A separation-type redundant dictionary learning algorithm for 3D signals

Abstract

The present application relates to the technical field of signal sparse representation, and specifically provides a separation type redundant dictionary learning algorithm for 3D signals, which comprises the following steps: S1: initialization process: initializing redundant dictionaries in three dimensions; S2: main iteration process: comprising a sparse coding stage and a dictionary updating stage; the sparse coding stage comprises: S21: reducing the sparse expression process to a two-dimensional matrix; S22: calculating the expression of samples in the sparse domain according to the initialized redundant dictionaries; S23: calculating the sparse expression error of all samples; the dictionary updating stage comprises: S24: arranging the 3D training sample data blocks into two-dimensional matrices after being respectively unfolded according to three dimensions; S25: updating the redundant dictionaries in three dimensions by solving a formula and performing atom normalization on the dictionaries. The learning algorithm in the present application reduces the time consumption of separation type dictionary training and reduces the sparse expression error of target signals.

Description

Technical Field

[0001] This invention relates to the field of signal sparse representation technology, and in particular to a separate redundant dictionary learning algorithm for 3D signals. Background Technology

[0002] The purpose of dictionary learning is to learn the common features of a target signal through a sample training process, thereby achieving a sparser and more accurate representation. It has wide applications in image compression, image denoising, and image restoration. However, due to the computational complexity and storage requirements of the training and application processes, dictionary learning is limited to low-dimensional, small-scale signals, such as the common use of block processing in two-dimensional image applications. For high-dimensional signals, their vectorized expansion is usually very large, which makes the computational complexity problem of traditional dictionary learning methods even more serious when dealing with high-dimensional signals.

[0003] To address this, scholars have proposed using separable dictionary canonical forms, sacrificing some of the freedom of sparse representation in exchange for the feasibility of processing high-dimensional signals. Currently, typical separable dictionary canonical forms include SeDiL, a separable dictionary training method proposed by Hawe et al., but this method's model is designed for two-dimensional signals and has a relatively complex computational process. With the development of science and technology, the demand for image information has shifted from traditional two-dimensional image matrices to higher-dimensional image cubes. Imaging systems, in addition to high spatial resolution, also need to possess high temporal resolution, high spectral resolution, and other indicators, collecting data to form combined data cubes such as space-space-spectrum, space-space-time, and space-spectrum-time.

[0004] In summary, in order to adapt to the development trend of high-dimensional images, how to design a separable dictionary learning method for high-dimensional signals is an urgent problem to be solved. Summary of the Invention

[0005] To address the aforementioned problems, this invention provides a separate redundant dictionary learning algorithm for 3D signals. This algorithm can adapt to the requirements of high-dimensional images, simplifies the training process of the separate dictionary, reduces the training time of the separate dictionary, and makes the training of large-scale 3D signal dictionaries feasible. The trained redundant dictionary can significantly reduce the error in the sparse representation of the target signal.

[0006] To achieve the above objectives, the present invention proposes the following technical solution: a separation-type redundancy dictionary learning algorithm for 3D signals, comprising the following steps:

[0007] S1: Initialization process: Initialize redundant dictionaries in three dimensions;

[0008] S2: Main Iteration Process: Includes the sparse coding stage and the dictionary update stage;

[0009] The sparse coding stage includes:

[0010] S21: Reduce the sparse representation process to a two-dimensional matrix;

[0011] S22: Calculate the representation of the sample in the sparse domain based on the initialized redundant dictionary;

[0012] S23: Calculate the sparse representation error for all samples;

[0013] The dictionary update phase includes:

[0014] S24: Expand the 3D training sample data blocks according to three dimensions and then organize them into a two-dimensional matrix;

[0015] S25: Update the redundant dictionary in three dimensions by solving the formula and perform atomic normalization on the dictionary.

[0016] Preferably, S1 includes the following sub-steps:

[0017] S11: Initialize redundant dictionaries in three dimensions ψ (0) ∈R No×N With ω (0) ∈R To×T Wherein, the training sample data x∈R M×N×T Mo, No, and To represent the number of redundant dictionary atoms;

[0018] S12: For each atom in the dictionary ψ (0) ω (0) Normalization is performed.

[0019] Preferably, in S21, the sparse representation process is reduced to a two-dimensional matrix operation form using formula (8):

[0020]

[0021] Where x3∈R (M×T)×N The matrix is ​​formed by expanding the 3D training sample data block along the third dimension; θ3∈R (Mo×To)×No Let ξ'(·)∈R be a sparse representation matrix. (M×T)×(Mo×T) ,δ'(·)∈R (Mo×T)×(Mo×To) It is a sparse basis deformation function.

