Implicit higher degree polynomials in x, y, z (or in x, y for curves in images) have considerable global and semiglobal representation power for objects in 3D space. (Spheres, cylinders, cones and planes are special c...
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ISBN:
(纸本)0819410314
Implicit higher degree polynomials in x, y, z (or in x, y for curves in images) have considerable global and semiglobal representation power for objects in 3D space. (Spheres, cylinders, cones and planes are special cases of such polynomials restricted to second degree). Hence, they have great potential for object recognition and position estimation and for object geometric-property measurement. In this paper we deal with four problems pertinent to using these polynomials in real world robust systems: (1) Characterization and fitting algorithms for the subset of these algebraic curves and surfaces that is bounded and exists largely in the vicinity of the data;(2) The aposteriori distribution for the possible polynomial coefficients given a data set. This measures the extent to which a data set constrains the coefficients of the best fitting polynomial;(3) Geometric Invariants for determining whether one implicit polynomial curve or surface is a rotation and translation of another, or whether one implicit polynomial curve is an affine transformation of another;(4) A Mahalanobis distance for comparing the coefficients or the invariants of two polynomials to determine whether the curves or surfaces that they represent are close over a specified region. In addition to handling objects with easily detectable features such as vertices, high curvature points, and straight lines, the polynomials and tools discussed in this paper are ideally suited to smooth curves and smooth curved surfaces which do no have detectable features.
Multiresolution and wavelets are being promoted as a possible aid in various numerical applications including signal and image processing. One of the theoretical advantages of wavelet bases is that they provide a simp...
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ISBN:
(纸本)0819410314
Multiresolution and wavelets are being promoted as a possible aid in various numerical applications including signal and image processing. One of the theoretical advantages of wavelet bases is that they provide a simple characterization of smoothness classes in terms of the coefficients in wavelet decompositions. In this presentation, we shall give examples of how this characterization can be used to advantage in various numerical applications.
Design and manipulation of curves and surfaces are required in many different areas of technology, for example in the definition of products such as cars and the casings for electrical devices. Fourier methods have lo...
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ISBN:
(纸本)0819410314
Design and manipulation of curves and surfaces are required in many different areas of technology, for example in the definition of products such as cars and the casings for electrical devices. Fourier methods have long been used in image processing but have seen much less use in computer aided design (CAD). This paper suggests the use of Fourier methods for the generation of blends, which provide smooth transitions between model surfaces, and fairings, which smooth large regions of the surface of an object. A description is given of the use of Fourier methods to allow predictable and simple control over curved shape in a two- dimensional system, and results of the system are presented and analyzed. This allows informed assertions to be made regarding the extension to a surface smoothing system, which, at the time of writing, has not been developed. In particular we note that under certain circumstances the Fourier smoothing methods reduce to spline production, and the connection with spline theory is made clear where applicable.
Reconstruction of shapes from partial information is a problem arising in many scientific and engineering applications. We present a method for reconstructing a two-dimensional manifold from an unstructured collection...
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ISBN:
(纸本)0819410314
Reconstruction of shapes from partial information is a problem arising in many scientific and engineering applications. We present a method for reconstructing a two-dimensional manifold from an unstructured collection of sampled points. The algorithm consists of two major steps. In the first step, we estimate the topological type of the manifold and also obtain a crude estimate of its geometry. In the second step, we improve the fit of the estimate to the data points, while keeping the topological type fixed.
For surfaces, such as Bezier or B-splines, or NURBS with positive weights, which are defined by networks of control points and for which identifiable pieces of surface are known to lie within the convex hull of a subs...
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ISBN:
(纸本)0819410314
For surfaces, such as Bezier or B-splines, or NURBS with positive weights, which are defined by networks of control points and for which identifiable pieces of surface are known to lie within the convex hull of a subset of the control points, recursive division is one of the most robust interrogation techniques available, guaranteeing except in very singular circumstances to find all components of the intersection. Offset surfaces, used where an object has a small non-zero thickness, and to determine cutter center loci where a cutting point must lie on a given surface, have not had this option. This paper describes a technique for applying recursive division interrogation to offset surfaces, where the offset is either constant or a function of surface normal. Sections I and II recapitulate the standard theory of recursive subdivision interrogation and the definition of offset surfaces. Section iii explores the concept of procedural interface, and section IV introduces that of a Quantized Hull. Sections V to VII suggest a naive method of recursive division interrogation, discover why it does not work, and show how it may be salvaged.
