CN107065006A - A Seismic Signal Coding Method Based on Online Dictionary Update - Google Patents

Abstract

A seismic signal coding method based on online dictionary updating belongs to a seismic signal data coding transmission method, solves the dictionary transmission problem caused by dictionary learning and sparse representation in seismic signal coding, and can be applied to various seismic signal measurement-based ground mapping. The invention comprises the following steps: (1) dividing input seismic signals into a plurality of groups according to a time sequence, and performing sparse coding on each group of data by using a dictionary in a cache to calculate a sparse coefficient; (2) quantizing and entropy coding the sparse coefficient in the step (1); (3) and reading the reconstructed data of the previous P transmitted groups from the cache, and performing dictionary learning by combining the sparse coefficients transmitted by the current group, thereby updating the dictionary required by sparse representation of the next group of data. According to the invention, by means of online dictionary updating, dictionary information does not need to be transmitted in real time under the premise of ensuring effective sparse representation of signals, so that data transmission data volume is effectively reduced, and the method can be suitable for various seismic signal high-speed acquisition application occasions.

Description

A Seismic Signal Coding Method Based on Online Dictionary Update

technical field

The invention belongs to a seismic signal data transmission method, in particular to a seismic signal lossy encoding method.

Background technique

Surveying and mapping technology based on seismic signal measurement is one of the effective methods for underground structure and mineral resource measurement. In each surveying and mapping, seismic signal measurement on the ground will generate more than 100T of data, and the bandwidth of signal transmission is extremely limited at present, so it is necessary to reduce the data volume of seismic signal through seismic signal coding technology before transmission. In the prior art, a seismic signal encoding method based on discrete cosine transform is proposed, which can obtain a compression factor close to 3 times. There is also a two-dimensional discrete cosine transform technology based on local seismic signal adaptation, so that the important features of the reconstructed seismic signal can be preserved. Furthermore, the seismic signal encoding technology using adaptive wavelet packets can obtain higher compression factor and better reconstruction quality. Due to its better direction-preserving characteristics, it is currently widely used in the feature extraction of seismic signals. The main idea of the above method is to use a suitable basis or redundant dictionary to represent the seismic signal, so that the representation of the signal is sparse. In recent years, sparse representation through dictionary learning has gained widespread attention, especially in image coding, and has been widely used in remote sensing images to learn dictionaries through double-sparse models, thereby obtaining better coding results. These research results all show the feasibility of applying dictionary learning and sparse representation in seismic signal coding.

Traditional encoding methods based on dictionary learning and sparse representation often include the following two main methods: (1) Sparse representation of online data acquired in real time through offline learned dictionary. For this method, an offline training set needs to exist in advance, and the required dictionary information is obtained through the offline training set. Therefore, whether the sparse representation of online data is effective depends heavily on the correlation between offline data and online data. For actual seismic signal measurements, it is difficult to obtain a general offline training set applicable to different situations. (2) Use the real-time online data to train the dictionary, and perform sparse representation on the real-time online data through the dictionary. In this method, it is necessary to transmit the dictionary so that the sparse representation process at the codec end can be synchronized. Therefore, the transmission of dictionary information will increase the size of the encoded code stream, thereby reducing the encoding performance.

Invention content:

The invention proposes a seismic signal encoding method based on online dictionary update, which belongs to the seismic signal data encoding transmission method, solves the problem of how to transmit the dictionary caused by using dictionary learning and sparse representation in seismic signal encoding, and can be applied to various seismic-based In subsurface mapping for signal measurements.

A method for encoding seismic signals based on online dictionary update of the present invention includes an encoding step and a decoding step; wherein,

The encoding steps include:

Step 1. Divide the input seismic signal into multiple groups according to time order, and use the dictionary in the cache to perform sparse coding for each group of data, specifically:

Step 11, divide the seismic signal data adjacent to T traces into one group, and process each group of data separately; assume that the current group of data is Z group of data, which is expressed as Y _z ; divide the data of each trace into several units, the length of each unit y _i is M×1, sort y _i according to the column; therefore, Y _z =[y ₁ ,...y _i ,...y _N ]; here it is assumed that each The length of the data recorded on the track is U, then there is the following relational expression: T×U=M×N;

Step 12. Read the dictionary D _z-1 in the cache, and given the sparsity of the sparse coefficient matrix W _Z as L, optimize and solve the following formula:

Step 2. Perform quantization and entropy coding on the sparse coefficients in step S1, specifically including:

