A multi-resolution sparse decomposition algorithm for overlapping ultrasound signal decomposition

Abstract

The application belongs to the technical field of ultrasonic nondestructive testing, and particularly relates to a multi-resolution sparse decomposition algorithm for ultrasonic signal separation and ultrasonic echo estimation in overlapping ultrasonic signal decomposition. The algorithm separates seriously overlapping ultrasonic echoes by continuously iterating the given ultrasonic signal, dividing the signal into two segments in each iteration, and decomposing each segment signal on a discrete Gabor dictionary with more refined discrete steps of parameters using a support matching pursuit algorithm. Meanwhile, a new discrete Gabor dictionary is generated by compressing the upper and lower boundaries of parameters in the Gabor function and refining the discrete steps of parameters in each iteration. Without increasing the number of atoms in the dictionary, the discrete interval of the dictionary parameters is gradually reduced to gradually improve the matching degree between the ultrasonic echoes and the obtained atoms after each algorithm iteration. The signal segments obtained by signal segmentation in each iteration are decomposed at a higher resolution than before, and the overlapping echoes that cannot be separated at a coarse scale can be separated at a higher scale, thereby gradually improving the accuracy of echo separation and echo estimation.

Description

Technical Field

[0001] This invention belongs to the field of ultrasonic nondestructive testing technology, and specifically relates to a multi-resolution sparse decomposition algorithm for ultrasonic signal separation and ultrasonic echo estimation in the decomposition of overlapping ultrasonic signals. Background Technology

[0002] Ultrasonic microscopy, a common non-destructive testing technique, is widely used for failure analysis in microelectronic packaging. The accurate estimation of reflected echoes is crucial for the detection and localization of internal defects in microelectronic packages. Incident ultrasonic waves reach various surfaces of the sample at different times and are partially reflected at each surface. Analyzing the reflected echoes can reveal information about defects within the package.

[0003] With the development of the chip industry, the thickness of the internal layer structure of modern 3D microelectronic packaging is gradually decreasing, which makes the overlap of reflected echoes more and more serious, and the imaging quality of ultrasound C-scans is also significantly reduced. The Support Matching Pursuit (SMP) algorithm using Gabor dictionaries is a widely used sparse decomposition method for decomposing ultrasound signals for echo separation and echo estimation.

[0004] Sparse decomposition describes the echo characteristics of ultrasonic signals in ultrasonic nondestructive testing using an overcomplete Gabor dictionary. A series of echoes constituting the Gabor dictionary are called atoms. The sparse decomposition problem can be defined as follows:

[0005]

[0006] Where ||x||0 is x (x∈R) M Let x be a given ultrasonic signal with a zero norm (i.e., the number of non-zero points in x), and y be an M×N matrix with M << N, where M is the number of atoms in the dictionary and N is the length of the signal y. Therefore, Ax = y has infinitely many solutions. However, if dictionary A is known, the entire sparse decomposition problem becomes finding the solution Ax = y with the fewest non-zero points in x. This is an NP-hard problem, so seeking suboptimal solutions becomes the preferred approach for sparse decomposition. Currently, convex relaxation, non-convex local optimization, and grey search algorithms are widely used in the development of sparse decomposition algorithms. Research on the application of existing sparse decomposition techniques in ultrasonic nondestructive testing shows that sparse decomposition has great potential in ultrasonic nondestructive testing.

[0007] However, existing sparse decomposition algorithms often struggle to find the optimal solution. Furthermore, they only find the optimal solution when the size of the overcomplete dictionary is less than a threshold due to cumulative coherence constraints; otherwise, the reliability and stability of the solution decrease significantly. In addition, to ensure that the atoms generated within the dictionary can more accurately match various ultrasonic echoes, the discretization step size of the Gabor dictionary parameters needs to be further refined, inevitably leading to an increase in dictionary size. The key to improving the performance of existing sparse decomposition algorithms in ultrasonic echo separation and estimation lies in refining the discretization step size of the dictionary parameters without increasing the dictionary size.

[0008] Ideally, each isolated atom should correspond to an echo in the original ultrasonic signal. However, due to the limited resolution of the SMP algorithm, when using SMP to iteratively decompose overlapping ultrasonic signals, the matching degree between the obtained atoms and ultrasonic echoes rapidly decreases or even becomes completely incorrect.

[0009] Therefore, this invention proposes a multi-resolution signal sparse decomposition algorithm to solve the above problems. Summary of the Invention

[0010] The technical problem to be solved by this invention:

[0011] To address the challenge of low accuracy and resolution in ultrasonic echo separation and estimation using existing sparse decomposition algorithms for chip ultrasonic testing, a multi-resolution signal sparse decomposition algorithm is proposed for accurate ultrasonic echo separation and estimation in overlapping ultrasonic signals. The proposed algorithm iterates through a given ultrasonic signal, dividing it into two segments in each iteration. Then, it uses the Support Matching Pursuit (SMP) algorithm to decompose each segment on a discrete Gabor dictionary with finer discrete step sizes to separate severely overlapping ultrasonic echoes. Simultaneously, in each iteration, a new discrete Gabor dictionary is generated by compressing the upper and lower boundaries of the Gabor function parameters and refining the step sizes. Without increasing the number of atoms in the dictionary, the discrete spacing of the dictionary parameters is gradually reduced, resulting in a progressively improving match between the ultrasonic echo and the obtained atoms after each algorithm iteration. Furthermore, in each iteration, the signal segments obtained through signal segmentation are decomposed at a higher resolution than before. Therefore, overlapping echoes that cannot be separated at a coarse scale can be separated at a higher scale, thereby gradually improving the accuracy of echo separation and echo estimation.