A well-known strength of the parametric representation of a curve or surface is the ease by which a piecewise-linear approximation is generated. This is often true in geometric design, where only smooth segments or pa...
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ISBN:
(纸本)0819410314
A well-known strength of the parametric representation of a curve or surface is the ease by which a piecewise-linear approximation is generated. This is often true in geometric design, where only smooth segments or patches are considered over a well-chosen polynomial basis. Visualizing an arbitrary and possibly discontinuous parametric surface is useful but non-trivial for algebraic surfaces defined by rational parameter functions. Such surfaces have pole curves in their domain, where the denominators of the parameter functions vanish, domain base points that correspond to entire curves on the surface, and other features that cause display algorithms to fail. These are ubiquitous problems occurring even among the natural quadrics. Sophisticated but unsuspecting display techniques (e.g. those implemented in Maple V, Mathmatica) produce completely unintelligible results. We provide a general solution and discuss our implementation. First, projective domain transformations are applied that map the entire surface from a finite domain region. Then, domain pole curves are identified and numerically approximated. Using Delaunay triangulation, a special decomposition of the domain is constructed that avoids discontinuities. This is then mapped onto the surface and clipped against a 3D box. The implementation can display very complicated surfaces, as we shall illustrate. Our techniques generalize in a straightforward way to rational varieties of any dimension.
A computational model for generating a graphical deformation process of a 3-D object from an initial state to a final state is proposed. The deformation process is represented by several intermediate states which are ...
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ISBN:
(纸本)0819410314
A computational model for generating a graphical deformation process of a 3-D object from an initial state to a final state is proposed. The deformation process is represented by several intermediate states which are interpolated from the two original states of an object. The generated processes can be used for filling the missing information between two given states, visualizing the changing process, or conveying the underlying idea more clearly through a pictorial sequence. The model includes a generic scheme for correspondence establishment and a physically-based process generator.
This paper presents an efficient technique to compress chain-encoded line drawings. The technique, called the address chain code, constructs a codebook containing vectors of chains which recur frequently in the chain-...
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ISBN:
(纸本)0819410314
This paper presents an efficient technique to compress chain-encoded line drawings. The technique, called the address chain code, constructs a codebook containing vectors of chains which recur frequently in the chain-encoded line drawings. The recurrent vectors are encoded as their corresponding addresses in the codebook preceded by header bits. The encoding procedure and an efficient way to organize and label the vectors, which turns out to be somewhat similar to the generalized chain codes, are described in this paper.
In surface reconstruction a mathematical description is created from sampled data points. While the displayed image may appear to be correct, there remains the analytical question as to whether it can be proven that t...
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ISBN:
(纸本)0819410314
In surface reconstruction a mathematical description is created from sampled data points. While the displayed image may appear to be correct, there remains the analytical question as to whether it can be proven that the stored surface model and the original artifact have the same topology. Previous approaches to investigate all possible topological configurations were cumbersome and offered little insight as to how those topological configurations were generated. An alternative algorithm is offered, where the mathematics of the algorithm does offer insight into the structure of these configurations.
It is shown how to construct G2-continuous spline with arcs of cubics. Each arc is a piece of the oval of a cubic and it is controlled locally by a triangle tangent to the arc at both endpoints. Formulas for mixed int...
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ISBN:
(纸本)0819410314
It is shown how to construct G2-continuous spline with arcs of cubics. Each arc is a piece of the oval of a cubic and it is controlled locally by a triangle tangent to the arc at both endpoints. Formulas for mixed interpolation of further points and tangents are given in terms of geometrically meaningful shape parameters. It is shown that under certain restrictions, the numerical values of the curvatures may be prescribed at the joints. Some new shape handles are developed for the local control of each arc of the spline. Intersection problems are easily handled. The main advantage of algebraic splines is that they are completely parametrization free.
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