[0022] Preferably, in S22, the Orthogonal Matching Pursuit (OMP) algorithm is used to solve for the representation of the sample in the sparse domain using formula (1):

[0023]

[0024] Among them ||·||F Denotes the Frobenius norm, x3 l Let θ3 be the data matrix of the l-th (1≤l≤L) training sample. l This represents the expression of the sample in the sparse domain. ψ (k-1) With ω (k-1) s1 and s2 are the dictionaries updated after the (k-1)th iteration, and s0 is the target sparsity upper limit;

[0025] In S23, each training sample image is obtained by solving formula (1). And calculate the sparse expression error ε for all samples. k .

[0026] Preferably, in S24, the 3D training sample data blocks are expanded and organized into a two-dimensional matrix operation format according to the third, second, and first dimensions respectively:

[0027]

[0028] Where x l 2∈R (T×N)×M x l 1∈R (N×M)×T The matrix is ​​formed by expanding the 3D training sample data block along the second and first dimensions, θ l 2∈R (To×No)×Mo θ l 1∈R (No×Mo)×To Let θ be the corresponding sparse representation matrix. l 2. θ l 1 can be made by Obtained by deformation.

[0029] Preferably, S25 includes the learning of a redundant dictionary ψ (k) The update and atom normalization process is performed, specifically by solving formula (2) to update the redundant learning dictionary ψ. (k) :

[0030]

[0031] fixed With ω (k-1) By applying equation (2) to ψ (k-1) The partial derivatives are then set to zero, and ψ is obtained through formula (3). (k) :

[0032]

[0033] For dictionary ψ (k) Perform atomic normalization.

[0034] Preferably, S25 includes the learning of a redundant dictionary. The update and atom normalization process is performed as follows: the redundant learning dictionary is updated using formula (4).

[0035]

[0036] Fixed ω (k-1) With ψ (k) By applying formula (4) to... The partial derivatives are taken, and the partial derivatives are set to zero. Then, the partial derivatives are obtained using formula (5).

[0037]

[0038] For dictionaries Perform atomic normalization.

[0039] Preferably, S25 includes the learning of a redundant dictionary ω. (k) The update and atomic normalization process is as follows: the redundant learning dictionary ω is updated using formula (6). (k) :

[0040]

[0041] Fixed ψ (k) and By applying formula (6) to ω (k-1) The partial derivatives are then set to zero, and ω is obtained using formula (7). (k) :

[0042]

[0043] For dictionary ω (k) Perform atomic normalization.

[0044] Preferably, a threshold is set, and the main iteration process S24-S25 is repeated until |ε k -ε k-1 | 2 When the value falls below a certain threshold, the main iteration process terminates.

[0045] Preferably, in S11, the Discrete Cosine Transform (DCT) basis or the Discrete Wavelet (DWT) basis is used as the initial values ​​for the three dictionaries.

[0046] Compared with existing technologies, the present invention achieves the following beneficial effects: The present invention reduces the sparse representation process to a two-dimensional matrix operation form through two stages: sparse coding and dictionary updating. The sparse representation process is organized into three two-dimensional matrix operation forms according to different expansion dimensions. Three redundant dictionaries are trained sequentially according to the three two-dimensional matrix forms and iterated until the change in sparse representation error is lower than a certain threshold. This greatly simplifies the training process of the separated dictionary, reduces the training time of the separated dictionary, and makes the training of 3D large-scale signal dictionary feasible. The redundant dictionary trained by this method significantly reduces the sparse representation error of the target signal. Attached Figure Description

[0047] Figure 1 This is an execution flowchart of the learning algorithm provided in an embodiment of the present invention.

[0048] Figure 2 This is a graph showing the decrease in sparse representation error with the number of iterations according to an embodiment of the present invention. Detailed Implementation

[0049] The appendix will be referenced below. Figure 1-2 Embodiments of the present invention are described below. In the following description, the same modules are denoted by the same reference numerals. Where the same reference numerals are used, their names and functions are also the same. Therefore, their detailed description will not be repeated.

[0050] To make the objectives, technical solutions, and advantages of this invention clearer, the following description is provided in conjunction with the appendix. Figure 1-2 The present invention will be further described in detail below with reference to specific embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and do not constitute a limitation thereof.

[0051] A separation-based redundancy dictionary learning algorithm for 3D signals includes the following steps:

[0052] S1: Initialization process: S1 includes the following sub-steps:

[0053] S11: Initialize redundant dictionaries in three dimensions ψ (0) ∈R No×N With ω (0) ∈R To×T The discrete cosine transform (DCT) basis or discrete wavelet (DWT) basis is used as the initial values ​​for the three dictionaries; where the training sample data x∈R M×N×T Mo, No, and To represent the number of redundant dictionary atoms.

[0054] S12: For each atom in the dictionary ψ (0) ω (0)Normalization is performed.

