From: Ian Scott <ian.m.scott@st...>  20061117 09:57:26

Tamir Yedidya wrote: > > Hi, > > I'm trying to use vnl_levenberg_marquardt in order to fit a circle to an > image. So, I'm having 3 parameters (x,y,r). > My initial guess is usually quite good. For example , my estimation can > be (157,157,100) and the best fit is (150,150,100). > > However, I found out that when using the default settings the algorithm > will converge to the initial guess. > I've tried setting the epsilon_function to 0.005 or higher values and > then the estimates start moving, the algorithm converges, > but not always to the correct answer. Then, on different settings I > might have to set the epsilon_function > to a different value for the algorithm to converge to the right value. > > My question is whether this is the right thing to do  setting epsilon? > or maybe the default parameters should > be good enough and the problem is with the cost function? If you have problems with a standard optimiser, the first place to look is always the cost function. In this case it is (the residual sum of squares) along all three axes. It is also worth plotting 2D surfaces from the response due to pairs of parameters. Since you are using a residualsbased optimiser, it is also worth checking that your residual behave correctly. Pick a residual vector r_1 at some position x_1, and then plot the projection of the residual onto r_1, for different parameter values. This way you get to see if your residual is behaving vaguely linearly around the minimum. Repeat for several different x_1. You may also need to look at the cost function at various scales of changes in the parameters. A function that looks smooth at x_0=[3:0.1:3] might start to look very noisy and noncontinuous at much smaller scales x_0=[3e4:1e5:3e4]. A useful compromise is to have a bilog scale in x. x=3; while (x < 1e5) { x=x/1.2; plot f(x); } x=1e5; while (x < 3) { x=x*1.2; plot f(x); } Whether you are explicitly calculating the Jacobian (rather than letting the cost function calculate it using finite differences), you may also need to check that your gradients behave correctly. Again, pick a gradient g_1 at a given x_1, and plot the magnitude of gradient for different x, projected onto g_1. Again repeat for different x_1. Finally (although it should have been first)  try the amoeba first. The amoeba is the most robust, hasslefree, reliable optimiser about. After it works, you can worry about using a faster optimiser (if speed turns out to be a problem.) By this time you will understand a lot more about your optimisation problem, allowing you to make more intelligent choices. Ian. 