A data-driven fractional wavelet transform denoising method

Abstract

The present invention proposes a data-driven fractional wavelet transform denoising method. Based on fractional wavelet denoising, the method replaces all filters with learnable convolution kernels and defines a learnable threshold function activation layer for signal denoising. The fractional scaling filters and fractional wavelet filters in the network and the positive and negative deviations in the threshold function are obtained through a neural network backpropagation algorithm. This method is a signal denoising method that is completely data-driven. Compared with conventional fractional wavelet denoising, data-driven fractional wavelet denoising eliminates the tedious threshold function design process and improves the efficiency of the denoising algorithm.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal and information processing, and in particular relates to a data-driven fractional-order wavelet transform denoising method. Background Art

[0002] Signal denoising is a critical issue in signal processing. In real life and scientific research, many signals contain noise, which can degrade signal quality, hindering analysis, diagnosis, and processing, and reducing signal reliability and accuracy. The goal of signal denoising is to separate noise from the signal, making it clearer and more accurate, thereby improving signal quality and reliability.

[0003] Among the many denoising methods, the denoising method based on fractional wavelet transform further improves the denoising effect on the basis of traditional wavelet denoising. Traditional wavelet denoising methods can effectively remove high-frequency noise in the signal by selecting a suitable wavelet basis and threshold processing method. However, for non-stationary signals, the complexity of their time-frequency characteristics limits the effectiveness of traditional wavelet denoising methods, while denoising methods based on fractional wavelet transform are more suitable for non-stationary signal denoising. Because the fractional wavelet transform has a free parameter, namely the rotation angle, by gradually increasing the rotation angle from zero to ninety degrees, the dynamic characteristics of the signal from the time domain to the time-frequency joint domain can be displayed. This characteristic provides a powerful tool for finely characterizing the time-frequency characteristics of non-stationary signals and accurately identifying and removing noise in the signal.

[0004] For the convenience of analysis, we first give the definition of fractional wavelet transform: for any signal with finite energy Its fractional wavelet transform is defined as:

[0005]

[0006] Among them, the scale parameter Displacement parameters Kernel function ψ α,a,b The expression of (t) is:

[0007]

[0008] In particular, when the rotation angle α = π / 2, the fractional wavelet transform will degenerate into the traditional wavelet transform. In practical applications, data is usually composed of finite values. Therefore, it is necessary to use a discrete form to represent the fractional wavelet transform, that is, to discretize the scale parameter and displacement parameter. Generally, take a = 2 -k , b=n2 -k , then the kernel function of fractional wavelet transform becomes:

[0009]

[0010] For this discrete form of fractional wavelet transform, the fractional wavelet decomposition and synthesis of the signal can be achieved through the convolution cascade algorithm structure. Its fractional scaling coefficients and fractional wavelet coefficients are:

[0011]

[0012]

[0013] If the coefficient If we know and Right now

[0014]

[0015]

[0016] The above process is called the decomposition algorithm of fractional wavelet transform. The following initialization is required:

[0017]

[0018] Among them, s(t) is the original signal to be decomposed.

[0019] Similarly, if the coefficients are known and Give the calculation coefficient The algorithm becomes the fractional wavelet transform synthesis algorithm, that is

[0020]

[0021] Wavelet coefficients obtained by fractional wavelet transform decomposition algorithm contains the details and noise in the signal, while the fractional scaling coefficient The smooth part of the signal is included. By filtering or thresholding the wavelet coefficients of the signal, the noise can be removed while retaining the signal details. The processed decomposition coefficients are synthesized to obtain the denoised signal.

[0022] While fractional wavelet denoising offers numerous advantages and promising applications, it requires manual determination of the appropriate fractional wavelet basis functions and threshold parameters for a specific application scenario, making it difficult to optimally match the data to the desired outcome. Furthermore, current fractional wavelet denoising methods often rely on empirical experience for parameter selection and optimization, failing to adapt to changing data. Summary of the Invention

[0023] The purpose of the present invention is to solve the problem that the fractional wavelet basis function and threshold parameters need to be manually set according to experience in the fractional wavelet denoising method and cannot be adjusted adaptively. A data-driven fractional wavelet transform denoising method is provided.

[0024] The present invention is implemented by the following technical solution. The present invention proposes a data-driven fractional wavelet transform denoising method, which includes the following steps:

[0025] Step 1: Collect training signals s(t) for the application scenario and build a training data set for the network;

[0026] Step 2: Establish a data-driven fractional wavelet transform denoising network, and use the data set to train the denoising network to obtain a trained denoising network;

[0027] The overall structure of the data-driven fractional wavelet transform denoising network consists of a signal decomposition module, a threshold function module, and a signal reconstruction module;

[0028] The signal decomposition module is responsible for decomposing the input signal to obtain fractional scaling coefficients and fractional wavelet coefficients. The signal decomposition module consists of an activation layer and a convolution layer. The signal decomposition module calculates the fractional scaling coefficients and fractional wavelet coefficients of this layer through formula (1) and formula (2).

