A multiscale decomposition method for undirected graph signals

Abstract

The application discloses a multiscale decomposition method of an undirected graph signal and belongs to the field of information processing. The application proposes a dynamic multiscale graph signal decomposition method based on a walk adjacency matrix for any graph signal. The method adopts a dynamic connection mode of an adjacent point based on a dynamic walk condition of the walk adjacency matrix, can realize dynamic definition of a current graph extreme point, provides a dynamic scale limit for multiscale decomposition of the graph signal, can realize multiscale decomposition of any graph signal, provides a necessary basis for multiscale processing of the graph signal, realizes dynamic controllable multiscale decomposition of the graph signal, and provides a technical basis for flexible decomposition and application of the graph signal.

Description

Technical Field

[0001] This invention pertains to the field of information processing, specifically to the decomposition of undirected graph signals, and relates to a multi-scale decomposition method for undirected graph signals. Background Technology

[0002] As is well known, the study of graphs can be traced back to Euler's Seven Bridges of Königsberg problem centuries ago. To date, graph theory, as an important branch of mathematical science and information theory, has proven to be a key area of ​​scientific research. With the rise of the Internet of Things (IoT), the interconnection of everything is becoming a future trend. Take urban temperature monitoring as an example: thermometers monitoring temperatures in various corners of the city are connected via a monitoring network. Each thermometer is a graph node, and the temperature measured by the thermometer is the data (signal) at that node. The temperature data measured by all the thermometers connected to the network (graph) in a city constitutes a complete graph signal (these signals no longer need to be arranged according to a time order rule like traditional one-dimensional signals, or according to a spatial order rule like an image). Furthermore, the rise of online social networks on the internet has made virtual relationships bridges for information dissemination, forming connections. The information of individual internet users becomes these discrete grid data points; the massive amounts of data sent by ubiquitous intelligent monitoring sensor networks are the flowing signals in the graph; and the increasingly sophisticated neural networks of the brain are also a natural graph domain, with the transmission of neural signals between nodes representing the natural flow of graph signals. In practice, traditional regular signals can be considered special cases of graph signals, and the ideas of graph signal processing can be applied back to traditional signal processing, including image processing. For example, constructing similarity graphs based on the similarity between data can enable some functions in artificial intelligence. In image processing, the idea of ​​constructing different similarity graphs based on the similarity of neighboring regions around pixels and then improving these graphs has become one of the most typical approaches in image processing.

[0003] Graph signals generally consist of three parts: graph nodes, graph edges, and data on the nodes. Based on the direction of the graph signal flow, they are further divided into directed graphs and undirected graphs. See [link to relevant documentation]. Figure 1 The example model of undirected graph signals is shown in the figure. Black dots represent the nodes of the graph, blue lines represent the edges of the graph, the nodes and edges constitute the graph (i.e., the domain of the graph signal), and the vertical red lines represent the numerical values ​​on the corresponding graph nodes (i.e., the graph signal).

[0004] Multiscale image decomposition plays a crucial role in image signal processing. In fact, as a special case of image signal decomposition, multiscale image decomposition (as a regularly sampled image signal) separates image components of different scales or image information with coarse contours and details. In recent years, various image decomposition methods have emerged. Among all these works, two types of multiscale image decomposition exist: fixed-basis image decomposition (FBID) and adaptive image decomposition (AID). FBID methods can be traced back to the original Laplacian pyramid decomposition and later wavelet decomposition (including traditional wavelet decomposition and variants such as ridgelet, bandlet, etc., to name just a few). These mentioned multiscale image decomposition methods are also known as multiresolution decomposition. The main idea of ​​this type of decomposition is to determine how much information (i.e., basis coefficients) is contained in the image. However, once the decomposition basis is given and fixed, the decomposition cannot be matched to various image scales.

[0005] Similarly, the ideas of these mentioned multi-scale image decompositions can be applied to the decomposition of general graph signals (e.g., irregularly sampled graph data). To avoid the drawbacks of fixed bases, several basis-free AID methods have been proposed. In other words, these AID decomposition methods are adaptive and data-driven. AID decomposition methods mainly include BEMD (Two-Dimensional Empirical Mode Decomposition) and its respective variants. BEMD is a direct extension of 1D EMD to 2D images, possessing both adaptability and data-driven characteristics. In classic BEMD methods, the interpolation functions, such as radius basis functions, spline functions, and trigonometric interpolation functions, are different. Apart from the different interpolation functions, a few BEMD methods are almost identical. Considering the shortcomings of these BEMD methods, some improved methods are introduced for different situations. For example, ASBEMD (Signal-Aided BEMD) is proposed to eliminate the drawback of insufficient extrema. SBEMD (Structure BEMD) aims to decompose images with strong structure through carefully designed structural extrema. DBEMD (Orientation BEMD) is used for oriented images. NLBEMD (Limited Neighborhood BEMD) aims to eliminate overshoot and speckles. GBEMD (Generalized BEMD) is used for decompositions with edge preservation and enhancement. In contrast, VMD (Variational Mode Decomposition) is used for smoother interpolation in image processing and other applications.

