Showing papers on "Bicubic interpolation published in 1989"

Two-Dimensional Semi-Lagrangian Transport with Shape-Preserving Interpolation

Abstract: The more attractive one dimensional, shape-preserving interpolation schemes as determined from a companion study are applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry. Hermite cubic and a rational cubic are considered for the interpolation form. Both require estimates of derivatives at data points. A cubic derivative form and the derivative estimates of Hyman and Akima are considered. The derivative estimates are also modified to ensure that the interpolant is monotonic. The modification depends on the interpolation form. Three methods are used to apply the interpolators to two-dimensional semi-Lagrangian advection. The first consists of fractional time steps or time splitting. The method has noticeable displacement errors and larger diffusion than the other methods. The second consists of two-dimensional interpolants with formal definitions of a two-dimensional monotonic surface and application of a two-dimensional monotonicity constraint. This approach is exam...

Method and apparatus for non-affine image warping

Abstract: A method and apparatus for two-pass image transformation, providing a general solution to execute arbitrary warping of an image. A bicubic mesh is created, by splines or other suitable means, and is used to create displacement tables for X and Y displacement. Alternatively, the displacement tables can be generated directly. The displacement tables represent the movement of each pixel from an original location in the source image to a new location in the destination image. One of the displacement maps is applied to the source image to create an intermediate image and to the other displacement map to create a resampled displacement map. The resampled map is then applied to the intermediate image to create the destination image. By resampling, compensation for altered location points is done automatically. In this manner, no inversion of the underlying equations and functions is required.

Scattered data interpolation in three or more variables

Abstract: This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.

An algorithm for monotone piecewise bicubic interpolation

Abstract: This paper describes an algorithm for monotone interpolation to monotone data on a rectangular mesh by piecewise bicubic functions. In [SIAM J. Numer. Anal. 22 (1985), pp. 386–400] the authors developed conditions on the Hermite derivatives that are sufficient for such a function to be monotonic. The present paper rewrites some of these conditions and presents a much simpler five-step algorithm for satisfying them that produces a visually pleasing monotone interpolant. The result of the algorithm does not depend on the order of the independent variables nor on whether the inequalities are swept left-to-right or right-to-left.

A generalized Ball curve and its recursive algorithm

Abstract: The use of Bernstein polynomials as the basis functions in Bezier's UNISURF is well known. These basis functions possess the shape-preserving properties that are required in designing free form curves and surfaces. These curves and surfaces are computed efficiently using the de Casteljau Algorithm. Ball uses a similar approach in defining cubic curves and bicubic surfaces in his CONSURF program. The basis functions employed are slightly different from the Bernstein polynomials. However, they also possess the same shape-preserving properties. A generalization of these cubic basis functions of Ball, such that higher order curves and surfaces can be defined and a recursive algorithm for generating the generalized curve are presented. The algorithm could be extended to generate a generalized surface in much the same way that the de Casteljau Algorithm could be used to generate a Bezier surface.

Rendering cubic curves and surfaces with integer adaptive forward differencing

Abstract: For most compute environments, adaptive forward differencing is much more efficient when performed using integer arithmetic than when using floating point. Previously low precision integer methods suffered from serious precision problems due to the error accumulation inherent to forward differencing techniques. This paper proposes several different techniques for implementing adaptive forward differencing using integer arithmetic, and provides an error analysis of forward differencing which is useful as a guide for integer AFD implementation. The proposed technique using 32 bit integer values is capable of rendering curves having more than 4K forward steps with an accumulated error of less than one pixel and no overflow problems. A hybrid algorithm employing integer AFD is proposed for rendering antialiased, texture-mapped bicubic surfaces.

Computer Transformation of Digital Images and Patterns

Abstract: This book is concerned with linear and nonlinear transformations of digitized images and patterns. Transformation models include linear, quadratic, cubic, bilinear, biquadratic, bicubic, Coons model and other nonlinear forms such as harmonic, projective, and perspective transformations. Discrete techniques have been developed to realize both forward and inverse transformations. The latter can be applied to normalize distorted images and to enhance the pattern recognition process. Efficient algorithms such as the splitting-shooting methods and splitting-integrating methods have been developed and analysed in this book for the first time. Graphical examples are given and compared with existing algorithms. This book is of interest to researchers in the areas of pattern recognition, character recognition, image processing, computer vision, computer graphics and other related fields.