[0029]

[0030]

[0031] The threshold function module is responsible for threshold processing of the decomposed fractional wavelet coefficients. The threshold function module is composed of a learnable threshold function activation layer, and its learnable threshold activation function is:

[0032]

[0033] in, are positive and negative learnable deviations, respectively, used to control the denoising range of the threshold function. and When , the threshold function module degenerates into a linear activation function;

[0034] The signal reconstruction module is responsible for reconstructing the denoised signal using the decomposition coefficients after threshold processing. The signal reconstruction module consists of a convolution layer and an activation layer. The signal reconstruction module reconstructs the fractional scaling coefficients of this layer using formula (4).

[0035]

[0036] Step 3: Collect the signal to be processed in the training data set constructed in step 1

[0037] Step 4: Use the trained denoising network to denoise the signal collected in step 3.

[0038] Furthermore, the specific steps of training the denoising network using the dataset are as follows:

[0039] Step 2.1: Decompose the angle parameter α in the network and the number of layers L to learn the convolution kernel. The threshold function activation layer Th(x) can be learned for initialization;

[0040] Step 22: Train the learnable parameters in the network.

[0041] Furthermore, the step 21 is specifically as follows:

[0042] (a) Initialize the angle parameter α of the denoising network according to the time-frequency characteristics of the training samples;

[0043] (b) Initialize the number of decomposition layers L of the denoising network based on the length of the training sample and the signal decomposition requirements in the application scenario;

[0044] (c) For all convolution kernels in the network Initialize and intuitively set the initial value of the convolution kernel according to the characteristics of the signal. The recommended initial values ​​are the scale filter and wavelet filter corresponding to the wavelet function;

[0045] (d) Parameter b in the learnable threshold activation function Th(x) in the network - and b + Initialize, the recommended initial value is 0.

[0046] Furthermore, the step 22 is specifically as follows:

[0047] (a) Initialize the signal. As input to the network;

[0048] (b) Use the signal decomposition module to calculate the fractional scaling coefficient c of this layer k and fractional wavelet coefficients d k ;

[0049] (c) If the number of decomposition layers has reached the requirement, the decomposition is completed and all the decomposition coefficients W = [d1, d2, ..., d L ,c L ]; otherwise, return to step (b) and continue to decompose the fractional scaling coefficient output by this layer as input;

[0050] (d) For the fractional wavelet coefficient d in the decomposition coefficient k Perform threshold processing to obtain the processed fractional wavelet coefficients And combine it with the fractional scaling coefficient to obtain the decomposition coefficient for reconstruction

[0051] (e) reconstructing the signal using a signal reconstruction module;

[0052] (f) If the reconstruction is complete, the final output is the denoised signal s[n]; otherwise, the fractional scaling coefficients obtained from the current layer reconstruction and the corresponding fractional wavelet coefficients in the decomposition coefficients are used as the input for the next layer, and the process returns to step (e) to continue reconstruction.

[0053] (g) Calculate the current objective function value using the objective function based on the decomposition coefficients and the denoised signal output from steps (d) and (f). The objective function is:

[0054]

[0055] in, are all the learnable parameters in the network;

[0056] (h) Based on the value of the objective function, all learnable parameters are updated according to formula (6);

[0057]

[0058] Among them, v is the current number of iterations, represents the i-th learnable parameter in the v-th iteration, μ is the learning rate, and λ is the momentum parameter, which is used to adjust the impact of the gradient generated by the previous iteration on the gradient of the current iteration. is the objective function right The gradient value of

[0059] (i) Determine whether the maximum number of iterations has been reached. If so, training is complete. Otherwise, return to step (a) to continue training.

