| Title: | Algorithms for Fitting Circles, Ellipses and Conics Based on the Work by Prof. Nikolai Chernov |
|---|---|
| Description: | Geometric circle fitting with Levenberg-Marquardt (a, b, R), Levenberg-Marquardt reduced (a, b), Landau, Spath and Chernov-Lesort. Algebraic circle fitting with Taubin, Kasa, Pratt and Fitzgibbon-Pilu-Fisher. Geometric ellipse fitting with ellipse LMG (geometric parameters) and conic LMA (algebraic parameters). Algebraic ellipse fitting with Fitzgibbon-Pilu-Fisher and Taubin. |
| Authors: | Jose Gama [aut, cre], Nikolai Chernov [aut, cph] |
| Maintainer: | Jose Gama <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 1.0.4 |
| Built: | 2024-12-01 07:58:49 UTC |
| Source: | CRAN |
AtoG converts algebraic parameters (A, B, C, D, E, F) to
geometric parameters (Center(1:2), Axes(1:2), Angle).
AtoG(ParA)AtoG(ParA)
ParA |
vector or array with geometric parameters (A, B, C, D, E, F) |
code is: -1 - degenerate cases 0 - imaginary ellipse 4 - imaginary parell lines 1 - ellipse 2 - hyperbola 3 - parabola
list(ParG, exitCode
|
list with algebraic parameters (Center(1:2), Axes(1:2), Angle) and exit code |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
AtoG(c(0.0551,-0.0908,0.1588,0.0489,-0.9669,0.1620))AtoG(c(0.0551,-0.0908,0.1588,0.0489,-0.9669,0.1620))
calculateCircle generates points from a circle
with many options, equally spaced, randomly spaced, with noise added
to the radius or limited to a segment of angle alpha.
calculateCircle(x, y, r, steps=50,sector=c(0,360),randomDist=FALSE, randomFun=runif, noiseFun = NA, ...)calculateCircle(x, y, r, steps=50,sector=c(0,360),randomDist=FALSE, randomFun=runif, noiseFun = NA, ...)
x |
center point x |
y |
center point y |
r |
radius |
steps |
number of points |
sector |
limited circular sector |
randomDist |
logical, TRUE = randomly spaced |
randomFun |
random function for the position of the points in the circle |
noiseFun |
random function for the noise |
... |
optional parameters to pass to randomFun |
points |
array n x 2 of point coordinates. |
Jose Gama
## Not run: # 100 points from a circle at c(0,0) with radius=200 a<-calculateCircle(0,0,200,100) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250)) par(new=T) # 12 points from a circle at c(0,0) with radius=190, points between 0 and 90 #degrees a<-calculateCircle(0,0,190,12,c(0,90)) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='red') par(new=T) # 12 points from a circle at c(0,0) with radius=180, points between 0 and 180 #degrees, uniform random distribution a<-calculateCircle(0,0,180,12,c(0,180),TRUE) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='green') par(new=T) # 12 points from a circle at c(0,0) with radius=170, points between 0 and 180 #degrees, normal random distribution a<-calculateCircle(0,0,170,12,c(0,180),TRUE,rnorm) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='blue') par(new=T) # 12 points from a circle at c(0,0) with radius=200, points between 0 and 180 #degrees, positioned by uniform random distribution, noise=normal random #distribution with sd=10 a<-calculateCircle(0,0,200,12,c(180,360),TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=10))) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='orange') ## End(Not run)## Not run: # 100 points from a circle at c(0,0) with radius=200 a<-calculateCircle(0,0,200,100) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250)) par(new=T) # 12 points from a circle at c(0,0) with radius=190, points between 0 and 90 #degrees a<-calculateCircle(0,0,190,12,c(0,90)) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='red') par(new=T) # 12 points from a circle at c(0,0) with radius=180, points between 0 and 180 #degrees, uniform random distribution a<-calculateCircle(0,0,180,12,c(0,180),TRUE) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='green') par(new=T) # 12 points from a circle at c(0,0) with radius=170, points between 0 and 180 #degrees, normal random distribution a<-calculateCircle(0,0,170,12,c(0,180),TRUE,rnorm) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='blue') par(new=T) # 12 points from a circle at c(0,0) with radius=200, points between 0 and 180 #degrees, positioned by uniform random distribution, noise=normal random #distribution with sd=10 a<-calculateCircle(0,0,200,12,c(180,360),TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=10))) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='orange') ## End(Not run)
calculateEllipse generates points from a ellipse
with many options, equally spaced, randomly spaced, with noise added
to the radius or limited to a segment of angle alpha.
