CN103345636A - Method for identifying foam working condition on copper flotation site based on wavelet multi-scale binaryzation - Google Patents

Abstract

The invention discloses a copper flotation on-site foam working condition recognition method based on wavelet multi-scale binarization. First, wavelet transform is performed on the foam grayscale image; then, wavelet approximation subgraphs of different scales are binarized; finally According to the space-frequency relationship of the two-dimensional wavelet transform, the white area area of each binary image is statistically calculated, and a new multi-scale statistical feature directly related to the foam appearance is obtained—the equivalent size feature. According to the obtained equivalent size characteristics, the equivalent size distribution map of the foam image can be further obtained, and the foam images of different working conditions can be directly distinguished from the distribution map. The invention is simple and effective, and has great significance for instructing the identification of foam working conditions on the spot of copper flotation.

Description

Recognition method of copper flotation field foam condition based on wavelet multi-scale binarization

technical field

The invention relates to a wavelet multi-scale binarization-based recognition method for copper flotation foam working conditions, which belongs to the fields of image processing technology, pattern recognition and the like.

Background technique

The flotation working condition is the working condition in the flotation production process, timely and accurate identification of the working condition is very important to guide the flotation production. The traditional way of relying on workers to observe the apparent changes of foam with naked eyes can no longer meet the needs of rapid and accurate identification of working conditions in today's flotation production. With the rapid development of technologies such as machine vision and image processing, great progress has been made in the intelligent identification of working conditions based on the foam characteristics of flotation sites. By quickly and accurately identifying the working conditions of the flotation site, the flotation production control system can adjust production parameters in time to keep the flotation production process in an optimal state.

Studies have shown that the visual characteristics of the foam surface such as the size and color of the foam directly observed by the naked eye (hereinafter referred to as the apparent characteristics of the foam) are a comprehensive reflection of the flotation conditions. How to accurately extract the foam appearance characteristics closely related to key production indicators in the flotation process is the key to realize the identification of flotation conditions. The traditional features applied to the recognition of copper flotation conditions are mainly obtained by segmentation methods, including foam color, size, loading rate, velocity and stability, etc., but these features obtained directly by segmenting the original image do not have multi-scale characteristics. A binary image is the simplest form of representing an image. By binarizing the foam grayscale image, the morphological statistical features such as the number and area of the foam can be obtained. Like the image segmentation method, the foam features obtained by the binarization method do not have multi-scale characteristics. Wavelet analysis has multi-scale characteristics, and can simulate the human visual system to analyze signals in multi-channel time-frequency domain, so as to obtain statistical features with more information. Although the traditional statistical features extracted by wavelet analysis have multi-scale characteristics, it is difficult to describe the appearance of foam that conforms to the visual habits of workers, thus directly affecting the accurate judgment of working conditions.

Therefore, it is necessary to design a method for identifying foam conditions in copper flotation based on wavelet multi-scale binarization.

Contents of the invention

The technical problem to be solved by the present invention is to provide a copper flotation on-site foam working condition identification method based on wavelet multi-scale binarization, which is easy to implement, The implementation cost is low and the recognition effect is good.

The technical solution of the invention is as follows:

A method for extracting foam image equivalent size features, comprising the following steps:

Step 1: Obtain a 3D foam image from the foam video obtained at the copper flotation site, convert the 3D foam image to grayscale, and then perform wavelet decomposition on the obtained 2D grayscale image to obtain wavelet subgraphs on different scales, Ignore the characteristics of each order detail subgraph, and only perform single-branch reconstruction on each order approximation subgraph to obtain the reconstructed approximation subgraph;

Step 2: According to the uncertainty principle and the discrete wavelet frequency domain relationship, calculate the range of foam diameter variation of subgraphs of different orders;

Step 3: Use the maximum inter-class variance method to binarize each reconstructed approximation sub-image to obtain a binary image, count the total area of the foam in each binary image, and then obtain the equivalent size characteristics of the foam; finally combine with step 2 to calculate Calculate the number of equivalent foams based on the obtained foam diameter variation range, and then obtain the foam equivalent size distribution map; [A foam image can be transformed into multiple approximation subgraphs after wavelet transformation, and then these approximation subgraphs (multiple images) for further processing to obtain the corresponding equivalent size distribution map. 】

Step 4: Compare the foam equivalent size distribution map with the benchmark distribution map to determine the copper flotation field conditions corresponding to the current foam image.