[0012] To achieve the above objectives, the present invention provides the following technical solution:

[0013] A multi-resolution sparse decomposition algorithm for decomposing overlapping ultrasound signals includes the following steps:

[0014] Step 1: Perform sparse decomposition on the initial dictionary D0 using the SMP algorithm for any signal y;

[0015] Step 2: Perform signal segmentation on the signal y using the sparse decomposition results obtained in Step 1 to obtain signal segment y1,1 and signal segment y1,2;

[0016] Step 3: Generate dictionaries D1,1 and D1,2 for signal segments y1,1 and y1,2 respectively, and update the initial dictionary D0 to D1,1 and D1,2;

[0017] Step 4: For each signal segment in Step 2, use the corresponding dictionary generated in Step 3 to perform sparse decomposition using the SMP algorithm;

[0018] Step 5: Repeat steps 2-4 until only one echo can be extracted from each individual signal segment.

[0019] A further technical solution of the present invention is as follows: the method for signal segmentation of signal y in step 2 is as follows:

[0020] Step 2.1: Arrange the m atoms obtained from the decomposition in Step 1 in ascending order of arrival time u;

[0021] Step 2.2: Calculate the arrival time difference between every two adjacent atoms, denoted as du = (du1, du2, ..., du...). m-1 );

[0022] Step 2.3: Denote the maximum value in du as du. q Determine signal segment y (n，i) The split point is located at the arrival time u q and u q+1 Between two atoms;

[0023] Step 2.4: Through atom u q and u q+1 The ultrasonic echo is reconstructed using its corresponding decomposition coefficients, denoted as echo(q) and echo(q+1), respectively.

[0024] Step 2.5: By determining whether echo(q) and echo(q+1) overlap, process the signal segment y. (n，i) Implement different signal segmentation strategies.

[0025] A further technical solution of the present invention is as follows: the dictionary generation process in step 3 is as follows:

[0026] Let the total number of atoms contained in the initial dictionary D0 be denoted as The size is calculated using formula (1) and used as a reference for generating the dictionary later.

[0027]

[0028] By employing the same discrete strategy and step size as dictionary D0, signal segment y (n，i) The corresponding dictionary D (n，i) The size is calculated from the new parameter range obtained in formula (2), denoted as

[0029]

[0030] Where α is the relaxation coefficient used to expand the parameter boundary;

[0031] Due to the narrowing of the dictionary parameter range, Less than Dictionary size shrinkage ratio k (n，i) for:

[0032]

[0033] A further technical solution of the present invention is as follows: In step 3, the process of determining the boundary of the dictionary update parameters is as follows:

[0034] After sparse decomposition of the signal segment y(n,i), m atoms are obtained, denoted as d. q = (q = 1, 2, ..., m), where the four parameters of each atom are defined as: s dq u dq f dq ω dq Then the lower bound sl of the scaling function s (n，i) It is estimated as min(sd1, sd2, ..., sd) m ), Upper Realm (n，i) It can be estimated as max(s) d1 s d2 , ..., s dm Similarly, the upper and lower bounds of the other three parameters are ul (n，i) ,uu (n，i) , fl (n，i) ,fu (n，i) ,ωl (n，i) ,ωu (n，i It can also be obtained using the same method.

[0035] A further technical solution of the present invention is as follows: In step 3, the dictionary update algorithm is as follows:

[0036] Assume the number of sampling points for the scale parameter s in the dictionary D0 is For dictionary D0 and Where N is the upper bound of the scaling parameter s; the scaling parameter s is optimized in the dictionary D by adjusting the coefficient a. (n，i) The sampling interval in the dictionary D, thus making the dictionary D (n，i) Size Approximately the size of the reference dictionary D0 The adjustment method for coefficient 'a' is as follows:

[0037]

[0038] The parameter boundaries are updated using formula (2) and the coefficients a are updated using formula (4), resulting in a new dictionary D. (n，i) Generated using a discretization scheme.

[0039] A further technical solution of the present invention is: in step 4, the algorithm iteration termination determination criteria are as follows:

[0040] Signal y (n，i) After n iterations of the algorithm, use dictionary D (n，i) The decomposition by the SMP algorithm is expressed as follows:

[0041]

[0042] By setting a threshold ε, a total of m atoms and the remaining residual R were obtained. m y (n，i) There are two possible scenarios, and the corresponding solutions are as follows:

[0043] 1. If m = 1, then the dictionary D (n，i) Updated to D with a more refined sampling interval (n+1，i) and the signal y (n，i) Use dictionary D (n+1，i)

[0044] Further decomposition will result in the following two scenarios:

[0045] 1.1 If the number of atoms obtained after more refined dictionary sparse decomposition is still 1, the algorithm iteration terminates and the sparse decomposition of the signal is completed;

[0046] 1.2 If the number of atoms obtained after more refined dictionary sparse decomposition is greater than 1, the algorithm continues to iterate until event 1.1 is achieved;

[0047] 2. If m > 1, the algorithm continues to iterate, and the dictionary is updated to a finer scale until event 1 is achieved.

