A. Schaum, "Theory and design of local interpolators", CVGIP: Graphical Models and Image Processing, Vol. 55, pp. 464-481, November 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....referred to as BC splines [196] was later studied by several authors [195] 196] 254] in the fields of visualization and computer graphics. An extension of the results of Park and Schowengerdt [188] concerning the previously discussed frequency domain error analysis was presented by Schaum [255]. Instead of the L2 norm, 25) he studied the performance metric I.fi .re (41) which summarizes the total interpolation error at a given shift s with respect to the original sampling grid. He found that in the case of oversampling, this error too is given by (26) where 2 and E may now ....

....medical image data have recently been carried out. Lehmann et al. 354] e.g. compared a total of 31 kernels, including the nearest neighbor, linear, and several quadratic [258] and cubic convolution kernels [187] 188] 192] 355] as well as the cubic B spline interpolator, various Lagrange [255] and Gaussian based [356] interpolators and truncated and Blackman Harris [357] windowed sinc kernels of different spatial support. From the results of computational cost analyses and forward backward transformation experiments carried out on CCD photographs, MRI sections, and X ray images, it ....

A. Schaum, "Theory and design of local interpolators," Graph. Models Image Process., vol. 55, no. 6, pp. 464'481, 1993.

....of discrete periodic signals is then equivalent to a circular convolution of the periodic sample set with the alternate kernel the Fourier series kernel. This method of interpolation of discrete periodic signals has been studied in [4] The Fourier series kernel has been reported in [5]. It contains several indeterminate points , i.e. points at which the function takes an indeterminate form (like 0 0) for which the interpolation cannot be numerically implemented directly. By applying the operation of Fourier series expansion on our signal, a numerically stable form of this ....

Schaum A. "Theory and Design of Local Interpolators", Comp. Vision Graph. Image Proc.: Models and Image Proc., Vol. 55, No. 6, pp 464-481, Nov. 1993.

....presence of noise. Practitioners have long considered finite support interpolators . They have designed them to be close to the , with the hope that this would bring good approximation characteristics [12] 14] Others observed that what we will call approximation order improves quality [15] [17]. An example is that of Keys cubic kernel which has a free parameter; its optimization turns out to be the one that provides the highest approximation order [18] In order to provide computationally efficient algorithms, these kernels were chosen to be piecewise polynomial and of short support. ....

....to be stored in byte format. Due to the size of the support of , the interpolation conditions add up to constraints if is integer, and to constraints otherwise; these equations match the degrees of freedom of the MOMS. In fact, these interpolators derive from Lagrange s interpolation formula [17], 33] 34] If we denote , then if otherwise, which can be shown to belong to the class of MOMS of order . We can obtain the polynomial defining these I MOMS directly, by expressing the interpolation condition in the Fourier Domain as Since is of order , we have that when . This precisely ....

A. Schaum, "Theory and design of local interpolators," Comput. Vis. Graph. Image Process., vol. 55, pp. 464--481, Nov. 1993.

....the synthesis functions have the same support and the same polynomial degree. 0 0.5 1 1.5 2 2. 5 3 70 60 50 40 30 20 10 0 Circular Frequency w Amplitude of E(w) dB) From worst to best Linear spline Dodgson s quadratic kernel Keys cubic kernel Schaum s cubic kernel [8] Cubic spline Optimal cubic kernel [7] Fig. 2. Fourier approximation kernels of interpolating (dashed line) and non interpolating (plain line) kernels. The results are significantly better (i.e. closer to the abscissa line) with generalized sampling. The gap between the performance of ....

A. Schaum, "Theory and design of local interpolators," Comput. Vis. Graph. Image Process., vol. 55, no. 6, pp. 464--481, November 1993.

