Hello Morhpometricians - Caveat: I am very new to the field of morphometrics. In fact, I am neither a biologist or an anthropologist. I am a PhD student of computer graphics and my work involves warping and deformation of objects, thus my interest in morphometrics. I hope this explains why I might be asking questions that may appear trivial as well as my repeated use of layman's terms and my occasional misuse of technical terms ;) .
Question: I have just implemented the 3D TPS method/algorithm described in Chapter 15 pp. 391-392 of "Geometric Morphometrics for Biologists (A Primer)" by M. L. Zelditch et. al. On visualizing the deformation, I noticed that the deformation/warp does not produce a smooth, curvature continuous object. For instance, the images below show a sphere (of 10 units radius), with the following 14 landmarks (? I suppose thats what the equivalent a "deformation point" would be): [The points are "moved" as indicated by the arrows] A: ( 10.0, 0.0, 0.0 ) -> A': ( 15.0, 0.0, 0.0 ) B: (-10.0, 0.0, 0.0 ) -> B': (-15.0, 0.0, 0.0 ) C: ( 0.0, 10.0, 0.0 ) -> C': ( 0.0, 15.0, 0.0 ) D: ( 0.0, -10.0, 0.0 ) -> D': ( 0.0, -15.0, 0.0 ) E: ( 0.0, 0.0, 10.0 ) -> E': ( 0.0, 0.0, 15.0 ) F: ( 0.0, 0.0, -10.0 ) -> F': ( 0.0, 0.0, -15.0 ) [These points are not moved] G: ( +7.071, +7.071, +7.071 ) H: ( -7.071, -7.071, -7.071 ) I: ( +7.071, +7.071, -7.071 ) J: ( +7.071, -7.071, +7.071 ) K: ( -7.071, +7.071, +7.071 ) L: ( -7.071, -7.071, +7.071 ) N: ( -7.071, +7.071, -7.071 ) O: ( +7.071, -7.071, -7.071 ) Sphere before deformation: http://i17.photobucket.com/albums/b52/videohead/sphere_before.jpg Sphere after deformation: http://i17.photobucket.com/albums/b52/videohead/sphere_after.jpg (Green crosses indicate the positions of the points in both images) The latter image highlights a few "unsightly" discontinuities in gradient/curvature of the deformed sphere at the points A' - F', which after some consideration wasn't much of a surprise considering that the linear combination of weights and the radial basis function U = |P - P[i]| is discontinuous where P = P[i]. That is to say if: F(P) = k0 + k1.P[x] + k1.P[y] + k1.P[z] + SUM_i( W[i].|P - P[i]| ) Where i = 1 .. N (N = no of points/landmarks). The derivative F'(P) (wrt to x, y, or z) is undefined at every P[i]. I suppose none of this is new to you experts, so my question is what would be the standard or recommended way of removing these discontinuities? I'm looking for some method that warps a 3D object but produces smooth i.e. continuous optimal deformations that minimize bending energy (something that makes the 2D TPS method so appealing) BUT for a 3D problem. Help, someone. Anyone? Thanks for bearing with me. - Olumide [EMAIL PROTECTED] -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