[0048] Beneficial effects

[0049] The multi-resolution sparse decomposition algorithm for ultrasonic signal separation and ultrasonic echo estimation in overlapping ultrasonic signal decomposition proposed in this invention has the following advantages compared with existing technologies:

[0050] 1. This invention provides a multi-resolution sparse decomposition algorithm for decomposing overlapping ultrasound signals. The algorithm separates the ultrasound signal into shorter signal segments, updates the dictionary by refining the parameter range, and applies parameter boundary compression to utilize information obtained from the last decomposition. This study tested the performance of this algorithm, using test data including different overlap percentages and different signal-to-noise ratios (SNR). Experimental results show that this method performs excellently in echo separation and echo estimation, maintaining good performance even with an overlap ratio as high as 67%.

[0051] 2. This invention provides a multi-resolution sparse decomposition algorithm for overlaid ultrasonic signal decomposition. Without increasing the Gabor dictionary size, the discretization spacing of the four parameters of the Gabor dictionary is further refined compared to previous dictionaries, thus improving detection accuracy. This resolves the contradiction that "reducing the discretization spacing of dictionary parameters inevitably increases the dictionary size." Furthermore, by formulating different discretization schemes for the four parameters of the Gabor dictionary, there are multiple methods to generate various customized Gabor dictionaries to meet the detection accuracy requirements at different parameters and scales.

[0052] 3. The multi-resolution sparse decomposition algorithm provided by this invention for the decomposition of overlapping ultrasound signals demonstrates excellent performance even in noisy environments, thanks to the improved signal-to-noise ratio and reduced root mean square error. Furthermore, experimental results show that the algorithm can capture the true structure of overlapping ultrasound signals, and the reconstructed ultrasound signal has a good match with the recorded ultrasound signal. Attached Figure Description

[0053] To more clearly illustrate the technical solution of the present invention, the accompanying drawings used in the embodiments will be briefly described below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope of protection of the present invention.

[0054] Figure 1 This is a flowchart of a multi-resolution sparse decomposition algorithm for decomposing overlapping ultrasound signals according to the present invention.

[0055] Figure 2 This is a schematic diagram of the start and end points of the signal when β = 0.05 according to the present invention;

[0056] Figure 3(a) is a schematic diagram of the overlapping signal of the present invention;

[0057] Figure 3(b) is a schematic diagram of the non-overlapping signal of the present invention;

[0058] Figure 4 This is a schematic diagram of the simulated ultrasonic signal of the present invention;

[0059] Figure 5 (a) is a diagram showing the effect of the atoms obtained by the sparse decomposition of the present invention in the frequency domain;

[0060] Figure 5 (b) is a diagram showing the matching effect between the reconstructed echo and the simulated ultrasonic echo of the present invention;

[0061] Figure 6 (a) is a diagram showing the effect of the atoms obtained by the first iterative sparse decomposition of this invention in the frequency domain;

[0062] Figure 6 (b) is a diagram showing the matching effect between the first iterative sparse decomposition reconstruction echo and the simulated ultrasonic echo of this invention;

[0063] Figure 7 (a) is a diagram showing the effect of the atoms obtained by the second iteration of sparse decomposition in the frequency domain of the present invention;

[0064] Figure 7 (b) is a diagram showing the matching effect between the reconstructed echo from the second iteration of sparse decomposition in this invention and the simulated ultrasonic echo.

[0065] Figure 8 This is a diagram illustrating the effect of the atoms obtained from the final iterative sparse decomposition of this invention in the frequency domain;

[0066] Figure 9 This is an energy error diagram under different overlap ratios according to the present invention;

[0067] Figure 10 This is a graph showing the changing trends of the signal-to-noise ratio and root mean square error of this invention.

[0068] Figure 11 This is a graph showing the relationship between the energy error and signal-to-noise ratio of the two overlapping echoes in this invention;

[0069] Figure 12 This invention utilizes a 100MHz transducer to implement an 8-layer stacked ultrasonic signal;

[0070] Figure 13 This is the decomposition result of the actual ultrasonic signal in this invention. Detailed Implementation

[0071] The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention.

[0072] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0073] Example 1

[0074] A multi-resolution sparse decomposition algorithm for decomposing overlapping ultrasound signals, such as Figure 1 As shown, it can be mainly divided into the following 5 steps:

[0075] Step 1: Initial Sparse Decomposition. For a given initial signal y, perform sparse decomposition on the standard overcomplete dictionary D0 as described in Equation 2 using the SMP algorithm.

[0076] Step 2: Signal Segmentation. The initial signal y is segmented into y1,1 and y1,2 in the time domain by the decomposition result obtained in Step 1.

