Time-varying frequency spectrum map compression surveying and mapping method based on dictionary learning and smooth regularization

Abstract

The invention provides a time-varying frequency spectrum map compression surveying and mapping method based on dictionary learning and smooth regularization, and the method comprises the steps: building a sparse base psi through the path loss between two points in a spatial frequency domain, carrying out the sparse representation of a signal in the spatial frequency domain, and building a compression sensing model of a three-dimensional space through combining with a measurement matrix; a measurement matrix in the compressed sensing model is optimized, singular value decomposition (SVD) is used for decomposing psi, sparse bases psi m corresponding to the first m maximum eigenvalues are selected, and an optimized measurement matrix is obtained after m times of iteration; the optimized measurement matrix is substituted into a compressed sensing model, an O-KSVD dictionary is adopted to learn and update a sparse base psi, and spectrum data are recovered; finally, sparse regularization and weighted smoothing regularization are combined to achieve the minimum Euclidean distance between adjacent points in the reconstructed spectrum map, smoothing regularization is used for driving the adjacent points in the reconstructed spectrum map to be similar, sparse representation in the spectrum map is updated through iteration till convergence, and actual spectrum mapping is obtained.

Description

Technical Field

[0001] The present invention relates to the technical field of spectrum mapping, and more specifically, to a time-varying spectrum map compression mapping method based on dictionary learning and smoothing regularization. Background Art

[0002] With the continuous increase of radio devices and the increasing scarcity of spectrum resources, how to effectively utilize spectrum resources has become widely valued. Spectrum mapping is an effective spectrum management technology that can infer and store spectrum data of the geographical area of ​​interest, including the spatial distribution of radio parameters such as signal strength, channel gain, modulation mode and access protocol, helping the communication system understand the current usage of spectrum resources and the communication status in the area, thereby promoting the effective allocation of spectrum resources.

[0003] Traditional spectrum mapping methods include model-based and data-based generation methods. Model-based generation methods usually assume known prior information such as propagation models and radiation source power. However, it is often difficult to obtain accurate information such as signal propagation models in complex electromagnetic environments. Data-based generation methods mainly include matrix completion and interpolation methods. They do not assume prior information such as known propagation models, but data-based generation methods are highly dependent on data structure. Matrix completion methods cannot be applied when an entire row or column of data is missing, and interpolation methods are only suitable for processing uniformly sampled data. Compared with traditional methods, spectrum mapping methods based on deep learning do not rely on prior information and data structure, and can achieve high-precision mapping, but they require a large amount of labeled data for training, and the model complexity is high.

[0004] Patent document CN118348326A proposes an electromagnetic spectrum mapping method for three-dimensional complex urban environments. Through the compressed sensing model and Bayesian theory, the electromagnetic target signal of the area to be measured is used to initially construct a spectrum map, and then the shadow fading in the environment is estimated by collecting spectrum data to correct the spectrum map and improve the accuracy of three-dimensional spectrum data recovery. This method uses drones to optimize the trajectory collection of sampling positions, which is relatively costly. According to the optimal sampling position, the nearest neighbor strategy is used to form the optimal sampling trajectory. The model method has many iterations, complex data generation, and is only based on the three-dimensional spectrum map recovery and mapping method in static scenes, which cannot meet the use of time-varying spectrum space. Summary of the invention

[0005] 1. Technical issues

[0006] In order to solve the problem of how to achieve accurate spectrum mapping in three-dimensional dynamic scenes, the present invention provides a time-varying spectrum map compression mapping method based on dictionary learning and smoothing regularization. This method dynamically adjusts the measurement matrix and sparse basis of the traditional three-dimensional compressed sensing model, further constrains the sparsity of the spectrum data, and improves the mapping accuracy of the space-time domain spectrum.

