Showing papers on "Bicubic interpolation published in 1995"
Book•
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TL;DR: This chapter discusses the construction of B-spline Curves and Surfaces using Bezier Curves, as well as five Fundamental Geometric Algorithms, and their application to Curve Interpolation.
Abstract: One Curve and Surface Basics.- 1.1 Implicit and Parametric Forms.- 1.2 Power Basis Form of a Curve.- 1.3 Bezier Curves.- 1.4 Rational Bezier Curves.- 1.5 Tensor Product Surfaces.- Exercises.- Two B-Spline Basis Functions.- 2.1 Introduction.- 2.2 Definition and Properties of B-spline Basis Functions.- 2.3 Derivatives of B-spline Basis Functions.- 2.4 Further Properties of the Basis Functions.- 2.5 Computational Algorithms.- Exercises.- Three B-spline Curves and Surfaces.- 3.1 Introduction.- 3.2 The Definition and Properties of B-spline Curves.- 3.3 The Derivatives of a B-spline Curve.- 3.4 Definition and Properties of B-spline Surfaces.- 3.5 Derivatives of a B-spline Surface.- Exercises.- Four Rational B-spline Curves and Surfaces.- 4.1 Introduction.- 4.2 Definition and Properties of NURBS Curves.- 4.3 Derivatives of a NURBS Curve.- 4.4 Definition and Properties of NURBS Surfaces.- 4.5 Derivatives of a NURBS Surface.- Exercises.- Five Fundamental Geometric Algorithms.- 5.1 Introduction.- 5.2 Knot Insertion.- 5.3 Knot Refinement.- 5.4 Knot Removal.- 5.5 Degree Elevation.- 5.6 Degree Reduction.- Exercises.- Six Advanced Geometric Algorithms.- 6.1 Point Inversion and Projection for Curves and Surfaces.- 6.2 Surface Tangent Vector Inversion.- 6.3 Transformations and Projections of Curves and Surfaces.- 6.4 Reparameterization of NURBS Curves and Surfaces.- 6.5 Curve and Surface Reversal.- 6.6 Conversion Between B-spline and Piecewise Power Basis Forms.- Exercises.- Seven Conics and Circles.- 7.1 Introduction.- 7.2 Various Forms for Representing Conics.- 7.3 The Quadratic Rational Bezier Arc.- 7.4 Infinite Control Points.- 7.5 Construction of Circles.- 7.6 Construction of Conies.- 7.7 Conic Type Classification and Form Conversion.- 7.8 Higher Order Circles.- Exercises.- Eight Construction of Common Surfaces.- 8.1 Introduction.- 8.2 Bilinear Surfaces.- 8.3 The General Cylinder.- 8.4 The Ruled Surface.- 8.5 The Surface of Revolution.- 8.6 Nonuniform Scaling of Surfaces.- 8.7 A Three-sided Spherical Surface.- Nine Curve and Surface Fitting.- 9.1 Introduction.- 9.2 Global Interpolation.- 9.2.1 Global Curve Interpolation to Point Data.- 9.2.2 Global Curve Interpolation with End Derivatives Specified.- 9.2.3 Cubic Spline Curve Interpolation.- 9.2.4 Global Curve Interpolation with First Derivatives Specified.- 9.2.5 Global Surface Interpolation.- 9.3 Local Interpolation.- 9.3.1 Local Curve Interpolation Preliminaries.- 9.3.2 Local Parabolic Curve Interpolation.- 9.3.3 Local Rational Quadratic Curve Interpolation.- 9.3.4 Local Cubic Curve Interpolation.- 9.3.5 Local Bicubic Surface Interpolation.- 9.4 Global Approximation.- 9.4.1 Least Squares Curve Approximation.- 9.4.2 Weighted and Constrained Least Squares Curve Fitting.- 9.4.3 Least Squares Surface Approximation.- 9.4.4 Approximation to Within a Specified Accuracy.- 9.5 Local Approximation.- 9.5.1 Local Rational Quadratic Curve Approximation.- 9.5.2 Local Nonrational Cubic Curve Approximation.- Exercises.- Ten Advanced Surface Construction Techniques.- 10.1 Introduction.- 10.2 Swung Surfaces.- 10.3 Skinned Surfaces.- 10.4 Swept Surfaces.- 10.5 Interpolation of a Bidirectional Curve Network.