Described step one comprises following sub-steps:

Step 1: Grayscale the original foam image;

The original three-dimensional RGB foam image K _(X×Y×3) becomes a two-dimensional grayscale image I _(X×Y) after being grayscaled;

Step 2: Select the sym4 wavelet to decompose the 2D grayscale image to the fifth order. At each decomposition order, the 2D wavelet transform will generate an approximation subimage and detail subimages in three different directions: horizontal, vertical, and diagonal ;

The wavelet transform formula at each decomposition order is:

和3个细节系数矩阵可由二维小波分解的Mallat算法获得：matrix of approximation coefficients in each decomposition order

and 3 detail coefficient matrices It can be obtained by the Mallat algorithm of two-dimensional wavelet decomposition:

$\left\{\begin{array}{c}{c}_{k,m}^j=\underset{l,\mathrm{no}}{\mathrm{\Σ}}{\stackrel{\‾}{h}}_{2k-l}{\stackrel{\‾}{h}}_{2m-\mathrm{no}}{c}_{l,\mathrm{no}}^{j+1}\\ d_{k,m}^{j,1}=\underset{l,\mathrm{no}}{\mathrm{\Σ}}{\stackrel{\‾}{h}}_{2k-l}{\stackrel{\‾}{g}}_{2m-\mathrm{no}}{c}_{l,\mathrm{no}}^{j+1}\\ d_{k,m}^{j,2}=\underset{l,\mathrm{no}}{\mathrm{\Σ}}{\stackrel{\‾}{g}}_{2k-l}{\stackrel{\‾}{h}}_{2m-\mathrm{no}}{c}_{l,\mathrm{no}}^{j+1}\\ d_{k,m}^{j,3}=\underset{l,\mathrm{no}}{\mathrm{\Σ}}{\stackrel{\‾}{g}}_{2k-l}{\stackrel{\‾}{g}}_{2m-\mathrm{no}}{c}_{l,\mathrm{no}}^{j+1}\end{array}\right.$ Formula 2

分别表示一维正交多分辨分析的两尺度方程和小波方程；j为分解层数，本例中进行五阶小波分解，则取j＝5，4，3，2，1；k，m∈Z分别代表系数矩阵的行和列，l，n∈Z分别代表系数矩阵

的行和列；【在进行第一阶小波变换时，l＝600，n＝800，k＝300，m＝400】系数序列h＝{h_i}是一个低通滤波器，有h＝{-0.076，-0.030，0.498，0.804，0.298，-0.100，-0.013，0.032}，

为h的时序反转

系数序列g＝{g_i}是一个高通滤波器且g_i＝(-1)ⁱh_1-i，为g的时序反转 The scaling function in the formula is: The three wavelet functions are: ψ ¹ (x, y) = φ(x)ψ(y), ψ ² (x, y) = ψ(x)φ(y), ψ ³ (x, y) = ψ (x)ψ(y), where and

Represent the two-scale equation and wavelet equation of one-dimensional orthogonal multiresolution analysis; j is the number of decomposition layers. In this example, for fifth-order wavelet decomposition, j=5, 4, 3, 2, 1; k, m∈ Z represents the coefficient matrix respectively The rows and columns of , l, n∈Z respectively represent the coefficient matrix

row and column; [when performing the first-order wavelet transform, l=600, n=800, k=300, m=400] the coefficient sequence h={h _i } is a low-pass filter, and h={ -0.076, -0.030, 0.498, 0.804, 0.298, -0.100, -0.013, 0.032},

Timing reversal for h

The coefficient sequence g={g _i } is a high-pass filter and g _i =(-1) ⁱ h _1-i , timing reversal for g

继续进行第二阶的二维小波变换，再次得到1个逼近子图和3个细节子图；依此类推，即可得到的五个分解阶次的小波变换；相应地，便得到灰度图像I_(X×Y)的多尺度表示；Firstly, formula 1 is used to perform the first-order two-dimensional wavelet transform, and one approximation subgraph and three detail subgraphs are obtained, and the coefficients are calculated by formula 2; then, the first-order approximation subgraph (low frequency part) is obtained by using formula 1

Continue to carry out the second-order two-dimensional wavelet transform to obtain 1 approximation subgraph and 3 detail subgraphs again; and so on, you can get The wavelet transform of the five decomposition orders of ; correspondingly, the multi-scale representation of the grayscale image I _(X×Y) is obtained;

Step 3: Ignore the characteristics of each order detail subgraph, and only perform single-branch reconstruction on each order approximation subgraph to obtain the reconstructed approximation subgraph:

The five approximation reconstruction subgraphs are denoted as S ^v , v=1, 2, ..., 5, which represent the contours of the foam subgraphs at different scales, and their coefficient matrix is obtained by the following reconstruction formula:

$c_{k,m}^{j+1}=\underset{l,\mathrm{no}}{\mathrm{\Σ}}{h}_{k-2l}{h}_{m-2\mathrm{no}}{c}_{l,\mathrm{no}}^j+\underset{l,\mathrm{no}}{\mathrm{\Σ}}{h}_{k-2l}{g}_{m-2\mathrm{no}}{d}_{l,\mathrm{no}}^{j,1}+\underset{l,\mathrm{no}}{\mathrm{\Σ}}{g}_{k-2l}{h}_{m-2\mathrm{no}}{d}_{l,\mathrm{no}}^{j,2}+\underset{l,\mathrm{no}}{\mathrm{\Σ}}{g}_{k-2l}{g}_{m-2\mathrm{no}}{d}_{l,\mathrm{no}}^{j,3}$ Formula 3.