[0007] 2. Technical solution

[0008] The object of the present invention is achieved by the following technical solution: a time-varying spectrum map compression surveying and mapping method based on dictionary learning and smoothing regularization, comprising the following steps:

[0009] S1 uses the path loss between two points in the three-dimensional frequency domain to construct a sparse basis ψ, combined with the measurement matrix φ for the time domain signal Perform sparse representation and establish a three-dimensional space spectrum map compressed sensing model

[0010] S2 optimizes the measurement matrix φ in the compressed sensing model, decomposes ψ using SVD singular value decomposition, and selects the sparse basis ψ corresponding to the first m largest eigenvalues m , calculate ψ m The column index value of the largest dominant column of the transposed matrix is ​​obtained after m iterations. The corresponding column index value is used to obtain the optimized measurement matrix

[0011] S3 collects sample data at time K, uses O-KSVD and the initialization dictionary D to constrain the sparse representation of sample data, and iteratively updates the optimized sparse basis Restore spectrum data through compressed sensing model and draw spectrum map at time K+1;

[0012] S4 combines sparse regularization and weighted smoothing regularization to optimize the data in the spectrum map to obtain sparse data and The Euclidean distance between adjacent points is minimized, and the sparse representation in the spectral map is iteratively updated Until convergence, get the time-varying spectrum map

[0013] Specifically, the power signal at any time in the spectrum map The strength can be expressed in terms of path loss as:

[0014]

[0015] Among them, χ represents the spatial frequency domain, χ j is a point inside χ, N χ Indicates the number of signal sources. l is the transmission power of the lth signal source, djl is the lth signal source and χ j The distance between them, η is the path loss exponent.

[0016] Specifically, the sparse spectrum representation in S1 can be transformed into the following function constraint problem through the compressed sensing model:

[0017]

[0018] in, Represents the spectrum data received by the sensor, φ∈R m×n is the measurement matrix, ψ is the sparse basis established by the point path loss, Indicates t i Time signal sparse representation of , ε represents Gaussian white noise; represents the sum of the absolute values ​​of the vector elements, Indicates the maximum reconstruction error of the spectrum.

[0019] Furthermore, the step of optimizing the measurement matrix in S2 is:

[0020] Select the sparse basis ψ corresponding to the first m largest eigenvalues m , perform singular value decomposition on the measurement matrix φ, introduce the left and right singular vector matrices, and obtain n eigenvalues ​​and corresponding eigenvectors;

[0021] Initialize ψ m And make ψ m Transpose matrix A0 and calculate the 2-norm maximum dominant column a in A0 γ1 , for a γ1 Perform Householder mirror transformation H1·a γ1 =(a γ1 |,0,…,0) T , get the dominant column index γ1;

[0022] After m operations, we get m corresponding leading column index values ​​{γ1,γ2,...,γ m}, get the optimized measurement matrix

[0023] Specifically, in S3, for the spectrum data collected at time K, the dictionary D=ψ is initialized using the O-KSVD dictionary learning method, and all data samples are calculated. Sparse representation of The constraint function is further expressed as:

[0024]

[0025] Among them, k0 is the sparse constraint parameter.

[0026] Specifically, each column vector in the dictionary D and the corresponding row in S are iteratively updated, and the minimum objective function is updated as follows:

[0027]

[0028] Among them, d k is the kth column of D, s k is the kth row of S.

[0029] Furthermore, for E k Perform singular value decomposition and define the matrix Ω to ensure data sparsity. k , remove s k All zero elements in , and retain non-zero values, further simplifying the objective function to:

[0030]

[0031] Among them, d k is the kth column of D, s k is the kth row of S, is a combination of non-zero elements.

[0032] Furthermore, the updated dictionary and sparse representation are iteratively updated until the error satisfies:

[0033]

[0034] Where ò is the threshold. When the error is less than or equal to the set threshold, the dictionary D = [d1, d2, …, d n ] is multiplied by the sparse basis ψ to obtain the updated sparse basis Used for spectrum mapping at the K+1th moment.

[0035] Specifically, S4 performs joint sparse regularization and weighted smoothing regularization on the K+1 time spectrum map:

[0036]

[0037] in, is the spectrum map obtained by compressed sensing after update, and is a sparse representation of the spectrum map; represents the updated sparse basis, W ij is a weight matrix used to ensure that the Euclidean distance of the sparse representation in the reconstructed map is minimized; J(s χ ) represents sparsity regularization, G(s χ ) represents weighted smoothing regularization.