[0006] However, all of the BEMD methods mentioned above, and their improved variants, share a limitation: while the extrema are data-driven, this fully adaptive data-driven approach ignores the dynamic changes in scale by only searching for local maxima and minima, meaning it cannot achieve data-driven decomposition of dynamic extrema. In other words, the extremum scale cannot be controlled in the above methods, so sometimes the decomposition cannot be effectively controlled as required. Summary of the Invention

[0007] To address the aforementioned problems, this invention proposes a multi-scale decomposition method for graph signals, which is applicable not only to regular graph signals such as images, but also to irregular graph signals such as grids.

[0008] The technical solution of the present invention is as follows:

[0009] In a first aspect, the present invention proposes a multi-scale decomposition method for undirected graph signals, comprising the following steps:

[0010] 1) Obtain the original undirected graph signal data v(t) to be decomposed; initialize the first loop threshold C, the second loop threshold Y, the first loop parameter c = 0, and the second loop parameter y = 0; initialize the decomposition parameter r. v (t) = v(t);

[0011] 2) Calculate r v All graphical minima and graphical maxima in (t);

[0012] 3) Interpolate the graph minimum and the graph maximum respectively to obtain the lower envelope and the upper envelope, and calculate the average envelope;

[0013] 4) From r v Subtracting the average envelope from (t) yields the decomposed component C. o (t);

[0014] 5) Let c ← c+1, and determine whether the first loop parameter c has reached the first loop threshold C:

[0015] If so, then decompose component C o (t) belongs to the graph IMF, let y←y+1, and execute step 6);

[0016] If not, update the decomposition parameter r. v (t)=C o (t), return to step 2);

[0017] 6) Calculate the IMF component C obtained from the y-th decomposition. g,y (t)=C o (t);

[0018] 7) Calculate the remaining amount r v (t)←r v (t)-C g,y (t);

[0019] 8) Determine if the second loop parameter y has reached the second loop threshold Y:

[0020] If so, output the multi-scale decomposition results;

[0021] If not, return to step 2).

[0022] Furthermore, the multi-scale decomposition result is expressed as follows:

[0023]

[0024] Among them, C g,y (t) represents the IMF component obtained from the y-th decomposition, r v (t) represents the final remaining amount.

[0025] Furthermore, step 2 specifically includes:

[0026] Based on the decomposition parameter r v Given an undirected graph structure (t), calculate the adjacency matrix for K walks.

[0027] Calculate the decomposition parameter r v All graph extrema points in (t) based on the adjacency matrix of K walks:

[0028] if Then v m It is a graph maximum point based on the adjacency matrix of K-step walks, where {v l},l≠m represents the set of nodes that are connected to node m in the adjacency matrix of the K-times walk by a number of walks ≥1;

[0029] if Then v m It is a graph minimum point based on the adjacency matrix of K-times walks.

[0030] Furthermore, the formula for calculating the adjacency matrix of K walks is as follows:

[0031] B K =A + ... + A K-1 +A K

[0032] Among them, B K Let A represent the adjacency matrix after K walks, and let A denote the decomposition parameter r. v The adjacency matrix corresponding to the undirected graph structure of (t), A K It represents the Kth power of the adjacency matrix A.

[0033] Furthermore, in step 3), a cubic interpolation function is used to interpolate the graph minimum and the graph maximum respectively.

[0034] Furthermore, the first loop threshold C ranges from 5 to 10.

[0035] Secondly, this invention proposes an application of the above-mentioned multi-scale decomposition method of undirected graph signals in the field of temperature monitoring. The original undirected graph signal data to be decomposed is the temperature signal measured by thermometers arranged at various locations in the target area. The connection relationship between each thermometer through the monitoring network cable constitutes the adjacency matrix of the undirected graph, and each thermometer constitutes the node set of the undirected graph.

[0036] Furthermore, the multi-scale decomposition results include temperature signals from high frequency to low frequency, which are then used for temperature pattern recognition in the target area after denoising and / or enhancement processing of the temperature signals at different frequencies.