In the second step,

The frequency domain relationship between discrete wavelet signals of different scales is: the frequency width of the approximation signal f _j (t) is only half of the frequency width of f _j+1 (t); the other half of the frequency width of f _j+1 (t) is given by Detail signal d _j (t) performance;

Table 1

approximation subgraph Frequency Range Spatial range A1 0～0.5π 0～1 A2 0～0.25π 0～2 A3 0～0.125π 0～4 A4 0～0.0625π 0～8 A5 0～0.03125π 0～16

【According to the uncertainty principle, the spatial frequency window area of wavelet multi-scale analysis is invariant. Therefore, the frequency widths of each reconstructed submap correspond to different spatial widths. 】According to the frequency domain relationship between discrete wavelet signals of different scales and the uncertainty principle, the spatial frequency relationship of the five approximation subgraphs is shown in Table 1. [The spatial range in the table is the equivalent diameter range in Figure 6]

Described step three comprises the following substeps:

Step 1: Calculate the optimal threshold value of each reconstructed submap according to the maximum between-class variance method, perform binarization on each reconstructed submap, and count the total area of foam in each binary image:

各灰度值出现的概率为

设以灰度L_th为门限将图像分割成2个区域，灰度级为1～L_th的像素区域A，灰度级为L_th+1～L-1的像素区域B：A、B出现的概率分别为：Suppose S ^v has L gray levels, and the number of pixels whose gray value is i is n _i , then the total number of pixels is

The probability of occurrence of each gray value is

Let the gray level L _th be used as the threshold to divide the image into two areas, the pixel area A with a gray level of 1 to L _th , and the pixel area B with a gray level of L _th +1 to L-1: A and B appear The probabilities are:

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; ;</span>$

The gray mean values of A and B are:

${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; =</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; ip</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; ip</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}}<span\; class="notranslate">\; ;</span>$

In this way, the total gray value of the image can be obtained:

${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; ip</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ;</span>$

From this, the between-class variance of A and B regions can be obtained:

σ ² =p _A (ω _A -ω ₀ ) ² +p _B (ω _B -ω ₀ ) ² ;

即为所求的最佳阈值：In order to obtain the optimal segmentation threshold, the variance between the two classes is used as the discriminant criterion, and the one with the largest ^σ2 value

That is the optimal threshold value sought:

${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; Arg</span>}\underset{\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; Max</span>}}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; V</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2,3,4,5</span>}<span\; class="notranslate">\; ,</span>$

再根据

分别对S^v进行二值化，得到五个二值图像，统计各二值图像的白色区域面积，即得到泡沫总体面积，记为A_v；Calculate the optimal threshold for each approximation reconstruction subgraph

Then according to

Carry out binarization to ^Sv respectively, obtain five binary images, count the white area area of each binary image, promptly obtain the total foam area, denote as _Av ;

Step 2: Subtract the total area of the foam in adjacent binary images to obtain the equivalent size feature of the foam;

Subtract the total foam area of adjacent binary images to obtain the equivalent area of the foam sub-image, which is denoted as E _q ;

Yes, E _q = A _q+1 -A _q , q = 1, 2, 3, 4;

E _q is the foam equivalent size feature obtained by multi-scale binarization, the subscript q is the number of equivalent size features, and q+1 is the decomposition order; since the decomposition order is 5, then 4 The equivalent size features correspond to four continuous foam diameter ranges: 1~2, 2~4, 4~8, 8~16; [this range is directly obtained by analyzing the space-frequency relationship between wavelet subgraphs, rather than by area. The diameters here are equivalent diameters, so units are not given. 】

Step 3: Calculate the number of equivalent foams according to the calculated foam diameter variation range, and then obtain the foam equivalent size distribution diagram;

Taking D _q as the average value of the range of foam diameter variation, the corresponding D _q of the four foam diameter ranges are 1.5, 3, 6, and 12 respectively, and the number of equivalent foams of each order is calculated according to the following formula:

【这个个数就是图6中的频数】

[This number is the frequency in Figure 6]

Finally, the equivalent size distribution diagram of the foam image under the current working condition is obtained.

In the step 4, the method for obtaining the reference distribution map is: first obtain a plurality of foam equivalent size distribution maps from a plurality of images of known working conditions through steps 1 to 3, and obtain a plurality of foam equivalent size distribution maps Perform statistics to obtain a reference distribution map; the statistics refer to counting the foam frequency of the equivalent diameter of each foam image under different working conditions, and the reference distribution map is obtained off-line.