Discontinuity preserving surface reconstruction

Abstract: A two-stage algorithm for visual surface reconstruction while preserving discontinuities is presented. The primary contribution is the second stage, a weighted bicubic spline as a surface descriptor. The weighted bicubic spline has a factor in the regularizing term that adapts the behavior of the spline across the discontinuity. The weighted bicubic interpolating spline is able to interpolate data with step discontinuities with no discernible distortion (such as Gibbs phenomena) in the interpolated surface. Since regularization is related to spline approximation, extensions to this work should suggest ideas for solving ill-posed problems in vision. Experiments with weighted bicubic spline interpolation are presented. >

A general algorithms for 3-D shape interpolation in a facet-based representation

Abstract: Concerned with interpolation between two objects defined using a faceted representation. The interpolation function is not examined, but the paper emphasizes the problem of correspondence between the two given key drawings. The problem is much more complex than interpolation between line drawings, because the two key drawings generally have a different total number of vertices, a different total number of facets and corresponding facets have a different number of vertices. A general algorithm is proposed based on a criterion of minimal dynamic displacement. There are numerous applications of the method: simulation of biological evolution, animation, portrait-robots

Display system interpolators

Abstract: An image interpolator implements an interpolation function providing real time, continuous zoom capability to an image display system. Output image pixels are obtained by interpolating the values of the colour or intensity of the 2x2 matrix of pixels surrounding the point on the input image. The disclosed arrangement employs a bi-linear interpolation algorithm implemented in the form of cascaded one-dimensional interpolation circuits. Magnification control is established so that a unit increment of the zoom controller, such as a cursor on a tablet, results in a constant increase in the degree of magnification. The coefficients required for the interpolation are generated in real time avoiding the need for time consuming table look-ups.

Two-parameter cubic convolution for image reconstruction

Abstract: This paper presents an analysis of a recently-proposed two-parameter piecewise-cubic convolution algorithm for image reconstruction. The traditional cubic convolution algorithm is a one-parameter, interpolating function. With the second parameter, the algorithm can also be approximating. The analysis leads to a Taylor series expansion for the average square error due to sampling and reconstruction as a function of the two parameters. This analysis indicates that the additional parameter does not improve the reconstruction fidelity - the optimal two-parameter convolution kernel is identical to the optimal kernel for the traditional one-parameter algorithm. Two methods for constructing the optimal cubic kernel are also reviewed.

Resampling Algorithms for Image Resizing and Rotation

Abstract: Resampling an image will be described as a low-pass filtering operation followed by sampling to a new coordinate system. To determine the interpolation function that gives the most visually appealing images, a comparison of common kernels is made. Linear, cubic, and windowed sinc functions are compared in terms of frequency response and with prints of images resized using separable extensions of these functions. While the windowed sinc gives the best approximation to an ideal low-pass filter, using this kernel results in objectionable ringing and jaggedness around edges in the image. Cubic interpolation is shown to provide the best compromise between image sharpness and these edge artifacts. For image rotation, and resizing by an arbitrary factor, the filter coefficients (samples of the interpolation function) need to be computed for each pixel of the new image. Alternatively, significant computation can be saved by dividing the distance between pixels of the original image into a number of intervals and precomputing a set of coefficients for each interval. Each new pixel is then computed by finding the interval in which it falls and using the corresponding set of coefficients. An analysis of the errors introduced with this resampling method is presented. This analysis shows the number of intervals required to produce high-quality resampled images.

The structure of inner matrices that satisfy multiple directional interpolation requirements

Abstract: The structure of an inner matrix that satisfies directional interpolation requirements for nondistinct values at the right half-plane is derived explicitly, in closed form, in terms of the interpolation directions. The simple expression that is obtained for the required inner transfer function matrix has the same structure as the one that has already been found for distinct interpolation requirements. As a result, this expression provides a comprehensive treatment, in the frequency domain, of interpolation problems that arise in many areas of linear system optimization. >

A direct procedure for interpolation on a structured curvilinear two-dimensional grid

Abstract: A direct procedure is presented for locally bicubic interpolation on a structured, curvilinear, two-dimensional grid. The physical (Cartesian) space is transformed to a computational space in which the grid is uniform and rectangular by a generalized curvilinear coordinate transformation. Required partial derivative information is obtained by finite differences in the computational space. The partial derivatives in physical space are determined by repeated application of the chain rule for partial differentiation. A bilinear transformation is used to analytically transform the individual quadrilateral cells in physical space into unit squares. The interpolation is performed within each unit square using a piecewise bicubic spline.

