Pose Space Deformation Notes - JP Lewis

Other related approaches
The origins of PSD are in an an approach I developed at ILM in 1995. That system was
used briefly in Jumanji (in the elephant-walks-over-car hero shot), and in Casper (it was
used to animate the dress in a scene where the fat ghost was dancing). ILM obtained
a patent on this approach in 1999. Compared with PSD, the this earlier approach used
a primitive form of scattered interpolation, and overall the interpolation was poorly
conceived. On the other hand, it did “invert” the skinning to do the interpolation in the
local space.
“Verbs and adverbs” interpolate motion of skeletons using RBFs (with added linear
trends). This paper really should have been cited by our 2000 Siggraph paper – however
I was working in production at the time and was not aware of it until later (in production
one has trouble keeping up with the academic literature, or anything else in life...).
Shape-by-Example is a technique presented by Sloan, Rose, and Cohen at the 2001
I3D conference; their technique has some broad similarities to PSD. Shape by example
interpolates shape using RBFs with linear added trends. However, they seem to be
interpolating the shape itself, rather than a delta from an underlying skinning system.
A quick analysis suggests that their technique requires less computation at the authoring
stage but is slower at runtime synthesis: Shape-by-example deforms a point as
x[j](p) = ∑ ∑
x k [j]r m,k R(p)
k m
whereas PSD is
x[j](p) = ∑ w k R(|p − p k |)
k
Since the sqrt involved in the distance can be folded into R, PSD deformation appears
to be more efficient.
The shape-by-example paper has some tips on improving RBF interpolation. One of
these improvements is arguable for the purpose of skinning however: they fit a polynomial
(e.g. linear) trend under the RBF deformation (as is commonly done in RBF
implementations of thin-plate splines). This is not always desirable, because it means
that a sculpted deformation does not die off far away from the original pose, but instead
continues to grow.
Regularization
If several of the sculpted poses are very similar then the matrix R can be close to singular.
There is an easy and effective approach to dealing (known by several names
including regularization, ridge regression, etc.). This technique is useful in other situations
too.
As motivation/background,) When two rows of a matrix are similar, the data can be
”explained” by setting the weight on one of the rows to zero, or by setting each to half