Beneficial effect:

The copper flotation on-site foam working condition recognition method based on wavelet multi-scale binarization of the present invention combines wavelet multi-scale analysis with image binarization, and utilizes wavelet multi-scale binarization to extract a new The multi-scale statistical feature directly related to the appearance and shape - the equivalent size feature. Identifying the production conditions of copper flotation based on this feature is of great significance for stabilizing the working conditions of copper flotation and realizing the optimal control of the copper flotation process.

The feature extracted by the present invention is the expression of the original image on different scales, and has multi-scale characteristics; at the same time, the feature is directly related to the apparent shape of the foam, and different foam images correspond to different equivalent size distributions. By comparing the equivalent size distribution diagrams, the foam images of different working conditions can be directly distinguished, so it is simple and effective, and it is of great significance to guide the identification of copper flotation working conditions.

Description of drawings

Figure 1 Foam images of three different working conditions;

Figure 2 The dendrogram of the fifth-order wavelet decomposition of the foam image;

Fig. 3 Foam image reconstruction approximation subgraph;

Figure 4 The frequency domain relationship between discrete wavelet signals of different scales;

The spatial frequency relationship of the five approximation subgraphs in Fig. 5;

Figure 6. Equivalent size distribution of foam images under three different working conditions.

Detailed ways

The present invention will be described in further detail below in conjunction with accompanying drawing and specific embodiment:

Example 1:

Below in conjunction with accompanying drawing the specific embodiment of the present invention is described, and certain copper flotation site has the foam of three kinds of different working conditions, is respectively normal foam, hydration foam and viscous foam, and the foam image of these three kinds of different working conditions is shown in Figure 1 shown.

In the first step, according to the foam video obtained at the copper flotation site, the foam image is obtained, and the three-dimensional image is grayscaled. Then wavelet decomposition is performed on the two-dimensional grayscale image to obtain wavelet subgraphs on different scales. Ignore the characteristics of each order detail subgraph, and only perform single-branch reconstruction on each order approximation subgraph to obtain the reconstructed approximation subgraph.

Step 1: Grayscale the original foam image;

The original three-dimensional RGB foam image K _(X×Y×3) becomes a two-dimensional grayscale image I _(X×Y) after being grayscaled;

Step 2: Select the sym4 wavelet to decompose the grayscale image to the fifth order. In each decomposition order, the two-dimensional wavelet transform will generate an approximation subgraph and three detail subgraphs in different directions: horizontal, vertical and diagonal. Fig. 2 is the fifth-order wavelet decomposition dendrogram of the foam image;

The wavelet transform formula at each decomposition order is:

和3个细节系数矩阵

可由二维小波分解的Mallat算法获得：matrix of approximation coefficients in each decomposition order

and 3 detail coefficient matrices

It can be obtained by the Mallat algorithm of two-dimensional wavelet decomposition:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; l</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; l</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; g</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; g</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; l</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; g</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; l</span>}}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; g</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

3个小波函数分别为：ψ¹(x，y)＝φ(x)ψ(y)，ψ²(x，y)＝ψ(x)φ(y)，ψ³(x，y)＝ψ(x)ψ(y)，其中和

分别表示一维正交多分辨分析的两尺度方程和小波方程；j为分解层数，本例中进行五阶小波分解，则取j＝5，4，3，2，1；k，m∈Z分别代表系数矩阵

的行和列，l，n∈Z分别代表系数矩阵

的行和列；【在进行第一阶小波变换时，l＝600，n＝800，k＝300，m＝400】系数序列h＝{h_i}是一个低通滤波器，有h＝{-0.076，-0.030，0.498，0.804，0.298，-0.100，-0.013，0.032}，为h的时序反转

系数序列g＝{g_i}是一个高通滤波器且g_i＝(-1)ⁱh_1-i，

为g的时序反转

The scaling function in the formula is:

The three wavelet functions are: ψ ¹ (x, y) = φ(x)ψ(y), ψ ² (x, y) = ψ(x)φ(y), ψ ³ (x, y) = ψ (x)ψ(y), where and

Represent the two-scale equation and wavelet equation of one-dimensional orthogonal multiresolution analysis; j is the number of decomposition layers. In this example, for fifth-order wavelet decomposition, j=5, 4, 3, 2, 1; k, m∈ Z represents the coefficient matrix respectively

The rows and columns of , l, n∈Z respectively represent the coefficient matrix

row and column; [when performing the first-order wavelet transform, l=600, n=800, k=300, m=400] the coefficient sequence h={h _i } is a low-pass filter, and h={ -0.076, -0.030, 0.498, 0.804, 0.298, -0.100, -0.013, 0.032}, Timing reversal for h

The coefficient sequence g={g _i } is a high-pass filter and g _i =(-1) ⁱ h _1-i ,

timing reversal for g

和3个细节子图(

)，其系数由公式(2)计算，此时

即为步骤1所获得的二维灰度图像矩阵I_(X×Y)(

的计算步骤为：对任一固定的列数n＝800，先用

与

的每个列向量作卷积，进行向下二抽样得到

然后再用

与

的每个行向量作卷积，进行向下二抽样得到

)；然后，再利用公式(1)对一阶逼近子图(低频部分)

