<br /> #<br /> # Subsequently improved by John Minter (2012)<br /> # <br /> # Arguments: <br /> # x, y - x and y coordinates of the data points.<br /> # If a single arg is provided it is assumed to be a<br /> # two column matrix.<br /> #<br /> # Returns a list with the following elements: <br /> #<br /> # coef - coefficients of the ellipse as described by the general <br /> # quadratic: ax^2 + bxy + cy^2 + dx + ey + f = 0 <br /> #<br /> # center - center x and y<br /> #<br /> # major - major semi-axis length<br /> #<br /> # minor - minor semi-axis length<br /> #<br /> EPS <- 1.0e-8 <br /> dat <- xy.coords(x, y) <br /> <br /> D1 <- cbind(dat$x * dat$x, dat$x * dat$y, dat$y * dat$y) <br /> D2 <- cbind(dat$x, dat$y, 1) <br /> S1 <- t(D1) %*% D1 <br /> S2 <- t(D1) %*% D2 <br /> S3 <- t(D2) %*% D2 <br /> T <- -solve(S3) %*% t(S2) <br /> M <- S1 + S2 %*% T <br /> M <- rbind(M[3,] / 2, -M[2,], M[1,] / 2) <br /> evec <- eigen(M)$vec <br /> cond <- 4 * evec[1,] * evec[3,] - evec[2,]^2 <br /> a1 <- evec[, which(cond > 0)] <br /> f <- c(a1, T %*% a1) <br /> names(f) <- letters[1:6] <br /> <br /> # calculate the center and lengths of the semi-axes <br /> #<br /> # see http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2288654/<br /> # J. R. Minter<br /> # for the center, linear algebra to the rescue<br /> # center is the solution to the pair of equations<br /> # 2ax + by + d = 0<br /> # bx + 2cy + e = 0<br /> # or<br /> # | 2a b | |x| |-d|<br /> # | b 2c | * |y| = |-e|<br /> # or<br /> # A x = b<br /> # or<br /> # x = Ainv b<br /> # or<br /> # x = solve(A) %*% b<br /> A <- matrix(c(2*f[1], f[2], f[2], 2*f[3]), nrow=2, ncol=2, byrow=T )<br /> b <- matrix(c(-f[4], -f[5]), nrow=2, ncol=1, byrow=T)<br /> soln <- solve(A) %*% b<br /><br /> b2 <- f[2]^2 / 4<br /> <br /> center <- c(soln[1], soln[2]) <br /> names(center) <- c("x", "y") <br /> <br /> 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]) <br /> den1 <- (b2 - f[1]*f[3]) <br /> den2 <- sqrt((f[1] - f[3])^2 + 4*b2) <br /> den3 <- f[1] + f[3] <br /> <br /> semi.axes <- sqrt(c( num / (den1 * (den2 - den3)), num / (den1 * (-den2 - den3)) )) <br /> <br /> # calculate the angle of rotation <br /> term <- (f[1] - f[3]) / f[2] <br /> angle <- atan(1 / term) / 2 <br /> <br /> list(coef=f, center = center, major = max(semi.axes), minor = min(semi.axes), angle = unname(angle)) <br />}<br /><br />Next here is a utility function which takes a fitted ellipse and returns a matrix of vertices for plotting:

<br />get.ellipse <- function( fit, n=360 ) <br />{<br /> # Calculate points on an ellipse described by <br /> # the fit argument as returned by fit.ellipse <br /> # <br /> # n is the number of points to render <br /> <br /> tt <- seq(0, 2*pi, length=n) <br /> sa <- sin(fit$angle) <br /> ca <- cos(fit$angle) <br /> ct <- cos(tt) <br /> st <- sin(tt) <br /> <br /> x <- fit$center[1] + fit$maj * ct * ca - fit$min * st * sa <br /> y <- fit$center[2] + fit$maj * ct * sa + fit$min * st * ca <br /> <br /> cbind(x=x, y=y) <br />}<br /><br />And finally, some demo code from John:

<br />create.test.ellipse <- function(Rx=300, # X-radius<br /> Ry=200, # Y-radius<br /> Cx=250, # X-center<br /> Cy=150, # Y-center<br /> Rotation=0.4, # Radians<br /> NoiseLevel=0.5) # Gaussian Noise level<br />{<br /> set.seed(42)<br /> t <- seq(0, 100, by=1)<br /> x <- Rx * cos(t)<br /> y <- Ry * sin(t)<br /> nx <- x*cos(Rotation)-y*sin(Rotation) + Cx<br /> nx <- nx + rnorm(length(t))*NoiseLevel <br /> ny <- x*sin(Rotation)+y*cos(Rotation) + Cy<br /> ny <- ny + rnorm(length(t))*NoiseLevel<br /> cbind(x=nx, y=ny)<br />}<br /><br />X <- create.test.ellipse()<br />efit <- fit.ellipse(X)<br />e <- get.ellipse(efit)<br />plot(X) <br />lines(e, col="red") <br /><br />print(efit)<br /><br />

^{1}Halíř 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)

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