calculateEllipse(x, y, a, b, angle = 0, steps = 50, sector = c(0, 360), randomDist = FALSE, randomFun = runif, noiseFun = NA, ...)calculateEllipse(x, y, a, b, angle = 0, steps = 50, sector = c(0, 360), randomDist = FALSE, randomFun = runif, noiseFun = NA, ...)
x |
center point x |
y |
center point y |
a |
axis a |
b |
axis b |
angle |
tilt angle |
steps |
number of points |
sector |
limited circular sector |
randomDist |
logical, TRUE = randomly spaced |
randomFun |
random function for the position of the points in the ellipse |
noiseFun |
random function for the noise |
... |
optional parameters to pass to randomFun |
points |
array n x 2 of point coordinates. |
Jose Gama
## Not run: # 50 points from an ellipse at c(0,0) with axis (200, 100), angle 45 degrees a<-calculateEllipse(0,0,200,100,45,50) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250)) par(new=T) # 10 points from an ellipse at c(0,0) with axis (200, 100), angle 45 degrees, #points between 0 and 180 # degrees, normal random distribution b<-calculateEllipse(0,0,200,100,45,10,c(0,90)) plot(b[,1],b[,2],xlim=c(-250,250),ylim=c(-250,250),col='red') par(new=T) # 50 points from an ellipse at c(0,0) with axis (200, 100), angle 45 degrees a<-calculateEllipse(0,0,200,100,45,50, randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=10))) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='cyan') ## End(Not run)## Not run: # 50 points from an ellipse at c(0,0) with axis (200, 100), angle 45 degrees a<-calculateEllipse(0,0,200,100,45,50) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250)) par(new=T) # 10 points from an ellipse at c(0,0) with axis (200, 100), angle 45 degrees, #points between 0 and 180 # degrees, normal random distribution b<-calculateEllipse(0,0,200,100,45,10,c(0,90)) plot(b[,1],b[,2],xlim=c(-250,250),ylim=c(-250,250),col='red') par(new=T) # 50 points from an ellipse at c(0,0) with axis (200, 100), angle 45 degrees a<-calculateEllipse(0,0,200,100,45,50, randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=10))) plot(a[,1],a[,2],xlim=c(-250,250),ylim=c(-250,250),col='cyan') ## End(Not run)
CircleFitByKasa applies the simple algebraic circle fit
(Kasa method)
CircleFitByKasa(XY)CircleFitByKasa(XY)
XY |
array of sample data |
vector(a, b, R)
|
vector with the values for the circle: center (a,b) and radius R |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c3 <- CircleFitByKasa(xy) xyc3<-calculateCircle(c3[1],c3[2],c3[3]) plot(xyc3[,1],xyc3[,2],xlim=c(-250,250),ylim=c(-250,250),col='green',type='l');par(new=TRUE)xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c3 <- CircleFitByKasa(xy) xyc3<-calculateCircle(c3[1],c3[2],c3[3]) plot(xyc3[,1],xyc3[,2],xlim=c(-250,250),ylim=c(-250,250),col='green',type='l');par(new=TRUE)
CircleFitByLandau applies the Geometric circle fit
(minimizing orthogonal distances) by Landau algorithm
CircleFitByLandau(XY,ParIni = NA, epsilon = 1e-06, IterMAX = 50)CircleFitByLandau(XY,ParIni = NA, epsilon = 1e-06, IterMAX = 50)
XY |
array of sample data |
ParIni |
initial guess (a, b, R) |
epsilon |
tolerance (small threshold) |
IterMAX |
maximal number of iterations, with a bad initial guess it may take >100 iterations |
vector(a, b, R)
|
vector with the values for the circle: center (a,b) and radius R |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Geometric circle fit (minimizing orthogonal distances) by Landau algorithm M. Landau, "Estimation of a circular arc center and its radius", Computer Vision, Graphics and Image Processing, Vol. 38, pages 317-326, (1987)