继续进行第二层的二维小波变换(此时j＝4，k＝150，m＝200，l＝300，n＝400)，再次得到1个逼近子图 $(c_{k,m}^j=c_{\mathrm{150,200}}^4)$ 和3个细节子图( $d_{k,m}^{j,1}=d_{\mathrm{150,200}}^{\mathrm{4,1}},$ $d_{k,m}^{j,2}=d_{\mathrm{150,200}}^{\mathrm{4,2}},$ )；依此类推，即得到

的五个分解阶次的小波变换，相应地，便得到灰度图像I_(X×Y)的多尺度表示；First apply the formula (1) to carry out the two-dimensional wavelet transform of the first layer, (take j=5, k=300, m=400, l=600, n=800,) to get an approximation subgraph

and 3 detail subgraphs (

), its coefficient is calculated by formula (2), at this time

That is, the two-dimensional grayscale image matrix I _(X×Y) obtained in step 1 (

The calculation steps of are: for any fixed number of columns n=800, first use

and

Each column vector of is convolved, and the second sampling is performed to obtain

and then use

and

Each row vector of is convolved, and the second sampling is performed to obtain

); Then, use the formula (1) to approximate the first-order subgraph (low frequency part)

Continue to carry out the two-dimensional wavelet transform of the second layer (at this time j=4, k=150, m=200, l=300, n=400), and obtain 1 approximation subgraph again $(c_{k,m}^j=c_{\mathrm{150,200}}^4)$ and 3 detail subgraphs ( $d_{k,m}^{j,1}=d_{\mathrm{150,200}}^{\mathrm{4,1}},$ $d_{k,m}^{j,2}=d_{\mathrm{150,200}}^{\mathrm{4,2}},$ ); and so on, we get

The wavelet transform of the five decomposition orders of , correspondingly, the multi-scale representation of the gray image I _(X×Y) is obtained;

Step 3: Ignore the characteristics of the detailed subgraphs of each order, and only perform single-branch reconstruction on the approximation subgraphs of each order to obtain the reconstructed approximation subgraph, as shown in Figure 3;

As the decomposition order increases, the wavelet subgraph becomes smaller gradually. In order to obtain subimages with the same size as the original image and maintain the frequency components of each subimage, single-branch reconstruction is required. When performing single-branch reconstruction, only the wavelet coefficients of a single subgraph are used for signal reconstruction, and the wavelet coefficients of other subgraphs are set to zero. Since the approximation subgraph reflects the contour of the foam image, and the detail subgraph reflects the detail changes of the foam image, when obtaining the foam size feature, the detail subgraph of each order can be ignored, and only the approximation subgraph can be reconstructed. The five approximation reconstruction subgraphs are respectively denoted as Sv ⁽ v=1, 2, ..., 5), which represent the contours of the foam subgraphs at different scales, and their coefficient matrix can be obtained by the reconstruction formula:

${\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; l</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; l</span>}}{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; l</span>}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; l</span>}}{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

In the second step, according to the uncertainty principle and the discrete wavelet frequency domain relationship, the range of foam diameter variation in subgraphs of different orders is deduced.

The frequency domain relationship between discrete wavelet signals of different scales is shown in Figure 4. Among them, f _j , f _j-1 , f _j-2 represent approximation signals at different scales, and d _j-1 , d _j-2 represent detail signals at different scales.

According to the uncertainty principle, the spatial frequency window area of wavelet multi-scale analysis is invariant. Therefore, the frequency widths of each reconstructed submap correspond to different spatial widths. According to Figure 4 and the uncertainty principle, the spatial frequency relationship of the five approximate subgraphs can be obtained, as shown in Figure 5.

In the third step, each reconstructed subimage is binarized by using the method of maximum variance between classes, and the total foam area of each binary image is calculated. Then calculate the foam equivalent size characteristics. Finally, combined with the variation range of the foam diameter calculated in the second step, the number of equivalent foams is calculated, and then the foam equivalent size distribution diagram is obtained.

Step 1: Calculate the optimal threshold value of each reconstructed subimage according to the maximum between-class variance method, perform binarization on it, and count the total area of the foam in each binary image;

各灰度值出现的概率为

显然p_i≥0，

设以灰度t为门限将图像分割成2个区域，灰度级为1～L_th的像素区域A，灰度级为L_th+1～L-1的像素区域B。A、B出现的概率分别为：Suppose S ^v has L gray levels, and the number of pixels whose gray value is i is n _i , then the total number of pixels is

The probability of occurrence of each gray value is

Obviously p _i ≥ 0,

Assume that the image is divided into two areas with the gray level t as the threshold, the pixel area A with the gray level of 1~ _Lth , and the pixel area B with the gray level of _Lth +1~L-1. The probabilities of A and B appearing are:

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

The gray mean values of the two classes A and B are:

${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; =</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; ip</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; ip</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

In this way, the total gray value of the image can be obtained:

${\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; ip</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

From this, the between-class variance of A and B regions can be obtained:

σ ² ＝p _A (ω _A -ω ₀ ) ² +p _B (ω _B -ω ₀ ) ² (7)

即为所求的最佳阈值：In order to obtain the optimal segmentation threshold, the variance between the two classes is used as the discriminant criterion, and the one with the largest ^σ2 value

That is the optimal threshold value sought:

${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="Arg\; Max"\; filenames="HDA00003394837700022.png,HDA00003394837700023.png"\; state="\{\{state\}\}">Arg</figure-callout></span><figure-callout\; id="0"\; label="Arg\; Max"\; filenames="HDA00003394837700022.png,HDA00003394837700023.png"\; state="\{\{state\}\}">\; </figure-callout>}\underset{}{\mathrm{<span\; class="notranslate">\; </span>}}\underset{\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; th</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">Max</span>}}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; V</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2,3,4,5</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

再根据

分别对S^v进行二值化，得到五个二值图像。统计各二值图像的白色区域面积，从而得到泡沫总体面积，记为A_v。Calculate the optimal threshold for each approximation reconstruction subgraph

Then according to

Binarize S ^v separately to obtain five binary images. The area of the white area of each binary image is counted to obtain the total area of the foam, which is denoted as A _v .

Step 2: Subtract the total area of the foam in adjacent binary images to obtain the equivalent size feature of the foam;

Subtract the total foam area of adjacent binary images to obtain the equivalent area of the foam sub-image, which is denoted as E _q . E _q is the foam equivalent size feature obtained by multi-scale binarization, q is the number of equivalent size features, and q+1 is the decomposition order. Since the decomposition order is 5, the 4 equivalent size features correspond to 4 continuous foam diameter ranges: 1~2, 2~4, 4~8, 8~16.

E _q =A _q+1 -A _q , q=1, 2, 3, 4 (9)

Step 3: Calculate the number of equivalent foams according to the calculated foam diameter variation range, and then obtain the foam equivalent size distribution diagram;

Since the shape of the foam is approximately a circle, its area can be calculated by π(D _q /2) ² , and D _q is taken as the average value of the range of foam diameters, which are 1.5, 3, 6, and 12 respectively, and can be obtained according to the following formula Calculate the number of equivalent foams of each order.

${\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}}{\mathrm{<span\; class="notranslate">\; \&pi;</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; D.</span>}}_{\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

Finally, the equivalent size distribution diagram of the foam image under the current working condition is obtained.

The fourth step is to compare the foam equivalent size distribution map obtained in real time with the reference distribution map to determine the foam working condition of the copper flotation site corresponding to the current foam image.

The method of obtaining the benchmark distribution map is as follows: firstly, manually classify the collected historical froth videos of the copper flotation site, so as to obtain multiple froth images of three different working conditions, and then process the three froth images respectively through steps 1 to 3 , to obtain multiple foam equivalent size distribution diagrams, and finally perform statistics on the obtained multiple foam equivalent size distribution diagrams to obtain a benchmark distribution diagram, as shown in FIG. 6 .

It can be seen from Figure 6 that the normal foam size distribution is relatively uniform, mainly concentrated in 2-10; the hydration foam is mostly small foam, and the size distribution is mainly concentrated in 1-4; the size distribution of viscous foam is similar to that of normal foam, but its small There is more foam than normal, and the large foam is less than normal. Compare the foam equivalent size distribution map obtained in real time with the reference distribution map, and observe (for example, use manual observation) the foam equivalent size distribution at this time, so that the current foam working conditions of the copper flotation site can be identified.

The results show that the features proposed in the present invention not only contain multi-scale information, but also are directly related to the foam appearance. Different foam images correspond to different equivalent size distributions. By comparing the distribution maps, the foam images of different working conditions can be directly distinguished. open. This method is simple and effective, and it is of great significance to guide the identification of foam conditions in copper flotation.

Claims (5)

1. a froth images equivalent dimension Feature Extraction method is characterized in that, may further comprise the steps:

Step 1: the foam video that obtains according to copper flotation scene obtains the three-dimensional foam image, the three-dimensional foam image is carried out gray processing, two dimensional gray image to gained carries out wavelet decomposition then, thereby obtain the small echo subgraph on the different scale, the feature of ignoring each rank details subgraph, only each rank ll channel is carried out single reconstruct, obtain the reconstruct ll channel;

Step 2: according to uncertainty principle and discrete wavelet frequency domain relation, extrapolate the foam diameter variation range of different order subgraphs;

Step 3: utilize maximum variance between clusters with each reconstruct ll channel binaryzation, obtain bianry image, count the foam total area of each bianry image, and then try to achieve foam equivalent dimension feature; The foam diameter variation range that last integrating step two is extrapolated calculates equivalent foam number, and then obtains foam equivalent dimension distribution plan;

Step 4: foam equivalent dimension distribution plan and benchmark distribution plan are compared, determine the copper flotation field working conditions of current froth images correspondence.