[0037] The beneficial effects of this invention are as follows: This invention proposes a dynamic multi-scale graph signal decomposition method based on a walking adjacency matrix for arbitrary graph signals. By using a dynamic walking pattern based on the walking adjacency matrix to dynamically connect adjacent points, it is possible to dynamically define the current graph extrema, providing dynamic scale constraints for multi-scale graph signal decomposition. This enables multi-scale decomposition of arbitrary graph signals, providing a necessary foundation for multi-scale graph signal processing, realizing dynamically controllable multi-scale decomposition of graph signals, and providing a technical basis for flexible graph signal decomposition and its applications. Attached Figure Description

[0038] Figure 1 This is an example model of an undirected graph signal shown in an embodiment of the present invention;

[0039] Figure 2 This is an undirected graph G obtained using urban temperature monitoring as an example, as shown in an embodiment of the present invention;

[0040] Figure 3 These are examples of different connection relationships of graph G after different walk steps of the adjacency matrix shown in the embodiments of the present invention (the blue numbers on the left indicate the number of walk connections), where (a) is the adjacency matrix connection relationship after one walk, (b) is the adjacency matrix connection relationship after two walks, and (c) is the adjacency matrix connection relationship after three walks.

[0041] Figure 4 This is an example of undirected graphical multi-scale decomposition of a temperature micro-change signal shown in an embodiment of the present invention, wherein (a) is the original graphical temperature signal, and (b) is the graphical component C. g,1 (c) Component C g,2 (d) Component C g,3 (e)Graph decomposition of residual quantity r v (t). Detailed Implementation

[0042] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of the present invention. However, the present invention can be practiced in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below. Technical features in the various embodiments of the present invention can be combined accordingly without mutual conflict.

[0043] This invention utilizes the dynamic graph signal extrema defined by the walking adjacency matrix to extend the traditional empirical pattern decomposition idea based on extrema points to graph signals to obtain the upper and lower envelopes of the graph signal, thereby iteratively obtaining an effective multi-scale decomposition of the graph signal.

[0044] Taking urban temperature monitoring as an example, thermometers monitoring temperatures in various corners of the city are connected via a monitoring network. The temperature data measured by all the thermometers connected to the network in a city (as shown in the diagram) constitutes a complete graphical signal, such as... Figure 2 As shown, it is an irregular grid diagram, where each thermometer is a node and the temperature measured by the thermometer is the data (signal) on that node.

[0045] from Figure 2 It can be seen that its adjacency matrix is ​​A:

[0046]

[0047] In this matrix, an element of 1 indicates that the two nodes (thermometers) are connected, while an element of 0 indicates that the two nodes (thermometers) are not connected.

[0048] The walking adjacency matrix is ​​defined as follows:

[0049] B K =A+A 2 +...+A K

[0050] Where K represents the number of steps. When K = 1, 2, and 3, we can obtain the adjacency matrix for a single-step walk, the adjacency matrix for a double-step walk, and the adjacency matrix for a triple-step walk, as follows:

[0051]

[0052]

[0053]

[0054] Wherein, the value b in the matrix mn This represents the number of times nodes m and n are connected by traversal.

[0055] Let v(t) = [v1(t)v2(t)...v M (t)] T Let G be the undirected graph to be decomposed signal, where the superscript T indicates transpose, and v m (t) represents the signal value of node m at time t. In this embodiment, the multi-scale decomposition method has the following steps:

[0056] Step 1: Define the extreme point.

[0057] For a given undirected graph G = {V} e ,A}, where V e (V e = {1, 2, ..., M}, where M is the number of nodes, is the set of nodes (vertices), and A is the adjacency matrix of G with K walks, and the adjacency matrix B is the adjacency matrix of G. K =A+A 2 +...+A K The graph shows the signal v(t) = [v1(t)v2(t)...v M (t)] T The node order in (time variable t can be omitted) is the same as in V. e ,but:

[0058] if Then v m It is B K The maximum value in a meaningful sense, where {v l}, l≠m indicates that the adjacency matrix B with node m in K walks is... K The set of nodes whose mid-travel connections have a count ≥ 1, i.e., v l B represents the adjacency matrix after K walks. K The graph signal values ​​of the non-zero elements in the m-th row;

[0059] Similarly, if Then v m It is B K The minimum value in a meaningful sense.

[0060] Step 2: Iteratively calculate the multi-scale decomposition results.

[0061] (1) Initialization: r v (t)=v(t), c=0.

[0062] (2) Determine r by defining the extreme points of the graph (step 1). v All graphical minima and graphical maxima in (t).

[0063] (3) Use a graph interpolation function (e.g., a cubic interpolation function) to interpolate the graph minimum and maximum values ​​respectively to obtain the lower envelope E. l(t) and upper envelope E u (t).

