Communications in Information and Systems

Volume 14 (2014)

Number 1

Constrained smoothing and interpolating spline surfaces using normalized uniform B-splines

Pages: 23 – 56



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

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

Clyde F. Martin (Department of Mathematics and Statistics, Texas Tech University, Lubbock, Tx., U.S.A.)


We consider the problem of constructing optimal smoothing and interpolating spline surfaces with equality and/or inequality constraints. By using normalized uniform B-splines as the basis functions, the problem of constructing optimal surfaces is to determine the so-called control points optimally. In particular, following a similar approach as in the case of curves, we formulate various types of equality and inequality constraints as linear functions of the control points. Included are constraints on the value at isolated points, those over intervals or over regions, or on integral value on a region, and their combinations. Concise expressions are derived for these constraints and it is shown that they can be incorporated easily to smoothing and interpolating spline problems. The splines can be of arbitrary degree, and the problem is reduced to convex quadratic programming (QP) problem. Some efficient algorithms are available for solving the QP problems numerically, thus the proposed method is useful for many applications. The performance is examined by numerical examples of interpolating function with boundary constraints, approximating probability density functions, and of smoothing digital image data.


B-splines, optimal smoothing splines, optimal interpolating splines, equality/inequality constraint, convex quadratic programming

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