Our approach to the LV surface non-rigid motion tracking is based on the 3D bending model, which takes advantages of ideas stemming from our previous efforts in 2D image sequence based LV motion tracking, and earlier versions of our approach were reported in . In this effort, the physical model of the LV is a thin-plate loosely constrained to deform in a predetermined way as will be shown next. The potential energy of an ideal thin flexible flat plate of elastic material which is a measure of the strain energy of the deformation is given in :
where , are the two principal curvatures of the surface. This measure is invariant to 3D rotation and translation. (Interestingly, Koenderink et al use the same quantity as the curvedness measure of a surface, and presents several properties, although they also emphasize the importance of the signs of the principal curvatures, which are currently under our investigation.)
We modify the above idea slightly to define the energy required to bend a curved plate or surface to a new deformed state as:
The principal curvatures of surface at time are given with no bar, while the same parameters at time are specified with bars. This equation assumes that the corresponding points on the surfaces are known, and arrives at a numerical value measuring the energy that was required to deform the surface at time to achieve a shape at time .
However, it is the goal of our motion tracking algorithm to find the corresponding points on different surfaces. Under the assumption that surface deforms only slightly and locally within a small time interval, for each sampled point on the first surface, we construct a search area on the second surface. The point within the search window on the second surface that is best matched(i.e. minimizing the bending energy) is chosen as the corresponding point to , while the bending energy for all the other points inside the window are also recorded to be used as a indicator of the uniqueness of the match. The result of this matching process yields a set of shape-based best-matched initial motion vectors Dfor pairs of surfaces derived from 3D image sequence, as well as information from within each search area as to how confident the match is.