Step 21. Quantize the sparse coefficient matrix using a uniform quantization method, specifically as follows:

w _Z (i, j) represents the value of the coefficient whose coordinates are (i, j) in the sparse coefficient matrix W _Z , and Δ represents the quantization step size, Represents the quantization result of the coefficient value of (i,j), and round(·) represents the rounding operation;

Step 22, create the non-zero coefficient position matrix PT that is made up of numerical value 0 and numerical value 1, creation method is as follows:

Among them, abs( ) represents the absolute value operation;

Step 23, using arithmetic coding for the non-zero coefficient position matrix PT;

Step 24, for non-zero coefficients (corresponding to PT (i, j) = 1 position Using Huffman coding;

Step 3. Read the reconstruction data of the previous P transmitted groups from the cache, and perform dictionary learning in combination with the sparse coefficients transmitted by the current group, so as to update the dictionary required for the sparse representation of the next group of data, specifically including:

Step 31, calculate P+1 group reconstruction data p∈[ZP,Z] (current group data is Z group), the calculation method is as follows:

in, unit in

Step 32, solve the required dictionary D _Z according to the following optimization process:

in, a _i represents a constant describing the correlation between groups, formula (2) is solved through the following steps of iterative operation:

Step 321, fixing D _Z , W' can be calculated by the aforementioned PS method;

Step 322, fixing W', D _Z can be updated according to the MOD method:

Step 323, repeating the above steps 321 and 322 to the specified number of iterations, updating the required dictionary D _Z ;

The decoding steps include:

Step 4. Perform inverse quantization and entropy decoding on the received sparse coefficients to generate a non-zero coefficient matrix W' _Z , as follows:

Step 41. Perform Huffman decoding on the non-zero coefficient encoded code stream to obtain the non-zero coefficient w _c ;

Step 42. Perform inverse quantization on the non-zero coefficient w _c to obtain the inverse quantization coefficient w' _c , specifically as follows:

w' _c =w _c ×Δ

Step 43: Perform arithmetic decoding on the coded code stream of the non-zero coefficient position matrix PT to obtain the non-zero coefficient position matrix PT, and combine the inverse quantization coefficient _w'c generated in step 42 to generate the non-zero coefficient matrix _W'Z ;

Step 5, carry out seismic signal reconstruction, specifically as follows:

Step 51. Read the dictionary D _z-1 in the cache to generate the reconstructed signal Y' _z , specifically as follows:

Y' _z ＝ D _z-1 × W' _Z

Step 52. Rearrange the reconstructed signal Y' _z =[y' ₁ ... y' _i ... y' _N ] (the length of each unit y' _i is M×1), and several adjacent units According to the way of columns, they are connected end to end to form a trace, therefore, the length of each trace is There are T marks in total;

Step 6. Generate the dictionary D _z in the cache for the reconstruction of the next set of data, that is, read the reconstructed data of the previous P transmitted groups from the cache, and perform dictionary learning in combination with the sparse coefficients transmitted by the current group, thereby updating The next set of dictionaries required for data sparse representation include:

Step 61, calculate P+1 group reconstruction data p∈[ZP,Z] (current group data is Z group), the calculation method is as follows:

in, unit in

Step 62, solve the required dictionary D _Z according to the following optimization process:

in, a _i represents a constant describing the correlation between groups, formula (2) is solved through the following steps of iterative operation:

Step 621, fixing D _Z , W' can be calculated by the aforementioned PS method;

Step 622, fixing W', D _Z can be updated according to the MOD method:

Step 623 , repeating the above steps 321 and 322 up to the specified number of iterations, and updating the required dictionary D _Z .

The present invention does not need real-time transmission of dictionary information by updating the online dictionary under the premise of ensuring effective sparse representation of signals, thereby effectively reducing the data volume of data transmission and being applicable to various high-speed acquisition applications of seismic signals.

Description of drawings

Fig. 1 is a flow chart of the present invention;

Fig. 2 is a part of the signal in the test seismic signal data;

Fig. 3 is the dictionary of learning;

Figure 4 is the result of the performance comparison of different methods.

detailed description

Example:

The present invention mainly comprises:

S1: Divide the input seismic signal into multiple groups according to time order, and use the dictionary in the cache to perform sparse coding for each group of data;

S2: Carry out quantization and entropy coding to the sparse coefficient in step S1;

S3: Read the reconstruction data of the previous P transmitted groups from the cache, and perform dictionary learning in combination with the sparse coefficients transmitted by the current group, so as to update the dictionary required for the sparse representation of the next group of data.

