Invalid data identification and repair method for near-field acoustic holography based on sparse modal expansion

Abstract

The present invention discloses a method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion. First, the acoustic radiation modal expansion formula of the measured data of the holographic surface is obtained by performing acoustic radiation modal expansion on the measured data; the acoustic radiation modal expansion formula is probabilistically described by using the Gaussian distribution assumption imposed on the observation noise; then, by applying Gaussian and two-point distribution priors to each modal coefficient and the measured data, the measurement point number and modal coefficient corresponding to the invalid data are obtained by Bayesian learning; the data at the measurement point number corresponding to the invalid data is reconstructed using the modal coefficient and replaces the original invalid data to obtain repaired data; the repaired data is input into the near-field acoustic holographic reconstruction algorithm to obtain the sound field reconstruction result of the reconstructed surface. As a pre-processing method for near-field acoustic holography technology, the present invention is not limited to real sound sources and sound field types, and can realize the identification and repair of invalid data within a large range of invalid data proportions.

Description

Technical Field

[0001] The present invention relates to a pre-processing method for improving the reliability of data measurement in near-field acoustic holography technology, and more specifically to a method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion. Background Art

[0002] Accurate sound source identification is a prerequisite for achieving noise control, low-noise design, and sound quality design for electromechanical products. Technically, this can be achieved through near-field acoustic holography, a sensor array measurement technology. In addition to using sound field reconstruction algorithms, data measurement is another essential step in near-field acoustic holography, and the quality of the measurement data significantly impacts the final sound source identification results. Currently, attention to measurement data quality primarily focuses on measurement noise, which is typically mitigated using methods such as wavelet denoising, low-pass wavenumber filters, and various regularization techniques used in reconstruction algorithms. It should be noted that noise reduction still relies on valid measurements, meaning that each measurement point must accurately record sound field information. However, invalid measurements, another serious issue that seriously impacts measurement data quality, have received little research to date. Invalid measurements can be caused by errors in sensor sensitivity calibration, improper contact in the data acquisition system, or sudden sensor failure. Ultimately, these invalid measurements result in erroneous recording of sound source information, reducing the reliability of the measurement data. The randomness and sudden nature of these factors can mislead the acoustic imaging results of the target sound source. Therefore, how to identify and repair invalid data needs to be urgently addressed and is of great significance, especially when using packaged sensor arrays for measurement, high measurement costs, harsh measurement environments, etc., where replacing bad sensors and re-measuring is unrealistic. Summary of the Invention

[0003] The present invention aims to avoid the shortcomings of the above-mentioned prior art and to solve the problem of invalid measurement by providing a near-field acoustic holographic invalid data identification and repair method based on sparse modal expansion. The method is not limited to the real sound source type and can realize the identification and repair of invalid data within a larger range of invalid data proportions.

[0004] The technical solution adopted by the present invention to solve the technical problem is:

[0005] The present invention provides a near-field acoustic holographic invalid data identification and repair method based on sparse modal expansion. The method is as follows: first, the measured data of the holographic surface are subjected to acoustic radiation modal expansion to obtain the acoustic radiation modal expansion of the measured data; the acoustic radiation modal expansion is described probabilistically using the Gaussian distribution assumption imposed on the observation noise; then, Gaussian and two-point distribution priors are imposed on each modal coefficient and the measured data, and Bayesian learning is used to obtain the measurement point number and modal coefficient corresponding to the invalid data; the modal coefficient is used to reconstruct the data at the measurement point number corresponding to the invalid data and replace the original invalid data to obtain the repaired data; the repaired data is input into the near-field acoustic holographic reconstruction algorithm to obtain the sound field reconstruction result of the reconstructed surface.

