Marine electromagnetic signal denoising method based on wavelet transform

Abstract

The invention discloses an ocean electromagnetic signal denoising method based on wavelet transform, which comprises the following steps of: performing wavelet decomposition on a noisy ocean electromagnetic signal to obtain a wavelet coefficient of each layer; carrying out threshold calculation on the high-frequency wavelet coefficient and obtaining soft and hard threshold functions for processing the wavelet coefficient; and performing self-adaptive threshold processing on the high-frequency wavelet coefficient, namely quantifying the high-frequency wavelet coefficient by adopting a self-adaptive threshold function which dynamically adjusts the characteristics of a soft threshold and a hard threshold by adjusting a parameter k belonging to [0, 1], and when k is equal to 0, the self-adaptive threshold function is equivalent to the soft threshold function, when k is equal to 1, the self-adaptive threshold function is equivalent to the hard threshold function, and when k is equal to 0, the self-adaptive threshold function is equivalent to the hard threshold function. To eliminate discontinuity and constant deviation at the threshold; 4, performing inverse wavelet transform on the processed wavelet coefficient, and reconstructing to obtain a de-noised ocean electromagnetic signal; not only are the signal-to-noise ratio and definition of the signals significantly improved, but also a more reliable and accurate data basis is provided for real-time monitoring, accurate analysis and subsequent scientific research of the marine electromagnetic signals.

Description

[0001] Methodology

[0002] The present invention relates to a method for denoising ocean electromagnetic signals, in particular to a method for denoising ocean electromagnetic signals based on wavelet transform.

[0003] Background Methods

[0004] Signal processing methods are a crucial cornerstone of today's information age. They involve the acquisition, transmission, analysis, and processing of various signals (such as sound, images, and electromagnetic signals). With advances in technology, signal processing methods have gained widespread application in fields such as communications, radar, sonar, seismic exploration, and medical imaging. However, in practical applications, signals are often interfered with by various noises, resulting in a degradation of signal quality and hindering subsequent analysis and processing. Therefore, signal denoising has become a crucial topic in signal processing.

[0005] In the field of signal processing, while traditional Fourier transforms can analyze the frequency components of a signal, they lack localization in the time domain, making it difficult to detect localized signal mutations. In contrast, the wavelet transform, as a new time-frequency analysis method, possesses multi-scale analysis capabilities, enabling it to characterize non-stationary features of a signal at different scales, such as edges, spikes, and discontinuities. This characteristic gives the wavelet transform a significant advantage in signal denoising.

[0006] Marine electromagnetic signals are a crucial physical quantity in the marine environment, containing a wealth of information about marine geology and ecology. However, due to the complexity and diversity of the marine environment, marine electromagnetic signals are often subject to interference from various sources, including background noise, equipment noise, and human interference. This noise not only reduces the signal-to-noise ratio and clarity of the signals, but also impairs subsequent data analysis and scientific research. Therefore, denoising marine electromagnetic signals to improve signal quality is a critical requirement in the field of marine electromagnetic detection and data analysis. Summary of the Invention

[0007] The present invention aims to solve the problem that the extraction of ocean electromagnetic signals is often interfered with by various noises, which not only reduces the signal-to-noise ratio and clarity of the signal, but also affects subsequent data analysis and scientific research.

[0008] A method for denoising ocean electromagnetic signals based on wavelet transform comprises the following steps:

[0009] Step 1: Perform wavelet decomposition on the noisy ocean electromagnetic signal to obtain the wavelet coefficients of each layer;

[0010] Step 2: Calculate the threshold of high-frequency wavelet coefficients and obtain the soft and hard threshold functions for wavelet coefficient processing;

[0011] Step 3: Adaptive threshold processing is performed on the high-frequency wavelet coefficients, including: quantizing the high-frequency wavelet coefficients using an adaptive threshold function, wherein the adaptive threshold function dynamically adjusts the characteristics of the soft threshold and the hard threshold by adjusting the parameter k∈[0,1]. When k=0, it is equivalent to the soft threshold function, and when k=1, it is equivalent to the hard threshold function, so as to eliminate discontinuity and constant deviation at the threshold.

[0012] Step 4: Perform inverse wavelet transform on the processed wavelet coefficients to reconstruct the denoised ocean electromagnetic signal.

