A Deep and Wide Two-Dimensional Subspace Tracking Method for Reconstructing Marine Environmental Monitoring Data

Abstract

The present invention relates to the field of ocean environmental monitoring data reconstruction. In the process of compressive sensing of ocean environmental monitoring data, existing greedy reconstruction methods are prone to problems such as being trapped in local optima and overfitting. A new sparse recovery method for ocean environmental monitoring data is proposed. It searches for the support set of ocean environmental monitoring data in two dimensions, depth and breadth. Each iteration examines multiple candidate support set estimates simultaneously, and finally selects the estimate that minimizes the reconstruction residual to ensure the accurate recovery of ocean environmental monitoring data from the measurement values.

Description

Technical Field

[0001] The present invention belongs to the technical field of marine environmental monitoring data reconstruction, and particularly relates to a method for reconstructing marine environmental monitoring data by deep and wide two-dimensional subspace pursuit. Background Art

[0002] Compressed Sensing (CS) can recover a high-dimensional K-sparse vector from a small number of linear measurements y = Φx Its support set is defined as T = {i|x i ≠0, i ∈ [1, N]}, where is usually referred to as the sensing matrix.

[0003] Although it is an underdetermined system of equations, due to the sparsity of marine environmental monitoring data, x can be accurately recovered from the measurement value y by solving the l0 norm minimization problem:

[0004]

[0005] When looking for a solution to this problem, it is possible to test whether all combinations of K atoms (columns in Φ) satisfy the equality constraint. There are theoretically candidate sets, which is obviously difficult to achieve for large N and K. Therefore, the key point of the CS technology of "less sampling, clever calculation" lies in how to recover the original marine environmental monitoring data to the greatest extent from only a small amount of observed data. Due to its simple implementation and good reconstruction performance, the greedy method has received extensive attention. Existing compressed sensing greedy reconstruction methods can be classified into two categories according to the support set search direction: depth search and breadth search.

[0006] Classical greedy methods expand only one candidate atom in each iteration. The method only relies on the depth of iteration to approximate the original marine environmental monitoring data, which is the depth direction search, such as Orthogonal Matching Pursuit (OMP), Orthogonal Least Squares (OLS) algorithms. These methods all adopt single-path one-way search for the support set and have no rejection mechanism. Once a wrong choice is made, the error will gradually expand. In response to this problem, the backtracking pruning strategy has emerged, such as Subspace Pursuit (SP), Compressive Sampling Matching Pursuit (CoSaMP), etc. These methods first expand atoms in batches and then eliminate possible wrong options from them. Although compared with the traditional OMP, they have added mechanisms for batch expansion and elimination of redundancy, but these methods still rely on the depth of iteration to approximate the original marine environmental monitoring data, that is, they search only in the depth direction, and there is still a large gap between them and the best achievable performance limit obtained by the Oracle estimator.

[0007] In recent years, Kwon et al. proposed Multi-Path Matching Pursuit (MMP), which improves the accuracy of sparse ocean environmental monitoring data recovery by tracking candidate atoms along multiple paths and expands a dimension for the support set search of the greedy method, namely breadth search. However, when the sparsity is slightly larger, problems will emerge, and the storage, computing, and search costs required by the method will increase exponentially. Summary of the Invention

[0008] To overcome the problems in the prior art, the present invention proposes a deep and wide two-dimensional subspace pursuit method for reconstructing ocean environmental monitoring data.

[0009] The technical solution of the present invention to solve the above technical problems is as follows:

[0010] A deep and wide two-dimensional subspace pursuit method for reconstructing ocean environmental monitoring data, comprising the following steps:

[0011] Step 1. Input the reduced-dimensional observation value y after compressive sensing of the ocean environmental monitoring data, the compressive sensing matrix Φ, the signal sparsity K, and the number of parallel search paths L; initialize the residual r 0 = y, initialize the depth-direction iteration number l, the parent node number i, and the breadth-direction iteration number j, let l = i = j = 0, and initialize the candidate support set

[0012] Step 2. Update the iteration number l = l + 1; search in the depth direction, let be the set of candidate support sets generated in the (l - 1)-th iteration, be the i-th element therein (i ∈ [1, |T l-1 |]), and it is also a candidate support set estimate. The correlation coefficient obtained in the l-th iteration is

[0013] Step 3. Traverse the parent node i = i + 1. If i ≤ |T l-1 |, loop to execute Steps 3 to 5;

[0014] Step 4. Sort the correlation coefficients obtained by iteration. The sorted coefficient index set is defined as T Δ = {π|sort(|v π |)}, where sort(|v π |) represents sorting the coefficients in v i in descending order, v π , π ∈ [1, |v i |] represents the elements in the vector v i ;