[0077] Step 3: Dictionary Update. After segmenting the initial signal y into two segments through signal segmentation, a customized Gabor dictionary is generated for each segment. The time width, center frequency, and spectral width are estimated from the atomic information obtained in Step 1. The upper and lower bounds of the newly generated dictionary parameters are also compressed accordingly based on the atomic information obtained in Step 1 to ensure that the dictionary generated for each signal segment has an independent parameter space. For signal segment y1,1, the newly generated dictionary is named D1,1. Similarly, for signal segment y1,2, the newly generated dictionary is named D1,2. The specific dictionary update steps will be described in detail in Section 3. By compressing the Gabor dictionary parameter space, the discrete length of the generated new dictionary will be reduced without increasing the dictionary size. Furthermore, due to the improved dictionary resolution, the matching degree between atoms in the dictionary and the ultrasonic echo is also improved, thereby increasing the accuracy of echo estimation.

[0078] Step 4: Perform sparse decomposition on the segmented signal segments using the newly generated dictionary. Each signal segment obtained in Step 2 is sparsely decomposed using the corresponding dictionary generated in Step 3 through the SMP algorithm.

[0079] Step 5: Repeat steps 2 and 3 until all overlapping echoes are separated. For ease of expression, let y(n,i) denote the signal segment i in the nth iteration, and its corresponding dictionary be denoted as D(n,i). In the nth iteration, a total of 2^n new signal segments will be obtained. The initial sparse decomposition described in step 1 is defined as the 0th iteration.

[0080] In each iteration, each signal segment is further divided into two shorter segments, and the corresponding Gabor dictionary parameters in both the time and frequency domains are further compressed. The matching degree between atoms and ultrasonic echoes is further optimized. Due to the improved dictionary resolution, overlapping echoes that were previously difficult to separate will eventually be separated at a higher precision scale.

[0081] The signal segmentation method is as follows:

[0082] Assume signal segment y (n，i) After the nth iteration of the algorithm, sparse decomposition using the SMP algorithm yields a total of m atoms. The signal segment y... (n，i) The steps, divided into two parts, are as follows:

[0083] 1) Arrange the m atoms in ascending order of arrival time u.

[0084] 2) Calculate the arrival time difference between every two adjacent atoms, denoted as du=(du1,du2,...,du m-1 ).

[0085] 3) Record the maximum value in du as du. q Determine signal segment y (n，i) The split point is located at the arrival time u q and u q+1 Between two atoms.

[0086] 4) Through atoms u q and u q+1 The corresponding decomposition coefficients are used to reconstruct the ultrasonic echo, denoted as echo(q) and echo(q+1), respectively.

[0087] 5) By determining whether echo(q) and echo(q+1) overlap, the signal segment y is processed. (n，i Different signal segmentation strategies are implemented. The specific segmentation strategies are as follows:

[0088] like Figure 2 As shown, in reconstructing the ultrasound echo, the signal start point is denoted as A, and the signal end point is denoted as B. Point A is the point where the signal amplitude first reaches the threshold from left to right. Point B is the point where the signal amplitude first reaches the threshold from right to left. These can be calculated using the following formulas:

[0089]

[0090]

[0091] Where β is a preset threshold. The starting points of echo(q) and echo(q+1) are denoted as A1 and A2, respectively, and the ending points are denoted as B1 and B2.

[0092] If A2 is less than B1 (e.g.) Figure 3a If the signals are not overlapping (as shown), then the two signals are considered to overlap; otherwise, they are considered not to overlap (as shown). Figure 3b (As shown).

[0093] If the two signals are determined to be non-overlapping, the signal split point is the midpoint between B1 and A2. Let u be the signal split point. s = (B1+A2) / 2;

[0094] If the two signals are determined to overlap, the signal segmentation points are B1 and A2, respectively. The first segment, obtained by dividing the signal at these two points, is from 0 to A2, and the second segment is from B1 to N. (n，i) , where N (n，i) For signal y (n，i The length of the segment is ( ). Obviously, the two signal segments obtained by this segmentation method overlap each other. This is to ensure that the segmented signal segments can completely cover the original signal and avoid the loss of signal information due to signal segmentation.

[0095] Dictionary update

[0096] Determining the boundary of dictionary update parameters

[0097] Suppose that the sparse decomposition of signal segment y(n,i) using the algorithm proposed in this paper yields m atoms, denoted as d. q (q = 1, 2, ..., m), the four parameters of each atom are defined as: s dq u dq f dq ω dq Then the lower bound sl of the scaling function s (n，i) It can be estimated as min(s) d1 s d2 , ..., s dm ), Upper Realm (n，i) It can be estimated as max(s) d1 s d2 , ..., s dm Similarly, the upper and lower bounds of the other three parameters are ul. (n，i) uv (n，i) , fl (n，i) ,fu (n，i) ,ωl(n，i) ,ωu (n，i) This can also be obtained using the same method. Therefore, the parameter bounds of the dictionary D(n,i) can be defined as:

[0098]

[0099] Here, α is the relaxation coefficient used to expand the parameter boundaries. This is because the m atoms are obtained by sparse decomposition at a scale with a relatively large parameter discretization step size, which will result in a matching error between them and the actual ultrasonic echo. The relaxation coefficient can help the signal segment y(n,i) to get closer to the actual signal echo in the (n+1)th iteration.

[0100] The generation of a new dictionary

[0101] Let the total number of atoms contained in the initial dictionary D0 be denoted as The dimensions can be calculated using the following formula and used as a reference for generating the dictionary later.