- 10.6 Coons Surfaces.- Eleven Shape Modification Tools.- 11.1 Introduction.- 11.2 Control Point Repositioning.- 11.3 Weight Modification.- 11.3.1 Modification of One Curve Weight.- 11.3.2 Modification of Two Neighboring Curve Weights.- 11.3.3 Modification of One Surface Weight.- 11.4 Shape Operators.- 11.4.1 Warping.- 11.4.2 Flattening.- 11.4.3 Bending.- 11.5 Constraint-based Curve and Surface Shaping.- 11.5.1 Constraint-based Curve Modification.- 11.5.2 Constraint-based Surface Modification.- Twelve Standards and Data Exchange.- 12.1 Introduction.- 12.2 Knot Vectors.- 12.3 Nurbs Within the Standards.- 12.3.1 IGES.- 12.3.2 STEP.- 12.3.3 PHIGS.- 12.4 Data Exchange to and from a NURBS System.- Thirteen B-spline Programming Concepts.- 13.1 Introduction.- 13.2 Data Types and Portability.- 13.3 Data Structures.- 13.4 Memory Allocation.- 13.5 Error Control.- 13.6 Utility Routines.- 13.7 Arithmetic Routines.- 13.8 Example Programs.- 13.9 Additional Structures.- 13.10 System Structure.- References.
4,552 citations
IBM1
TL;DR: The results show that tetrahedral interpolation, with close to half the computational cost of tnlinear interpolations, is capable of providing better accuracy, and one diagonal extraction from cubic packing is useful as a general-purpose color space interpolator.
Abstract: Three-dimensionalinterpolation is suitable for many kinds of color space transformations. We examine and analyze several linear interpolation schemes-some standard, some known, and one novel. An interpolation algorithm design is divided into three parts: packing (filling the space of the input variable with sample points), extraction (selecting from the constellation of sample points those appropriate to the interpolation of a specific input point), and calculation (using the extracted values and the input point to determine the interpolated approximation to the outputpoint). We focus on regular (periodic) packing schemes. Seven principles govern the design of linear interpolation algorithms: 1) Each sample point should be used as a vertex of as many polyhedra as possible; 2) the polyhedra should completely fill the space; 3) polyhedra that share any part of
a face must share the entire face; 4) the polyhedra used should have the fewest vertices possible; 5) polyhedra should be small; 6) in the absence of information about cuivature anisotropy, polyhedra should be close to regular in shape; and 7) polyhedra should be of similar size. A test for interpolation algorithm performance in performing actual color space conversions is described, and results are given
for an example color space conversion using several linear interpolation methods. The extractions from cubic, body-centered-cubic, and face-centered-cubic lattices are described and analyzed. The results confirm Kanamori's claims for the accuracy of PRISM interpolation; it comes close to the accuracy of trilinear interpolation with roughly three-quarters the computations. The results show that tetrahedral
interpolation, with close to half the computational cost of tnlinear interpolation, is capable of providing better accuracy. Of the tetrahedral interpolation techniques, one diagonal extraction from cubic packing is useful as a general-purpose color space interpolator...
159 citations
Book•
30 Sep 1995
TL;DR: The structure of the Handbook is illustrated with examples of Sweep Method for Tridiagonal and Pentadiagonal Matrices and its applications in discrete geometry and discrete geometry.
