fitting two ellipses to outline of a body

Question

I'm trying to fit two ellipses to what happens to be the top view / outline of a body. For simplicity let's use the following example:

As you can see, this simple body is made up of a long core (blue) and a head (red). In reality this outline would be in one color, I'm just using two colors for visualization purposes here.

I know how to fit a single ellipse to either parts of that outline, but I do not know how to do the fit to two ellipses, given the constraint that these two ellipses are actually connected. In this particular case the constraint is that the two ellipses will never part and that there can only be a certain angle between ellipse 1 and ellipse 2.

I'm grateful for any pointers that tell me how to write a function, so that after calling magic_fitting_function(body_outline) the program returns to me the coordinates of the two underlying ellipses:

EDIT1: what is are the minimal requirements that could make solving this problem easier? E.g. if I were given one point, two points, etc, how would that possibly simplify the problem?

EDIT2: I'm looking for a programming language independent solution.

EDIT3: any hints on how to formulate the constraint of these two ellipses being located in a certain relationship to each other programmatically? E.g.: I know that the small ellipse will always be located at one end of the major axis of the big ellipse. Plus the small ellipse can only rotate by +- 90 degrees relative to the big ellipse.

Answer 1

1. Find the two points that are furthest from each other. One will belong to Ellipse1, the other to Ellipse2.
2. Check the nearest neighbors of these points to get 5 points belonging to Ellipse1, and 5 to Ellipse2.
3. Check out Wikipedia, and choose your favorite equation of an ellipse.
4. Using junior high algebra, for each ellipse, plug in the points to get 5 simultaneous equations, and solve these to get the 5 parameters that define your ellipse.

EDIT

I didn't realize this was tagged "matlab". In that case, once you have identified some points for each ellipse, there are matlab functions for fitting the points to an ellipse.

Answer 2

You can try isolating ellipses using the Hough transform. There are some FEX tools out there that are worth trying, for example this Ellipse Detection Using 1D Hough Transform.

Answer 3

I have never solved this problem, so I'm just throwing out a suggestion.

First, generate a bounding ellipse for the entire figure in order to determine what are the topmost and the bottommost points. (This step may not be necessary if you have a better way of finding these points.)

Next, detect the location of the "neck" by using a modified binary search. (Here I'm assuming that your bounding ellipse has a vertical orientation, as though the figure were standing up or standing on its head.) Generate two sets of bounding ellipses: one with an ellipse from the top of the figure to the 1/4 point of the figure (meaning if you draw a line through the bounding ellipse then the 1/4 point is between the top-left point and the middle) and with an ellipse from the 1/4 point to the bottom of the figure, and one with an ellipse from the top of the figure to the 3/4 point and with an ellipse from the 3/4 point to the bottom of the figure; the set of ellipses with the smaller total area is the one that is better encapsulating the head. Continue the search (e.g. next test an ellipse from the top to the 1/8 point / 7/8 point, and/or from the top to the 3/8 point / 5/8 point) until you've minimized the total bounding area of the set of ellipses; the point at which the ellipses meet is the neck. (No need to be too precise with this, it probably doesn't make much difference if you put the neck at the 34/256ths point or at the 35/256ths point.)

To detect the neck you may want to use bounding boxes instead of bounding ellipses.

Finally, adjust the two bounding ellipses in order to meet their angle constraints, e.g. by moving their extreme points in 5% increments (so assuming that the head ellipse's extreme points are on the y-coordinates 0 and 50 and the body ellipse's extreme points are on the y-coordinates 50 and 200, adjust them so that their extreme y-coordinates are on 0 and 60 and on 40 and 200).

Answer 4

If you have the complete outline, you can find where the two ellipses intersect - just look for the two sharp corners where the first derivative of your outline becomes discontinuous. Then, draw a straight line between those corners.

Everything on one side of the line is in ellipse A, everything on the other side is in ellipse B. The sharp corners are in both ellipses. Now, just fit a single ellipse to each of the two ellipses you've found and recalculate the points where the fitted ellipses intersect.