[0102]

[0103] By employing the same discrete strategy and step size as dictionary D0, signal segment y (n，i) The corresponding dictionary D (n，i) The size is calculated from the new parameter range obtained from the formula, denoted as . Due to the narrowing of the dictionary parameter range, Less than Dictionary size shrinkage ratio k (n，i) for:

[0104]

[0105] Using the above method, without increasing the Gabor dictionary size, the discretization spacing of the four parameters of the Gabor dictionary is further refined compared to previous dictionaries, thus improving detection accuracy. This resolves the contradiction that "reducing the discretization spacing of dictionary parameters inevitably increases the dictionary size." Furthermore, by developing different discretization schemes for the four parameters of the Gabor dictionary, there are multiple methods to generate various customized Gabor dictionaries to meet the detection accuracy requirements at different parameters and scales.

[0106] In ultrasonic nondestructive testing, the polarity of the reflected echo only changes when the ultrasonic wave travels from a high-density material to a low-density material, and this change occurs at both interfaces. Therefore, the phase ω of the ultrasonic echo changes very little. In this application, the refinement of the discrete spacing of the phase parameter ω is ignored when generating the customized Gabor dictionary. Based on the discretization scheme, a dictionary update algorithm is proposed to generate a customized Gabor dictionary by refining the discretization spacing of the scale parameter s and the center frequency f.

[0107] Assume the number of sampling points for the scale parameter s in the dictionary D0 is For dictionary D0 and Here, N is the upper bound of the scaling parameter s. Therefore, the scaling parameter s in the dictionary D can be optimized by adjusting the coefficient a. (n，i) The sampling interval in the dictionary D, thus making the dictionary D (n，i) Size Approximately the size of the reference dictionary D0 The adjustment method for coefficient 'a' is as follows:

[0108]

[0109] By updating the parameter boundaries and by updating the coefficient a, the new dictionary D is obtained. (n，i) It can be generated using a discretization scheme.

[0110] An ultrasonic signal is a bandwidth pulse modulated at the transducer's center frequency, typically simulated using a Gabor function. A Gabor dictionary, generated across the entire time-frequency plane using the four parameters (f, s, u, ω) of the Gabor function in discretization equation 2, is widely used in ultrasonic nondestructive testing.

[0111]

[0112] Where f is the center frequency of the atom, s is the scaling function, u is the arrival time, and w is the phase. This dictionary is the most primitive Gabor dictionary, and the discretization scheme used to generate it is as follows:

[0113]

[0114] 0 < j < log₂N, 0 ≤ p < N² -j+1 0≤k<2 j+1 ,

[0115] Where N is the length of the signal, this is a concise yet complete Gabor dictionary.

[0116] Algorithm Iteration Termination Criteria

[0117] Signal y(n，i) After n iterations of the algorithm, use dictionary D (n，i) The decomposition by the SMP algorithm can be expressed as:

[0118]

[0119] Assume that by setting a threshold ε, a total of m atoms are obtained and the remaining residual R is calculated. m y (n，i) There will be two scenarios, and the corresponding solutions are as follows:

[0120] 1. If m = 1, then the dictionary D (n，i) Updated to D with a more refined sampling interval (n+1，i) and the signal y (n，i) Use dictionary D (n+1，i)

[0121] Further decomposition may result in the following two possibilities:

[0122] 1.1 If the number of atoms obtained after more refined dictionary sparse decomposition is still 1, the algorithm iteration terminates and the sparse decomposition of the signal is completed.

[0123] 1.2 If the number of atoms obtained after more refined dictionary sparse decomposition is greater than 1, the algorithm continues to iterate until event 1.1 is achieved.

[0124] 2. If m > 1, the algorithm continues to iterate, and the dictionary is updated to a finer scale until event 1 is achieved.

[0125] Example 2

[0126] The above method was simulated, and the simulation results are as follows:

[0127] 2.1 Multiple overlapping ultrasound signals

[0128] The purpose of the simulation is to verify the performance of the proposed sparse decomposition algorithm. Figure 4 The image shows an ultrasonic signal y composed of the superposition of four ultrasonic echoes. Each ultrasonic echo is generated by the Gabor function described in section 2.1, and its parameters are shown in the table below:

[0129] Table 1. Parameters used to generate simulated ultrasonic echoes

[0130]

[0131] 1) Initial sparse decomposition

[0132] First, the SMP algorithm is used to perform sparse decomposition of the signal y on dictionary D0. By setting the threshold ε to 0.2, four atoms are obtained, and their parameters are shown in Table 2. To demonstrate the algorithm's performance, Figure 5a is a schematic diagram of these four atoms in the frequency domain. Figure 5 b represents the degree of matching between the reconstructed echo of these four atoms and the simulated ultrasonic echo.

[0133] Table 2. Relevant parameters of the four atoms obtained from the initial sparse decomposition

[0134]

[0135] In the signal segmentation stage, after arranging the atoms in Table 1 in ascending order of arrival time u, the arrival time difference du between adjacent atoms is calculated. The results show that the maximum value is du2 = 64. Therefore, atoms 2 and 3 are selected to segment the signal y into y2. (1，1) and y (1，2) .

[0136] The starting and ending points of atoms 2 and 3 can be calculated using the formula:

[0137] Given A1 = 0, B1 = 153, A2 = 94, and B2 = 256, it is clear that A2 < B1. Therefore, atoms 2 and 3 are determined to be overlapping. Using the overlapping signal segmentation method proposed in Section 4, the signal y is segmented into y2 and y3 using B1 and A2. (1，1) and y (1，2) Two segments, the ranges of the two segments are y (1，1) ∈[0, 153], y (1，2) ∈[94,256].