[0006] The method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion of the present invention is carried out in the following steps:

[0007] Step a: Perform acoustic radiation modal expansion on the measured data of the holographic surface to obtain the acoustic radiation modal expansion of the measured data; and use the Gaussian distribution assumption imposed on the observation noise to perform a probabilistic description of the acoustic radiation modal expansion. The process is as follows:

[0008] Formula (1) satisfies the measured sound pressure p at position x on the holographic surface h (x) equivalent source sound radiation expression:

[0009]

[0010] In formula (1):

[0011] E(x) represents the measurement deviation at position x on the holographic surface. When the measurement deviation is not 0, it represents the corresponding measured sound pressure p h (x) is invalid;

[0012] n represents any equivalent source among the Q equivalent sources arranged on the equivalent source surface, n = 1, 2, ..., Q;

[0013] y n represents the position of the nth equivalent source;

[0014] q represents the equivalent source strength; q(y n ) represents the equivalent source strength of the nth equivalent source;

[0015] G(x,y n ) indicates that the n Sound pressure transfer function to position x;

[0016] r(x) represents the measurement noise at position x on the holographic surface;

[0017] Using formula (1), the acoustic radiation mode expansion formula of the measured data when the number of measurement points on the holographic surface is M is obtained as formula (2):

[0018] p h -E=Uα+R (2)

[0019] In formula (2):

[0020] p h represents the measured sound pressure vector with dimension M×1;

[0021] E represents the measurement deviation vector with dimension M×1;

[0022] U represents the sound radiation mode matrix with dimension M×M;

[0023] α represents the modal coefficient vector with dimension M×1;

[0024] R represents the measurement noise vector with dimension M×1;

[0025] Taking advantage of the unusability of invalid data, the invalid data identification and repair model is obtained through formula (2) as shown in formula (3):

[0026] k⊙p h =[K⊙U]α+R (3)

[0027] In formula (3):

[0028] The symbol “⊙” represents the Hadamard product;

[0029] k represents a binary invalid data identification vector with a dimension of M×1;

[0030] K represents a binary mask matrix consisting of M k and with a dimension of M×M;

[0031] By introducing a noise vector R with a mean of 0 and a variance of σ -1 Gaussian distribution, we can get the probability description p(k⊙p h |α,σ) as in formula (4):

[0032]

[0033] In formula (4):

[0034] k i represents the i-th element in the binary invalid data identification vector k, i = 1, 2, ..., M;

[0035] p hi Represents the measured sound pressure vector p h The i-th element in , i = 1, 2, ..., M;

[0036] (K⊙U) i represents the i-th row of the matrix K⊙U, i=1,2,…,M;

[0037] represents Gaussian distribution;

[0038] By substituting formula (3) into formula (4), we obtain formula (5):

[0039]

[0040] Using formula (5), we can obtain formula (6):

[0041]

[0042] Step b: applying Gaussian and two-point distribution priors to each modal coefficient and measurement data, respectively, and using Bayesian learning to obtain the measurement point number and modal coefficient corresponding to the invalid data; using the modal coefficient to reconstruct the data at the measurement point number corresponding to the invalid data and replace the original invalid data to obtain repaired data. The process is as follows:

[0043] By applying a two-point distribution to each element in vector k, the prior probability description p(k|ξ) of vector k is obtained as follows:

[0044]

[0045] In formula (7):

[0046] ξ i Represents the element k i The expectation of the two-point distribution imposed, 0<ξ i <1;

[0047] ξ is represented by ξ i The dimension of the expected vector is M×1;

[0048] Using equations (6) and (7), we can obtain the logarithmic posterior probability lnp(k) of vector k as equation (8):

[0049] lnp(k)∝ln[p(k|ξ)p(k⊙p h |α,k,σ)] (8)

[0050] By substituting equations (6) and (7) into equation (8), we can obtain the value of element k in vector k: i The logarithmic posterior probability lnp(k i ) as in formula (9):

[0051]

[0052] In formula (9):

[0053] H represents the conjugate transpose, and e represents the base of the exponent;

[0054] Using formula (9) to obtain element k i Expectations As shown in formula (10):

[0055]

[0056] The element k is obtained by formula (10) i The value rule of is as follows:

[0057]

[0058] In formula (11):

[0059] t represents a predetermined threshold;

[0060] The set Index consisting of the measurement point numbers corresponding to the invalid data identified by formula (11) is as follows:

[0061] Index={i|k i =0} (12)

[0062] The sparsity of vector α is promoted by applying a Gaussian distribution to each element of vector α. The prior probability description p(α|γ) of vector α is represented by formula (13):

[0063]

[0064] In formula (13):

[0065] α j represents the jth element in vector α, j = 1, 2, …, M;

[0066] Represents the element α j the variance of the imposed Gaussian distribution;

[0067] γ represents the variance The reciprocal of is a vector of dimension M×1;

[0068] Using equations (6) and (13), we can obtain the logarithmic posterior probability lnp(α) of vector α as shown in equation (14):

[0069] lnp(α)∝ln[p(α|γ)p(k⊙p h |α,k,σ)] (14)

[0070] By substituting equations (6) and (13) into (14), we obtain equation (15):

[0071]

[0072] In formula (15):

[0073] diag(γ) represents the diagonal matrix composed of the elements in vector γ;

[0074] diag(k) represents the diagonal matrix consisting of the elements in vector k;

[0075] Use formula (15) to obtain the expectation of vector α It is represented by formula (16):

[0076]

[0077] Use Equation (16) to obtain the estimate of vector α It is represented by formula (17):

[0078]

[0079] The repair data is obtained by using equations (2) and (17), which is represented by equation (18):

[0080]

[0081] In formula (18):

[0082] represents the repair data of the i-th measurement point;

[0083] Represents a vector The i-th element of ;

[0084] Step c: inputting the repaired data into a near-field acoustic holographic reconstruction algorithm to obtain a sound field reconstruction result of the reconstruction surface.

[0085] The near-field acoustic holographic invalid data identification and repair method based on sparse modal expansion of the present invention is not limited to the sound field type and can be a free field or a non-free field.

[0086] The near-field acoustic holography invalid data identification and repair method based on sparse modal expansion of the present invention is not limited to the selection of the near-field acoustic holography algorithm.

[0087] Compared with the existing technology, the beneficial effects of the present invention are embodied in:

[0088] 1. The method of the present invention directly performs acoustic radiation expansion on the measured data, and is therefore not limited to the selection of sound source type, sound field type, and near-field acoustic holography algorithm.

[0089] 2. The method of the present invention is a pre-processing method for improving the reliability of data measurement in near-field acoustic holography technology. Therefore, its repaired data is not limited to the selection of subsequent near-field acoustic holography sound field reconstruction algorithm.

[0090] 3. The method of the present invention is based on sparse modal expansion and can use measured data with fewer than the number of measurement points to repair invalid data. Therefore, it can be effective within a certain range of invalid data proportion. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 is the spatial distribution of the sound source, measurement point, and equivalent source in the present invention;

[0092] Figure 2a and Figure 2b This is the recognition and repair result of the method of the present invention when the invalid data accounts for 5% to 20%;

[0093] Figure 3 The modal coefficient amplitude calculated by the method of the present invention when the frequency is 500 Hz;

[0094] Figure 4 The modal coefficient amplitude calculated by the method of the present invention when the frequency is 1000 Hz;

[0095] Figure 5 It is divided into the modal coefficient amplitude calculated by the method of the present invention when the frequency is 1500Hz; DETAILED DESCRIPTION

[0096] In this embodiment, the method for identifying and repairing invalid data of near-field acoustic holography based on sparse modal expansion is as follows: first, by performing acoustic radiation modal expansion on the measured data of the holographic surface, that is, based on the transfer matrix modal expansion of the sound pressure, the acoustic radiation modal expansion of the measured data is obtained; the acoustic radiation modal expansion is described probabilistically by assuming a Gaussian distribution on the observation noise, that is, the likelihood function of the acoustic radiation modal expansion; then, by applying Gaussian and two-point distribution priors to each modal coefficient and the measured data, the measurement point number and modal coefficient corresponding to the invalid data are obtained by Bayesian learning; at the same time, the data at the measurement point number corresponding to the invalid data is reconstructed using the obtained modal coefficient and the original invalid data is replaced, that is, the obtained modal coefficient is substituted into the acoustic radiation modal expansion to obtain the repaired data; finally, the sound field reconstruction result of the reconstructed surface is obtained by inputting the repaired data into the near-field acoustic holographic reconstruction algorithm. As a pre-processing method of near-field acoustic holography technology, the present invention is not limited to real sound sources and sound field types, and can realize the identification and repair of invalid data within a large range of invalid data proportions.