Abstract: Preface Why Splines? On the Structure of the Handbook Spline Functions Spline Functions of One Variable Interpolating Cubic Splines Smoothing Cubic Splines Other Types of Splines Spline Functions of Two Variables Interpolating Bicubic Splines Smoothing Bicubic Splines Geometric Splines Spline Curves Main Facts of the Curve Theory Bezier Curves B-Spline Curves Beta-Spline Curves Other Spline Curves Spline Surfaces Main Facts of the Surface Theory Bezier Surfaces B-Spline Surfaces Beta-Spline Surfaces Other Spline Surfaces Appendices Programs of Sweep Method for Tridiagonal and Pentadiagonal Matrices Description of Diskette Bibliography Index
109 citations
21 Oct 1995
TL;DR: Six different rotation methods for generating projections and reconstructions of simulated and clinical data are analyzed and it is suggested that the method of shears with cubic interpolation offers a computationally efficient approach with accuracy between that of bilinear and bicubic interpolation.
Abstract: Rotation-based approaches to iterative reconstruction algorithms offer unique trade-offs between accuracy, computational complexity, and smoothing. We analyzed six different rotation methods for generating projections and reconstructions of simulated and clinical data. Nearest neighbor, upsampled nearest neighbor, bilinear interpolation, and bicubic interpolation were all used to determine the values at rotated grid points. A decomposition of the rotation transformation matrix into three components was also investigated. Linear and cubic interpolation were used in the three pass method. For all of the methods, mean normalized square errors were computed as was a measure of smoothness. Our results demonstrate the trade-offs associated with the different methods and suggest that the method of shears with cubic interpolation offers a computationally efficient approach with accuracy between that of bilinear and bicubic interpolation.
67 citations
TL;DR: The problem of interpolation subject to simple linear constraints is addressed, looking at the problem of constructing a piecewise bicubic function u(x, y) from data on a rectangular mesh, such that u( x, y] is nonnegative (positive).
63 citations
TL;DR: A new method, based on mathematical morphology, is presented here to implement the interpolation by means of a combined operation of weighted dilation and erosion, which successfully resolves the interpolations problem when there is no overlapping area between the two objects.
51 citations
09 May 1995
TL;DR: A nonlinear method for image interpolation is presented based on spatial domain directional interpolation, which is extended to the whole image by interpolating in multiple directions using directional weights.
Abstract: A nonlinear method for image interpolation is presented based on spatial domain directional interpolation. Existing directional interpolation algorithms, which only consider edge regions, are extended to the whole image by interpolating in multiple directions. The interpolated values along various directions are combined using directional weights, which depend on the variation in that direction. The interpolation value for each direction is assigned based on the magnitude of its directional derivative.
41 citations
21 Nov 1995
TL;DR: A geometric interpolation technique to overcome limitations of subpixel interpolation by not blurring across discontinuities, and explicitly and accurately placing them, which is shown to be better than spline-like interpolations in smooth regions, and far better in discontinuous ones.
Abstract: Subpixel interpolation methods often use local surface fits or structural models in a local neighborhood to obtain the interpolated curve. Whereas their performance is good in smooth regions of the curve, it is typically poor in the vicinity of singularities. Similarly, when geometric estimates are regularized, discontinuities are often blurred over, leading to poor estimates in their vicinity. We propose a geometric interpolation technique to overcome these limitations by: 1) not blurring across discontinuities, and 2) explicitly and accurately placing them. The essential idea is to prevent the propagation of information across singularities by explicitly placing a "shock"; information is only allowed to propagate from the smoother side. The placement of shocks is guided by geometric continuity constraints, resulting in subpixel interpolation with accurate geometric estimates. The interpolations are shown to be better than spline-like interpolations in smooth regions, and far better in discontinuous ones. We demonstrate the usefulness of the technique in capturing not only smooth evolving curves, but also discontinuous ones, even when multiple or entire curves are present in the same pixel.
37 citations
TL;DR: In this paper, an algorithm for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible is described. But this method requires the grid points of the partition to be contained in these sets, and no large linear systems have to be solved.
