Communications in Information and Systems

Volume 16 (2016)

Number 3

Trivariate and $n$-variate optimal smoothing splines with dynamic shape modeling of deforming object

Pages: 147 – 183



Hiroyuki Kano (School of Science and Engineering, Tokyo Denki University, Saitama, Japan)

Hiroyuki Fujioka (Department of System Management, Fukuoka Institute of Technology, Fukuoka, Japan)


We develop a method of constructing multi-variate optimal smoothing splines using normalized uniform B-spline as the basis functions. First we consider trivariate splines in details, which are useful particularly for modeling dynamic shape of 3-dimensional deformable object by using two variables for 3D shape and one for time evolution. The splines are constructed as a tensor product of three B-splines, and an optimal smoothing spline problem is solved together with typical examples of constraints as periodicity and boundary conditions. The algorithms are developed so that various types of constraints can be incorporated easily and existing numerical solvers for convex quadratic programming (QP) can be readily applicable for numerical solutions. The theory and algorithms are then extended to the general $n$-variate case. Using trivariate splines, we demonstrate usefulness of the method by two examples; vibration of rectangular membrane and dynamic 3D shape modeling of red blood cell. We will see that relatively small number of observation data with noises yield satisfactory results.


splines, smoothing, multi-variate splines, vibration of rectangular membrane, dynamic shape modeling, red blood cell

Received 27 November 2016

Accepted 21 November 2017

Published 30 November 2017