[0097] In specific implementation, the method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion is carried out as follows:

[0098] Step a: Perform acoustic radiation modal expansion on the measured data of the holographic surface, i.e., perform modal expansion based on the transfer matrix of the sound pressure, and obtain the acoustic radiation modal expansion of the measured data; use the Gaussian distribution assumption imposed on the observation noise to describe the acoustic radiation modal expansion probabilistically, i.e., the likelihood function of the acoustic radiation modal expansion, and the process is as follows:

[0099] Formula (1) satisfies the measured sound pressure p at position x on the holographic surface h (x) equivalent source sound radiation expression:

[0100]

[0101] In formula (1):

[0102] E(x) represents the measurement deviation at position x on the holographic surface. When the measurement deviation is not 0, it represents the corresponding measured sound pressure p h (x) is invalid;

[0103] n represents any equivalent source among the Q equivalent sources arranged on the equivalent source surface, n = 1, 2, ..., Q;

[0104] y n represents the position of the nth equivalent source;

[0105] q represents the equivalent source strength; q(y n ) represents the equivalent source strength of the nth equivalent source;

[0106] G(x,y n ) indicates that the n Sound pressure transfer function to position x;

[0107] r(x) represents the measurement noise at position x on the holographic surface;

[0108] Using formula (1), we can obtain the sound radiation mode expansion of the measured data when the number of measurement points on the holographic surface is M, that is, the transfer matrix mode expansion based on the sound pressure, as shown in formula (2):

[0109] p h -E=Uα+R (2)

[0110] In formula (2):

[0111] p h represents the measured sound pressure vector with dimension M×1;

[0112] E represents the measurement deviation vector with dimension M×1;

[0113] U represents the sound radiation mode matrix with dimension M×M;

[0114] α represents the modal coefficient vector with dimension M×1;

[0115] R represents the measurement noise vector with dimension M×1;

[0116] Taking advantage of the unusability of invalid data, the invalid data identification and repair model is obtained through formula (2) as shown in formula (3):

[0117] k⊙p h =[K⊙U]α+R (3)

[0118] In formula (3):

[0119] The symbol “⊙” represents the Hadamard product;

[0120] k represents a binary invalid data identification vector with a dimension of M×1;

[0121] K represents a binary mask matrix consisting of M k and with a dimension of M×M;

[0122] By introducing a noise vector R with a mean of 0 and a variance of σ -1 Gaussian distribution, we can get the probability description p(k⊙p h |α,σ), which is the likelihood function of the acoustic radiation mode expansion, as shown in formula (4):

[0123]

[0124] In formula (4):

[0125] k i represents the i-th element in the binary invalid data identification vector k, i = 1, 2, ..., M;

[0126] p hi Represents the measured sound pressure vector p h The i-th element in , i = 1, 2, ..., M;

[0127] (K⊙U) i represents the i-th row of the matrix K⊙U, i=1,2,…,M;

[0128] represents Gaussian distribution;

[0129] By substituting formula (3) into formula (4), we obtain formula (5):

[0130]

[0131] Using formula (5), we can obtain formula (6):

[0132]

[0133] Step b: By applying Gaussian and two-point distribution priors to each modal coefficient and measurement data, Bayesian learning is used to obtain the measurement point number and modal coefficient corresponding to the invalid data; at the same time, the obtained modal coefficient is used to reconstruct the data at the measurement point number corresponding to the invalid data and replace the original invalid data. The obtained modal coefficient is substituted into the sound radiation mode expansion to obtain the repaired data. The process is as follows:

[0134] By applying a two-point distribution to each element in vector k, the prior probability description p(k|ξ) of vector k is obtained as follows:

[0135]

[0136] In formula (7):

[0137] ξ i Represents the element k i The expectation of the two-point distribution imposed, 0<ξ i <1;

[0138] ξ is represented by ξ i The dimension of the expected vector is M×1;

[0139] Using equations (6) and (7), we can obtain the logarithmic posterior probability lnp(k) of vector k as equation (8):

[0140] lnp(k)∝ln[p(k|ξ)p(k⊙p h |α,k,σ)] (8)

[0141] By substituting equations (6) and (7) into equation (8), we can obtain the value of element k in vector k: i The logarithmic posterior probability lnp(k i ) as in formula (9):

[0142]

[0143] In formula (9):

[0144] H represents the conjugate transpose, and e represents the base of the exponent;

[0145] Using formula (9) to obtain element k i Expectations As shown in formula (10):

[0146]

[0147] The element k is obtained by formula (10) i The value rule of element k i The limiting conditions when taking 0 or 1 are as follows:

[0148]

[0149] In formula (11):

[0150] t represents a predetermined threshold;

[0151] The set Index consisting of the measurement point numbers corresponding to the identified invalid data is obtained through formula (11), that is, the identification result of invalid data, as shown in formula (12):

[0152] Index={i|k i =0} (12)

[0153] The sparsity of vector α is promoted by applying a Gaussian distribution to each element of vector α. The prior probability description p(α|γ) of vector α is represented by formula (13):

[0154]

[0155] In formula (13):

[0156] α j represents the jth element in vector α, j = 1, 2, …, M;

[0157] Represents the element α j the variance of the imposed Gaussian distribution;

[0158] γ represents the variance The reciprocal of is a vector of dimension M×1;

[0159] Using equations (6) and (13), we can obtain the logarithmic posterior probability lnp(α) of vector α as shown in equation (14):

[0160] lnp(α)∝ln[p(α|γ)p(k⊙p h |α,k,σ)] (14)

[0161] By substituting equations (6) and (13) into (14), we obtain equation (15):

[0162]

[0163] In formula (15):

[0164] diag(γ) represents the diagonal matrix composed of the elements in vector γ;

[0165] diag(k) represents the diagonal matrix consisting of the elements in vector k;

[0166] Use formula (15) to obtain the expectation of vector α It is represented by formula (16):

[0167]

[0168] Use Equation (16) to obtain the estimate of vector α That is the expectation of vector α It is represented by formula (17):

[0169]

[0170] The repair data is obtained by using equations (2) and (17), and the obtained modal coefficients are substituted into the acoustic radiation modal expansion, which is represented by equation (18):

[0171]

[0172] In formula (18):

[0173] represents the repair data of the i-th measurement point;

[0174] Represents a vector The i-th element of ;

[0175] Step c: inputting the repaired data into a near-field acoustic holographic reconstruction algorithm to obtain a sound field reconstruction result of the reconstruction surface.

[0176] This embodiment is not limited to the sound field type, which can be a free field or a non-free field; and is not limited to the choice of near-field acoustic holography algorithm.

[0177] The method of the present invention was tested as follows:

[0178] Figure 1 As shown, a size of 0.5×0.5×0.003m 3 The vibration aluminum plate b1 is placed on the plane of Z = 0m to generate a sound field with a frequency range of 200 to 1500Hz. The center position of the vibration aluminum plate b1 is (0,0,0)m. The center position and size of the vibration aluminum plate b1 are set on the plane of Z = 0.08m, respectively, to (0,0,0.08)m and 0.5×0.5m. 2 The holographic surface b2 is discretized at intervals of 0.05 m, and measurement points are arranged at the discrete points. An equivalent source surface b3 with a center position of (0, 0, 0.06) m is set on the plane Z = 0.06 m, and the equivalent source surface b3 is discretized at intervals of 0.05 m; equivalent sources are arranged at the discrete points. The number of equivalent sources Q can be obtained by formula (19):

[0179]

[0180] Acquire the sound field information in an environment with a signal-to-noise ratio of 30 dB, and generate invalid data using formula (20):

[0181]

[0182] In formula (20):

[0183] max[·] represents the maximum value operation.