[0013] Furthermore, the wavelet decomposition in step 1 includes: selecting wavelet basis functions with orthogonality, compact support and symmetry, and determining the number of decomposition layers according to noise distribution characteristics, and performing multi-scale decomposition on the signal;

[0014] The basis for determining the number of decomposition layers is as follows: if the noise in the signal is mainly concentrated in the high-frequency part, choose a larger number of decomposition layers to better separate the signal and noise; conversely, if the noise is also largely distributed in the low-frequency part, then choose a smaller number of decomposition layers to avoid excessive decomposition and signal distortion.

[0015] Furthermore, the method for calculating the threshold value of the high-frequency wavelet coefficients in step 2 is as follows: the square of the coefficients after wavelet decomposition has the dimension of energy. Therefore, the decomposed wavelet coefficients are squared and sorted, a threshold is given, the corresponding risk value is calculated, that is, its likelihood estimate is obtained, and non-likelihood minimization is performed to obtain the selected threshold value. The specific method is:

[0016] (1) Arrange the wavelet transformed coefficients of each layer from small to large by squaring them to obtain a vector w = [w1, w2, ..., w n ], where w1≤w2≤…≤w n , n is the number of wavelet coefficients;

[0017] (2) Calculate the risk vector R = [r1, r2, ..., r n ],but Among them, r i To introduce the risk vector element, the above formula is iterated multiple times to obtain the minimum r i , recorded as r0, and find the corresponding w i Denoted as w0;

[0018] (3) Calculate the threshold λ = σ(w0) 1 / 2 .

[0019] Furthermore, the method for obtaining the hard threshold function for processing the wavelet coefficients in step 2 is:

[0020] The hard threshold function is:

[0021]

[0022] Where: w j,k represents the wavelet coefficients of signal decomposition, represents the estimated value of the wavelet coefficient obtained by the threshold method, and λ is the threshold;

[0023]

[0024] Where σ is the noise standard deviation, which is estimated using the following empirical formula:

[0025]

[0026] The absolute value of the coefficients after wavelet decomposition is compared with the threshold λ. Points less than the threshold become 0, and points greater than or equal to the threshold retain their original values. During the hard threshold processing, since the hard threshold function is discontinuous in the entire wavelet area (there may be discontinuities at λ and -λ, causing oscillation of the reconstructed signal), it is necessary to improve the continuity of the wavelet coefficient values.

[0027] Furthermore, the noise standard deviation σ can also be estimated by using the median absolute deviation:

[0028] Right now where median(|w j，k |) is the median of the absolute values of the wavelet coefficients, and 0.6745 is the normalization constant.

[0029] Furthermore, the method for obtaining the soft threshold function for processing the wavelet coefficients in step 2 is:

[0030]

[0031] w j,k represents the wavelet coefficients of signal decomposition, It represents the estimated value of the wavelet coefficient obtained by the threshold method. After the soft threshold method is processed, the wavelet coefficient value has good continuity and no discontinuity problem, which is easy to process. However, when the wavelet coefficient is large, the estimated wavelet coefficient value has a fixed deviation from the original wavelet coefficient value, which will also bring inevitable errors to the reconstructed signal. In addition, the soft threshold adopts a constant value compression for the wavelet coefficient greater than the threshold, which is inconsistent with the trend that the noise component gradually decreases as the wavelet coefficient increases.

[0032] Furthermore, the adaptive threshold function in step 3 is:

[0033]

[0034] Where w j,k represents the wavelet coefficients of signal decomposition, represents the estimated value of wavelet coefficients obtained by the threshold method, m, n, k are adjustment factors of the improved threshold function, which enhance the flexibility of the threshold function;

[0035] Parameters m and n determine the form of the threshold function. The value of parameter k is between 0 and 1. If k is 0, the threshold function is equivalent to a soft threshold function. If k is 1, the threshold function is equivalent to a hard threshold function.

[0036] The adjustable parameter k can overcome the discontinuity of the hard threshold function and the constant deviation of the soft threshold function when processing wavelet coefficients, while also retaining the original advantages of soft and hard thresholds;

[0037] The improved threshold function has infinite order continuous derivatives, which provides a basis for the selection of wavelet adaptive threshold.