Abstract: We describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible. The splines are defined on rectangular partitions adding one or two diagonals to each rectangle. The interpolation sets are selected in such a way that the grid points of the partition are contained in these sets, and no large linear systems have to be solved. Our method is to generate a net of line segments and to choose point sets in these segments which satisfy the Schoenberg-Whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied. In order to include the grid points in the interpolation sets, we give a sufficient Schoenberg-Whitney type condition for interpolation by bivariate splines supported in certain cones. This approach is completely different from the known interpolation methods for bivariate splines of degree at most three. Our method is illustrated by some numerical examples.
20 citations
Patent•
13 Apr 1995TL;DR: In this article, the level difference between two pixels on an interpolation line is calculated for all interpolation lines, likelihoods of correlation are calculated from the level differences obtained using a membership function.
Abstract: Interpolation method and apparatus for interpolating pixels using levels of pixels surrounding a pixel to be interpolated is disclosed wherein a level difference between two pixels on an interpolation line is calculated for all interpolation lines, likelihoods of correlation are calculated from the level differences obtained using a membership function, an interpolation direction is determined based on the likelihoods of correlation and directions of the interpolation lines and an interpolation value for the pixel to be interpolated is calculated based on levels of two pixels on the interpolation line having the interpolation direction determined.
19 citations
Patent•
15 Feb 1995TL;DR: An interpolation filter that enables interpolation without changing the values of the interpolated points coinciding with the sampled data points, the signal content of the original image, or the variance of the image as discussed by the authors.
Abstract: An interpolation filter that enables interpolation without changing the values of the interpolated points coinciding with the sampled data points, the signal content of the original image, or the variance of the original image.
TL;DR: Using this method, solid objects bounded with bicubic Bezier patches can be G 1 smoothed with bICubic patches and can be used for constructing smooth closed surfaces whose modeling is important for CAD/CG.
TL;DR: A Lagrangian parameter approach to problems of best constrained approximation in Hilbert space is reviewed, which is applied to the problem of interpolation of data in a plane by a cubic spline function which is subject to obstacles.
Abstract: We review a Lagrangian parameter approach to problems of best constrained approximation in Hilbert space. The variable is confined to a closed convex subset of the Hilbert space and is also assumed to satisfy linear equalities. The technique is applied to the problem of interpolation of data in a plane by a cubic spline function which is subject to obstacles. The obstacles may be piecewise cubic polynomials over the original knot set. A characterization result is obtained which is used to develop a Newton-type algorithm for the numerical solution.
23 Oct 1995
TL;DR: A new morphological interpolation technique is presented that makes intensive use of geodesic dilation, a morphological operator that may be implemented by means of FIFO queues, which results in a very efficient process compared to those that perform interpolation by linear filtering on the initial image.
Abstract: In this paper a new morphological interpolation technique is presented. It is applied to the coding of the smooth (primary) component in a sketch-based image compression approach for very low bit-rates. The interpolation technique is intended to perform two dimensional interpolation from any set of initial pixels and, in particular, from sketch data. It makes intensive use of geodesic dilation, a morphological operator that may be implemented by means of FIFO queues. This results in a very efficient process compared to those that perform interpolation by linear filtering on the initial image. For the application of this method to interpolative image coding, the sketch data is extracted as a set of maximum curvature lines by means of the watershed algorithm. From such information, the interpolation technique obtains a fair reconstruction of both the smooth texture component and the main transitions of the image signal at low bit-rate cost.
01 Jan 1995
TL;DR: Most of the published spline algorithms are for polynomial splines and the vast preponderance are for cubic splines as mentioned in this paper, and there is an even smaller literature on splines with more or less arbitrary nodal functions.
Abstract: Spline functions are well known and are widely used for practical approximation of functions or more commonly for fitting smooth curves through preassigned points. Spline techniques have the advantage over most approximation and interpolation techniques in that they are computationally feasible. Most of the published spline algorithms are for polynomial splines and the vast preponderance are for cubic splines. There is a small but excellent literature on the so called exponential splines and there is an even smaller literature on splines with more or less arbitrary nodal functions,[9, 3].