[0184] In order to evaluate the performance of the method of the present invention, two indicators are introduced: invalid data identification accuracy IA and invalid data repair accuracy RMSE. The identification accuracy IA is represented by formula (21):

[0185]

[0186] In formula (21):

[0187] b represents the number of non-zero elements in the binary invalid data identification vector k;

[0188] {a l} represents the set of measurement point numbers corresponding to invalid data, i.e. {a l |a l ∈{1,2,...,M},l=1,2,...,b};

[0189] It is represented as the set of measurement point numbers corresponding to the identified invalid data, that is, The symbol “num(·)” indicates a counting operation;

[0190] The restoration accuracy RMSE is represented by formula (22):

[0191]

[0192] In formula (22):

[0193] Represented by the measurement point The repaired sound pressure vector is constructed, i = 1, 2, ..., M;

[0194] Expressed as the holographic surface theoretical sound pressure vector;

[0195] The symbol “||·||2” means taking the two-norm operation;

[0196] Figure 2a and Figure 2b The identification and repair results of the present invention method when the invalid data accounts for 5% to 20% are shown in the figure. Figure 2a It is about the invalid data identification accuracy IA, Figure 2b It is about the invalid data repair accuracy RMSE.

[0197] Figure 2a and Figure 2b The solid line a1, long dashed line a2, thin dashed line a3 and dot-dash line a4 correspond to the results when the invalid data accounts for 5%, 10%, 15% and 20%, respectively. As can be seen from the figure, within the range of invalid data proportion investigated, the method of the present invention can accurately identify invalid data and can obtain a data repair accuracy with an overall error of less than 5%.

[0198] Figure 3 、 Figure 4 and Figure 5 The modal coefficient amplitudes at frequencies of 500, 1000 and 1500 Hz are calculated for the method of the present invention respectively. It can be seen that the method of the present invention can promote the sparsity of the modal coefficients of the established sound radiation mode expansion.

Claims

1. A method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion, characterized by: First, the acoustic radiation modal expansion of the measured data of the holographic surface is performed to obtain the acoustic radiation modal expansion of the measured data; the Gaussian distribution assumption imposed on the observation noise is used to describe the acoustic radiation modal expansion probabilistically; then, by applying Gaussian and two-point distribution priors to each modal coefficient and the measured data, the measurement point number and modal coefficient corresponding to the invalid data are obtained using Bayesian learning; the modal coefficients are used to reconstruct the data at the measurement point number corresponding to the invalid data and replace the original invalid data to obtain the repaired data; The repaired data is input into the near-field acoustic holography reconstruction algorithm to obtain the sound field reconstruction result of the reconstructed surface.

