View Full Version : least-square fitting

02-01-2006, 06:38 PM
Dear Biomch-L user

I got problems understanding the use of least-square fitting. I red several papers dealing with that approach and it seem to be a very nice tool. For example one uses it to fit a perfect sphere into the femoral head to get the centre and model the femoral head. Another person uses this technique to fit a cylinder into the femoral condyle - this it what I would like to do. Another persons use the technique to extract the linear portion of a non-lineal force-displacement curve. That is cool - I would like to do that too ;-)

However when I try to fit a sphere in to the femoral head or a cylinder into the femoral condyle I get a fitting of the object which represents the "mean square distance" from all points involved (femoral head and neck; non cylindrical parts of the condyle). That is the important point: how do you explain to the algorithm, which point is relevant and which not. In fact the simplest least square approach is to compute a regression line, like we do it using basic statistics. When all the data points lie along a line you will get a good representation using a least-square approach. However if you have data points which are far from the ideal line, they will influence the algorithm so that the final line will be a "global average" of all the points and the "optimal" line will be left out. In fact that is the problem for the force-displacement curve. If you already know which part is the linear on - ease to get the slope, however if you do not know where it starts and where exactly ends you will get the "average" slope of the curve- not only the linear one.

Using the approach for the cylinder fitting procedure I get a minimization of the least-square distance - resulting in a very small, cylinder which is smashed against the condyle wall. Sure, from a mathematical point of view an optimal result - but not what I expected. I wonder if anyone could help me out and explain me the way to convince the algorithm doing what I want: finding the cylindrical shaped portion of the condyle and fitting an ideal cylinder onto it. I assume the secret will be the same for all three problems: sphere, cylinder and force-displacement curve?

Sincerely yours, Peter Böttcher.

Unfortunately the BIOMCH-L server do not accepts attachments. However, if someone is interested in some screen shots illustrating my problem I would be happy to email it directly.

P. Böttcher, Dr med vet, DipECVS
European Veterinary Specialist in Surgery
Klinik für Kleintiere
Universität Leipzig
An den Tierkliniken 23
D-04103 Leipzig (Germany)
Tel: +49-341-9738700
Fax: +49-341-9738799
email: boettcher@kleintierklinik.uni-leipzig.de