[0038] Furthermore, the inverse wavelet transform in step 4 includes: combining the high-frequency coefficients after threshold processing with the unprocessed low-frequency coefficients, and restoring the signal layer by layer through multi-scale recursive reconstruction, specifically including: using multi-scale noise suppression and signal enhancement to suppress ocean electromagnetic noise

[0039] Perform wavelet transform on the signal and decompose it into J layers;

[0040] Perform thresholding on the wavelet coefficients:

[0041]

[0042] Where: is the wavelet coefficient of the subband x position in the j-scale d-direction, is the multi-scale product of the noisy wavelet coefficients at position x in the sub-band of scale j and direction d; T jd is the threshold for the multi-scale product; f(x) is the original signal to be processed; the threshold of this method does not act directly on the wavelet coefficients, but on the multi-scale product.

[0043] Furthermore, the inverse wavelet transform in step 4 further includes reconstructing the signal using an inverse wavelet transform, including: the formula of the inverse wavelet transform is as follows:

[0044]

[0045] Among them, j represents the scale parameter, c j-1，m is the scale coefficient of the scale space, h is the low-pass filter coefficient; g is the high-pass filter coefficient, c j，k is the low-frequency coefficient at scale j, d j，k is the high-frequency coefficient at scale j; k is the current scale translation parameter, and m is the previous scale translation parameter;

[0046] Performing inverse wavelet transform specifically includes the following aspects:

[0047] (1) Start from the highest layer: Start from the highest layer (lowest frequency) of the wavelet decomposition and use the wavelet coefficients of this layer to perform inverse transform; this layer usually represents the low-frequency components or contours of the signal;

[0048] (2) Layer-by-layer recursion: Next, the inverse transform result of the previous layer is combined with the wavelet coefficients of the current layer (slightly higher frequency), and the inverse transform is performed again; this process is repeated layer by layer until the lowest layer (highest frequency) of the decomposition is reached;

[0049] (3) Reconstructing detail coefficients: In each layer, in addition to low-frequency coefficients, there are also high-frequency detail coefficients (such as horizontal, vertical, and diagonal details); these coefficients also need to be reconstructed through inverse transformation and combined with the low-frequency components;

[0050] (4) Superimpose the reconstruction results: Superimpose the results obtained from each layer of inverse transformation to form the final denoised signal; this process ensures that the information of the signal at all scales is accurately restored.

[0051] In general, the above method scheme conceived by the present invention can achieve the following beneficial effects compared with the existing methods:

[0052] (1) The ocean electromagnetic signal denoising method based on wavelet transform proposed in the present invention, through the ocean electromagnetic noise analysis method based on wavelet transform proposed in this patent, by utilizing the multi-scale analysis characteristics of wavelet transform, the complex ocean electromagnetic signal is finely decomposed into wavelet coefficients of different frequencies and positions. This process not only reveals the detailed characteristics of the signal at different scales, but also provides a solid foundation for subsequent noise removal.

[0053] (2) The ocean electromagnetic signal denoising method based on wavelet transform of the present invention can effectively filter out the noise components while ensuring that the key useful information in the signal is fully preserved by implementing precise threshold processing or other advanced denoising strategies on these wavelet coefficients.

[0054] (3) The present invention's wavelet-based ocean electromagnetic signal denoising method utilizes the powerful capability of inverse wavelet transform to accurately reconstruct the processed wavelet coefficients into denoised ocean electromagnetic signals. This process not only significantly improves the signal-to-noise ratio and clarity of the signal, but also provides a more reliable and accurate data foundation for real-time monitoring, accurate analysis, and subsequent scientific research of ocean electromagnetic signals. Furthermore, the method's wide application potential also makes it show great value and prospects in multiple fields such as geophysical exploration, seismic monitoring, and image processing. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1This is a flow chart of a method for denoising ocean electromagnetic signals based on wavelet transform according to an embodiment of the present invention;

[0056] Figure 2 This is a flow chart of wavelet decomposition of ocean electromagnetic signals according to an embodiment of the present invention;

[0057] Figure 3 This is a flow chart of performing inverse wavelet transform according to an embodiment of the present invention;

[0058] Figure 4 is a noisy signal diagram according to an embodiment of the present invention;

[0059] Figure 5 This is a hard threshold denoising image according to an embodiment of the present invention;

[0060] Figure 6 This is a denoised image using the soft thresholding method according to an embodiment of the present invention;

[0061] Figure 7 This is a denoising image using the adaptive threshold method according to an embodiment of the present invention. DETAILED DESCRIPTION

[0062] In order to make the objects, methods, solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are only intended to illustrate the present invention and are not intended to limit the present invention. In addition, the method features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0063] In order to make the purpose, method and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0064] It should be noted that the illustrations provided in the embodiments of the present invention are only schematic illustrations of the basic concept of the present invention. Therefore, the drawings only show components related to the present invention and are not drawn according to the number, shape and size of components in actual implementation. In actual implementation, the type, quantity and proportion of each component can be changed at will, and the component layout type may also be more complicated.