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Geometric circle fit (minimizing orthogonal distances) by Landau algorithm M. Landau, "Estimation of a circular arc center and its radius", Computer Vision, Graphics and Image Processing, Vol. 38, pages 317-326, (1987)
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c6 <- CircleFitByLandau(xy) xyc6<-calculateCircle(c6[1],c6[2],c6[3]) plot(xyc6[,1],xyc6[,2],xlim=c(-250,250),ylim=c(-250,250),col='purple',type='l');par(new=TRUE)xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c6 <- CircleFitByLandau(xy) xyc6<-calculateCircle(c6[1],c6[2],c6[3]) plot(xyc6[,1],xyc6[,2],xlim=c(-250,250),ylim=c(-250,250),col='purple',type='l');par(new=TRUE)
CircleFitByPratt applies the Algebraic circle fit by Pratt
CircleFitByPratt(XY)CircleFitByPratt(XY)
XY |
array of sample data |
vector(a, b, R)
|
vector with the values for the circle: center (a,b) and radius R |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c2 <- CircleFitByPratt(xy) xyc2<-calculateCircle(c2[1],c2[2],c2[3]) plot(xyc2[,1],xyc2[,2],xlim=c(-250,250),ylim=c(-250,250),col='blue',type='l');par(new=TRUE)xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c2 <- CircleFitByPratt(xy) xyc2<-calculateCircle(c2[1],c2[2],c2[3]) plot(xyc2[,1],xyc2[,2],xlim=c(-250,250),ylim=c(-250,250),col='blue',type='l');par(new=TRUE)
CircleFitBySpath applies the Geometric circle fit by Spath
CircleFitBySpath(XY, ParIni = NA, epsilon = 1e-06, IterMAX = 50)CircleFitBySpath(XY, ParIni = NA, epsilon = 1e-06, IterMAX = 50)
XY |
array of sample data |
ParIni |
initial guess (a, b, R) |
epsilon |
tolerance (small threshold) |
IterMAX |
maximal number of iterations, with a bad initial guess it may take >100 iterations |
vector(a, b, R)
|
vector with the values for the circle: center (a,b) and radius R |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c5 <- CircleFitBySpath(xy) xyc5<-calculateCircle(c5[1],c5[2],c5[3]) plot(xyc5[,1],xyc5[,2],xlim=c(-250,250),ylim=c(-250,250),col='magenta',type='l');par(new=TRUE)xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c5 <- CircleFitBySpath(xy) xyc5<-calculateCircle(c5[1],c5[2],c5[3]) plot(xyc5[,1],xyc5[,2],xlim=c(-250,250),ylim=c(-250,250),col='magenta',type='l');par(new=TRUE)
CircleFitByTaubin applies the simple algebraic circle fit
(Taubin method)
CircleFitByTaubin(XY)CircleFitByTaubin(XY)
XY |
array of sample data |
vector(a, b, R)
|
vector with the values for the circle: center (a,b) and radius R |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c1 <- CircleFitByTaubin(xy) xyc1<-calculateCircle(c1[1],c1[2],c1[3]) plot(xyc1[,1],xyc1[,2],xlim=c(-250,250),ylim=c(-250,250),col='red',type='l');par(new=TRUE)xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c1 <- CircleFitByTaubin(xy) xyc1<-calculateCircle(c1[1],c1[2],c1[3]) plot(xyc1[,1],xyc1[,2],xlim=c(-250,250),ylim=c(-250,250),col='red',type='l');par(new=TRUE)
EllipseDirectFit applies the algebraic ellipse fit method
by Fitzgibbon-Pilu-Fisher
EllipseDirectFit(XY)EllipseDirectFit(XY)
XY |
array of sample data |
vector(A, B, C, D, E, F)
|
vector of algebraic parameters of the fitting ellipse: ax^2 + bxy + cy^2 +dx + ey + f = 0 |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
A. W. Fitzgibbon, M. Pilu, R. B. Fisher, 1999 Direct Least Squares Fitting of Ellipses IEEE Trans. PAMI, Vol. 21, pages 476-480
A. W. Fitzgibbon, M. Pilu, R. B. Fisher, "Direct Least Squares Fitting of Ellipses", IEEE Trans. PAMI, Vol. 21, pages 476-480 (1999) Halir R, Flusser J (1998) Proceedings of the 6th International Conference in Central Europe on Computer Graphics and Visualization, Numerically stable direct least squares fitting of ellipses (WSCG, Plzen, Czech Republic), pp 125–132.
