Indeed. But there are different ways of defining smoothness, and the TPS criterion is only one of them. In biological morphopmetrics it's quite possibly the best one, because it's simple. But if you want to make a topographic model, for example, you may find that another smoothness criterion will work better, such as third-order differentials. My point just being that TPS is not the only method that maximizes "smoothness" - depending on how you define smoothness.
Oyvind Hammer Geological Museum University of Oslo On Fri, 3 Dec 2004, morphmet wrote: > Another thought. There is nothing 'special' about the thin-plate splines in > the sense that they are not based on any particular biological model. On the > other hand they that the important property of describing a difference > between two shapes as a smooth deformation. Smoothness is important because > that means that there will not be some bend in the deformation unless there > is some displacement of landmarks that requires it. As an extreme > alternative, one could use some polynomial function (like an early > suggestion of fitting a trend surface). Such functions are notorious for > implying deformations out beyond any of the data points used to define the > function. That is clearly not desirable. It would be better to have a model > but if one does not have a model then one would like to use the smoothest > possible functions that can account for the observed differences in shape > and do not impose structure where there is no evidence for a shape change. > > -------------------- > F. James Rohlf, Distinguished Professor > SUNY Stony Brook, NY 11794-5245 > www: life.bio.sunysb.edu/ee/rohlf > > > > > -----Original Message----- > > From: morphmet [mailto:[EMAIL PROTECTED] > > Sent: Friday, December 03, 2004 1:03 AM > > To: morphmet > > Subject: Re: Why Thin-Plate Splines? > > > > > > > > > > Thin-plate splines is a special case of RBF, in the sense > > that there > > > is an RBF kernel function > > > (r^2 log r, or something like that) which will give you the TPS > > > solution. > > > > > > In one sense, the TPS minimizes second-order differentials. > > It might > > > be argued that this is the simplest curvature measure. > > Certainly, you > > > could use other kernel functions to minimize third-order > > differentials > > > or anything else, and this certainly has many applications, > > but this > > > whole concept of minimizing curvature has no intrinsic biological > > > meaning, and we just need something relatively simple that we can > > > discuss and understand easily. > > > > Thanks! I love simple and direct answers ;-) > > > > > Another thing is that morphometricians have developed a > > large toolbox > > > of useful analysis methods based on the TPS (bending > > energies, partial > > > and relative warps, etc.). No similar theory and methods have been > > > developed for other kernel functions. > > > > I've heard of those i.e. partial and relative warps. Please > > shed some light on them. > > > > Thanks again, > > > > Olumide > > (email: [EMAIL PROTECTED]) > > > > PS: Do you mean that the use of TPS in the morphometrics just > > as a matter of habit? > > > > -- > > Replies will be sent to the list. > > For more information visit http://www.morphometrics.org > > > > -- > Replies will be sent to the list. > For more information visit http://www.morphometrics.org > > -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