[0038] Specifically, the sparse representation of the spectrum map is iteratively updated until convergence, and the fixed sparse representation is optimized. The spectrum map x is: in, represents the updated sparse basis of the spectrum map, is a sparse representation of the three-dimensional space χ.

[0039] 3. Beneficial effects

[0040] The present invention uses SVD and Householder mirror transformation to obtain an optimized measurement matrix by processing a sparse basis, thereby obtaining sampling point positions, thereby avoiding the problem of low spectrum mapping performance caused by uneven data distribution under random sampling.

[0041] The present invention uses dictionary learning based on the online K-singular value decomposition algorithm, dynamically adjusts the sparse basis, updates the dictionary, effectively enhances the sparse representation ability of the sparse basis for the spectrum, improves the signal reconstruction ability of compressed sensing and the reconstruction ability of the spectrum map, thereby meeting the time-varying spectrum mapping in three-dimensional dynamic scenes.

[0042] The present invention combines sparse regularization and weighted smoothing regularization on the spectrum reconstruction map, restores the spatial structure data signal of the three-dimensional spectrum map by making adjacent points in the spectrum map similar, solves the problem that the sparse representation process in compressed sensing ignores the spatial structure of the spectrum map, and improves the accuracy of spectrum mapping. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a schematic diagram of the process of spectrum map compression surveying and mapping in a three-dimensional dynamic scene in one embodiment of the present invention;

[0044] Figure 2 A schematic diagram of a spectrum map compression surveying and mapping process in an embodiment of the present invention;

[0045] Figure 3 A schematic diagram of a compressed sensing process in an embodiment of the present invention;

[0046] Figure 4 It is a schematic diagram of spectrum map compression mapping performance of different algorithms under different noise powers and sampling rates in one embodiment of the present invention;

[0047] Figure 5 It is a schematic diagram of the spectrum map compression surveying and mapping performance of the algorithm of the present invention under different numbers of mobile signal sources and moving speeds in one embodiment of the present invention. DETAILED DESCRIPTION

[0048] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0049] 1. The invention proposes a time-varying spectrum map compression surveying and mapping method based on dictionary learning and smoothing regularization, the method comprising the following steps:

[0050] 1. Establish a 3D space compressed sensing model

[0051] for Figure 2 A three-dimensional spectrum map is displayed for mapping, and the three-dimensional space time domain and frequency domain contain fixed and moving signal sources. Compared with the actual three-dimensional space domain, the number of collected signal sources is sparse, which means that the original signal is sparse under a specific sparse basis. The spectrum sparse basis orthogonal matrix is ​​constructed through the two-point path loss in space, and the signal is represented as a linear combination of the basis, which is used to realize the actual spectrum map mapping in three-dimensional space. Here, the sparse basis ψ of three-dimensional spectrum mapping can be derived as follows:

[0052]

[0053] in, represents the path loss between two points in the spatial frequency domain, d mn represents the distance between two points, and η is the path loss exponent.

[0054] Based on the path loss, the power intensity at any location in the three-dimensional spectrum map can be obtained as follows:

[0055]

[0056] Among them, χ represents three-dimensional space, j is a point inside χ, and N χ Represents the number of signal sources, P l represents the transmission power of the lth signal source, represents the lth signal source and χ j The distance between them, η is the path loss exponent.

[0057] like Figure 3 As shown, in three-dimensional space compressed sensing, the signal In the sparse basis ψ∈R n×n is sparse, so It can be expressed as in Indicates t i The spectrum data at the moment, For spectrum data Sparse representation of n=n x ×n y ×n z .

[0058] Therefore, the three-dimensional space compressed sensing model can be represented as the spectrum data received by the sensor

[0059]

[0060] Among them, φ∈R m×n is the measurement matrix, ψ is the sparse basis, ε represents Gaussian white noise, Indicates t i The spectrum data at the moment, For spectrum data sparse representation of .