[0065] Reference Figure 1 , the overall analysis method of the embodiment of the present invention is as follows:

[0066] (1) Select appropriate wavelet basis functions and decomposition layers, perform wavelet decomposition on the noisy ocean electromagnetic signal, and obtain the wavelet coefficients of each layer.

[0067] Step 1: Select wavelet basis function

[0068] Ensure the orthogonality of the wavelet basis functions. Orthogonality minimizes the correlation between the frequency bands obtained from wavelet packet decomposition, eliminating information redundancy. Furthermore, the wavelet packet decomposition of the vibration signal should ensure that all frequency bands combined cover the entire frequency band occupied by the original signal, and that there should be no overlap between frequency bands in the frequency domain. Orthogonality simplifies analysis and facilitates accurate signal reconstruction. In wavelet transforms, orthogonality ensures that wavelet coefficients at different scales do not interfere with each other, thereby improving the accuracy of signal reconstruction.

[0069] Ensure the localization of the wavelet basis function in the time domain. This requires the wavelet basis function to have a high vanishing moment and compact support. A high vanishing moment indicates that the wavelet basis function decays to zero quickly, resulting in better resolution in the time domain. Compact support is also an important property of the wavelet basis function; the narrower the compact support width, the faster the decay, and the better the localization of the wavelet basis function.

[0070] Clarify the lossless nature of signal reconstruction. Wavelet packet decomposition allows for the decomposition of the original signal into frequency bands without redundancy or omission. Symmetrical basis functions ensure linear phase in wavelet filtering, preventing signal distortion and increasing the method's speed. Symmetry is crucial for maintaining the signal's phase characteristics, particularly in image processing, where edge information must be preserved. When processing marine electromagnetic signals, it is important to select wavelet basis functions that can effectively characterize both signal abrupt changes and detailed information.

[0071] Step 2: Determine the number of decomposition levels

[0072] The number of decomposition levels significantly impacts the effectiveness of wavelet decomposition. A larger number of decomposition levels improves the accuracy of frequency band division and facilitates precise energy extraction of signals within each frequency band. However, increasing the number of decomposition levels also increases computational complexity. Therefore, a trade-off between frequency band division accuracy and computational complexity is necessary when selecting the number of decomposition levels.

[0073] For noisy ocean electromagnetic signals, the appropriate number of decomposition layers can be determined by observing the signal's characteristics and the distribution of the noise. Generally speaking, if the noise in the signal is primarily concentrated in the high-frequency portion, a larger number of decomposition layers can be selected to better separate the signal from the noise. Conversely, if the noise is also significantly distributed in the low-frequency portion, a smaller number of decomposition layers may be necessary to avoid excessive decomposition and signal distortion.

[0074] Step 3: Perform wavelet decomposition

[0075] After determining the wavelet basis functions and the number of decomposition layers, wavelet decomposition is performed on the noisy ocean electromagnetic signal. The decomposition process typically involves the following steps: 1. Preprocessing the original signal by removing the DC component and normalizing it. 2. Based on the selected wavelet basis functions and number of decomposition layers, the preprocessed signal is subjected to wavelet decomposition to obtain the wavelet coefficients of each layer. 3. The obtained wavelet coefficients are further processed, such as thresholding and quantization, to remove noise and extract useful signal information. During the wavelet decomposition process, it is important to select the appropriate decomposition method and parameter settings to ensure the accuracy and reliability of the decomposition results.

[0076] Step 4: Get the wavelet coefficients of each layer

[0077] After the above steps, the wavelet coefficients of the noisy ocean electromagnetic signal at each layer can be obtained. These coefficients contain information about the signal at different frequency bands and can be used for subsequent signal analysis, processing, and reconstruction.

[0078] (2) Take appropriate thresholds to quantize the high-frequency wavelet coefficients.

