(I am new to this subject, so sorry if the question is too trivial).
I would like to find a method, possibly using polyharmonic splines, that would be a smooth mapping of $\mathbb{R}^3 \mapsto \mathbb{R}^3$ and would also allow shape manipulation via control handles.
General polyharmonic spline interpolation consists of evaluating a function [1] of the nature of $$f(\mathbf{x}) = \sum_{i=1}^N w_i \phi(|\mathbf{x} - \mathbf{c}_i|) + \mathbf{v}^{\top}(1,\mathbf{x})$$
where $\mathbf{x} \in \mathbb{R}^d$ is the point to be interpolated, $\mathbf{c} \in \mathbb{R}^d$ denotes the control handles and the weights $\mathbf{w}$ and $\mathbf{v}$ are computed directly. But I am not sure if/how the mapping $f:\mathbb{R}^d \mapsto \mathbb{R}$ can be adapted to perform a $\mathbb{R}^3 \mapsto \mathbb{R}^3$ mapping, as well as being used for shape manipulation via control handles, without computing the weight matrices with every update of the control handles.
On a similar note, [2] offers a possible method for using a similar Thin Plate Spline (TPS) formulation for $\mathbb{R}^3 \mapsto \mathbb{R}^3$ mapping, but I am not sure how to manipulate the target/destination control handles without having to re-compute the weights in a closed form.
Any help/suggestion in this matter would be hugely appreciated.
[1] wikipedia.org/wiki/Polyharmonic_spline
[2] Hu, Y., Zhou, M. and Wu, Z., 2009. A dense point-to-point alignment method for realistic 3D face morphing and animation. International Journal of Computer Games Technology, 2009.
The standard reference on interactive deformation of 3D space is a paper by Sederberg and Parry, SIGGRAPH 1986. There are many ways to modify and extend their method, but it will give you a place to start.
Actually, Bézier himself described this technique in his doctoral thesis, in 1977, but people almost always cite the Sederberg/Parry paper, instead.
More here: https://en.wikipedia.org/wiki/Free-form_deformation