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
A. W. Fitzgibbon, M. Pilu, R. B. Fisher, 1999 Direct Least Squares Fitting of Ellipses IEEE Trans. PAMI, Vol. 21, pages 476-480
A. W. Fitzgibbon, M. Pilu, R. B. Fisher, "Direct Least Squares Fitting of Ellipses", IEEE Trans. PAMI, Vol. 21, pages 476-480 (1999) Halir R, Flusser J (1998) Proceedings of the 6th International Conference in Central Europe on Computer Graphics and Visualization, Numerically stable direct least squares fitting of ellipses (WSCG, Plzen, Czech Republic), pp 125–132.
xy<-calculateEllipse(0,0,200,100,45,50, randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250),col='magenta');par(new=TRUE) ellipDirect <- EllipseDirectFit(xy) ellipDirectG <- AtoG(ellipDirect)$ParG xyDirect<-calculateEllipse(ellipDirectG[1], ellipDirectG[2], ellipDirectG[3], ellipDirectG[4], 180/pi*ellipDirectG[5]) plot(xyDirect[,1],xyDirect[,2],xlim=c(-250,250),ylim=c(-250,250),type='l', col='cyan');par(new=TRUE)xy<-calculateEllipse(0,0,200,100,45,50, randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250),col='magenta');par(new=TRUE) ellipDirect <- EllipseDirectFit(xy) ellipDirectG <- AtoG(ellipDirect)$ParG xyDirect<-calculateEllipse(ellipDirectG[1], ellipDirectG[2], ellipDirectG[3], ellipDirectG[4], 180/pi*ellipDirectG[5]) plot(xyDirect[,1],xyDirect[,2],xlim=c(-250,250),ylim=c(-250,250),type='l', col='cyan');par(new=TRUE)
EllipseFitByTaubin applies the Algebraic ellipse fit by
Taubin
EllipseFitByTaubin(XY)EllipseFitByTaubin(XY)
XY |
array of sample data |
vector(A, B, C, D, E, F)
|
vector with the values for the ellipse |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
xy<-calculateEllipse(0,0,200,100,45,50, randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250),col='magenta');par(new=TRUE) ellipTaubin <- EllipseFitByTaubin(xy) ellipTaubinG <- AtoG(ellipTaubin)$ParG xyTaubin<-calculateEllipse(ellipTaubinG[1], ellipTaubinG[2], ellipTaubinG[3], ellipTaubinG[4], 180/pi*ellipTaubinG[5]) plot(xyTaubin[,1],xyTaubin[,2],xlim=c(-250,250),ylim=c(-250,250),type='l', col='red');par(new=TRUE)xy<-calculateEllipse(0,0,200,100,45,50, randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250),col='magenta');par(new=TRUE) ellipTaubin <- EllipseFitByTaubin(xy) ellipTaubinG <- AtoG(ellipTaubin)$ParG xyTaubin<-calculateEllipse(ellipTaubinG[1], ellipTaubinG[2], ellipTaubinG[3], ellipTaubinG[4], 180/pi*ellipTaubinG[5]) plot(xyTaubin[,1],xyTaubin[,2],xlim=c(-250,250),ylim=c(-250,250),type='l', col='red');par(new=TRUE)
ellipticity ellipticity = flattening factor
ellipseEccentricity eccentricity of the ellipse
ellipseFocus focus of the ellipse
ellipseRa radius at apoapsis (the farthest distance)
ellipseRp radius at periapsis (the closest distance)
ellipse.l semi-latus rectum l
ellipticity(minorAxis,majorAxis)ellipticity(minorAxis,majorAxis)
minorAxis |
minor ellipse axis |
majorAxis |
major ellipse axis |
scalar result
Jose Gama
Wikipedia Ellipse http://en.wikipedia.org/wiki/Ellipse#Mathematical_definitions_and_properties
Wikipedia Ellipse http://en.wikipedia.org/wiki/Ellipse#Mathematical_definitions_and_properties
estimateInitialGuessCircle estimates initial guess values
for the center and radius of the circle
estimateInitialGuessCircle(XY)estimateInitialGuessCircle(XY)
XY |
array of sample data |
vector(a, b, R)
|
vector with the estimates for the circle: center (a,b) and radius R |
Jose Gama
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) estimateInitialGuessCircle(xy)xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) estimateInitialGuessCircle(xy)
fit.conicLMA fits a conic to a given set of points
(Implicit method) using algebraic parameters. Conic: Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0
fit.conicLMA(XY, ParAini, LambdaIni, epsilonP = 1e-10, epsilonF = 1e-13, IterMAX = 2e+06)fit.conicLMA(XY, ParAini, LambdaIni, epsilonP = 1e-10, epsilonF = 1e-13, IterMAX = 2e+06)
XY |
array of sample data |
ParAini |
initial parameter vector c(A,B,C,D,E,F) |
LambdaIni |
initial value of the control parameter Lambda |
epsilonP |
tolerance (small threshold) |
epsilonF |
tolerance (small threshold) |
IterMAX |
maximum number of (main) iterations, usually 10-20 will suffice |
list(ParA, RSS, iters
|
list with algebraic parameters (Center(1:2), Axes(1:2), Angle), Residual Sum of Squares and number of iterations |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
N. Chernov and H. Ma, 2011 Least squares fitting of quadratic curves and surfaces In: Computer Vision, Editor S. R. Yoshida, Nova Science Publishers; pp. 285-302.