[0055] S2: Main Iteration Process: Includes the sparse coding stage and the dictionary update stage.

[0056] The sparse coding stage includes:

[0057] S21: Reduce the sparse representation process to a two-dimensional matrix.

[0058] In S21, the sparse representation process is reduced to a two-dimensional matrix operation form using formula (8):

[0059]

[0060] Where x3∈R (M×T)×N The matrix is ​​formed by expanding the 3D training sample data block along the third dimension; θ3∈R (Mo×To)×No Let ξ'(·)∈R be a sparse representation matrix. (M×T)×(Mo×T) ,δ'(·)∈R (Mo×T)×(Mo×To) It is a sparse basis deformation function.

[0061] S22: Calculate the representation of the sample in the sparse domain based on the initialized redundant dictionary.

[0062] In S22, the Orthogonal Matching Pursuit (OMP) algorithm is used to solve for the representation of the sample in the sparse domain using formula (1):

[0063]

[0064] Among them ||·|| F Denotes the Frobenius norm, x3 l Let θ3 be the data matrix of the l-th (1≤l≤L) training sample. l This represents the expression of the sample in the sparse domain. ψ (k-1) With ω (k-1) s0 represents the dictionary updated after the (k-1)th iteration, and s0 is the upper limit of the target sparsity.

[0065] S23: Calculate the sparse representation error ε for all samples. k In S23, each training sample image is obtained by solving formula (1). And calculate the sparse expression error ε for all samples. k .

[0066] The dictionary update phase includes:

[0067] S24: Expand the 3D training sample data block along three dimensions and then organize it into a two-dimensional matrix; specifically, expand the 3D training sample data block along the third, second, and first dimensions and then organize it into a two-dimensional matrix operation format:

[0068]

[0069] Where x l 2∈R (T×N)×M x l 1∈R (N×M)×T The matrix is ​​formed by expanding the 3D training sample data block along the second and first dimensions, θ l 2∈R (To×No)×Mo θ l 1∈R (No×Mo)×To Let θ be the corresponding sparse representation matrix. l 2. θ l 1 can be made by Obtained by deformation.

[0070] S25: Update the redundant dictionary in three dimensions by solving the formula and perform atomic normalization on the dictionary.

[0071] First, this includes the redundant learning dictionary ψ (k) The update and atom normalization process is performed, specifically by solving formula (2) to update the redundant learning dictionary ψ. (k) :

[0072]

[0073] fixed With ω (k-1) By applying equation (2) to ψ (k-1) The partial derivatives are then set to zero, and ψ is obtained through formula (3). (k) :

[0074]

[0075] For dictionary ψ (k) Perform atomic normalization.

[0076] Secondly, this includes the learning of redundant dictionaries. The update and atom normalization process is performed as follows: the redundant learning dictionary is updated using formula (4).

[0077]

[0078] Fixed ω (k-1) With ψ (k) By applying formula (4) to... The partial derivatives are taken, and the partial derivatives are set to zero. Then, the partial derivatives are obtained using formula (5).

[0079]

[0080] For dictionaries Perform atomic normalization.

[0081] Furthermore, this includes the redundant learning dictionary ω (k) The update and atomic normalization process is as follows: the redundant learning dictionary ω is updated using formula (6). (k) :

[0082]

[0083] Fixed ψ (k) and By applying formula (6) to ω (k-1) The partial derivatives are then set to zero, and ω is obtained using formula (7). (k) :

[0084]

[0085] For dictionary ω (k) Perform atomic normalization.

[0086] Set a threshold and repeat the main iteration process S24-S25 until |ε k -ε k-1 | 2 When the value falls below a certain threshold, the main iteration process terminates.

[0087] Depend on Figure 2 As shown in the curve of sparse representation error decreasing with the number of iterations, the error decreases significantly with the increase of the number of iterations.

[0088] It should be understood that the various forms of processes shown above can be used to reorder, add, or delete steps. For example, the steps described in this invention disclosure can be executed in parallel, sequentially, or in different orders, as long as the desired result of the technical solution disclosed in this invention can be achieved, and this is not limited herein.

[0089] The specific embodiments described above do not constitute a limitation on the scope of protection of this invention. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this invention should be included within the scope of protection of this invention.