....k ) the aspects that have been emphasized are: 1) its maximal degree N ; 2) the width W of its support; 3) its regularity int 2 C R ; and to some extent 4) its order of approximation L. This search for adequate interpolators is still active today; recent contributions include those of Schaum [1], Appledorn [2] German [3] Dodgson [4] or Meijering [5] Unfortunately, it appears that the improvements of each new proposal have been less and less substantial. Recently, we showed that one reason for this saturation of design is that the interpolation constraint is too strong; only by ....

.... than B splines with L = W ; those are called MOMS (Maximal Order Minimal Support) and all are members of the family (x) L 1 X m=0 am d m dx m L 1 (x) where famg is a set of constants [7] Members of this family include the symmetric and interpolating functions discussed by Schaum [1]. Let us apply the design methodology proposed in this paper to rederive the Lagrange like quadratic Schaum interpolator S 2 f2; 3; 1; 3g. We first verify that the existence conditions are satisfied, and determine that there are two degrees of freedom. Since N W , these two degrees of freedom ....

A. Schaum, "Theory and design of local interpolators," CVGIP: Graphical Models and Image Processing, vol. 55, no. 6, pp. 464--481, November 1993.

.... authors examined the more restrictive least squares case and identified a quantity that is the same as the first error term in (13) Considering only the interpolating case , another version of the second part of Theorem 1 (restricted to Shannon bandlimited functions) also appeared in [37] and [38]. As we have shown in [25] Theorem 1 is a quite powerful result that has many interesting implications for approximation theory. We will further investigate this aspect in Section IV. In particular, we will exploit Theorem 1 to derive a variety of very accurate error estimates that can be ....

A. Schaum, "Theory and design of local interpolators," Comput. Vision Graph. Image Process., vol. 55, no. 6, pp. 464--481, Nov. 1993. BLU AND UNSER: QUANTITATIVE FOURIER ANALYSIS OF APPROXIMATION TECHNIQUES: PART I 2795

....interpolating kernel and Keys [1981] used a cubic spline. An analysis and comparison of these image interpolation methods is given by Parker et al. 1983] There have been recent studies into the development of the theory of interpolating kernels [Lucke, 1993; Lanzavecchia and Bellon, 1994, 1995; Schaum, 1993; Appledorn, 1996; Unser et al. 1992; Knoll, 1991] These works focus on the developing of an optimal single interpolation kernel as a substitute for the sinc kernel. The kernels to develop are local. The assumption made in these works is that the signal has been sampled, at least, at the ....

....implemented. In particular, kernels have been composed of trigonometric functions [Lucke, 1993; Lanzavecchia and Bellon, 1994 and 1995] linear sums of Gaussian functions [Appledorn, 1996] polynomial B splines [Unser et al. 1992] and trigonometric as well as low order polynomial functions [Schaum, 1993]. It is interesting to note that some of the developed local kernels approach the sinc function in the limit of an infinite data sequence [Unser et al. 1992; Lucke, 1993] Interpolation filters for restoring highly subsampled images have also been studied [Knoll, 1991] The method is based on the ....

Schaum, A. (1993), "Theory and Design of Local Interpolators," Comp. Vis., Graph., Image Process.:Graphical Models and Image Process., Vol. 55, No. 6, pp. 464-481.

....of discrete periodic signals is then equivalent to a circular convolution of the periodic sample set with the alternate kernel the Fourier series kernel. This method of interpolation of discrete periodic signals has been studied in [4] The Fourier series kernel has been reported in [5]. It contains several indeterminate points , i.e. points at which the function takes an indeterminate form (like 0 0) for which the interpolation cannot be numerically implemented directly. By applying the operation of Fourier series expansion on our signal, a numerically stable form of this ....

Schaum A. "Theory and Design of Local Interpolators", Comp. Vision Graph. Image Proc.: Models and Image Proc., Vol. 55, No. 6, pp 464-481, Nov. 1993.

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A. Schaum, "Theory and design of local interpolators", CVGIP: Graphical Models and Image Processing, Vol. 55, pp. 464-481, November 1993.

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