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
N. Chernov and H. Ma, 2011 Least squares fitting of quadratic curves and surfaces In: Computer Vision, Editor S. R. Yoshida, Nova Science Publishers; pp. 285-302.
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParAini <- matrix(c(0.2500,0, 1.0000, 0, 0, -1.0000),ncol=1) LambdaIni=0.1 fit.conicLMA(XY,ParAini,LambdaIni)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParAini <- matrix(c(0.2500,0, 1.0000, 0, 0, -1.0000),ncol=1) LambdaIni=0.1 fit.conicLMA(XY,ParAini,LambdaIni)
fit.ellipseLMG Fits an ellipse to a given set of points
(Implicit method) using geometric parameters. Conic:
fit.ellipseLMG(XY,ParGini,LambdaIni = 1, epsilon = 1e-06, IterMAX = 200, L = 200)fit.ellipseLMG(XY,ParGini,LambdaIni = 1, epsilon = 1e-06, IterMAX = 200, L = 200)
XY |
array of sample data |
ParGini |
initial parameter vector c(Center(1:2), Axes(1:2), Angle) |
LambdaIni |
initial value of the control parameter Lambda |
epsilon |
tolerance (small threshold) |
IterMAX |
maximum number of (main) iterations, usually 10-20 will suffice |
L |
boundary for major/minor axis |
list(ParG, RSS, iters, TF)
|
list with geometric parameters (A,B,C,D,E,F), Residual Sum of Squares, number of iterations and TF==TRUE if the method diverges |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
N. Chernov and H. Ma, 2011 Least squares fitting of quadratic curves and surfaces In: Computer Vision, Editor S. R. Yoshida, Nova Science Publishers; pp. 285-302.
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
N. Chernov and H. Ma, 2011 Least squares fitting of quadratic curves and surfaces In: Computer Vision, Editor S. R. Yoshida, Nova Science Publishers; pp. 285-302.
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParGini <- matrix(c(0,0,2,1,0),ncol=1) LambdaIni=0.1 fit.ellipseLMG(XY,ParGini,LambdaIni)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParGini <- matrix(c(0,0,2,1,0),ncol=1) LambdaIni=0.1 fit.ellipseLMG(XY,ParGini,LambdaIni)
fitbookstein Linear ellipse fit using bookstein constraint
conic2parametric Diagonalise A - find Q, D such at A = Q' * D * Q
fitggk Linear least squares with the Euclidean-invariant constraint Trace(A) = 1
fitbookstein(x)fitbookstein(x)
x |
array of sample data |
list(z, a, b, alpha)
|
list with fitted ellipse parameters |
Jose Gama
Richard Brown, May 28, 2007 http://www.mathworks.com/matlabcentral/fileexchange/15125-fitellipse-m/content/demo/html/ellipsedemo.html
Richard Brown, May 28, 2007 http://www.mathworks.com/matlabcentral/fileexchange/15125-fitellipse-m/content/demo/html/ellipsedemo.html
W. Gander, G. H. Golub, R. Strebel, 1994 Least-Squares Fitting of Circles and Ellipses BIT Numerical Mathematics, Springer
GtoA converts geometric parameters (A, B, C, D, E, F) to
algebraic parameters (Center(1:2), Axes(1:2), Angle).