Claims

1. A method for learning a discrete redundancy dictionary for 3D signals, characterized in that, 3D signals are three-dimensional images. The separated redundant dictionary learning method includes the following steps: S1: Initialization process: Initialize redundant dictionaries in three dimensions; S2: Main Iteration Process: This includes the sparse coding stage and the dictionary update stage; The sparse coding stage includes: S21: Reduce the sparse representation process to a two-dimensional matrix; S22: Calculate the representation of the sample in the sparse domain based on the initialized redundant dictionary; In S22, the Orthogonal Matching Pursuit (OMP) algorithm is used to solve for the representation of the sample in the sparse domain using formula (1): (1) Among them ||·|| F Denotes the Frobenius norm. x 3 l For the first l (1≤ l ≤ L ) training sample data matrix, θ 3 l This represents the expression of the sample in the sparse domain. φ (k-1) , ψ (k-1) and ω (k-1) They are respectively the ( k -1) iterations of the updated dictionary s 0 represents the upper limit of the target sparsity; S23: Calculate the sparse representation error for all samples; In S23, each training sample image is obtained by solving formula (1). θ 3 l (1≤ l ≤ L ), and calculate the sparse expression error for all samples. ε k ; The dictionary update phase includes: S24: Expand the 3D training sample data blocks according to three dimensions and then organize them into a two-dimensional matrix; S25: Update the redundant dictionary in three dimensions by solving the formula and perform atomic normalization on the dictionary.

2. The method for learning a discrete redundant dictionary for 3D signals according to claim 1, characterized in that, S1 includes the following sub-steps: S11: Initialize redundant dictionaries in three dimensions φ (0) ∈R Mo×M , ψ (0) ∈R No×N and ω (0) ∈R To×T Among them, training sample data x ∈R M×N×T , Mo , No , To This represents the number of redundant dictionary atoms. S12: For each atom in the dictionary φ (0) , ψ (0) , ω (0) Normalization is performed.

3. The method for learning a discrete redundant dictionary for 3D signals according to claim 1 or 2, characterized in that, In S21, the sparse representation process is reduced to a two-dimensional matrix operation form using formula (8): （8） in, x 3∈R (M×T)×N The matrix formed by expanding a 3D training sample data block along the third dimension; θ 3∈R (Mo×To) ×No It is a sparse representation matrix. ξ' (·)∈R (M×T)×(Mo×T) , δ' (·)∈R (Mo×T)×(Mo×To) For sparse basis deformation functions, Mo , No , To This represents the number of redundant dictionary atoms.

4. The method for learning a discrete redundant dictionary for 3D signals according to claim 1, characterized in that, In S24, the 3D training sample data blocks are expanded and rearranged into a two-dimensional matrix operation format according to the third, second, and first dimensions respectively: in x l 2∈R (T×N)×M , x l 1∈R (N×M)×T The matrix is ​​formed by expanding the 3D training sample data block along the second and first dimensions. θ l 2∈R (To×No)×Mo , θ l 1∈R (No×Mo)×To This is the corresponding sparse representation matrix. θ l 2. θ l 1 can be made by θ 3 l Obtained by deformation.

5. The method for learning a separated redundant dictionary for 3D signals according to claim 4, characterized in that, S25 includes a redundant learning dictionary. ψ (k) The update and atom normalization process is performed, specifically by solving formula (2) to update the redundant learning dictionary. ψ (k) : (2) fixed φ (k-1) and ω (k-1) By taking the equation (2) ψ (k-1) The partial derivatives are taken, and the partial derivatives are set to zero. Then, the partial derivatives are obtained using formula (3). ψ (k) : (3) For dictionaries ψ (k) Perform atomic normalization.

6. The method for learning a separated redundant dictionary for 3D signals according to claim 5, characterized in that, S25 includes a redundant learning dictionary. φ (k) The update and atomic normalization process is performed as follows: the redundant learning dictionary is updated using formula (4). φ (k) : (4) fixed ω (k-1) and ψ (k) By applying formula (4) to... φ (k-1) The partial derivatives are taken, and the partial derivatives are set to zero. Then, the partial derivatives are obtained using formula (5). φ (k) : (5) For dictionaries φ (k) Perform atomic normalization.

7. The method for learning a separated redundant dictionary for 3D signals according to claim 6, characterized in that, S25 includes a redundant learning dictionary. ω (k) The row update and atomic normalization process is as follows: the redundant learning dictionary is updated using formula (6). ω (k) : (6) fixed ψ (k) and φ (k) By applying formula (6) to... ω (k-1) The partial derivatives are taken, and the partial derivatives are set to zero. Then, the partial derivatives are obtained using formula (7). ω (k) : (7) For dictionaries ω (k) Perform atomic normalization.

8. The method for learning a separated redundant dictionary for 3D signals according to claim 7, characterized in that, Set a threshold, and repeat the main iteration process S24-S25 until | ε k - ε k-1 | 2 When the value falls below a certain threshold, the main iteration process terminates.

9. The method for learning a discrete redundant dictionary for 3D signals according to claim 8, characterized in that, S11 uses either the Discrete Cosine Transform (DCT) basis or the Discrete Wavelet (DWT) basis as the initial values ​​for the three dictionaries.

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