[0060] The beneficial effects of the present invention are:

[0061] This paper proposes a data-driven fractional wavelet transform denoising method. Building on fractional wavelet transform denoising, this method defines a learnable threshold activation function layer, replacing the manually configured filters and threshold functions with learnable parameters. These parameters are updated via a backpropagation algorithm, enabling adaptive adjustment of the fractional wavelet basis functions and threshold parameters based on the input signal's characteristics and corresponding indicators, thereby achieving better signal denoising. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 A block diagram of the steps of the signal denoising method of the present invention;

[0063] Figure 2 This is a structural diagram of the data-driven fractional wavelet transform denoising method;

[0064] Figure 3 This is the principle block diagram of the signal decomposition module k in the denoising network;

[0065] Figure 4 This is the principle block diagram of the threshold processing module k in the denoising network;

[0066] Figure 5 Schematic diagram of the learnable threshold function in the threshold processing module;

[0067] Figure 6 This is the principle block diagram of the signal reconstruction module k in the denoising network;

[0068] Figure 7 The figure is a comparison diagram of the signal after denoising using the method of the present invention and the original signal. DETAILED DESCRIPTION

[0069] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0070] See Figure 1-Figure 7 The present invention proposes a data-driven fractional wavelet transform denoising method, which includes the following steps:

[0071] Step 1: Collect training signals s(t) for the application scenario and build a training data set for the network;

[0072] Step 2: Establish a data-driven fractional wavelet transform denoising network, and use the data set to train the denoising network to obtain a trained denoising network;

[0073] like Figure 2 As shown, the overall structure of the data-driven fractional wavelet transform denoising network consists of a signal decomposition module, a threshold function module, and a signal reconstruction module;

[0074] The signal decomposition module is responsible for decomposing the input signal to obtain fractional scaling coefficients and fractional wavelet coefficients. Figure 3As shown in FIG, the signal decomposition module is composed of an activation layer and a convolution layer; the signal decomposition module calculates the fractional-order scaling coefficient and the fractional-order wavelet coefficient of this layer through formula (1) and formula (2);

[0075]

[0076]

[0077] The threshold function module is responsible for threshold processing of the fractional wavelet coefficients obtained by decomposition. The threshold function module is composed of a learnable threshold function activation layer, such as Figure 4 As shown, its learnable threshold activation function is:

[0078]

[0079] in, are positive and negative learnable deviations, respectively, used to control the denoising range of the threshold function. and When , the threshold function module degenerates into a linear activation function;

[0080] The signal reconstruction module is responsible for reconstructing the denoised signal using the decomposition coefficients after threshold processing. Figure 5 As shown in Figure 1, the signal reconstruction module consists of a convolution layer and an activation layer; the signal reconstruction module reconstructs the fractional scaling coefficient of this layer through formula (4);

[0081]

[0082] Step 3: Collect the signal to be processed in the training data set constructed in step 1

[0083] Step 4: Use the trained denoising network to denoise the signal collected in step 3. The denoising result is as follows: Figure 7 shown.

[0084] The specific steps for training the denoising network using the dataset are as follows:

[0085] Step 2.1: Decompose the angle parameter α in the network and the number of layers L to learn the convolution kernel. The threshold function activation layer Th(x) can be learned for initialization;

[0086] Step 22: Train the learnable parameters in the network.

[0087] The step 21 is specifically as follows:

[0088] (a) Initialize the angle parameter α of the denoising network according to the time-frequency characteristics of the training samples;

[0089] (b) Initialize the number of decomposition layers L of the denoising network based on the length of the training sample and the signal decomposition requirements in the application scenario;

[0090] (c) For all convolution kernels in the network Initialize and intuitively set the initial value of the convolution kernel according to the characteristics of the signal. The recommended initial values ​​are the scaling filter and wavelet filter corresponding to the wavelet function; commonly used wavelets include Morlet wavelet, Mexoicanhat wavelet and Daubechies series wavelet;

[0091] (d) Parameter b in the learnable threshold activation function Th(x) in the network - and b + Initialize, the recommended initial value is 0.

[0092] The step 22 is specifically as follows:

[0093] (a) Initialize the signal. As input to the network;

[0094] (b) Use the signal decomposition module to calculate the fractional scaling coefficient c of this layer k and fractional wavelet coefficients d k ;

[0095] (c) If the number of decomposition layers has reached the requirement, the decomposition is completed and all the decomposition coefficients W = [d1, d2, ..., d L ,c L ]; otherwise, return to step (b) and continue to decompose the fractional scaling coefficient output by this layer as input;

[0096] (d) For the fractional wavelet coefficient d in the decomposition coefficient k Perform threshold processing to obtain the processed fractional wavelet coefficients And combine it with the fractional scaling coefficient to obtain the decomposition coefficient for reconstruction

[0097] (e) reconstructing the signal using a signal reconstruction module;

[0098] (f) If the reconstruction is complete, the final output is the denoised signal s[n]; otherwise, the fractional scaling coefficients obtained from the current layer reconstruction and the corresponding fractional wavelet coefficients in the decomposition coefficients are used as the input for the next layer, and the process returns to step (e) to continue reconstruction.