A frequency-domain characterization of interpolation from nonuniformly spaced data

Abstract: A linear time-varying (LTV) framework for modeling interpolation from nonuniformly spaced samples is described. This LTV model is used to obtain a frequency-domain description of the interpolation process. The frequency-domain characteristics of a number of commonly used interpolators are presented. In addition, this frequency-domain description permits the design of one-dimensional interpolators through the use of two-dimensional digital filter design techniques. A two-dimensional frequency sampling design approach is explored in an attempt to better approximate the ideal interpolator. >

Measuring the surface area of the face

Abstract: A noncontact, biologically safe imaging system for measuring surface area and volume is described. It uses a technique known as structured light, a method in which the surfaces of 3-D objects are measured by projecting a regular geometric pattern onto its surface and viewing the illuminated surface using one or more cameras at different angles. The 3-D locations of points on the surface of the skin are reconstructed from the structured light, and the region of interest is segmented. After the reconstructed 3-D locations and the segmented image are combined, cubic spline interpolation is used to represent the space curves and bicubic surface patches to represent the surface area. Gaussian quadrature double integration is used to calculate the surface area. The steps necessary to do the segmentation, curve and patch representation, and surface area computation are outlined. It is found that the surface area estimates are as accurate as estimates based on visual inspection. >

Computer-controlled 5-axis machine tool based on trochoidal interpolation.

Abstract: In this paper, a new interpolation formula is proposed to control the cutting motion of 5-axis machine tools in straight and circular motions. This interpolation formula is named of Trochoidal Interpolation. It is a combination of straight and circular interpolations with variable blending function, which are carried out by utilizing DDA algorithm. The proposed interpolation formula makes it possible to shorten a required NC command tape and to simplify the motion control algorithm when the 5-axis NC machine tools was applied to complex shape machinings.

A tension-compatible patch for shape-preserving surface interpolation

Abstract: Several issues and techniques related to the problem of constructing shape-preserving interpolants for discrete gridded data are discussed. For choosing slopes and twists it is shown that the method of parabolic blending is intuitively appealing and easy to implement, and that it produces results consistent with standard notions of implied shape. It is shown by examples that standard bicubic patches fail to respect the shape implied by this augmented data, and two alternative patching schemes based on adaptive tension techniques, which remedy the problem, are provided. The constructions are general enough to encompass many different tensioning methods, including the classical tension splines of D.G. Schweikert (1966) and rational splines. Both schemes used univariate techniques along grid lines to produce a network of shape-preserving curves. One scheme then makes further use of tensioned interpolation to provide the extra information that allows use of a general-purpose Coons patch. The other scheme involves the use of a novel patch 'customized' for blending the independently tensioned boundary curves. The latter method performs comparably with the former, at considerably reduced cost. >

Digital interpolation beamforming using recursive filters

Abstract: The problem of steering delay quantization in beamforming can be circumvented by using digital interpolation and digital interpolation can be implemented using recursive filters. The recursive interpolation scheme offers considerable computational savings over conventional finite impulse response (FIR) digital interpolation methods.

A VLSI architecture for bicubic surface patch image generation

Abstract: A VLSI architecture is presented for generating ray-tracing images of bicubic surface in Bezier form by using the subdivision algorithm. It uses a set of tree transverse operators to find the nearest intersection between a surface patch and a ray by traversing the entire subdivision tree in preorder. This scheme retains the advantage of subdivision and substantially reduces the circuit area by eliminating the need for a stack to store control points of intermediate bicubic patches. It also eliminates the multiplications needed in the subdivision iteration. The time to traverse additional nodes is offset by the removal of the stack memory access time. The performance of the architecture is analyzed and compared with the alternative approaches. >

Fast algorithms for Nevanlinna-Pick interpolation and H infinity optimization

Abstract: Fast algorithms are derived for the scalar and tangential directional Nevanlinna-Pick interpolation problems These fast algorithms require fewer multiplications than standard algorithms for these problems The scalar interpolation problem, which arises in SISO (single input single output) H infinity optimization, is solved using a three-term recurrence similar to that of the so-called split algorithms for linear prediction The tangential directional interpolation problem, which arises in MIMO (multiple input multiple output) H infinity optimization, is solved using an algorithm similar to a series of Householder transformations Applications to H infinity optimization are noted >