2005 Fiscal Year Final Research Report Summary
Control Theoretic Optimal Splines and Their Applications
| Project/Area Number |
16560393
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Control engineering
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| Research Institution | Tokyo Denki University |
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Principal Investigator |
KANO Hiroyuki Tokyo Denki University, College of Science and Engineering, Professor, 理工学部, 教授 (00246654)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIOKA Hiroyuki Tokyo Denki University, College of Science and Engineering, Assistant Professor, 理工学部, 講師 (10349798)
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| Project Period (FY) |
2004 – 2005
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| Keywords | optimal smoothing splines / B-splines / spline surfaces / linear control systems / character font / contour modeling / image processing / numerical analysis |
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| Research Abstract |
This study is on the theory and applications of designing optimal spline curves and surfaces. The basic problem is that, for given data points in a plane or space, we design optimal curves or surfaces that pass through or pass near the points and that are smooth. For such problems and their variants, our first aim is to develop novel theories and concise algorithms mainly from the viewpoints of systems and control theory. The second aim is to apply the results to real problems in various fields such as engineering, science, etc. In below, we summarize the results obtained in this study.
We established the methods for designing optimal smoothing curves and surfaces using B-splines as the basis functions, and derived theorems for the existence and uniqueness of solutions, for asymptotical properties when the number of data increases, and for the statistical properties when the data noises are present. Employing optimal smoothing splines and dynamic font model for generating characters, we
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developed a scheme for generating cursive characters and character strings as seen in Japanese calligraphy. Imposing periodicity boundary conditions, we developed theories and algorithms of periodic smoothing spline curves and surfaces. The results are applied successfully to the problem of contour and shape modeling of wet material such as jellyfish, red blood cell etc., and the models are used for analyzing the motion and shape changes. Moreover, utilizing the fact that the designed splines are continuously differentiable piecewise polynomials, we derived a systematic method for detecting and computing all the extrema, hence including global maximum and minimum, of the splines. This method is shown to be very useful for edge detection in digital images, in particular when there are data noises. On the other hand, employing linear control systems, we derived the methods for designing optimal splines and periodic splines, and moreover the method for multi-level smoothing splines that correspond to Hermite interpolations. These methods are shown to be useful for the problems in numerical analyses such as numerical solutions of ordinary differential equations, numerical integrations, etc. Less
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