[0099] (g) Calculate the current objective function value using the objective function based on the decomposition coefficients and the denoised signal output from steps (d) and (f). The objective function is:

[0100]

[0101] in, are all the learnable parameters in the network;

[0102] (h) Based on the value of the objective function, all learnable parameters are updated according to formula (6);

[0103]

[0104] Among them, v is the current number of iterations, represents the i-th learnable parameter in the v-th iteration, μ is the learning rate, and its common value is 0.001. λ is the momentum parameter, and its common value is 0.9, which is used to adjust the influence of the gradient generated by the previous iteration on the gradient of the current iteration. is the objective function right The gradient value of

[0105] (i) Determine whether the maximum number of iterations has been reached. If so, training is complete. Otherwise, return to step (a) to continue training.

Claims

1. A data-driven fractional wavelet transform denoising method, characterized by: The method comprises the following steps: Step 1: Collect training signals s(t) for the application scenario and build a training data set for the network; Step 2: Establish a data-driven fractional wavelet transform denoising network, and use the data set to train the denoising network to obtain a trained denoising network; The overall structure of the data-driven fractional wavelet transform denoising network consists of a signal decomposition module, a threshold function module, and a signal reconstruction module; The signal decomposition module is responsible for decomposing the input signal to obtain fractional scaling coefficients and fractional wavelet coefficients. The signal decomposition module consists of an activation layer and a convolution layer. The signal decomposition module calculates the fractional scaling coefficients and fractional wavelet coefficients of this layer through formula (1) and formula (2). The threshold function module is responsible for threshold processing of the decomposed fractional wavelet coefficients. The threshold function module is composed of a learnable threshold function activation layer, and its learnable threshold activation function is: in, They are positive and negative learnable deviations, respectively, used to control the denoising range of the threshold function. When b k - =0 and When , the threshold function module degenerates into a linear activation function; The signal reconstruction module is responsible for reconstructing the denoised signal using the decomposition coefficients after threshold processing. The signal reconstruction module consists of a convolution layer and an activation layer. The signal reconstruction module reconstructs the fractional scaling coefficients of this layer using formula (4). Step 3: Collect the signal to be processed in the training data set constructed in step 1 Step 4: Use the trained denoising network to denoise the signal collected in step 3.

2. The method according to claim 1, characterized in that The specific steps for training the denoising network using the dataset are as follows: Step 2.1: Decompose the angle parameter α in the network and the number of layers L to learn the convolution kernel. The threshold function activation layer Th(x) can be learned for initialization; Step 22: Train the learnable parameters in the network.

3. The method according to claim 2, characterized in that The step 21 is specifically as follows: (a) Initialize the angle parameter α of the denoising network according to the time-frequency characteristics of the training samples; (b) Initialize the number of decomposition layers L of the denoising network based on the length of the training sample and the signal decomposition requirements in the application scenario; (c) For all convolution kernels in the network Initialize and intuitively set the initial value of the convolution kernel according to the characteristics of the signal. The recommended initial values ​​are the scale filter and wavelet filter corresponding to the wavelet function; (d) Parameter b in the learnable threshold activation function Th(x) in the network - and b + Initialize, the recommended initial value is 0.

4. The method according to claim 3, characterized in that The step 22 is specifically as follows: (a) Initialize the signal. As input to the network; (b) Use the signal decomposition module to calculate the fractional scaling coefficient c of this layer k and fractional wavelet coefficients d k ; (c) If the number of decomposition layers has reached the requirement, the decomposition is completed and all the decomposition coefficients W = [d1, d2, ..., d L ,c L ]; Otherwise, return to step (b) and continue to decompose the fractional scaling coefficient output by this layer as input; (d) For the fractional wavelet coefficient d in the decomposition coefficient k Perform threshold processing to obtain the processed fractional wavelet coefficients And combine it with the fractional scaling coefficient to obtain the decomposition coefficient for reconstruction (e) reconstructing the signal using a signal reconstruction module; (f) If the reconstruction is complete, the final output is the denoised signal s[n]; otherwise, the fractional scaling coefficients obtained from the current layer reconstruction and the corresponding fractional wavelet coefficients in the decomposition coefficients are used as the input for the next layer, and the process returns to step (e) to continue reconstruction. (g) Calculate the current objective function value using the objective function based on the decomposition coefficients and the denoised signal output from steps (d) and (f). The objective function is: in, are all the learnable parameters in the network; (h) Based on the value of the objective function, all learnable parameters are updated according to formula (6); Among them, v is the current number of iterations, represents the i-th learnable parameter in the v-th iteration, μ is the learning rate, and λ is the momentum parameter, which is used to adjust the impact of the gradient generated by the previous iteration on the gradient of the current iteration. is the objective function right The gradient value of (i) Determine whether the maximum number of iterations has been reached. If so, training is complete. Otherwise, return to step (a) to continue training.

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