New results on monotone cubic splines and hierarchical scattered data interpolation.
Item
-
Title
-
New results on monotone cubic splines and hierarchical scattered data interpolation.
-
Identifier
-
AAI9986297
-
identifier
-
9986297
-
Creator
-
Alfy, Yitzhak.
-
Contributor
-
Adviser: George Wolberg
-
Date
-
2000
-
Language
-
English
-
Publisher
-
City University of New York.
-
Subject
-
Engineering, Electronics and Electrical | Computer Science | Mathematics
-
Abstract
-
Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C2 continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. It is possible for a set of monotonically increasing (or decreasing) control points to yield a curve that is not monotonic, i.e., the spline may oscillate. In such cases, it is necessary to sacrifice some smoothness in order to preserve monotonicity.;One goal of this thesis is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We derive a novel optimization-based approach and analyze its merits against other work in the field. The following related topics have been investigated: (1) fitting monotone cubic spline to data with changing monotonicity; (2) construction of monotone and convex/concave interpolating cubic splines; (3) construction of C2 interpolating cubic splines with extra knots inserted between each pair of data points; (4) smoothing noisy data with C2 cubic splines; and (5) fitting monotone piecewise bicubic interpolation to scattered data and data on rectangular meshes.;Surface fitting to scattered data is another important problem of great interest to many fields of science and technology. Although reconstruction from uniform samples is a well-known subject, the theory of reconstruction from nonuniform samples is not equally mature. Various local and global methods abound. In local methods, the value of the surface at some position depends only on the data points that are in its neighborhood. In global methods, the value depends upon all data points. Generally, the construction of an approximating surface by local methods requires less computational effort than global methods, especially when dealing with large data sets. Surfaces produced by global methods, however, have better approximation properties.;In recent work, Wolberg and his colleagues introduced a multiresolution cubic B-spline approximation (MBA) algorithm. The hierarchical MBA algorithm, which is both fast and global, was demonstrated to produce high fidelity reconstruction in diverse applications such as image reconstruction, image warping, and object reconstruction. This thesis presents an analysis of key mathematical properties of the MBA algorithm. The resulting analysis suggests an alternate global hierarchical approach which we call multiresolution filtering approximation (MFA). The MFA scheme is intuitive and simple to implement. Numerical experiments show that the MBA and MFA schemes yield similar quantitative performance for synthetic test functions. The applicability of the MFA algorithm for image processing applications has been examined. We applied the MFA algorithm to image reconstruction from nonuniform samples.
-
Type
-
dissertation
-
Source
-
PQT Legacy CUNY.xlsx
-
degree
-
Ph.D.
