BS EN ISO 16610-29 pdf free download
BS EN ISO 16610-29-2020 pdf free download.Geometrical product specifications (GPS) – Filtration – Part 29: Linear profile filters: Wavelets (ISO 16610-29:2020).
4 General wavelet description
4.1 General A cubic prediction wavelet claiming to conform with this document shall satisfy the procedure given in Annex A. A cubic spline wavelet claiming to conform with this document shall satisfy the procedure given in Annex B. NOTE The relationship to the filtration matrix model is given in Annex C.
4.2 Basic usage of wavelets Wavelet analysis consists of decomposing a profile into a linear combination of wavelets g a,b (x), all generated from a single mother wavelet [4] . This is similar to Fourier analysis, which decomposes a profile into a linear combination of sinewaves, but unlike Fourier analysis, wavelets are finite in both spatial and frequency domain. Therefore, they can identify the location as well as the scale of a feature in a profile. As a result, they can decompose profiles where the small-scale structure in one portion of the profile is unrelated to the structure in a different portion, such as localized changes (i.e. scratches, defects or other irregularities). Wavelets are also ideally suited for non-stationary profiles. Basically, wavelets decompose a profile into building blocks of constant shape, but of different scales.
4.3 Wavelet transform The discrete wavelet transform [5] of a profile, s(x), given as height values, s(x i ), at uniformly sampled positions, x i = (i−1) Δx (where Δx is the sampling interval, i = 1, …, n and n being the number of sampling points), with the wavelet function g((x−b)/a), is given by the differences (or details), d k (i), and the smoothed data, s k (i), and a subsequent decimation (down-sampling) for each level or rung, k, of decomposition. The smoothed data and differences are obtained by convolving the signal with the scaling function, h, and the wavelet, g, as shown in Formula (2a) and Formula (2b):
where j = −m, …, −2, −1, 0, 1, 2, …, m; (m is the number of coefficients of the filter on one side from the centre).
The dilation parameter, a, is determined by the level of decomposition, k, and by down-sampling the smoothed data commonly by a factor of two, i.e. a = 2 −k , respectively. a = 1/(2 k Δx), such that for each step of the decomposition ladder the number of smoothed data points reduces by a factor of two.
The decomposition starts with the original signal values, s(x i ), denoted as s 0 (i). The mother wavelet of the discrete wavelet transform is defined as a set of discrete high-pass filter coefficients, g j , and the scaling function as a set of discrete low-pass filter coefficients, h j . As the decimation is carried out by keeping every second value of the smooth and every second of the difference signal, the total number of data points is conserved, such that n/2 of the s 1 (i) are saved and n/2 of the d 1 (i) and the distance between the i-th and the (i+1)-th is then 2Δx. For the second decomposition step, the set of n/2 differences, d 1 (i), will be kept until termination but the set of the s 1 (i) is subdivided half and half, such that n/4 values s 2 (i) and n/4 values d 2 (i) are obtained. For the k-th step of decomposition and decimation n/2 k of s k (i) and n/2 k values d k (i) are evaluated and the distance between the i-th and the (i+1)-th is then 2 k Δx. Therefore, the dilation is done by down-sampling, i.e. managing the indices of the signal rather than changing the wavelet and scaling functions. Thus, for discrete wavelet transformations only the two sets of filter coefficients, the set {h j , j = −m,..,0,..m} for the low-pass and {g j , j = −m,..,0,..m} for the high- pass, define the analysis filter.BS EN ISO 16610-29 pdf download.