[0061] Using the compressed sensing theory, the spectrum sparse representation is optimized as follows, and the minimum function of the sparse representation within the maximum reconstruction error is solved as follows:

[0062]

[0063] in, is the sum of the absolute values ​​of the sparse representation vector elements, is the maximum reconstruction error, φ∈R m×n is the measurement matrix, ψ is the sparse basis, t i The spectrum data at the moment, For spectrum data sparse representation of .

[0064] 2. Measurement matrix optimization

[0065] First, SVD (singular value decomposition) is used to preprocess the sparse basis ψ to obtain n eigenvalues ​​and corresponding eigenvectors, and the sparse basis corresponding to the first m largest eigenvalues ​​is selected as ψ m , construct the vector matrix:

[0066] [U n×n ,Δ n×n ,V n×n ]=svds(ψ,n)

[0067] V n×m =V n×n (:,1:m)

[0068] ψ m =U n×n ×Δ n×n ×V n×m

[0069] Among them, svds(ψ,n) represents the singular value decomposition of the matrix ψ, U n×n With V n×n are the left singular vector matrix and the right singular vector matrix respectively, Δ n×n is a diagonal matrix with singular values.

[0070] initialization is mTranspose and calculate the dominant column with the largest 2-norm in A0 right Perform a Householder mirror transformation: Where γ1 is the leading column index;

[0071] H1 is calculated as:

[0072]

[0073] Among them, e is a vector whose first element is 1 and the rest are 0. The length of e is Same; introduce the permutation matrix P1, swap the γ1th column with the 1st column, and multiply by diag(I0,H1), which can eliminate the dominant column The contribution generated after the mirror transformation is performed, so that the subsequent dominant column calculation is not affected.

[0074] The result of the first operation of A0 is:

[0075] A1=diag(I0,H1)·A0·P1;

[0076] After m operations, m corresponding column index values ​​{γ1,γ2,...,γ m}, respectively corresponding to the sampling point positions in the three-dimensional space χ, and the optimized measurement matrix is ​​obtained The process of optimizing the measurement matrix avoids the problems of uneven data distribution and incomplete data generated in random sampling.

[0077] 3. Dictionary learning updates sparse basis

[0078] The sparse basis is updated using a dictionary learning method based on the online K-Singular Value Decomposition (O-KSVD) algorithm. The specific steps are as follows:

[0079] The compressed sensing model is used to restore the spectrum data. The compressed sensing uses an optimized measurement matrix And a fixed sparse basis, select K sample data, sort by time to get

[0080] Initialize the dictionary D=ψ and use the O-KSVD dictionary learning method. The optimization strategy is to minimize the following objective function:

[0081]

[0082] in, is the sparse representation of data X under dictionary D, k0 is the sparse constraint parameter, requiring each sparse representation There should be non-zero values ​​less than k0.

[0083] The dictionary is fixed and the sparse representation of all data samples is calculated, with the goal of finding a set of as few non-zero elements as possible to represent the spectral data. There are corresponding sparse representations The objective function is updated as follows:

[0084]

[0085] Iteratively update each column vector in D and the corresponding row in S. The optimization goal at this stage is:

[0086]

[0087] where d k is the kth column of D, s k is the kth row of S, optimizing each d k To better fit the error of sparse representation.

[0088] For E k Update d using singular value decomposition k and k , because at this time s k It is not sparse, directly to E k Performing singular value decomposition will affect s k The sparsity of

[0089] So we define the matrix Ω k , remove s k All zero elements in, keep the non-zero values, and then keep E k The corresponding column vector. Let Ω k ∈R K×B Middle position The element at is set to 1, and the rest of the elements are set to 0. B represents s k The number of non-zero elements in , from which we can get:

[0090]

[0091] in is a combination of non-zero elements.