[0015] Step 5. Search in the breadth direction j = j + 1, including the following sub-steps:

[0016] If \(j\leq L\), loop and execute:

[0017] Take the sorted coefficient index set \(T\) Δ The first \(S\) items in it are denoted as \(\Omega=T\) Δ (1:S), and combine \(\Omega\) with the estimated mother node support set to obtain the augmented set Try to reconstruct to obtain

[0018] If then find the largest first \(K\) coefficients to form a new estimated support set That is, pruning is completed, and use to reconstruct the marine environmental monitoring data to obtain and update the residual and the set of restored support sets Let \(T\) Δ \(=T\) Δ \(-\Omega\); Prepare to start the reconstruction of the \((j + 1)\)-th sub-path until \(j = L\). Then, a total of \(LS\) atoms are selected from the \(L\) sub-paths under the \(i\)-th mother node, that is, \(T\) Δ (1:LS);

[0019] Step 6. When the iteration stops, determine the candidate support set that minimizes the residual as the final support set, and output the marine environmental monitoring data

[0020] Furthermore, in the breadth-direction search, take the first \(S\) items in \(T\) Δ Specifically, they are the \(S\) items with the largest correlation coefficients.

[0021] Furthermore, the marine environmental monitoring data includes, but is not limited to, temperature, pressure, and humidity.

[0022] Compared with the prior art, the present invention has the following technical effects:

[0023] The present invention proposes a new sparse recovery method for marine environmental monitoring data, searches for the support set of marine environmental monitoring data in two dimensions of depth and breadth, examines multiple candidate support set estimates simultaneously in each iteration, and finally selects the estimate that minimizes the reconstruction residual to ensure the accurate recovery of marine environmental monitoring data from the measured values. Brief Description of the Drawings

[0024] Figure 1 It is a schematic diagram of the method idea of the deep and wide two-dimensional subspace tracking marine environmental monitoring data reconstruction method (\(L = 2, K = 4\)) (× represents the atoms removed in the pruning process). Detailed Embodiment

[0025] The principles and features of the present invention will be described below in conjunction with the accompanying drawings. The examples given are only used to explain the present invention and are not intended to limit the scope of the present invention.

[0026] Based on the breadth-first search criterion and the batch atom expansion strategy, the present invention proposes a deep and wide two-dimensional subspace tracking method for reconstructing marine environmental monitoring data, which parallelly expands multiple support set search paths. Each path expands candidate atoms in batches, selects sub-paths with decreasing residuals, and prunes the miscellaneous branches to approximate the best achievable performance limit of the Oracle estimator.

[0027] Briefly describe the notation used in this article. y is the dimension-reduced observation value of the marine environmental monitoring data after compressive sensing, Φ is the compressive sensing matrix, and x represents a certain marine environmental monitoring data (such as temperature, pressure, humidity, etc.). Φ S is a sub-matrix composed of the columns in Φ indexed by S, which is column full rank, and its pseudo-inverse is span(Φ S ) is the space spanned by the columns of Φ S , is the operator for projecting onto span(Φ S ), is the operator for projecting onto the orthogonal complement space of span(Φ S ), where I is the identity matrix.

[0028] First, the deep and wide two-dimensional subspace tracking method for reconstructing marine environmental monitoring data searches in the depth direction. Let be the set of candidate sets generated in the (l - 1)-th iteration, is the i-th element in it (i ∈ [1, |T l-1 |]), and it is also a candidate support set estimate. In the l-th iteration, first obtain the correlation coefficient , and then sort it. v π , π ∈ [1, |v i |] represents the elements in the vector v i . sort(|v π |) means sorting the coefficients in v i in descending order. The sorted coefficient index set is defined as T Δ = {π|sort(|v π |)}.

[0029] Next, start searching in the breadth direction. Take the first S terms in T Δ (i.e., the S terms with the largest correlation coefficients) and denote them as Ω = T Δ (1:S). Combine Ω with the parent node support set to obtain the expansion set denoted as (the j-th sub-path). Try to reconstruct to obtain If Then find the largest top-K coefficients to form a new support set estimate That is, pruning is completed, and use Reconstruct the marine environmental monitoring data to obtain And update the residual And the set of restored support sets Let T Δ = T Δ - Ω, to avoid expanding duplicate atoms for each sub-path. Prepare to start reconstructing the (j + 1)-th sub-path until j = L. Then, a total of LS atoms are selected for the L sub-paths under the i-th parent node, that is, T Δ (1:LS).

[0030] When the iteration terminates, the candidate set that minimizes the residual is determined as the final support set. Figure 1 The schematic diagram of the method idea when L = 2 and K = 4 is shown.