[0079] According to Bashevar's theorem, the square of the coefficients after wavelet decomposition has the dimension of energy. Therefore, the decomposed wavelet coefficients are squared and sorted, and a threshold is given to find the corresponding risk value, that is, to obtain its likelihood estimate. Then, the non-likelihood minimization is performed to obtain the selected threshold. The specific method is as follows:

[0080] 1. Arrange the wavelet transformed coefficients of each layer from small to large by squaring them to obtain a vector w = [w1, w2, ..., w n ], where w1≤w2≤…≤w n , n is the number of wavelet coefficients;

[0081] 2. Calculate the risk vector R = [r1, r2, ..., r n ],but Among them, r i To introduce the risk vector element, the above formula is iterated multiple times to obtain the minimum r i , recorded as r0, and find the corresponding w i Denoted as w0;

[0082] 3. Calculate the threshold λ = σ(w0) 1 / 2 .

[0083] Wavelet transform analysis is based on a denoising method based on wavelet thresholds. This method can achieve near-optimality in the sense of minimum mean square error. Based on this method, soft and hard threshold functions for wavelet coefficient processing can be derived.

[0084] The hard threshold function is:

[0085]

[0086] Where: w j,k represents the wavelet coefficients of signal decomposition, It represents the estimated value of wavelet coefficients obtained by the threshold method, and λ is the threshold.

[0087]

[0088] Where σ is the noise standard deviation, which is estimated using the following empirical formula:

[0089]

[0090] The absolute value of the coefficients after wavelet decomposition is compared with the threshold λ. Points less than the threshold are set to 0, and points greater than or equal to the threshold retain their original value. During hard thresholding, since the hard threshold function is discontinuous throughout the entire wavelet region, there may be discontinuities at λ and -λ, causing oscillations in the reconstructed signal. Therefore, it is necessary to improve the continuity of the wavelet coefficient values.

[0091] The soft threshold function is:

[0092]

[0093] Although the soft threshold method produces good continuity in the wavelet coefficients, without discontinuities, and thus easy to process, when the wavelet coefficients are large, the estimated wavelet coefficients have a fixed deviation from the original wavelet coefficients, which inevitably introduces errors into the reconstructed signal. Furthermore, the soft threshold method compresses wavelet coefficients greater than the threshold at a constant value, which is inconsistent with the trend that the noise component gradually decreases as the wavelet coefficients increase.

[0094] In order to solve the above problems, the present invention introduces an adaptive threshold processing function, and the control equation of the method adopted is:

[0095]

[0096] Where m, n, and k are adjustment factors for the improved threshold function, which enhance the flexibility of the threshold function.

[0097] Parameters m and n determine the form of the threshold function. The value of parameter k is between 0 and 1. If k is 0, the threshold function is equivalent to a soft threshold function. If k is 1, the threshold function is equivalent to a hard threshold function.

[0098] The adjustable parameter k can overcome the discontinuity of the hard threshold function and the constant deviation of the soft threshold function when processing wavelet coefficients, while also retaining the original advantages of soft and hard thresholds.

[0099] The improved threshold function has infinite order continuous derivatives, which provides a basis for the selection of wavelet adaptive threshold.

[0100] (3) Perform inverse wavelet transform on the processed wavelet coefficients to reconstruct the denoised ocean electromagnetic signal.

[0101] The inverse wavelet transform is the inverse process of the wavelet transform. It converts the processed wavelet coefficients back to the original signal space to obtain the denoised signal. In wavelet denoising, the inverse wavelet transform is used to reconstruct the high-frequency wavelet coefficients and low-frequency wavelet coefficients (usually unchanged) after threshold quantization to restore the clean signal. The specific steps are as follows:

[0102] Step 1: Use multi-scale noise suppression and signal enhancement methods to further suppress ocean electromagnetic noise

[0103] Perform wavelet transform on the signal and decompose it into J layers;

[0104] Perform thresholding on the wavelet coefficients:

[0105]

[0106] Where: is the wavelet coefficient of the subband x position in the j-scale d-direction, is the multi-scale product of the noisy wavelet coefficients at position x in the d-direction subband of scale j. The threshold of this method does not act directly on the wavelet coefficients, but on the multi-scale product, which can greatly weaken the noise and improve the smoothness of the denoised signal. Specifically, it includes the following three aspects:

[0107] 1. Principle of the Method and Threshold Application. This method proposes a new approach in the field of wavelet denoising. Rather than directly applying thresholding to the wavelet coefficients, it instead applies the threshold to the multiscale product. This innovation lies in the fact that the multiscale product can reflect the correlation of signals at different scales. By calculating the product of wavelet coefficients at adjacent scales or specific scales, the scale differences between the effective and noise components of the signal can be captured. Therefore, applying the threshold to the multiscale product can more accurately identify and remove noise while preserving the effective characteristics of the signal.