[0092] Therefore, the objective function can be simplified to:

[0093]

[0094] At this time, s k The sparsity of will not be affected, since only the non-zero elements of the data are used in the calculation. Use singular value decomposition to update dk and

[0095]

[0096] d k =U(:,1)

[0097]

[0098] Matrix U n×n The first column is d k The solution is to use the matrix V B×B The first row of n×B Multiply the first diagonal element of to get the sparse representation Then add the zero elements corresponding to the original positions to In , the dictionary and sparse representation are iteratively updated until the error satisfies:

[0099]

[0100] Where, ò is the threshold. When the error is less than or equal to the threshold, the calculated dictionary D=[d1,d2,…,d n ] is multiplied by the sparse basis ψ to obtain the updated sparse basis Spectrum mapping for the K+1th moment:

[0101]

[0102] The specific steps of using weighted smoothing regularization on the recovered spectral map are:

[0103] Specifically, the reconstructed spectrum map is kept sparse by joint sparse regularization and weighted smoothing regularization, while the Euclidean distance between adjacent points is minimized, thereby restoring the spatial correlation of the spectrum map. The joint regularization model is:

[0104]

[0105] Among them, among them, is the spectrum map obtained by compressed sensing after update, and is a sparse representation of the spectrum map; represents the updated sparse basis, W ij is the weight matrix, J(s χ ) represents sparsity regularization, which is used to maintain the sparsity of data; G(s χ ) represents the weighted smoothing regularization, which is used to drive the neighboring points in the reconstructed spectrum map to be similar.

[0106] Weight matrix W ij The calculation method is:

[0107]

[0108] Among them, the value of β satisfies σ is the influence factor, d0 is the maximum distance between adjacent points;

[0109] calculate and The Euclidean distance between ij express:

[0110]

[0111] in, and Respectively represent the spectrum map and The spatial coordinates of .

[0112] To solve the optimization problem in joint regularization, we iteratively update Until convergence, when When optimized, the rest is fixed.

[0113] After the above optimization, the spectrum map x is drawn as:

[0114]

[0115] in, is the updated spectral sparse basis, is the sparse representation of the χ spectrum in three-dimensional space;

[0116] Finally, the MSE error analysis is performed based on the drawn spectrum map and the real spectrum map:

[0117]

[0118] Where i=1,...,n is the data point in the updated spectrum map, and They are the real spectrum map and the spectrum map obtained by compressed mapping.

[0119] 2. Simulation Contents of the Present Invention

[0120] The simulation scenario of the present invention is an outdoor open area of ​​1km×1km×1km, and the sensor sampling interval is 100m. The signal source transmission power is 30W, the center frequency is 2000MHz, the bandwidth is 200KHz, and the time step is 1s. The sample K value is 3, the distance d0 of adjacent points in the spectrum map is 300m, and the weight matrix calculation influence factor σ is set to 0.8.

[0121] like Figure 4 As shown, the horizontal axis is the sampling rate, and the vertical axis is the mean square error of spectrum mapping under different methods; the performance of the three-dimensional spectrum map compression mapping method provided by the present invention is improved by 2.67dB, 3.31dB and 17.72dB respectively compared with the MOD, SFI and Random methods when the noise power is -30dBm and the sampling rate is 0.5.

[0122] like Figure 5 As shown, the horizontal axis is the moving speed of the signal source, and the vertical axis is the mean square error generated by the spectrum map compression mapping of the present invention under different numbers of mobile signals. The spectrum mapping performance decreases with the increase of the number of mobile signal sources. When the speed of the mobile signal source is 15m / s, the spectrum mapping performance under a single mobile signal source is improved by 3.45dB, 5.58dB, 6.91dB and 7.85dB compared with 2, 4, 8 and 16 mobile signal sources, respectively.

[0123] The above schematically describes the invention and its implementation methods, which is not restrictive. Without departing from the spirit or basic features of the invention, the invention can be implemented in other specific forms. What is shown in the accompanying drawings is only one of the implementation methods of the invention. The actual structure is not limited to this, and any figure mark in the claims should not limit the claims involved. Therefore, if a person of ordinary skill in the art is inspired by it, without departing from the purpose of the invention, a structural method and an embodiment similar to the technical solution are designed without creativity, which should all fall within the scope of protection of this patent. In addition, the word "including" does not exclude other elements or steps, and the word "one" before the element does not exclude the inclusion of "multiple" elements. The multiple elements stated in the product claim can also be implemented by one element through software or hardware. The words first, second, etc. are used to indicate names, and do not indicate any specific order.