01 Jan 1995
TL;DR: In a recent lecture at the NATO Advanced Studies Institute as discussed by the authors, the authors tried to acquaint the audience with the very recent progress that has been made in the approximation of functions defined on spheres.
Abstract: The purpose of this lecture at the NATO Advanced Studies Institute was purely expository. I tried to acquaint the audience with the very recent progress that has been made in the approximation of functions defined on spheres. For practical purposes, the sphere of greatest interest would appear to be S2, since it serves as a (flawed) model of the Earth’s surface. There is every reason to believe that the new approximation methods now being developed will be found useful in geological exploration and meteorological modelling, to name just two areas of application.
TL;DR: The authors propose a novel two-dimensional cubic-spline interpolation scheme that guarantees the absence of spurious ripples, and simultaneously provides interpolation of both the measured function and its first derivative.
Abstract: The use of device models directly based on measured information is well established in nonlinear microwave circuit analysis, and may provide significant advantages over analytic models relying on parameter extraction techniques. These ‘database’ models always include some sort of interpolation mechanism for overcoming the inherently discrete nature of measured data. This is a delicate aspect, since interpolation may produce unwanted ripples or other ill-conditioning that may considerably degrade the model performance. The authors propose a novel two-dimensional cubic-spline interpolation scheme that guarantees the absence of spurious ripples, and simultaneously provides interpolation of both the measured function and its first derivative.
01 Jan 1995
TL;DR: The global constraint of the control parameters associated with each control vertex is relaxed and the new local shape-control parameters with their larger range of usability further enhance the power of this free-form surface design scheme.
Abstract: In free-form modeling a closed smooth piecewise surface is highly desirable when the smoothness across the boundaries of patches can be represented within the formulation. Closed, smooth, piecewise bicubic surfaces, defined on simple quadrilateral mesh. (SQM), may be defined as SQM surfaces. We have extended previous work on SQM surfaces and described the surface representation in rational form. The global constraint of the control parameters associated with each control vertex is relaxed. The new local shape-control parameters with their larger range of usability further enhance the power of this free-form surface design scheme. We have also provided more B-spline functions. A complete set of B-spline functions for various topologies of the SQM is now available. Examples demonstrate that editing of shapes for reasonably complex objects can be carried out on an SGI Personal Iris machine at an interactive rate.
18 Jun 1995
TL;DR: In this article, a plate, under the action of an appropriately chosen set of localized forces, will deform smoothly and can be made to pass through any reasonable set of coordinate points, hence, this deformed plate can be used as an interpolation tool.
Abstract: Due to the advantages of shaping reflector antennas for high gain or contour beam applications, reflector surfaces have departed from being described by simple equations, and often require the assistance of some interpolation scheme for their characterization Various interpolation tools are available for describing an arbitrary surface, but most can only handle a grid of equally spaced points (ie, uniform grid) To reduce errors associated with under sampling a highly shaped reflector region, while maintaining a reasonable computational efficiency, it is desirable to have an interpolation scheme capable of handling a non-uniform grid, so that a more dense sampling can be used only where needed We investigate the capabilities and limitations of the surface spline The idea behind their method is that a plate, under the action of an appropriately chosen set of localized forces, will deform smoothly and can be made to pass through any reasonable set of coordinate points Hence, this deformed plate can be used as an interpolation tool A related interpolation method, the pseudo-spline interpolation, is also considered
01 Mar 1995
TL;DR: In this article, an elliptic grid generation method is presented which generates excellent boundary conforming grids in domains in 2D physical space based on the composition of an algebraic and elliptic transformation.