Further, step S1 is specifically:

S11: Divide the seismic signal data of adjacent T traces into one group, and process each group of data separately; assuming that the current group of data is Z group of data, it is expressed as Y _z . The data of each trace is equally divided into several units, and the length of each unit y _i is M×1, and the y _i are sorted by columns. Therefore, Y _z =[y ₁ ,...y _i ,...y _N ]. Assume here that the length of data recorded on each track is U, then there is the following relationship: T×U=M×N.

S12: Read the dictionary D _z-1 in the cache, given the sparsity of the sparse coefficient matrix W _Z as L, optimize and solve the following formula:

For the solution of formula (1), we intend to use the PS method (“Partial search vector selection for sparse signal representation,” in NORSIG-03). The PS method is based on the OMP algorithm (Orthogonal Matching Pursuit, "Comparison of basis selection methods," in Signals, Systems and Computers, 1996. Conference Record of the Thirtieth Asilomar Conference on), therefore, the flow of the OMP algorithm is first given:

The PS method modifies the above step (1) from the search process of only the maximum correlation dictionary unit to the search of several maximum correlation dictionary units, thus providing more search decisions to obtain better sparse vectors.

Step S2 is specifically:

S21: Use a uniform quantization method to quantize the sparse coefficient matrix, as follows:

w _Z (i, j) represents the value of the coefficient whose coordinates are (i, j) in the sparse coefficient matrix W _Z , and Δ represents the quantization step size, Represents the quantization result of the coefficient value of (i, j), and round(·) represents the rounding operation.

S22: Create a non-zero coefficient matrix PT composed of a value 0 and a value 1, the creation method is as follows:

Among them, abs(·) represents the absolute value operation.

S23: Arithmetic coding is used for the non-zero coefficient matrix PT.

S24: Apply Huffman coding to the non-zero coefficients (corresponding to w ^r _i,j at the position of PT(i,j)=1).

Step S3 is specifically:

S31: Calculating P+1 group reconstruction data p∈[ZP,Z] (current group data is Z group), the calculation method is as follows:

in, unit in

S32: Solve required dictionary D _Z according to following optimization process:

in, a _i represent constants describing the correlation between groups.

Equation (2) can be solved by iterative operation as follows:

1) Fixed D _Z , W' can be calculated by the aforementioned PS method;

2) Fixed W', D _Z can be updated according to the MOD method ("Method of Optimal Directions for FrameDesign," in 1999 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)):

3) Repeat the above step 1) and step 2) up to the specified number of iterations to generate the required updated dictionary D _Z .

Example 1:

1. The test seismic signal data comes from the UTAM image database (http://utam.gg.utah.edu/SeismicData/SeismicData.html), we choose the Find-Trapped-miners data as the test data, which contains 72 sensors, each Each sensor contains 135 traces;

2. Take 1600 time-length samples for each trace, and the data of every 10 traces is a group, and some test data are shown in Figure 1;

1. Assume that the current group is the third group (the first 2 groups of data have been encoded and the relevant data has been output to the decoder and cache), read the dictionary D ₂ in the cache and perform sparse coding. In this embodiment, the cache The size of the dictionary is 16×64, and the sparsity is 1/16 (the ratio of non-zero coefficients to the overall coefficients). Some data in W ₃ obtained by calculation are as follows;

0 0 0 0 0 0 0 0 0 0 0 0 0 191086.69 0 0 0 0 0 0 0 0 0 69516.63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 263371.03 0 0 0 0 0 -275961.58 248225.21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2. Quantize W ₃ with a quantization step size of 1024; use Huffman coding for non-zero data after quantization, and use arithmetic coding for non-zero element positions. We use SNR to measure the difference between the original signal and the reconstructed signal. At this time The SNR is 21.2dB.

3. The dictionary is updated by reading the first two sets of reconstruction data and W ₃ , and the updated dictionary D ₃ is shown in FIG. 3 .