2. froth images equivalent dimension Feature Extraction method according to claim 1 is characterized in that the described step

Rapid one comprises following substep:

Step 1: original froth images gray processing;

Original three-dimensional RGB froth images K _{(X * Y * 3)}Become the two dimensional gray image I behind the gray processing _{(X * Y)}

Step 2: select the sym4 small echo that the two dimensional gray image is carried out five rank and decompose, decompose order at each, two-dimensional wavelet transformation will produce the details subgraph of a ll channel and level, vertical, three different directions of diagonal line;

At each wavelet transformation formula that decomposes order be:

Formula 1

Decompose the matrix of coefficients that approaches in order at each

With 3 detail coefficients matrixes

Can be obtained by the Mallat algorithm that 2-d wavelet decomposes:

$\left\{\begin{array}{c}{c}_{k,m}^j=\underset{l,n}{\mathrm{\&Sigma;}}{\stackrel{\&OverBar;}{h}}_{2k-l}{\stackrel{\&OverBar;}{h}}_{2m-n}{c}_{l,n}^{j+1}\\ d_{k,m}^{j,1}=\underset{l,n}{\mathrm{\&Sigma;}}{\stackrel{\&OverBar;}{h}}_{2k-l}{\stackrel{\&OverBar;}{g}}_{2m-n}{c}_{l,n}^{j+1}\\ d_{k,m}^{j,2}=\underset{l,n}{\mathrm{\&Sigma;}}{\stackrel{\&OverBar;}{g}}_{2k-l}{\stackrel{\&OverBar;}{h}}_{2m-n}{c}_{l,n}^{j+1}\\ d_{k,m}^{j,3}=\underset{l,n}{\mathrm{\&Sigma;}}{\stackrel{\&OverBar;}{g}}_{2k-l}{\stackrel{\&OverBar;}{g}}_{2m-n}{c}_{l,n}^{j+1}\end{array}\right.$ Formula 2

Formula mesoscale function is: 3 wavelet functions are respectively: ψ ¹(x, y)=φ (x) ψ (y), ψ ²(x, y)=ψ (x) φ (y), ψ ³(x, y)=ψ (x) ψ (y), wherein

With

Two yardstick equations and the little wave equation of representing an orthogonal dimension multiresolution analysis respectively; J carries out five rank wavelet decomposition for decomposing the number of plies in this example, then get j=5,4,3,2,1; K, m ∈ Z is the system of representatives matrix number respectively

Row and column, l, n ∈ Z be the system of representatives matrix number respectively

Row and column; Coefficient sequence h=(h _iBe a low-pass filter, and h={-0.076 is arranged ,-0.030,0.498,0.804,0.298 ,-0.100 ,-0.013,0.032},

Sequential counter-rotating for h

Coefficient sequence g={g _iBe a Hi-pass filter and g _i=(1) ⁱh _1-i,

Sequential counter-rotating for g

At first application of formula 1 is carried out the two-dimensional wavelet transformation on first rank, obtains 1 ll channel and 3 details subgraphs, and its coefficient is calculated by formula 2; Then, utilize 1 pair of single order ll channel of formula (low frequency part)

Proceed the two-dimensional wavelet transformation on second rank, obtain 1 ll channel and 3 details subgraphs again; The rest may be inferred, can obtain Five wavelet transformations that decompose orders; Correspondingly, just obtain gray level image I _{(X * Y)}Multiple dimensioned expression;

Step 3: ignore the feature of each rank details subgraph, only each rank ll channel is carried out single reconstruct, obtain the reconstruct ll channel:

Approaching the reconstruct subgraph with five remembers respectively and makes S ^v, v=1,2 ..., 5, they have represented the outline portion of foam subgraph under the different scale, and its matrix of coefficients is tried to achieve by following reconstruction formula:

$c_{k,m}^{j+1}=\underset{l,n}{\mathrm{\&Sigma;}}{h}_{k-2l}{h}_{m-2n}{c}_{l,n}^j+\underset{l,n}{\mathrm{\&Sigma;}}{h}_{k-2l}{g}_{m-2n}{d}_{l,n}^{j,1}+\underset{l,n}{\mathrm{\&Sigma;}}{g}_{k-2l}{h}_{m-2n}{d}_{l,n}^{j,2}+\underset{l,n}{\mathrm{\&Sigma;}}{g}_{k-2l}{g}_{m-2n}{d}_{l,n}^{j,3}$ Formula 3.