Abstract: An elliptic grid generation method is presented which generates excellent boundary conforming grids in domains in 2D physical space. The method is based on the composition of an algebraic and elliptic transformation. The composite mapping obeys the familiar Poisson grid generation system with control functions specified by the algebraic transformation. New expressions are given for the control functions. Grid orthogonality at the boundary is achieved by modification of the algebraic transformation. It is shown that grid generation on a minimal surface in 3D physical space is in fact equivalent to grid generation in a domain in 2D physical space. A second elliptic grid generation method is presented which generates excellent boundary conforming grids on smooth surfaces. It is assumed that the surfaces are parametrized and that the grid only depends on the shape of the surface and is independent of the parametrization. Concerning surface modeling, it is shown that bicubic Hermite interpolation is an excellent method to generate a smooth surface which is passing through a given discrete set of control points. In contrast to bicubic spline interpolation, there is extra freedom to model the tangent and twist vectors such that spurious oscillations are prevented.
TL;DR: A parallel implementation of Chebyshev method is presented for the B-spline surface interpolation problem and it is shown that this algorithm can successfully be used in finite precision floating-point arithmetic.
01 Jan 1995
TL;DR: The abstract theory of linear parabolic problems, developed in the first part of this treatise, rests on two cornerstones: on the theory of analytic semigroups and on interpolation theory.
Abstract: The abstract theory of linear parabolic problems, developed in the first part of this treatise, rests on two cornerstones: on the theory of analytic semigroups and on interpolation theory. In the first section of this chapter we discuss in some detail the classes of generators of analytic semigroups that are used throughout. In the second section we review the fundamentals of interpolation theory. We also introduce the class of ‘admissible interpolation functors’ that, on the one hand, is flexible enough to build an easy and general abstract theory of evolution equations on it, and, on the other hand, is general enough to cover all applications occurring in practice.
01 Jan 1995
TL;DR: In this paper, the authors discuss multivariate interpolation of a function defined on certain triangular lattices of points and extend these ideas to lattices on a general simplex, and introduce some notation.
Abstract: This paper makes no attempt to survey the vast literature on multivariate interpolation. (See, for example, the fine survey by Lorentz [7].) Here we shall discuss interpolation of a function defined on certain triangular lattices of points and extend these ideas to lattices on a general simplex. We begin by introducing some notation.
Journal Article•
TL;DR: A new method-the method of B-spline generatix is proposed, by which interpolation surfaces of C2 continuity can be designed, and, the surface design in mLinufacturing such as die-machining t ship-building, etc.
Abstract: Maily popular design methods of free-form surface in CAD/CAM, such as Bsplins surface, Bener surface, Ball stlrface, etc, only can be used to clesigll fitting, not interpolation surface, and are not fit for designing interpolation surfaces in engneering. In this work, a new method-the method of B-spline generatix is proposed, by which interpolation surfaces of C2 continuity can be designed. And, the surface design in mLinufacturing such sa die-machining t ship-building, etc.
TL;DR: In this article, a method for solving the problem of surface/surface intersection is suggested, which uses an adaptive subdivision and a facetted model based on triangles, and a bounding box and a different dividing method are newly developed.
Abstract: A method for solving the problem of surface/surface intersection in suggested. This method uses an adaptive subdivision and a facetted model based on triangles. A bounding box and a different dividing method are newly developed. Instead of the calculation of the characteristic points, a new linking technique is developed. This method solves the intersection problem between bicubic or higher grade surfaces. The desired caculation-precision is specified by the input parameter.
TL;DR: In this article, the applicability of cubic pseudosplines is explored via numerical comparison with previously obtained results from the implementation of the quintic ones for ray-optical field evaluations, when the antenna secondary field is calculated by integration of physical-optics-induced currents.
Abstract: A recent study demonstrated the superior stability of quintic pseudosplines, when compared with usual oscillatory series expansions, for global interpolation of numerically defined reflector surfaces presenting critical behavior of their principal curvatures. Although interpolating functions with continuous derivatives up to second order are necessary for ray-optical field evaluations, C-class surface descriptions are sufficient when the antenna secondary field is calculated by integration of physical-optics-induced currents. In this connection, the applicability of cubic pseudosplines is explored via numerical comparison with previously obtained results from the implementation of the quintic ones. © 1995 John Wiley & Sons. Inc.