2. The method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion according to claim 1 is characterized by: Proceed as follows: Step a: Perform acoustic radiation modal expansion on the measured data of the holographic surface to obtain the acoustic radiation modal expansion of the measured data; and use the Gaussian distribution assumption imposed on the observation noise to perform a probabilistic description of the acoustic radiation modal expansion. The process is as follows: Formula (1) satisfies the measured sound pressure p at position x on the holographic surface h (x) equivalent source sound radiation expression: In formula (1): E(x) represents the measurement deviation at position x on the holographic surface. When the measurement deviation is not 0, it represents the corresponding measured sound pressure p h (x) is invalid; n represents any equivalent source among the Q equivalent sources arranged on the equivalent source surface, n = 1, 2, ..., Q; y n represents the position of the nth equivalent source; q represents the equivalent source strength; q(y n ) represents the equivalent source strength of the nth equivalent source; G(x,y n ) indicates that the n Sound pressure transfer function to position x; r(x) represents the measurement noise at position x on the holographic surface; Using formula (1), the acoustic radiation mode expansion formula of the measured data when the number of measurement points on the holographic surface is M is obtained as formula (2): p h -E=Uα+R (2) In formula (2): p h represents the measured sound pressure vector with dimension M×1; E represents the measurement deviation vector with dimension M×1; U represents the sound radiation mode matrix with dimension M×M; α represents the modal coefficient vector with dimension M×1; R represents the measurement noise vector with dimension M×1; Taking advantage of the unusability of invalid data, the invalid data identification and repair model is obtained through formula (2) as shown in formula (3): k⊙p h =[K⊙U]α+R (3) In formula (3): The symbol "⊙" represents the Hadamard product; k represents a binary invalid data identification vector with a dimension of M×1; K represents a binary mask matrix consisting of M k and with a dimension of M×M; By introducing a noise vector R with a mean of 0 and a variance of σ -1 Gaussian distribution, we can get the probability description p(k⊙p h |α,σ) as in formula (4): In formula (4): k i represents the i-th element in the binary invalid data identification vector k, i = 1, 2, ..., M; Represents the measured sound pressure vector p h The i-th element in , i = 1, 2, ..., M; (K⊙U) i represents the i-th row of the matrix K⊙U, i=1,2,…,M; represents Gaussian distribution; By substituting formula (3) into formula (4), we obtain formula (5): Using formula (5), we can obtain formula (6): Step b: applying Gaussian and two-point distribution priors to each modal coefficient and measurement data, respectively, and using Bayesian learning to obtain the measurement point number and modal coefficient corresponding to the invalid data; using the modal coefficient to reconstruct the data at the measurement point number corresponding to the invalid data and replace the original invalid data to obtain repaired data. The process is as follows: By applying a two-point distribution to each element in vector k, the prior probability description p(kξ) of vector k is obtained as follows: In formula (7): ξ i Represents the element k i The expectation of the two-point distribution imposed, 0<ξ i <1; ξ is represented by ξ i The dimension of the expected vector is M×1; Using equations (6) and (7), we can obtain the logarithmic posterior probability lnp(k) of vector k as equation (8): lnp(k)∝ln[p(k|ξ)p(k⊙p h |α,k,σ)] (8) By substituting equations (6) and (7) into equation (8), we can obtain the value of element k in vector k: i The logarithmic posterior probability lnp(k i ) as in formula (9): In formula (9): H represents the conjugate transpose, and e represents the base of the exponent; Using formula (9) to obtain element k i Expectations As shown in formula (10): The element k is obtained by formula (10) i The value rule of is as follows: In formula (11): t represents a predetermined threshold; The set Index consisting of the measurement point numbers corresponding to the invalid data identified by formula (11) is as follows: Index={i|k i =0} (12) The sparsity of vector α is promoted by applying a Gaussian distribution to each element of vector α. The prior probability description p(α|γ) of vector α is represented by formula (13): In formula (13): α j represents the jth element in vector α, j = 1, 2, …, M; Represents the element α j the variance of the imposed Gaussian distribution; γ represents the variance The reciprocal of is a vector of dimension M×1; Using equations (6) and (13), we can obtain the logarithmic posterior probability lnp(α) of vector α as shown in equation (14): lnp(α)∝ln[p(α|γ)p(k⊙p h |a,k,s)] (14) By substituting equations (6) and (13) into (14), we obtain equation (15): In formula (15): diag(γ) represents the diagonal matrix composed of the elements in vector γ; diag(k) represents the diagonal matrix consisting of the elements in vector k; Use formula (15) to obtain the expectation of vector α It is represented by formula (16): Use Equation (16) to obtain the estimate of vector α It is represented by formula (17): The repair data is obtained by using equations (2) and (17), which is represented by equation (18): In formula (18): represents the repair data of the i-th measurement point; Represents a vector The i-th element of ; Step c: inputting the repaired data into a near-field acoustic holographic reconstruction algorithm to obtain a sound field reconstruction result of the reconstruction surface.

3. The method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion according to claim 1 is characterized by: The sound field type is not limited and can be free field or non-free field.

4. The method for identifying and repairing invalid data in near-field acoustic holography based on sparse modal expansion according to claim 1 is characterized by: Not limited to the choice of near-field acoustic holography algorithm.

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