GtoA(ParG)GtoA(ParG)
ParG |
list with geometric parameters (A, B, C, D, E, F) |
ParA |
vector or array with algebraic parameters (Center(1:2), Axes(1:2), Angle) |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
GtoA(c(0,0,20,60,45))GtoA(c(0,0,20,60,45))
JmatrixLMA Computes the Jacobian matrix(Implicit method)
using algebraic parameters
JmatrixLMA(XY,ParA,XYproj)JmatrixLMA(XY,ParA,XYproj)
XY |
array of sample data |
ParA |
initial parameter vector c(Center(1:2), Axes(1:2), Angle) |
XYproj |
corresponding n projection points on the conic |
list(Res, J)
|
list with the Residual Sum of Squares and the Jacobian matrix |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParA <- matrix(c(0.250000000000000,0,1,0,0,-1),ncol=1) XYproj=matrix(c(0.394467220216675,0.980356518335872,0.833315950425981, 0.909063326557293,1.40466123643977,0.711850899213363, 1.70601340510202,0.521899957274429,1.89925244997324,0.313384799914835, 1.06482258038841,0.846485805004280,1.95308457257492, 0.215325713960169,1.91319150256275,0.291418202297698),8,2,byrow=TRUE) JmatrixLMA(XY,ParA,XYproj)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParA <- matrix(c(0.250000000000000,0,1,0,0,-1),ncol=1) XYproj=matrix(c(0.394467220216675,0.980356518335872,0.833315950425981, 0.909063326557293,1.40466123643977,0.711850899213363, 1.70601340510202,0.521899957274429,1.89925244997324,0.313384799914835, 1.06482258038841,0.846485805004280,1.95308457257492, 0.215325713960169,1.91319150256275,0.291418202297698),8,2,byrow=TRUE) JmatrixLMA(XY,ParA,XYproj)
JmatrixLMG Computes the Jacobian matrix (Implicit method)
using geometric parameters
JmatrixLMG(XY,A,XYproj)JmatrixLMG(XY,A,XYproj)
XY |
array of sample data |
A |
initial parameter vector c(Xc,Yc,a,b,alpha) |
XYproj |
corresponding n projection points on the conic |
list(Res, J)
|
list with the Residual Sum of Squares and the Jacobian matrix |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) A <- matrix(c(0,0,2,1,0),ncol=1) XYproj=matrix(c(0.394467220216675,0.980356518335872,0.833315950425981, 0.909063326557293,1.40466123643977,0.711850899213363, 1.70601340510202,0.521899957274429,1.89925244997324,0.313384799914835, 1.06482258038841,0.846485805004280,1.95308457257492, 0.215325713960169,1.91319150256275,0.291418202297698),8,2,byrow=TRUE) JmatrixLMG(XY,A,XYproj)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) A <- matrix(c(0,0,2,1,0),ncol=1) XYproj=matrix(c(0.394467220216675,0.980356518335872,0.833315950425981, 0.909063326557293,1.40466123643977,0.711850899213363, 1.70601340510202,0.521899957274429,1.89925244997324,0.313384799914835, 1.06482258038841,0.846485805004280,1.95308457257492, 0.215325713960169,1.91319150256275,0.291418202297698),8,2,byrow=TRUE) JmatrixLMG(XY,A,XYproj)
LMcircleFit applies a Geometric circle fit (minimizing
orthogonal distances) based on the standard Levenberg-Marquardt scheme
LMcircleFit(XY, ParIni, LambdaIni = 1, epsilon = 1e-06, IterMAX = 50)LMcircleFit(XY, ParIni, LambdaIni = 1, epsilon = 1e-06, IterMAX = 50)
XY |
array of sample data |
ParIni |
initial guess (a, b, R) |
LambdaIni |
initial value for the correction factor lambda |
epsilon |
tolerance (small threshold) |
IterMAX |
maximum number of (main) iterations, usually 10-20 will suffice |
vector(a, b, R)
|
vector with the estimates for the circle: center (a,b) and radius R |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c4 <- LMcircleFit(xy) xyc4<-calculateCircle(c4[1],c4[2],c4[3]) plot(xyc4[,1],xyc4[,2],xlim=c(-250,250),ylim=c(-250,250),col='cyan',type='l')xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c4 <- LMcircleFit(xy) xyc4<-calculateCircle(c4[1],c4[2],c4[3]) plot(xyc4[,1],xyc4[,2],xlim=c(-250,250),ylim=c(-250,250),col='cyan',type='l')
LMreducedCircleFit applies a Geometric circle fit