[0064] (4) Calculate the lower envelope E l (t) and upper envelope E u The average envelope EN of (t) avg (t), that is

[0065] (5) From r v Subtract the average envelope EN from (t) avg (t), yielding the decomposed components: C o (t)=r v (t)-EN avg (t), and let c←c+1.

[0066] (6) Determine whether c is greater than or equal to the threshold C. If so, then the component C... o (t) belongs to the graph IMF (Intrinsic Mode Function), let r v (t)=C o If (t), proceed to step (2); otherwise, proceed to step (7). In this embodiment, C is generally set to 5-10.

[0067] (7) Obtain IMF components: C g (t)=C o (t).

[0068] (8) Calculate the remaining amount: r v (t)←r v (t)-C g (t), repeat steps (2) to (8) until the iteration loop is Y times (Y is the number given in advance) and then proceed to step (9).

[0069] (9) Output the multi-scale decomposition results: Where C g,y (t) represents the IMF component obtained in the y-th decomposition, where Y is the number of iterative decompositions, and r v (t) represents the remaining amount.

[0070] Down Figure 4This example illustrates the decomposition of an undirected graph signal at multiple scales in this embodiment. The graph G is a simulated urban temperature sensing network, with physical network lines as connections. The noise simulates minute temperature changes. A fixed walk count of 3 (K=3) is used, and the number of components (loop count Y=3) is empirically determined. The multi-scale decomposition result contains temperature signals from high to low frequencies. After decomposition, various subsequent operations can be performed on the graph signal, including denoising, enhancement, and recognition. For example, denoising and / or enhancing temperature signals of different frequencies can be used for temperature pattern recognition in the target area.

[0071] This invention is applicable to any graphical signal, and is not limited to the above-mentioned temperature monitoring field.

[0072] The above examples are merely specific embodiments of the present invention. Obviously, the present invention is not limited to the above embodiments and many variations are possible. All variations that can be directly derived or conceived by those skilled in the art from the disclosure of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A multi-scale decomposition method for undirected graph signals, characterized in that, Includes the following steps: 1) Obtain the original undirected graph signal data to be decomposed. ; Initialize the first loop threshold C, the second loop threshold Y, the first loop parameter c=0, and the second loop parameter y=0; initialize the decomposition parameters. ; 2) Calculation All the graph minima and graph maxima in the graph are as follows: Based on decomposition parameters For an undirected graph structure, the adjacency matrix for K-step walks is calculated using the following formula: ； in, Let A represent the adjacency matrix after K walks, and let A represent the decomposition parameters. The adjacency matrix corresponding to the undirected graph structure. Represents the Kth power of the adjacency matrix A; Calculate decomposition parameters All graph extrema based on the adjacency matrix of K-order walks: if ,but These are the graph maxima points based on the adjacency matrix of K-times walks, where... Let m be the set of nodes that are connected to node m in the adjacency matrix of the K-times walk by a number of times greater than or equal to 1. if ,but These are graph minima based on the adjacency matrix of K-step walks; 3) Interpolate the graph minimum and the graph maximum respectively to obtain the lower envelope and the upper envelope, and calculate the average envelope; 4) From Subtracting the average envelope from the middle yields the decomposed components. ; 5) Order Determine whether the parameter c of the first loop reaches the threshold C of the first loop: If so, then decompose into components. Belongs to the IMF chart, making Execute step 6). If not, update the decomposition parameters. (Return to step 2) 6) Calculate the IMF components obtained from the y-th decomposition. ; 7) Calculate the remaining amount ; 8) Determine if the second loop parameter y reaches the second loop threshold Y: If so, output the multi-scale decomposition results; If not, return to step 2).

2. The multi-scale decomposition method for undirected graph signals according to claim 1, characterized in that, The multi-scale decomposition results are expressed as follows: ； in, Let y be the IMF component obtained from the y-th decomposition. This represents the final remaining amount.

3. The multi-scale decomposition method for undirected graph signals according to claim 1, characterized in that, In step 3), a cubic interpolation function is used to interpolate the graph minimum and the graph maximum respectively.

4. The multi-scale decomposition method for undirected graph signals according to claim 1, characterized in that, The first cycle threshold C ranges from 5 to 10.

5. An application of the multi-scale decomposition method for undirected graph signals according to any one of claims 1-4 in the field of temperature monitoring, characterized in that, The original undirected graph signal data to be decomposed consists of temperature signals measured by thermometers placed at various locations in the target area. The connection relationship between each thermometer through the monitoring network cable forms the adjacency matrix of the undirected graph, and each thermometer constitutes the node set of the undirected graph.

6. The application according to claim 5, characterized in that, The multi-scale decomposition results include temperature signals from high frequency to low frequency. After denoising and / or enhancing the temperature signals of different frequencies, they are used for temperature pattern recognition in the target area.

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