Abstract: An algorithm and the corresponding computer program for solution of the scattered data interpolation problem is described. Given points ( x k , y k , f k ), k = 1,…, N a locally defined function F ( x , y ) which has the property F ( x k , y k ) = f k , k = 1,…, N is constructed. The algorithm is based on a weighted sum of locally defined thin plate splines, and yields an interpolation function which is differentiable. The program is available from the author.

Abstract: We present an algorithm for constructing a basis of the ideal of all polynomials, which vanish at a preassigned set of points {y1,...,ym} ⊂ Kn, K a field. The algorithm yields also Newton-type polynomials for pointwise interpolation. These polynomials admit an immediate construction of interpolating polynomials and allow to shorten the algorithm, if it is applied to an enlarged set {y1,...,ym1} ⊂ Kn, m1>m.

Abstract: Computational theories of structure-from-motion and stereo vision only specify the computation of three-dimensional surface information at special points in the image. Yet the visual perception is clearly of complete surfaces. To account for this a computational theory of the interpolation of surfaces from visual information is presented. The problem is constrained by the fact that the surface must agree with the information from stereo or motion correspondence, and not vary radically between these points. Using the image irradiance equation, an explicit form of this surface consistency constraint can be derived. To determine which of two possible surfaces is more consistent with the surface consistency constraint, one must be able to compare the two surfaces. To do this, a functional from the space of possible functions to the real numbers is required. In this way, the surface most consistent with the visual information will be that which minimizes the functional. To ensure that the functional has a unique minimal surface, conditions on the form of the functional are derived. In particular, if the functional is a complete semi-norm that satisfies the parallelogram law, or the space of functions is a semi-Hilbert space and the functional is a semi-inner product, then there is a unique (to within possibly an element of the null space of the functional) surface that is most consistent with the visual information. It can be shown, based on the above conditions plus a condition of rotational symmetry, that there is a vector space of possible functionals that measure surface consistency, this vector space being spanned by the functional of quadratic variation and the functional of square Laplacian. Arguments based on the null spaces of the respective functionals are used to justify the choice of the quadratic variation as the optimal functional. Possible refinements to the theory, concerning the role of discontinuities in depth and the effects of applying the interpolation process to scenes containing more than one object, are discussed.

Abstract: We establish equations of non linear filtering, prediction (extrapolation) and smoothing (interpolation) in the case where the signal is a non degenerate diffusion process, and the observation is a noisy functional of the signal. We consider both the case of observation noise correlated with the signal, and the opposite case where we establish “robust” form of the equations. We study finally the case of unbounded coefficients, and the case where there is a feedback from the observation to the signal.

Abstract: We have studied the ability of observers to discriminate between suprathreshold vertical sinusoidal spatial-frequency gratings on the basis of spatial frequency. The results show that spatial-frequency discrimination is not a smooth function of spatial frequency but instead appears regularly segmented. Similar results were also obtained in an experiment in which observers discriminated between pairs of narrow vertical lines on the basis of their separation. Angular resolutions achieved for both discrimination tasks were less than the spacing between photoreceptors, requiring some type of neural interpolation. The similarity between the two sets of data indicates that discrimination between spatial-frequency gratings is probably based on the separation between two features exactly one cycle apart. We suggest that the segmentation reflects the existence of neural-image representations with discrete levels of spatial accuracy.

Abstract: An algorithm for the numerical factorization of very high degree but well-conditioned polynomials is developed. This is used to factor the z-transform of finite-length signals, and the zeros are used to calculate the unwrapped phase. The method has been tested on signals up to 512 points in length. A complete Fortran 77 program is given for the case of a real-valued signal. Two related analytical issues are treated. First, the interpretation of phase unwrapping as an interpolation problem is discussed. Second, an explanation is given for the observed numerical difficulties in the method of phase unwrapping using adaptive integration of the phase derivative. The trouble is due to the clustering of the zeros of high degree polynomials near the unit circle.

Abstract: Computational modules early in the human vision system typically generate sparse information about the shapes of visible surfaces in the scene. Moreover, visual processes such as stereopsis can provide such information at a number of levels spanning a range of resolutions. In this paper, we extend this multi-level structure to encompass the subsequent task of reconstructing full surface descriptions from the sparse information. The mathematical development proceeds in three steps. First, the surface most consistent with the sparse constraints is characterized as the solution to an equilibrium state of a thin flexible plate. Second, local, finite element representations of surfaces are introduced and, by applying the finite element method, the continuous variational principle is transformed into a discrete problem in the form of a large system of linear algebraic equations whose solution is computable by local-support, cooperative mechanisms. Third, to exploit the information available at each level of resolution, a hierarchy of discrete problems is formulated and a highly efficient multi-level algorithm, involving both intra-level relaxation processes and bi-directional inter-level algorithm, involving both intra- level relaxation processes and bidirectional inter-level local interpolation processes is applied to their simultaneous solution.. Examples of the generation of hierarchies of surface representations from stereo constraints are given. Finally, the basic surface approximation problem is revisited in a broader mathematical context whose implications are of relevance to vision.

Abstract: In the present paper we study a general form of Peetre's J- and K-methods of interpolation. Special emphasis is given to the equivalence theorem for J- and K-spaces and to reiteration theorems.

Abstract: Sufficient conditions in terms of interpolation variances are given for a Gaussian process to have a jointly continuous local time. In the stationary case these conditions can be verified in terms of the spectral density and are seen to be within logarithmic factors of the best possible conditions. A bound for the modulus of continuity in the space variable is also obtained.