[0108] 2. Effect and mechanism of noise reduction. Because noise often behaves randomly at different scales, its multiscale product is typically small. However, the effective components of a signal often exhibit a certain degree of correlation at different scales, resulting in a relatively large multiscale product. This method sets an appropriate threshold and treats multiscale products below the threshold as noise and removes them. This effectively reduces the impact of noise, making the denoised signal clearer and more accurate. Furthermore, because thresholding is performed on the multiscale product, it avoids signal distortion that can occur when directly processing the wavelet coefficients.

[0109] 3. Improved smoothness of the denoised signal. Another significant advantage of applying a threshold to the multiscale product is its ability to improve the smoothness of the denoised signal. Because the multiscale product reflects the correlation of the signal at different scales, retaining the larger multiscale product and reconstructing it using the inverse wavelet transform can produce a smoother denoised signal. This smoothness is reflected not only in the signal's appearance but also in its intrinsic quality. The denoised signal retains the original signal's characteristics while removing the interference and fluctuations caused by noise, making the signal more stable and reliable. This is of great significance for subsequent signal processing and analysis.

[0110] Step 2: Reconstruct the signal using inverse wavelet transform

[0111] Determine the wavelet basis function used for reconstruction. It should be consistent with the wavelet basis function selected when decomposing the original signal. Determine the number of wavelet decomposition layers, which determines the level of wavelet coefficients required to reconstruct the signal. The wavelet coefficients of each layer after thresholding and denoising.

[0112] The formula for inverse wavelet transform is as follows:

[0113]

[0114] Among them, c j-1，m is the scale coefficient of the scale space, h and g are the filter coefficients.

[0115] Performing inverse wavelet transform specifically includes the following aspects:

[0116] 1. Start from the highest layer: Start from the highest layer (lowest frequency) of the wavelet decomposition and use the wavelet coefficients of this layer to perform the inverse transform. This layer usually represents the low-frequency components or the general outline of the signal.

[0117] 2. Recursion layer by layer: Combine the inverse transform result of the previous layer with the wavelet coefficients of the current layer (slightly higher frequency) and perform the inverse transform again. This process continues layer by layer until the lowest layer (highest frequency) of the decomposition is reached.

[0118] 3. Reconstructing detail coefficients: In each layer, in addition to low-frequency coefficients, there are also high-frequency detail coefficients (such as horizontal, vertical, and diagonal details). These coefficients also need to be reconstructed through inverse transformation and combined with the low-frequency components.

[0119] 4. Superimpose the reconstruction results: The results of each layer of inverse transformation are superimposed to form the final denoised signal. This process ensures that the signal information at all scales is accurately restored.

[0120] Step 3: Get the denoised signal

[0121] 1. Output reconstructed signal: After the wavelet coefficients of all layers are inverse transformed and superimposed, the output signal is the denoised signal.

[0122] 2. Verify the reconstruction effect: Although this step is not a direct part of the inverse transform, the reconstructed signal is usually verified to ensure the denoising effect and signal quality improvement.

[0123] The comprehensive application and performance evaluation system effectively evaluates the denoising effect by comparing the signal-to-noise ratio and standard deviation of the denoised signal.

[0124] Experimental verification: To verify the denoising effect of the improved threshold method, a noisy signal with a length of is constructed by using the Wnoise function in Matlab, and its noise standard deviation is 2. Then, the Sym8 wavelet is used as the wavelet function, the decomposition layer is 5, and the adaptive threshold function method is used for denoising. The results are compared with the traditional soft and hard threshold function denoising methods. Figure 2 、 3 As shown in Figures 4 and 5, it can be found that the signal after denoising using the adaptive threshold method is smoother.

