Calculation Nv(x) of complex order v numerically, we must calculate Df{JN+ε(x)}. When Df{JN+ε(x)} is calculated by the recurrence method, this letter will analyze the error of Df{JN+ε(x)}, and will determine the optimum number of recurrences.

We frequently use a polynomial to approximate a complex function. This study shows a method which determines the optimum coefficients and the number of terms of the polynomial, and the error of the polynomial is estimated.

Hankel's asymptotic expansions are frequently used for numerical calculation of cylindrical functions of complex order. We beforehand need to estimate the precisions of the cylindrical functions calculated with Hankel's asymptotic expansions in order to use these expansions. This letter presents comparatively simple expressions for rough estimations of the errors of the cylindrical functions calculated with the asymptotic expansions, and features of the errors are discussed.

Roundoff error due to iterative computation with finite wordlength degrades the quality of decoded images in fractal image coding that employs a deterministic iterated function system. This paper presents a state-space approach to roundoff error analysis of fractal image coding for grey-scale images. The output noise variance matrix and the noise matrix are derived for the measures of error and the output noise variance is newly defined as the pixel mean of diagonal elements of the output noise matrix. A quantitative comparison of experimental roundoff error with analytical result is made for the output noise variance. The result shows that our analysis method is valid for the fractal image coding. Our analysis method is useful to design a real-time and low-cost decoding hardware with finite wordlength for fractal image coding.

First, the necessity of examining the numerical calculation of the Bessel function Jν(x) of complex order ν is explained. Second, the possibility of the numerical calculation of Jν(x) of arbitrary complex order ν by the use of the recurrence formula is ascertained. The rounding error of Jν(x) calculated by this method is investigated next by means of theory and numerical experiments when the upper limit of recurrence is sufficiently large. As a result, it was known that there is the possibility that the rounding error grows considerably when ν is complex. Counterplans against the growth of the rounding error will be described.

Introducing a general statistical model of image noise, we present an optimal algorithm for computing 3-D motion from two views without involving numerical search: () the essential matrix is computed by a scheme called renormalization; () the decomposability condition is optimally imposed on it so that it exactly decomposes into motion parameters; () image feature points are optimally corrected so that they define their 3-D depths. Our scheme not only produces a statistically optimal solution but also evaluates the reliability of the computed motion parameters and reconstructed points in quantitative terms.

Radar signals fluctuate because of the incoherent scattering of raindrops. Dual-polarization radar estimates rainfall rates from differential reflectivity (ZDR) and horizontal reflectivity (ZH). Here, ZDR and ZH are extracted from fluctuating radar signals by averaging. Therefore, instrumentally measured ZDR and ZH always have errors, so that estimated rainfall rates also have errors. This paper evaluates rainfall rate errors caused by signal fluctuation. Computer simulation based on a physical raindrop model is used to investigate the standard deviation of rainfall rate. The simulation considers acquisition time, and uses both simultaneous and alternate sampling of horizontal and vertical polarizations for square law and logarithmic estimators at various rainfall rates and elevation angles. When measuring rainfall rates that range from 1.0 to 10.0mm/h with the alternate sampling method, using a logarithmic estimator at a relatively large elevation angle, the estimated rainfall rates have significant errors. The simultaneous sampling method is effective in reducing these errors.

It is important for LSI system designers to estimate computational errors when designing LSI's for numeric computations. Both for the prediction of the errors at an early stage of designing and for the choice of a proper hardware configuration to achieve a target performance, it is desirable that the errors can be estimated in terms of a minimum of parameters. This paper presents a theoretical error analysis of multiply-accumulation implemented by distributed arithmetic(DA) and proposes a new method for estimating the mean-squared error. DA is a method of implementing the multiply-accumulation that is defined as an inner product of an input vector and a fixed coefficient vector. Using a ROM which stores partial products. DA calculates the output by accumulating the partial products bitserially. As DA uses no parallel multipliers, it needs a smaller chip area than methods using parallel multipliers. Thus DA is effectively utilitzed for the LSI implementation of a digital signal processing system which requires the multiply-accumulation. It has been known that, if the input data are uniformly distributed, the mean-squared error of the multiply-accumulation implemented by DA is a function of only the word lengths of the input, the output, and the ROM. The proposed method for the error estimation can calculate the mean-squared error by using the same parameters even when the input data are not uniformly distributed. The basic idea of the method is to regard the input data as a combination of uniformly distributed partial data with a different word length. Then the mean-squared error can be predicted as a weighted sum of the contribution of each partial data, where the weight is the ratio of the partial data to the total input data. Finally, the method is applied to a two-dimensional inverse discrete cosine transform (IDCT) and the practicability of the method is confirmed by computer simulations of the IDCT implemented by DA.

Logarithmic number systems (LNS) provide a very fast computational method. Their exceptional speed has been demonstrated in signal processing and then in computer graphics. But the precision problem of LNS in computer graphics has not been fully examined. In this paper analysis is made for the problem of LNS in picture generation, in particular for circle drawing. Theoretical error analysis is made for the circle drawing. That is, some expressions are developed for the relative error variances. Then they are examined by simulation experiments. Some comparisons are also done with floating point arithmetic with equivalent word length and dynamic range. The results show that the theory and the experiments agree reasonably well and that the logarithmic arithmetic is superior to or at least comparable to the corresponding floating point arithmetic with equivalent word length and dynamic range. Those results are also verified by visual inspections of actually drawn circles. It also shows that the conversion error (from integer to LNS), which is inherent in computer graphics with LNS, does not make too much influence on the total computational error for circle drawing. But it shows that the square-rooting makes the larger influence.