Abstract: A linear interpolating method and apparatus for signals in a memory wherein first signals are converted into second signals by addressing in at least four-dimensional fashion, for use in a color picture processing machine such as a color scanner, a color facsimile, a color T.V. monitor, and other signal processing machine converting coordinates of signals with at least four variables, are disclosed. The second signals corresponding to certain stepped values of the first signals are read out of the memory, and then the values of the read signals are interpolated at an interpolation point positioned therebetween. At least four-dimensional interpolation unit space is divided into at least four-dimensional dissection spaces, and then it is discriminated which of the dissection spaces includes the interpolation point. Then, the interpolated value at the interpolation point is obtained as a weighted sum of the values at the vertices of the discriminated dissection space.

Abstract: This paper is a sequel to an earlier work in which we applied classical Novnnlinna-Pick interpolation to the stabilization of a single input-output plant, with uncertainly in the gain factor by means of an asymptotically stable control and feedback sensor. In this present work, we show that for the proper design we need a modification of the classical interpolation procedure, a point not clearly brought out in our previous paper.

Abstract: A three-dimensional table look-up MOSFET modeling technique is described. The table, which is able to deal with future submicron devices, is constructed with a few thousand work memory capacity requirement by suppressing data redundancy. Sufficiently high accuracy, with less than point several percent error, is achieved by using a special interpolation, which is called curve shape fitting technique. Computational time to perform the interpolation from the table is much less than that for the analytical model.

Abstract: A system for describing three-dimensional surfaces in a form suitable for finite element analysis is described. The system makes extensive use of real-time interactive computer graphics techniques for both input and display. Discrete transfinite mappings are used as the mathematical basis for the surface representation. The mathematical basis and the reasons for choosing this form of representation are discussed. Explicit forms of the mappings based on Lagrange polynomial interpolation functions are presented. Finally, the interactive graphics procedures for defining finite element meshes are described.

Abstract: In this paper an exact interpolation formula forms the basis for reconstructing computerized tomographic (CT) imagery by direct Fourier methods. Practical variations of exact interpolation are compared with other interpolation methods (i.e., nearest neighbor, etc.) and are shown to yield superior imagery. Images produced by the direct Fourier approach using near-exact interpolation are shown to be equal in quality with those produced by filtered convolution backprojection (FCBP). Moreover, the direct Fourier approach computes an image in O(N2 log N) time versus O(N3) for the FCBP method.

Abstract: Recently, interest has revived in the classical problems of vernier and stereoscopic acuity. The precision with which observers can align two bars in a vernier task is as high as 2 sec arc, which is much finer than the grain of the retinal receptor mosaic, where the cones are separated by 20–30 sec arc even at their most densely packed. Thus, it seems that the visual system may be able to interpolate between the discrete samples provided by the retinal mosaic; recently, several possible mechanisms have been proposed1,2. Mathematically there is no mystery about interpolation. The optical system of the eye acts as a low-pass filter removing all frequencies above ∼60 cycles deg−1, so it follows from the sampling theorem that the retinal image can be fully represented by sampling at a frequency of 120 cycles deg−1, which is approximately that of the receptor mosaic. In principle, therefore, discrete sampling by the retinal mosaic does not remove information from the image. Indeed, the low-pass filtering action of the eye implies that acuity would not be lost, even if the signal itself were divided into discrete samples, before observation. We report here that human observers can locate the spatial position of a periodic visual pattern with a precision as high as 5–10 sec arc, even though the pattern is coarsely sampled at an interval over 10 times that amount. This suggests that the human visual system can construct a surprisingly accurate representation of a pattern from discrete samples, despite the samples being sufficiently widely spaced to be visually resolved.

Abstract: Apparatus for interpolating data along radial lines so that it can be displayed at display points arranged in orthogonal rows and columns by recursively adding stored values to derive signals indicative of the radial position of each display point along the radial data lines and its angular position between the radial data lines. Interpolation is done along each radial line to derive first and second intermediate interpolated values, and these values are interpolated so as to derive the final data value for the display point at its angular position. Alternatively, the intermediate interpolated values can be attained by angular interpolation and the final data value by radial interpolation.

Abstract: A system for altering the size of an image by a desired magnification after optical scanning thereof provides a plurality of output scan values differing in number from the number of sampled scan signals produced during the optical scanning. The output scan values are representative of the density of the image at pixel locations spaced in rows and columns across the image, with the spacing of the pixel locations being dependent upon a selected magnification value. The system includes scanning means for providing a series of sequences of sampled scan signals, and converter means for converting the sequences of sampled scan signals to corresponding sequences of digital scan values. A control circuit is responsive to a selected magnification value for providing a plurality of interpolation control values in synchronism with the successive sequences of digital scan values. The interpolation control values indicate the spatial relationship of output scan values with respect to pairs of sequences of digital scan values. An interpolation circuit is responsive to the pairs of sequences of digital scan values and to the interpolation control values and interpolates between pairs of sequences of the digital scan values to provide the output scan values.

Abstract: The combined interpolation scheme is used to study energy bands in nickel. Parameters are determined by results of photoemission experiments. The Fermi surface is computed and compared with experiment. Explicit formulas are given for some bands along symmetry directions.

Abstract: In a control device for an industrial articulated robot, in the linear interpolation between two specified points, the distance between the two points is divided into segments, and the interpolation increments of the articulation drive axes for each interpolation internal, which correspond to the segments, are calculated for interpolation so that the interpolation increments are distributed uniformly with time.