Example 2:

1. Sparsely represent the fourth set of data with the updated dictionary D ₃ of the third set of data, and update the dictionary D ₄ ;

2. Sparsely represent the fifth set of data with the updated dictionary D ₄ of the fourth set of data, and update the dictionary D ₅ ;

3. Quantize and encode each set of data above, calculate the code rate and measure the distortion by SNR;

4. Adjust different sparsity and repeat the above 3 steps to get the distortion under different code rates.

5. In order to prove the effectiveness of the algorithm, we also experimented on the seismic signal encoding algorithm based on DCT, Curvelet and offline dictionary learning method (K-SVD+ORMP), and obtained the rate-distortion of different algorithms. The relevant comparative experimental results are as follows Figure 4 shows.

The specific embodiments described in this technology are merely illustrative of the spirit of the invention. Those skilled in the art to which the present invention belongs can make various modifications or supplements to the described specific embodiments or adopt similar methods to replace them, but they will not deviate from the spirit of the present invention or go beyond the definition of the appended claims range.

Claims (1)

1. A seismic signal encoding method based on online dictionary update, characterized in that, comprising an encoding step and a decoding step; wherein, The encoding steps include: Step 1. Divide the input seismic signal into multiple groups according to time order, and use the dictionary in the cache to perform sparse coding for each group of data, specifically: Step 11, divide the seismic signal data adjacent to T traces into one group, and process each group of data separately; assume that the current group of data is Z group of data, which is expressed as Y _z ; divide the data of each trace into several units, the length of each unit y _i is M×1, sort y _i according to the column; therefore, Y _z =[y ₁ ,...y _i ,...y _N ]; here it is assumed that each The length of the data recorded on the track is U, then there is the following relational expression: T×U=M×N; Step 12. Read the dictionary D _z-1 in the cache, and given the sparsity of the sparse coefficient matrix W _Z as L, optimize and solve the following formula: <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mi>Z</mi> </msub> <mo>=</mo> <munder> <mi>argmin</mi> <mi>W</mi> </munder> <mo>|</mo> <mo>|</mo> <msub> <mi>Y</mi> <mi>Z</mi> </msub> <mo>-</mo> <msub> <mi>D</mi> <mrow> <mi>Z</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mi>W</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <mi>W</mi> <mo>|</mo> <msub> <mo>|</mo> <mn>0</mn> </msub> <mo>&amp;le;</mo> <mi>L</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> Step 2. Perform quantization and entropy coding on the sparse coefficients in step S1, specifically including: Step 21. Quantize the sparse coefficient matrix using a uniform quantization method, specifically as follows: <mrow> <msup> <msub> <mi>w</mi> <mi>Z</mi> </msub> <mi>r</mi> </msup> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>r</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>w</mi> <mi>Z</mi> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> <mi>&amp;Delta;</mi> </mfrac> <mo>)</mo> </mrow> </mrow> w _Z (i, j) represents the value of the coefficient whose coordinates are (i, j) in the sparse coefficient matrix W _Z , and Δ represents the quantization step size, Represents the quantization result of the coefficient value of (i,j), and round(·) represents the rounding operation; Step 22, create the non-zero coefficient position matrix PT that is made up of numerical value 0 and numerical value 1, creation method is as follows: <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mi>T</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mi>f</mi> <mi> </mi> <mi>a</mi> <mi>b</mi> <mi>s</mi> <mrow> <mo>(</mo> <msubsup> <mi>w</mi> <mi>Z</mi> <mi>r</mi> </msubsup> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>&gt;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>P</mi> <mi>T</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mi>f</mi> <mi> </mi> <mi>a</mi> <mi>b</mi> <mi>s</mi> <mrow> <mo>(</mo> <msubsup> <mi>w</mi> <mi>Z</mi> <mi>r</mi> </msubsup> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>=</mo> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> Among them, abs( ) represents the absolute value operation; Step 23, using arithmetic coding for the non-zero coefficient position matrix PT; Step 24, for non-zero coefficients (corresponding to PT (i, j) = 1 position ) adopts Huffman coding; Step 3. Read the reconstruction data of the previous P transmitted groups from the cache, and perform dictionary learning in combination with the sparse coefficients transmitted by the current group, so as to update the dictionary required for the sparse representation of the next group of data, specifically including: Step 31, calculate P+1 group reconstruction data p∈[ZP,Z] (current group data is Z group), the calculation method is as follows: <mrow> <msub> <mover> <mi>Y</mi> <mo>~</mo> </mover> <mi>p</mi> </msub> <mo>=</mo> <msub> <mi>D</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>W</mi> <mi>p</mi> <mi>q</mi> </msubsup> </mrow> in, unit in Step 32, solve the required dictionary D _Z according to the following optimization process: <mrow> <mtable> <mtr> <mtd> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>D</mi> <mi>Z</mi> </msub> <mo>,</mo> <msup> <mi>W</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;rsqb;</mo> <mo>=</mo> <munder> <mi>argmin</mi> <mrow> <mi>D</mi> <mo>,</mo> <mi>W</mi> </mrow> </munder> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>Y</mi> <mo>~</mo> </mover> <mrow> <mi>Z</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mi>D</mi> <mi>W</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <mi>W</mi> <mo>|</mo> <msub> <mo>|</mo> <mn>0</mn> </msub> <mo>&amp;le;</mo> <mi>L</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> in, a _i represents a constant describing the correlation between groups, formula (2) is solved through the following steps of iterative operation: Step 321, fixing D _Z , W' can be calculated by the aforementioned PS method; Step 322, fixing W', D _Z can be updated according to the MOD method: <mrow> <msub> <mi>D</mi> <mi>Z</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>Y</mi> <mo>~</mo> </mover> <mrow> <mi>Z</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>W</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>W</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>W</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> Step 323, repeating the above steps 321 and 322 to the specified number of iterations, updating the required dictionary D _Z ; The decoding steps include: Step 4. Perform inverse quantization and entropy decoding on the received sparse coefficients to generate a non-zero coefficient matrix W' _Z , as follows: Step 41. Perform Huffman decoding on the non-zero coefficient encoded code stream to obtain the non-zero coefficient w _c ; Step 42. Perform inverse quantization on the non-zero coefficient w _c to obtain the inverse quantization coefficient w' _c , specifically as follows: w′ _c ＝w _c ×Δ Step 43: Perform arithmetic decoding on the coded code stream of the non-zero coefficient position matrix PT to obtain the non-zero coefficient position matrix PT, and combine the inverse quantization coefficient _w'c generated in step 42 to generate the non-zero coefficient matrix _W'Z ; Step 5, carry out seismic signal reconstruction, specifically as follows: Step 51. Read the dictionary D _z-1 in the cache to generate the reconstructed signal Y' _z , specifically as follows: Y' _z ＝ D _z-1 × W' _Z Step 52. Rearrange the reconstructed signal Y' _z =[y' ₁ ... y' _i ... y' _N ] (the length of each unit y' _i is M×1), and several adjacent units According to the way of columns, they are connected end to end to form a trace, therefore, the length of each trace is There are T marks in total; Step 6. Generate the dictionary D _z in the cache for the reconstruction of the next set of data, that is, read the reconstructed data of the previous P transmitted groups from the cache, and perform dictionary learning in combination with the sparse coefficients transmitted by the current group, thereby updating The next set of dictionaries required for data sparse representation include: Step 61, calculate P+1 group reconstruction data p∈[ZP,Z] (current group data is Z group), the calculation method is as follows: <mrow> <msub> <mover> <mi>Y</mi> <mo>~</mo> </mover> <mi>p</mi> </msub> <mo>=</mo> <msub> <mi>D</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>W</mi> <mi>p</mi> <mi>q</mi> </msubsup> </mrow> in, unit in Step 62, solve the required dictionary D _Z according to the following optimization process: <mrow> <mtable> <mtr> <mtd> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>D</mi> <mi>Z</mi> </msub> <mo>,</mo> <msup> <mi>W</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;rsqb;</mo> <mo>=</mo> <munder> <mi>argmin</mi> <mrow> <mi>D</mi> <mo>,</mo> <mi>W</mi> </mrow> </munder> <mo>|</mo> <mo>|</mo> <msub> <mover> <mi>Y</mi> <mo>~</mo> </mover> <mrow> <mi>Z</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mi>D</mi> <mi>W</mi> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <mo>|</mo> <mo>|</mo> <mi>W</mi> <mo>|</mo> <msub> <mo>|</mo> <mn>0</mn> </msub> <mo>&amp;le;</mo> <mi>L</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> in, a _i represents a constant describing the correlation between groups, formula (2) is solved through the following steps of iterative operation: Step 621, fixing D _Z , W' can be calculated by the aforementioned PS method; Step 622, fixing W', D _Z can be updated according to the MOD method: <mrow> <msub> <mi>D</mi> <mi>Z</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>Y</mi> <mo>~</mo> </mover> <mrow> <mi>Z</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>W</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>W</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>W</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> Step 623 , repeating the above steps 321 and 322 up to the specified number of iterations, and updating the required dictionary D _Z .