[0138] Set the relaxation coefficient to α = 1.5. Then dictionary D (1，1) and D (1，2) The compressibility boundary of the scale parameter s can be calculated using Equation 3: Compared to dictionary D0, dictionary D (1，1) and D (1，2) The mesoscale parameter s has a more compact value range, which reduces the number of possible values ​​for s in sparse decomposition. Therefore, to improve the algorithm's resolution, the value of the coefficient 'a' is optimized to refine the range of the scale parameter s in the dictionary D. (1，1) and D (1，2) The sampling interval is determined by the dictionary D. The optimization of coefficient 'a' is based on the dictionary D. (1，1) and D (1，2) Size and Should be related to the size of the initial dictionary D0 Keep similarity. (For dictionary D) (1，1) and D (1，2) The updated coefficient 'a' is:

[0139]

[0140]

[0141] Since the updated 'a' is smaller than the 'a' in dictionary D0, the scale parameter 's' will have more sampling points for the same signal length, thus improving the algorithm's detection accuracy. (Dictionary D after coefficient update) (1，1) and D (1，2) The size can be calculated using formula 4: Compared to the size of the initial dictionary D0 The dictionaries are of similar size.

[0142] 2) First iteration of the algorithm

[0143] In the first iteration of the algorithm, the signal segment y (1，1) and y (1，2) The updated dictionary D was used respectively through the SMP algorithm. (1，1) and D (1，2) Sparse decomposition is performed, and the threshold ε is set to be the same as the initial sparse decomposition. Figure 6 The degree of matching between the image of the atoms obtained from the decomposition in the frequency domain and the corresponding reconstructed echo and the simulated ultrasonic echo.

[0144] Due to signal segment y (1，1) and y (1，2) Each only decomposes into 2 atoms, therefore the signal segment y (1，1) Based on the arrival time of its two atoms, the signal is segmented into y. (2，1) and y (2，2) Two parts. Similarly, signal segment y (1，2) Divide it into y based on the arrival time of its two atoms. (2，3) and y (2，4) Two parts. Among them, signal segment y (1，1) The starting and ending points of the two atoms obtained from the decomposition can be calculated using Formula 7:

[0145] A1=0, B1=85, A2=84, B2=153

[0146] Signal segment y (1，2) The starting and ending points of the two atoms obtained from the decomposition can also be calculated using Formula 7:

[0147] A1=94, B1=220, A2=137, B2=247

[0148] In signal segment y (1，1) and y (1，2) In the signal segmentation, A2 is always less than B1, so the two groups of atoms decomposed from each of these two signal segments are determined to be overlapping. Using the signal segmentation strategy proposed in Section 4, signal segment y... (1，1) Divided into y by points B1 and A2 of its atomic group (2，1) and y (2，2)Two segments. Signal segment y (1，2) Divided into y by points B1 and A2 of its atomic group (2，3) and y (2，4) Two segments. The ranges of the four segmented signals are as follows:

[0149] y (2，1) ∈[0, 85], y (2，2) ∈[84, 153]

[0150] y (2，3) ∈[94, 220], y (2，4) ∈[137, 247]

[0151] Meanwhile, in the dictionary update part, four customized dictionaries D need to be generated separately. (2，1) D (2，2) D (2，3) , and D (2，4) The corresponding use is in the new number segment y (2，1) ,y (2，2) ,y (2，3) and y (2，4) Similar to the initial sparse decomposition, the relaxation coefficient is set to α = 1.5. The dictionary D can be calculated using the formula. (2，1) D (2，2) D (2，3) , and D (2，4) The range of values ​​for the compressed scale parameter s.

[0152]

[0153]

[0154] Compression of the value range makes dictionary D (2，1) D (2，2) D (2，3) , and D (2，4) The number of sampling points for the scale parameter s is reduced. In order to ensure that the newly generated dictionary maintains a similar size to the initial dictionary D0, the sampling interval of the parameter a is optimized by a formula to increase the number of sampling points for the scale parameter s within its value range.

[0155]

[0156]

[0157]

[0158]

[0159] The newly generated dictionary D (2，1) D (2，2) D(2，3) , and D (2，4) The dimensions can be calculated using formulas.

[0160]

[0161] It can be seen that by optimizing parameter a, the detection accuracy of scale parameter s is improved without increasing the dictionary size. However, in y (2，1) ,y (2，2) ,y (2，3) and y (2，4) There are still 2 atoms in each of them. According to the second case in the algorithm termination condition, the algorithm continues and performs the second iteration.

[0162] 3) Second iteration of the algorithm

[0163] In this algorithm iteration, signal segment y (2，1) ,y (2，2) ,y (2，3) and y (2，4) Using the SMP algorithm in its corresponding dictionary D (2，1) D (2，2) D (2，3) , and D (2，4) Perform sparse decomposition. The decomposition result and the reconstructed echo are as follows: Figure 7 As shown, by setting the residual threshold ε to be the same as the initial sparse decomposition, each signal segment is decomposed into only one atom. This achieves the first case of the algorithm termination condition in Section 5, therefore, the algorithm continues and proceeds to the termination verification stage.