3. froth images equivalent dimension Feature Extraction method according to claim 1 is characterized in that, in the described step 2,

Frequency domain between the discrete wavelet different scale signal closes: approximation signal f _j(t) band width only is f _J+1(t) half of band width; f _J+1(t) second half of band width is by detail signal d _j(t) performance;

Table 1

Ll channel Frequency range Spatial dimension A1 0～0.5π 0～1 A2 0～0.25π 0～2

A3 0～0.125π 0～4 A4 0～0.0625π 0～8 A5 0～0.03125π 0～16

The spatial frequency that obtains five ll channels according to the frequency domain between discrete wavelet different scale signal relation and uncertainty principle concerns as shown in table 1.

4. froth images equivalent dimension Feature Extraction method according to claim 1 is characterized in that described step 3 comprises following substep:

Step 1: calculate the optimal threshold of each reconstruct subgraph according to maximum variance between clusters, each reconstruct subgraph is carried out binaryzation, and count the foam total area of each bianry image:

If S ^vL gray level arranged, and gray-scale value is that the pixel count of i is n _i, then total pixel count is

The probability that each gray-scale value occurs is

If with gray scale L _ThFor thresholding is divided into 2 zones with image, gray level is 1～L _ThPixel region A, gray level is L _ThThe probability that pixel region B:A, the B of+1～L-1 occurs is respectively:

$p_A=\underset{i=0}{\overset{L_{\mathrm{th}}}{\mathrm{\&Sigma;}}}{p}_i,$ $p_B=\underset{i=L_{\mathrm{th}}+1}{\overset{L-1}{\mathrm{\&Sigma;}}}{p}_i=1-p_A;$

The gray average of A and B two classes is respectively:

${\mathrm{\&omega;}}_A=\frac{\underset{i=0}{\overset{L_{\mathrm{th}}}{\mathrm{\&Sigma;}}}{\mathrm{ip}}_i}{p_A},$ ${\mathrm{\&omega;}}_B=\frac{\underset{i=L_{\mathrm{th}}+1}{\overset{L-1}{\mathrm{\&Sigma;}}}{\mathrm{ip}}_i}{p_B};$

Thereby can obtain the total gray average of image:

${\mathrm{\&omega;}}_0=p_A{\mathrm{\&omega;}}_A+p_B{\mathrm{\&omega;}}_B=\underset{i=L_{\mathrm{th}}=1}{\overset{L-1}{\mathrm{\&Sigma;}}}{\mathrm{ip}}_i;$

Can obtain the inter-class variance in A, B two zones thus:

σ ²＝p _A(ω _A-ω ₀) ²+p _B(ω _B-ω ₀) ²；

For obtaining the optimum segmentation threshold value, as criterion, make σ with the inter-class variance of two classes ²The value maximum

Be the optimal threshold of asking:

$L_{\mathrm{th},v}^{*}=\mathrm{Arg}\underset{0\&le;L_{\mathrm{th}}\&le;L-1}{\mathrm{Max}}[p_A{({\mathrm{\&omega;}}_A-{\mathrm{\&omega;}}_0)}^2+p_B{({\mathrm{\&omega;}}_B-{\mathrm{\&omega;}}_0)}^2],V=\mathrm{1,2,3,4,5},$

Calculate the optimal threshold that respectively approaches the reconstruct subgraph Basis again Respectively to S ^vCarry out binaryzation, obtain five bianry images, add up the white portion area of each bianry image, namely obtain the foam total area, be designated as A _v

Step 2: adjacent binary image foam total area is subtracted each other, and obtains foam equivalent dimension feature;

The foam total area of adjacent binary image is subtracted each other, obtain foam subgraph equivalent area, be designated as E _qHave,

E _q＝A _q+1-A _q，q＝1，2，3，4；

E _qBe the foam equivalent dimension feature of trying to achieve by multiple dimensioned binaryzation, subscript q is the number of equivalent dimension feature, and q+1 is for decomposing order; Because decomposing order is 5, then 4 equivalent dimension features are distinguished corresponding 4 continuous foam diameter variation ranges: 1～2,2～4,4～8,8～16; [this scope is that the space-frequency relation between the analysis wavelet subgraph directly obtains, rather than is obtained by area.The diameter here is equivalent diameter, does not therefore give unit.】

Step 3: according to the foam diameter variation range of extrapolating, calculate equivalent foam number, and then obtain foam equivalent dimension distribution plan;

With D _qBe the mean value of foam diameter variation range, the D of 4 foam diameter variation range correspondences _qBe respectively 1.5,3,6,12, obtain the equivalent foam number of each order according to following formula:

Finally obtain the equivalent dimension distribution plan of current working froth images.

5. froth images equivalent dimension Feature Extraction method according to claim 1, it is characterized in that, in the described step 4, the acquisition methods of benchmark distribution plan is: the image by several known operating modes obtains a plurality of foam equivalent dimension distribution plans through step 1 to three earlier, a plurality of foam equivalent dimension distribution plans to gained are added up, thereby obtain the benchmark distribution plan; Described statistics refers to count the foam frequency of each the froth images equivalent diameter under different operating modes, and the benchmark distribution plan is that off-line obtains.

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