[0031] In a specific embodiment of the deep and wide two-dimensional subspace tracking marine environmental monitoring data reconstruction method of the present invention, the following steps are included:

[0032] Step 1. Input the reduced-dimensional observation value y after compressive sensing of the marine environmental monitoring data, the compressive sensing matrix Φ, the signal sparsity K, and the number L of parallel search paths; initialize the residual r 0 = y, initialize the depth-direction iteration number l, the parent node number i, and the breadth-direction iteration number j, let l = i = j = 0, and initialize the candidate support set

[0033] Step 2. Update the iteration number l = l + 1;

[0034] Search in the depth direction, let be the set of candidate support sets generated in the (l - 1)-th iteration, is the i-th element (i ∈ [1, |T l-1 |]) among them, and is also a candidate support set estimate. The correlation coefficients obtained in the l-th iteration are

[0035] Step 3. Traverse the parent node i = i + 1. If i ≤ |T l-1 |, loop to execute Steps 3 to 5;

[0036] Step 4. Sort the correlation coefficients obtained by iteration. The sorted coefficient index set is defined as T Δ = {π|sort(|v π |)}, where sort(|v π |) represents sorting the coefficients in v i in descending order, v π , π ∈ [1, |vi |] represents the elements of vector v i in;

[0037] Step 5. Search in the breadth direction with j = j + 1, including the following sub-steps:

[0038] If j ≤ L, loop and execute:

[0039] Take the first S terms of the sorted coefficient index set T Δ and denote them as Ω = T Δ (1:S), and merge Ω with the estimated support set of the parent node to obtain the augmented set Try to reconstruct to obtain

[0040] If then find the largest first K terms of the coefficients to form a new estimated support set i.e., complete pruning, and use to reconstruct the marine environmental monitoring data to obtain and update the residual and the set of restored support sets T Δ = T Δ - Ω;

[0041] Step 6. If go to Step 2; otherwise, stop the iteration, determine the candidate support set that minimizes the residual as the final support set, and output the marine environmental monitoring data

[0042]

[0043]

[0044] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A deep and wide two-dimensional subspace tracking method for reconstructing ocean environmental monitoring data, characterized in that including the following steps: Step 1. Input the reduced-dimensional observation value y after compressive sensing of ocean environmental monitoring data, the compressive sensing matrix Φ, the signal sparsity K, and the number L of parallel search paths; initialize the residual r 0 = y, initialize the iteration sequence number l in the depth direction, the parent node sequence number i, and the iteration sequence number j in the breadth direction, set l = i = j = 0, and initialize the candidate support set Step 2. Update the depth direction iteration number \(l = l + 1\); search in the depth direction, and let be the set of candidate support sets generated in the \((l - 1)\)-th iteration, where \(T_i\) is the \(i\)-th element (\(i\in[1,|T l-1 |]\)), and is also a candidate support set estimate. The correlation coefficient \(v i =\varPhi T r i l-1 is obtained in the \(l\)-th iteration; Step 3. Traverse the parent node \(i = i + 1\). If \(i\leq|T|\), l-1 loop and execute Steps 3 to 5; Step 4. Sort the correlation coefficients obtained by iteration, and the sorted coefficient index set is defined as T Δ ={π|sort(|v π |)}, where sort(|v π |) represents descending order of the coefficients in v i , v π , π ∈ [1, |v i |] represents the elements in the vector v i ; Step 5. Search in the breadth direction with j = j + 1, including the following sub-steps: If j ≤ L, loop and execute: Take the sorted coefficient index set T Δ The first S terms in it are denoted as Ω = T Δ (1:S), and merge Ω with the mother node support set estimate to obtain the augmented set Try to reconstruct to obtain If find the largest K coefficients to form a new support set estimate i.e., complete pruning, and use reconstruct the marine environmental monitoring data to obtain and update the residual and the set of restored support sets Let T Δ = T Δ - Ω; Prepare to start reconstructing the (j + 1)-th sub-path until j = L. Then, a total of LS atoms are selected from the L sub-paths under the i-th parent node, i.e., T Δ (1:LS); Step 6. When the iteration stops, determine the candidate support set that minimizes the residual as the final support set, and output the marine environmental monitoring data 2. The method for reconstructing marine environmental monitoring data by deep and wide two-dimensional subspace tracking according to claim 1, wherein Take the first T items in the breadth-direction search Δ The first S items in Δ are specifically the S items with the largest correlation coefficients.

3. A method for reconstructing ocean environmental monitoring data by deep and wide two-dimensional subspace tracking according to claim 1, characterized in that The marine environment monitoring data includes but is not limited to temperature, pressure, and humidity.

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