[0125] The denoising effects of the adaptive thresholding method, soft thresholding method, and hard thresholding method proposed in this paper were evaluated using both signal-to-noise ratio and standard deviation. The evaluation results are shown in the table below. As can be seen from the table, the proposed method has a higher signal-to-noise ratio and a smaller standard deviation, resulting in better denoising effect.

[0126] Table 1 Comparison of signal-to-noise ratio and standard deviation

[0127]

[0128] The technical solution of the present invention utilizes the multi-scale analysis properties of the wavelet transform to decompose ocean electromagnetic signals into wavelet coefficients of different frequencies and locations. Then, by applying thresholding or other denoising strategies to these wavelet coefficients, the noise component can be effectively removed while retaining the useful information in the signal. Finally, an inverse wavelet transform is used to reconstruct the processed wavelet coefficients into a denoised ocean electromagnetic signal.

[0129] The wavelet transform-based ocean electromagnetic signal denoising method not only has a profound theoretical background and methodological advantages, but also meets the practical needs of the fields of ocean electromagnetic exploration and data analysis. The application of this method will help improve the signal-to-noise ratio and clarity of ocean electromagnetic signals, providing a more accurate and reliable signal foundation for subsequent data analysis and scientific research.

[0130] In summary, the denoising effect of the adaptive threshold method proposed in the present invention is better than that of traditional soft threshold denoising and hard threshold denoising. It can effectively overcome the denoising error caused by the constant deviation between the estimated value and the true value in the soft threshold denoising method, and can also effectively suppress the signal oscillation phenomenon that is prone to occur in the hard threshold denoising method, thereby better preserving the details of the signal.

[0131] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for denoising ocean electromagnetic signals based on wavelet transform, characterized in that: The steps include: Step 1: Perform wavelet decomposition on the noisy ocean electromagnetic signal to obtain the wavelet coefficients of each layer; Step 2: Calculate the threshold of high-frequency wavelet coefficients and obtain the soft and hard threshold functions for wavelet coefficient processing; Step 3: Adaptive threshold processing is performed on the high-frequency wavelet coefficients, including: quantizing the high-frequency wavelet coefficients using an adaptive threshold function, wherein the adaptive threshold function dynamically adjusts the characteristics of the soft threshold and the hard threshold by adjusting the parameter k∈[0,1]. When k=0, it is equivalent to the soft threshold function, and when k=1, it is equivalent to the hard threshold function, so as to eliminate discontinuity and constant deviation at the threshold. Step 4: Perform inverse wavelet transform on the processed wavelet coefficients to reconstruct the denoised ocean electromagnetic signal.

2. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 1, characterized in that: The wavelet decomposition in step 1 includes: selecting wavelet basis functions with orthogonality, compact support and symmetry, and determining the number of decomposition layers according to noise distribution characteristics to perform multi-scale decomposition on the signal; The basis for determining the number of decomposition layers is as follows: if the noise in the signal is mainly concentrated in the high-frequency part, choose a larger number of decomposition layers to better separate the signal and noise; conversely, if the noise is also largely distributed in the low-frequency part, then choose a smaller number of decomposition layers to avoid excessive decomposition and signal distortion.

3. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 1, characterized in that: The method for calculating the threshold value of the high-frequency wavelet coefficients in step 2 is as follows: the square of the coefficients after wavelet decomposition has the dimension of energy. Therefore, the decomposed wavelet coefficients are squared and sorted, a threshold is given, the corresponding risk value is calculated, that is, its likelihood estimate is obtained, and the non-likelihood minimization is performed to obtain the selected threshold value. The specific method is as follows: (1) Arrange the wavelet transformed coefficients of each layer from small to large by squaring them to obtain a vector w = [w1, w2, ..., w n ], where w1≤w2≤…≤w n , n is the number of wavelet coefficients; (2) Calculate the risk vector R = [r1, r2, ..., r n ],but Among them, r i To introduce the risk vector element, the above formula is iterated multiple times to obtain the minimum r i , recorded as r0, and find the corresponding w i Denoted as w0; (3) Calculate the threshold λ = σ(w0) 1 / 2 .

4. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 1, characterized in that: The method for obtaining the hard threshold function for processing the wavelet coefficients in step 2 is: The hard threshold function is: Where: w j,k represents the wavelet coefficients of signal decomposition, represents the estimated value of the wavelet coefficient obtained by the threshold method, and λ is the threshold; Where σ is the noise standard deviation, which is estimated using the following empirical formula: The absolute value of the coefficients after wavelet decomposition is compared with the threshold λ. Points less than the threshold become 0, and points greater than or equal to the threshold retain their original values. In the hard threshold processing process, since the hard threshold function is discontinuous in the entire wavelet area, it is necessary to improve the continuity of the wavelet coefficient value.

5. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 4, characterized in that: The noise standard deviation σ can also be estimated by using the median absolute deviation: Right now where median(|w j，k |) is the median of the absolute values of the wavelet coefficients, and 0.6745 is the normalization constant.

6. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 1, characterized in that: The method for obtaining the soft threshold function for processing the wavelet coefficients in step 2 is: w j,k represents the wavelet coefficients of signal decomposition, It represents the estimated value of the wavelet coefficient obtained by the threshold method. After the soft threshold method is processed, the wavelet coefficient value has good continuity and no discontinuity problem, which is easy to process. However, when the wavelet coefficient is large, the estimated wavelet coefficient value has a fixed deviation from the original wavelet coefficient value, which will also bring inevitable errors to the reconstructed signal. In addition, the soft threshold adopts a constant value compression for the wavelet coefficient greater than the threshold, which is inconsistent with the trend that the noise component gradually decreases as the wavelet coefficient increases.

7. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 1, characterized in that: The adaptive threshold function in step 3 is: Where w j,k represents the wavelet coefficients of signal decomposition, represents the estimated value of wavelet coefficients obtained by the threshold method, m, n, k are adjustment factors of the improved threshold function, which enhance the flexibility of the threshold function; Parameters m and n determine the form of the threshold function. The value of parameter k is between 0 and 1. If k is 0, the threshold function is equivalent to a soft threshold function. If k is 1, the threshold function is equivalent to a hard threshold function. The adjustable parameter k can overcome the discontinuity of the hard threshold function and the constant deviation of the soft threshold function when processing wavelet coefficients, while also retaining the original advantages of soft and hard thresholds; The improved threshold function has infinite order continuous derivatives, which provides a basis for the selection of wavelet adaptive threshold.

8. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 1, characterized in that: The inverse wavelet transform in step 4 includes: combining the high-frequency coefficients after threshold processing with the unprocessed low-frequency coefficients, and restoring the signal layer by layer through multi-scale recursive reconstruction, specifically including: using multi-scale noise suppression and signal enhancement to suppress ocean electromagnetic noise Perform wavelet transform on the signal and decompose it into J layers; Perform thresholding on the wavelet coefficients: Where: is the wavelet coefficient of the subband x position in the j-scale d-direction, is the multi-scale product of the noisy wavelet coefficients at position x in the sub-band of scale j and direction d; T jd is the threshold for the multi-scale product; f(x) is the original signal to be processed; the threshold of this method does not act directly on the wavelet coefficients, but on the multi-scale product.

9. The method for denoising ocean electromagnetic signals based on wavelet transform according to claim 8, characterized in that: The inverse wavelet transform in step 4 further includes reconstructing the signal using an inverse wavelet transform, including: the formula of the inverse wavelet transform is as follows: Among them, j represents the scale parameter, c j-1，m is the scale coefficient of the scale space, h is the low-pass filter coefficient; g is the high-pass filter coefficient, c j，k is the low-frequency coefficient at scale j, d j，k is the high-frequency coefficient at scale j; k is the current scale translation parameter, and m is the previous scale translation parameter; Performing inverse wavelet transform specifically includes the following aspects: (1) Start from the highest level: Start from the highest level of wavelet decomposition and use the wavelet coefficients of this level to perform inverse transform; this level usually represents the low-frequency components or contours of the signal; (2) Layer-by-layer recursion: Then, the inverse transform result of the previous layer is combined with the wavelet coefficients of the current layer, and the inverse transform is performed again; this process is repeated layer by layer until the lowest layer of decomposition is reached. (3) Reconstructing detail coefficients: In each layer, in addition to low-frequency coefficients, there are also high-frequency detail coefficients; these coefficients also need to be reconstructed through inverse transformation and combined with the low-frequency components; (4) Superimpose the reconstruction results: Superimpose the results obtained from each layer of inverse transformation to form the final denoised signal; this process ensures that the information of the signal at all scales is accurately restored.

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