Background: My research interests are primarily concerned with the theory of splines and their applications to computer aided geometric design (CAGD). The design of curves and surfaces plays an important role not only in the construction of different products such as car bodies, ship hulls, airplane fuselages and wings, etc., but also in the description of geological, physical and even medical phenomena. New areas of CAGD applications include computer vision and inspection of manufactured parts, medical research (software for digital diagnostic equipment), image analysis, high resolution TV systems, cartography, the film industry, etc.
Exact Solutions of Finite-Difference Schemes: At the first stage of my scientific career I was involved in the numerical solution of problems of mathematical physics in several variables. It was the time when the method of fractional steps, better known as the method of splitting, gained popularity in many areas of scientific computing. I constructed high-order accuracy finite-difference schemes for Laplace and biharmonic equations. Using the idea of fractional steps, I suggested efficient iterative algorithms for the numerical treatment of the corresponding finite-difference schemes. In the case of a difference analogue of the Dirichlet problem for Laplace and biharmonic equations on a 2-D square domain and uniform mesh with N knots in both directions one obtains the exact finite-difference solution after N iterations. This is a much more efficient approach than the traditional successive over-relaxation technique. Later these results were extended by different authors to the equations of elasticity and plasticity and to multiply connected domains.
Optimal Error Bounds for Spline Approximation: In the year 1980 I completed a book on Methods of Spline Functions in collaboration with Professor Yuri S. Zav'yalov and Dr. Valeri L. Miroshnichenko (Sobolev Institute of Mathematics in Novosibirsk, Russia), published by Nauka in Moscow. This book is a comprehensive survey of different efficient algorithms for computing 1-D and 2-D splines. Our own main results relate to a new technique which we developed to obtain optimal error bounds for polynomial spline interpolation. This method is based on an integral representation of the error estimate. In many cases it gives minimal values of constants in error bounds for interpolation splines. The same approach is also used to find optimal error bounds for local spline approximation methods. This book has become a standard textbook for students, researchers and engineers in Russia and over eleven thousands copies were sold.
Shape-Preserving Spline Interpolation: Standard methods of spline functions do not preserve shape properties of the data. By introducing shape control parameters into the spline structure, one can preserve various characteristics of the initial data including positivity, monotonicity, convexity, linear and planar sections. Based on interpolating splines, methods with shape control are usually called methods of shape-preserving spline interpolation. Here the main challenge is to develop algorithms that choose shape control parameters automatically. The majority of such algorithms solve the problem only for some special data. To solve the problem in a general setting, I gave a classification of the initial data and reduced the problem of shape-preserving interpolation to the problem of Hermite interpolation with constraints of inequality type. The solution is a C2 local generalized tension spline with additional knots. This allows for development of a local algorithm of shape-preserving spline interpolation where shape control parameters are selected automatically to meet the monotonicity and convexity constraints for the data. Its application makes it possible to give a complete solution to the shape-preserving interpolation problem for arbitrary data and isolate the sections of linearity, the angles, etc.
Tension GB-Splines: In my opinion, my most significant contribution to the theory of splines, involves the development of "direct methods" for constructing explicit formulae for tension generalized basis splines (GB-splines for short) and finding recursive algorithms for the calculation of GB-splines. This approach has yielded new local bases for various tension splines including among others rational, exponential, hyperbolic, variable order splines, and splines with additional knots. These results were expanded to GB-splines of arbitrary order. I investigated the main properties of GB-splines and their series such as shape-preserving properties, invariance with respect to affine transformations, etc. I have shown that the GB-spline series is a variation diminishing function and the systems of GB-splines are weak Chebyshev systems.
Shape-Preserving Local Spline Approximation: The method of local spline approximation due to Lyche and Schumaker [J. Aprox. Theory 15(1975)], combined with recursive algorithms for computing polynomial splines, was found to be efficient in practical applications. However it gives a curve which only approximates the data, but changes the data shape substantially. For data with specified tolerances, I developed algorithms of shape-preserving local spline approximation based on GB-splines with automatic selection of the shape control parameters. These parameters are chosen to satisfy the given tolerances and the monotonicity and convexity conditions for the initial data. Such methods are successfully used in the shape-preserving approximation of multivalued surfaces where the initial data is assumed to be given as a set of pointwise-assigned non-intersecting curvilinear sections of a 3-D solid. Additionally, my algorithms of shape-preserving parametrization may be used to improve the quality of representation of the shape-preserving spline curves and surfaces. This technique is described in full detail in my book Methods of Shape-Preserving Spline Approximation. World Scientific Publ. Co. Pte. Ltd., Singapore, 2000.
Research plans: My current research interests include shape-preserving spline approximation, difference methods for constructing splines, surface reconstruction from cross-sections, and scientific visualization.
Shape-Preserving Curve and Surface Approximation: A fundamental requirement of all CAGD software is to be able to approximate data in a manner which produces either a curve or surface which has the same "shape" as the data. The need for such methods is essential for a wide range of applications in industrial design, and in the visualization of scientific data. The known algorithms of shape-preserving approximation with automatic selection of shape control parameters are mainly comonotone, i.e. the spline on the i-th data interval is increasing or decreasing with the data on that interval. Such splines have the disadvantage that they must have slope zero at a point where the neighboring secant lines have a sign change in their slope; hence, any local extrema of the underlying approximation are assumed to be in the data sample. I am working on developing algorithms of shape-preserving approximation with automatic choice of shape control parameters which are free of this disadvantage and generalize comonotone approximations.
Difference Methods for Constructing Splines: Spline theory is mainly grounded on two approaches: the algebraic one (where splines are understood as smooth piecewise functions) and the variational one (where splines are obtained via minimization of quadratic functionals with equality and/or inequality constraints). Although less common, a third approach, where splines are defined as the solutions of differential multipoint boundary value problems (DMBVP for short), has been considered. Even though some of the important classes of splines can be obtained from all three schemes, specific features sometimes make the last one an important tool in practical settings. I want to develop difference methods for the construction of shape-preserving splines. In the 1-D case the basic DMBVP gives hyperbolic splines. In the 2-D case, I plan to consider a generalization of thin plate splines. This approach permits to avoid evaluation of hyperbolic and other special functions. It can be generalized to smoothing splines and to the approximation of scattered data in a straightforward manner.
Surface Reconstruction From Cross-Sections: This application is widely used in automobile and aircraft manufacturing, shipbuilding, and in Computer Aided Tomography. The aim is to provide a method for reconstruction a complete surface from a sequence of cross-sectional images or contours. In tomography one has the problem of matching appropriate points on adjacent contours, and when branching occurs within the object, i.e. a single contour at one level branches into two or more at the next. Research is in progress to devise algorithms to perform reconstruction automatically with local shape control.
Scientific Visualization: Visualization involves the use of computer graphics tools to explore data resulting from measurements, or data produced by simulation models. Visualization of complex data is an excellent way to gain a better understanding of the system under study, and it is also valuable as a means of communication. This is an exciting area which combines many skills: graphics, knowledge of the system under study, interface design, and imagination. Visualisation has widespread applications whenever a complex system is being investigated. The methods of shape-preserving spline approximation of curves and surfaces described in my above mentioned book were realized as a package of computer programs which enable one to construct and render complex multivariate surfaces. I plan to improve this package for purposes of scientific visualization and aim to include it in standard software Matlab.
Page Last Updated on February 12, 2008