[0164] 4) Algorithm termination verification

[0165] Since the first case of the algorithm's termination condition was achieved in the second iteration, it is necessary to determine which of the two sub-cases of the first condition was achieved. The dictionary is then further updated to D using the same method. (3，1) D (3，2) D (3，3) , and D (3，4) , signal segment y (2，1) ,y (2，2) ,y (2，3) and y (2，4) Using the SMP algorithm in a more refined dictionary D (3，1) D (3，2) D (3，3) , and D (3，4) Sparse decomposition is performed on the above, with the residual threshold set to ε = 0.2. For example... Figure 8 As shown, each signal segment can still only be decomposed into one atom. Therefore, condition 1.2 in the algorithm's termination condition is met. The algorithm iteration stops, and the sparse decomposition of signal y is complete.

[0166] 2.2 Comparison of echo estimation performance between traditional SMP algorithm and algorithm of this application

[0167] The accuracy of echo estimation is described by three metrics: energy error, coefficient error, and amplitude error, where the energy error E error Defined as:

[0168]

[0169] Where y i The original signal, This is a signal for reconstruction.

[0170] The coefficient error is used to evaluate the similarity of sparse decomposition results, and is defined as:

[0171]

[0172] in For the estimated reflection coefficient, c i This is the reflection coefficient.

[0173] Amplitude error is an indicator used to evaluate amplitude-polarity AMI systems, and is defined as:

[0174]

[0175] Where A is the maximum intensity in the reconstructed echo.

[0176] Table 3 lists Figure 4 The results for the three metrics of the simulated signal are shown above. It can be seen that after each iteration, the energy error, coefficient error, and amplitude error are significantly improved.

[0177] Table 3. Performance demonstration of the echo estimation algorithm in this application.

[0178]

[0179]

[0180] 2.3 Sparse decomposition of ultrasound signals under different overlap ratios

[0181] Two echoes of the same shape were generated at different time locations to simulate frequency-dependent attenuation in ultrasonic signals that does not exist over very short propagation distances. One echo gradually moves to meet the other to simulate different percentages of overlap. Figure 9At each percentage point shown, the algorithm proposed in this application decomposes the resulting overlapping signal and examines the energy difference with the original echo. On average, the energy error is very small, less than 0.5% when the overlap percentage is below 67%. However, when the overlap percentage increases to above 82%, the algorithm fails to correctly resolve the two echoes because the two echoes are merging and appear as a single echo.

[0182] 2.4 Noise robustness test of the algorithm in this application

[0183] by Figure 9 For reference, two ultrasonic echoes with the same shape and an overlap ratio of 50% were selected and Gaussian white noise (AWGN) was added.

[0184] To more intuitively demonstrate the algorithm's performance in the face of noise, signal-to-noise ratio (SNR) and root mean square error (RMSE) are introduced to evaluate the algorithm's noise robustness. Their definitions at the input before sparse decomposition are as follows:

[0185]

[0186]

[0187] Among them, f signal For a noise-free signal, f noise The added Gaussian white noise is N, where N is the signal length. Similarly, at the output, the signal-to-noise ratio and root mean square error can be defined as follows:

[0188]

[0189]

[0190] in To reconstruct the signal, and considering the randomness of Gaussian white noise, the algorithm... Figure 10 and Figure 11 Each SNR point was run 100 times, and the average value was taken as the output result.

[0191] according to Figure 10 It can be concluded that for any input SNR in the range of [0dB, 25dB], the output signal has an approximate SNR. out -SNR in A 9dB improvement. The output RMSE is on average twice as good as the input RMSE. From Figure 11 It can be seen that even when the signal-to-noise ratio is close to 0 dB and the two echoes overlap by 50%, the energy error remains within 40% of the total energy. These results demonstrate that the proposed algorithm exhibits good performance in separating overlapping ultrasonic signals in noisy environments.

[0192] Example 3

[0193] The test sample used in the experimental phase was an 8-layer stacked chip package. Data acquisition equipment was a Sonoscan Gen6™ C-scanning system. Ultrasonic signals were generated using a 100MHz PZT transducer. The ultrasonic signals were recorded at a sampling frequency of 1GHz. The signals from the first 6 layers are shown below. Figure 12 As shown.

[0194] right Figure 12 The ultrasonic signal was separated using the sparse decomposition algorithm proposed in this paper, and the results are as follows: Figure 13 As shown. Figure 13 As shown, by applying the algorithm proposed in this paper, we are able to effectively separate the dominant echoes that retain most of the signal energy of the ultrasonic echoes. These echoes appear as six black shadows on the phase plane. Furthermore, by compressing the boundary dictionary and using adaptive spatial compression techniques, the number of atoms used to reconstruct the signal is reasonably reduced, and the reconstructed signal exhibits a high degree of matching with the original signal.

