• Proceedings of the Korean Society of Near Infrared Spectroscopy Conference
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• 2001.06a
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• pp.1041-1041
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• 2001

Removal of variability in spectra data before the application of chemometric modeling will generally result in simpler (and presumably more robust) models. Particularly for sparsely sampled data, such as typically encountered in diode array instruments, the use of Savitzky-Golay (S-G) derivatives offers an effective method to remove effects of shifting baselines and sloping or curving apparent baselines often observed with scattering samples. The application of these convolution functions is equivalent to fitting a selected polynomial to a number of points in the spectrum, usually 5 to 25 points. The value of the polynomial evaluated at its mid-point, or its derivative, is taken as the (smoothed) spectrum or its derivative at the mid-point of the wavelength window. The process is continued for successive windows along the spectrum. The original paper, published in 1964 [1] presented these convolution functions as integers to be used as multipliers for the spectral values at equal intervals in the window, with a normalization integer to divide the sum of the products, to determine the result for each point. Steinier et al. [2] published corrections to errors in the original presentation [1], and a vector formulation for obtaining the coefficients. The actual selection of the degree of polynomial and number of points in the window determines whether closely situated bands and shoulders are resolved in the derivatives. Furthermore, the actual noise reduction in the derivatives may be estimated from the square root of the sums of the coefficients, divided by the NORM value. A simple technique to evaluate the actual convolution factors employed in the calculation by the software will be presented. It has been found that some software packages do not properly account for the sampling interval of the spectral data (Equation Ⅶ in [1]). While this is not a problem in the construction and implementation of chemometric models, it may be noticed in comparing models at differing spectral resolutions. Also, the effects on parameters of PLS models of choosing various polynomials and numbers of points in the window will be presented.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.241-261
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• 2013

In this paper, we find the coefficients for the Weierstrass ${\wp}(x)$ and ${\wp}^{{\prime}{\prime}}(x)$($x=\frac{1}{2}$, $\frac{\tau}{2}$, $\frac{\tau+1}{2}$)-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ([23]). Finally, we reprove congruences for the functions ${\mu}(n)$ and ${\nu}(n)$ in Hahn's article [11, Theorems 6.1 and 6.2].

• Journal of the Korea Computer Graphics Society
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• v.21 no.5
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• pp.1-10
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• 2015

This study proposes a method of developing authoring tool that can easily and intuitively generate diverse digital leaves that compose virtual landscape. The main system of the proposed authoring tool consists of deformation method for the contour of leaf blade based on image warping, procedural modeling of leaf vein and visualization method based on mathematical model that expresses the color and brightness of leaves. First, the proposed authoring tool receives leaf input image and searches for contour information on the leaf blades. It then designs leaf blade deformation method that can generate diverse shapes of leaf blades in an intuitive structure using feature-based image warping. Based on the computed leaf blade contour, the system implements the generalized procedural modeling method suitable for the authoring tool that generates natural vein patterns appropriate for the leaf blade shape. Finally, the system applies visualization function that can express color and brightness of leaves and their changes over time using a mathematical model based on convolution sums of divisor functions. This paper provides texture support function so that the digital leaves that were generated using the proposed authoring tool can be used in a variety of three-dimensional digital contents field.