(minimizing orthogonal distances) based on the standard Levenberg-Marquardt scheme in the "reduced" (a,b) parameter space
LMreducedCircleFit(XY, ParIni, LambdaIni = 1, epsilon = 1e-06, IterMAX = 50)LMreducedCircleFit(XY, ParIni, LambdaIni = 1, epsilon = 1e-06, IterMAX = 50)
XY |
array of sample data |
ParIni |
initial guess (a, b) |
LambdaIni |
initial value for the correction factor lambda |
epsilon |
tolerance (small threshold) |
IterMAX |
maximum number of (main) iterations, usually 10-20 will suffice |
vector(a, b, R)
|
vector with the estimates for the circle: center (a,b) and radius R |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
Nikolai Chernov, 2010 Circular and linear regression: Fitting circles and lines by least squares Chapman & Hall/CRC, Monographs on Statistics and Applied Probability, Volume 117
xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c7 <- LMreducedCircleFit(xy) xyc7<-calculateCircle(c7[1],c7[2],c7[3]) plot(xyc7[,1],xyc7[,2],xlim=c(-250,250),ylim=c(-250,250),col='pink',type='l')xy<-calculateCircle(0,0,200,50,randomDist=TRUE,noiseFun=function(x) (x+rnorm(1,mean=0,sd=50))) plot(xy[,1],xy[,2],xlim=c(-250,250),ylim=c(-250,250));par(new=TRUE) c7 <- LMreducedCircleFit(xy) xyc7<-calculateCircle(c7[1],c7[2],c7[3]) plot(xyc7[,1],xyc7[,2],xlim=c(-250,250),ylim=c(-250,250),col='pink',type='l')
Residuals.ellipse projects a given set of points onto an
ellipse and computing the distances from the points to the ellipse
Residuals.ellipse(XY,ParG)Residuals.ellipse(XY,ParG)
XY |
array of sample data |
ParG |
vector 5x1 of the ellipse parameters (Center(1:2), Axes(1:2), Angle) |
list(Res, J)
|
list with the Residual Sum of Squares and the Jacobian matrix |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) Residuals.ellipse(XY,ParG)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) Residuals.ellipse(XY,ParG)
Residuals.hyperbola projects a given set of points onto an
hyperbola and computing the distances from the points to the hyperbola
Residuals.hyperbola(XY,ParG)Residuals.hyperbola(XY,ParG)
XY |
array of sample data |
ParG |
vector 5x1 of the hyperbola parameters (Center(1:2), Axes(1:2), Angle) |
list(RSS, XYproj)
|
list with the Residual Sum of Squares and the array of coordinates of projections |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) Residuals.hyperbola(XY,ParG)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) Residuals.hyperbola(XY,ParG)
Residuals.parabola projects a given set of points onto an
parabola and computing the distances from the points to the parabola
Residuals.parabola(XY,ParG)Residuals.parabola(XY,ParG)
XY |
array of sample data |
ParG |
vector 4x1 of the parabola parameters (Vertex(1:2), p, Angle) |
list(RSS, XYproj)
|
list with the Residual Sum of Squares and the array of coordinates of projections |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) Residuals.parabola(XY,ParG)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) Residuals.parabola(XY,ParG)
ResidualsG projects a given set of points onto an ellipse
and computing the distances from the points to the ellipse
ResidualsG(XY,ParG)ResidualsG(XY,ParG)
XY |
array of sample data |
ParG |
vector 5x1 of the ellipse parameters (Center(1:2), Axes(1:2), Angle) |
list(RSS, XYproj)
|
list with the Residual Sum of Squares and the array of coordinates of projections |
Jose Gama
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
Nikolai Chernov, 2014 Fitting ellipses, circles, and lines by least squares http://people.cas.uab.edu/~mosya/cl/
N. Chernov, Q. Huang, and H. Ma, 2014 Fitting quadratic curves to data points British Journal of Mathematics & Computer Science, 4, 33-60.
XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) ResidualsG(XY,ParG)XY <- matrix(c(1,7,2,6,5,8,7,7,9,5,3,7,6,2,8,4),8,2,byrow=TRUE) ParG <- matrix(c(0,0,2,1,0),ncol=1) ResidualsG(XY,ParG)