[0195] This paper proposes a multi-resolution analysis sparse decomposition algorithm for decomposing overlapping ultrasound signals. First, the algorithm separates the ultrasound signal into shorter signal segments. Then, it updates the dictionary by refining the parameter range and applies parameter boundary compression to utilize information obtained from the last decomposition. The algorithm's performance was tested with different overlap percentages and signal-to-noise ratios (SNRs). Experimental results show that the proposed method performs excellently in echo separation and echo estimation, maintaining good performance even with an overlap ratio as high as 67%. Furthermore, the improved SNR and reduced root mean square error demonstrate excellent performance even in noisy environments. In addition, experimental results show that the algorithm can capture the true structure of overlapping ultrasound signals, and the reconstructed ultrasound signal has a good match with the recorded ultrasound signal.

[0196] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A multi-resolution sparse decomposition algorithm for decomposing overlapping ultrasound signals, characterized in that: Includes the following steps: Step 1: Perform sparse decomposition on the initial dictionary D0 using the SMP algorithm for any signal y; Step 2: Using the sparse decomposition results obtained in Step 1, segment the signal y to obtain signal segments y1,1 and y1,2. The method for segmenting the signal y is as follows: Step 2.1: Arrange the m atoms obtained from the decomposition in Step 1 in ascending order of arrival time u; Step 2.2: Calculate the arrival time difference between every two adjacent atoms, denoted as . ; Step 2.3: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require The maximum value in is denoted as Determine the signal segment The location of the dividing point is the arrival time. and Between two atoms; Step 2.4: Through atoms and The ultrasonic echo is reconstructed using its corresponding decomposition coefficients, denoted as echo(q) and echo(q+1), respectively. Step 2.5: By determining whether echo(q) and echo(q+1) overlap, the signal segment... Implement different signal segmentation strategies; Step 3: Generate dictionaries D1,1 and D1,2 for signal segments y1,1 and y1,2 respectively, and update the initial dictionary D0 to D1,1 and D1,2; the dictionary generation process is as follows: Let the total number of atoms contained in the initial dictionary D0 be denoted as The value is calculated using formula (1) and used as a reference for the size of the dictionary to be generated later. （1） in, N is the signal length; k is the discrete index of the atom's center frequency f; p is the discrete index of the time displacement arrival time u; i is the discrete index of the phase ω; j is the discrete level index of the scale function s; By employing the same discrete strategy and step size as dictionary D0, the signal segment Corresponding dictionary The size is calculated from the new parameter range obtained in formula (2), denoted as ; （2） Where f is the center frequency of the atom, and s is the scaling function. ω is the arrival time, α is the phase, α is the relaxation coefficient used to expand the parameter boundaries, l is the number of iterations of the algorithm, and n is the segment number of the signal. Step 4: For each signal segment in Step 2, use the corresponding dictionary generated in Step 3 to perform sparse decomposition using the SMP algorithm; Step 5: Repeat steps 2-4 until only one echo can be extracted from each individual signal segment.

2. The multi-resolution sparse decomposition algorithm for overlapping ultrasound signal decomposition according to claim 1, characterized in that: In step 3, the shrinkage ratio of the dictionary size for: （3）； in, Defined as signal segment Corresponding dictionary The size.

3. The multi-resolution sparse decomposition algorithm for overlapping ultrasound signal decomposition according to claim 1, characterized in that: In step 3, the process of determining the boundaries of the dictionary update parameters is as follows: After sparse decomposition of the signal segment y(n,i), m atoms are obtained, denoted as m. The four parameters of each atom are defined as follows: Then the lower bound of the scaling function s Estimated as Upper Realm It can be estimated as Similarly, the upper and lower bounds of the other three parameters... Obtained using the same method.

4. The multi-resolution sparse decomposition algorithm for decomposing overlapping ultrasound signals according to claim 1, characterized in that: In step 3, the dictionary update algorithm is as follows: Assume the number of sampling points for the scale parameter s in the dictionary D0 is For dictionary D0, ,and Where N is the signal length, i.e., the upper bound of the scaling function s; the scaling function s is optimized by adjusting the coefficient a in the dictionary. The sampling interval in the dictionary, and thus the dictionary Size Approximately the size of the reference dictionary D0 The adjustment method for coefficient 'a' is as follows: 4） The new dictionary is generated by updating the parameter boundaries using formula (2) and updating the coefficients a using formula (4). Generated using a discretization scheme.

5. The multi-resolution sparse decomposition algorithm for overlapping ultrasound signal decomposition according to claim 1, characterized in that: In step 4, the criteria for determining the termination of algorithm iteration are as follows: Signal After n algorithm iterations, use a dictionary The decomposition by the SMP algorithm is expressed as follows: （5） in, For the remaining residual, For the k-th selected Gabor atom; By setting a threshold ε, a total of m atoms and the remaining residual were obtained. There are two possible scenarios, and the corresponding solutions are as follows:

1. If m=1, then the dictionary will be... Updated to have a more refined sampling interval and the signal Use a dictionary Decomposed again; 2. m > 1. The algorithm continues to iterate, and the dictionary is updated to a finer scale until event 1 is achieved.

6. The multi-resolution sparse decomposition algorithm for overlapping ultrasound signal decomposition according to claim 5, characterized in that: The results of the further decomposition can be in the following two scenarios: 1.1 If the number of atoms obtained after more refined dictionary sparse decomposition is still 1, the algorithm iteration terminates and the sparse decomposition of the signal is completed; 1.2 If the number of atoms obtained after more refined dictionary sparse decomposition is greater than 1, the algorithm continues to iterate until event 1.1 is achieved.

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