Industry 4.1. Группа авторов
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Название: Industry 4.1

Автор: Группа авторов

Издательство: John Wiley & Sons Limited

Жанр: Техническая литература

Серия:

isbn: 9781119739913

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СКАЧАТЬ efficiently by using a multi‐resolution analysis (MRA) of fast wavelet transform invented by Mallat [9].

      MRA computationally decomposes images into a two‐scale relation with various time and frequency resolutions by DWT, which is composed of a series of low‐pass and high‐pass filters.

      An L‐level wavelet decomposition is illustrated in detail. The process starts with inputting images of length N=2L into a low‐pass filter g[k] and a high‐pass filter h[k], then images is convolved with g[k] and h[k] for generating two vectors images and images of length N/2, respectively.

      The contents of images and images are approximation and detail coefficients of DWT at the first level, respectively. Further, images is used as an input for obtaining wavelet coefficients images and images at the second level of resolution with the length of coefficients being N/4. In other words, a recursive relationship exists between the approximation and detail coefficients at successive levels of resolution. Therefore, the general decomposition form of wavelet coefficients of length N at the j level can be expressed as follows:

      (2.2)equation

      (2.3)equation

       jscale parameters j = 1, 2, …, L;

       Lnumber of decomposition levels;

       N data length of a discrete signal;

       n translation parameter ;

        nth approximation coefficient at level j;

        nth detail coefficient at level j;

       g[k] DWT low‐pass filters; and

       h[k] DWT high‐pass filters.

 Schematic illustration of three-level decomposition tree of the DWT.

       Thresholding

      The performance of wavelet de‐noising depends on the determination of two factors, the threshold value λ and the threshold function. In this section, the wavelet de‐noising algorithm adopts the soft thresholding method to adaptively filter the specific spectrums of the noisy signals to obtain modified wavelet coefficients images. Let images at level j.

      The soft thresholding method sets every wavelet coefficient cj[n] to zero if |cj[n]| is less than or equal to a chosen threshold λ; otherwise, the threshold is subtracted from any cj[n]. Then, all modified coefficients images = images at level j can be defined as:

      (2.4)equation

      (2.5)equation

       cj[n] nth wavelet coefficients at level j;

        nth modified wavelet coefficients at level j;

       λ threshold of cj[n]; and

       MAD(cL − 1[n]) mean absolute deviation of cL − 1[n].

       Reconstruction

      The de‐noised sensor data images can be reconstructed from all modified wavelet coefficients at the level of resolution L. The reconstruction process is in the opposite direction to the decomposition process; that is, the process proceeds with level j=L, L−1, …, till 1. As such, intermediate modified approximation coefficients images at each level of resolution, j−1 can be recovered by up‐sampling modified approximation and detail coefficients at the level of resolution j. Finally, images is obtained with j = 1 as follows:

      (2.6)equation

      2.3.3 Feature Extraction

      Feature extraction [6–8] is the process to generate a smaller linear or nonlinear combination set to represent the original high‐dimensional data set. Thus, the de‐noised sensing signals need to be transformed into meaningful signal features (SFs), which can adequately describe the physical meaning of the signal and maintain relevant information of the machining operations [6]. However, monitoring machining conditions based on a single SF is not enough [7]. To properly describe machining precisions, a set of multiple SFs is required to provide further insight into coordination [8].

      This section introduces feature extraction approaches СКАЧАТЬ