Claims

1. A time-varying spectrum map compression mapping method based on dictionary learning and smoothing regularization, characterized in that: include: S1 uses the path loss between two points in the three-dimensional frequency domain to construct a sparse basis ψ, combined with the measurement matrix φ for the time domain signal Perform sparse representation and establish a three-dimensional space spectrum map compressed sensing model S2 optimizes the measurement matrix φ in the compressed sensing model, decomposes ψ using SVD singular value decomposition, and selects the sparse basis ψ corresponding to the first m largest eigenvalues m , calculate ψ m The column index value of the largest dominant column of the transposed matrix is ​​obtained after m iterations. The corresponding column index value is used to obtain the optimized measurement matrix S3 collects sample data at time K, uses O-KSVD and the initialization dictionary D to constrain the sparse representation of sample data, and iteratively updates the optimized sparse basis Restore spectrum data through compressed sensing model and draw spectrum map at time K+1; S4 combines sparse regularization and weighted smoothing regularization to optimize the data in the spectrum map to obtain sparse data and The Euclidean distance between adjacent points is minimized, and the sparse representation in the spectral map is iteratively updated Until convergence, get the time-varying spectrum map 2. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 1, characterized in that: The power signal at any time in the spectrum map The strength can be expressed in terms of path loss as: Among them, χ represents the spatial frequency domain, χ j is a point inside χ, N χ Indicates the number of signal sources. l is the transmission power of the lth signal source, d jl is the lth signal source and χ j The distance between them, η is the path loss exponent.

3. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 1, characterized in that: The sparse spectrum representation in S1 can be transformed into the following function constraint problem through the compressed sensing model: in, Represents the spectrum data received by the sensor, φ∈R m×n is the measurement matrix, ψ is the sparse basis established by the point path loss, Indicates t i Time signal sparse representation of , ε represents Gaussian white noise; represents the sum of the absolute values ​​of the vector elements, Indicates the maximum reconstruction error of the spectrum.

4. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 1, characterized in that: The steps for optimizing the measurement matrix in S2 are: Select the sparse basis ψ corresponding to the first m largest eigenvalues m , perform singular value decomposition on the measurement matrix φ, introduce the left and right singular vector matrices, and obtain n eigenvalues ​​and corresponding eigenvectors; Initialize ψ m And make ψ m Transpose matrix A0 and calculate the 2-norm maximum dominant column a in A0 γ1 , for a γ1 Perform Householder mirror transformation Get the dominant column index γ1; After m operations, we get m corresponding leading column index values ​​{γ1,γ2,...,γ m }, get the optimized measurement matrix 5. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 1, characterized in that: In S3, for the spectrum data collected at time K, the dictionary learning method based on O-KSVD is used to initialize the dictionary D = ψ and calculate all data samples Sparse representation of The constraint function is further expressed as: Among them, k0 is the sparse constraint parameter.

6. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 5, characterized in that: Iteratively update each column vector in the dictionary D and the corresponding row in S, and the minimum objective function is updated as follows: Among them, d k is the kth column of D, s k is the kth row of S.

7. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 6, characterized in that: For E k Perform singular value decomposition and define the matrix Ω to ensure data sparsity. k , remove s k All zero elements in , and retain non-zero values, further simplifying the objective function to: Among them, d k is the kth column of D, s k is the kth row of S, is a combination of non-zero elements.

8. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 7, characterized in that: Iterate the updated dictionary and sparse representation until the error satisfies: Where ò is the threshold. When the error is less than or equal to the set threshold, the dictionary D = [d1, d2, …, d n ] is multiplied by the sparse basis ψ to obtain the updated sparse basis Used for spectrum mapping at the K+1th moment.

9. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 8, characterized in that: S4 performs joint sparse regularization and weighted smoothing regularization on the K+1 time spectrum map: in, is the spectrum map obtained by compressed sensing after update, and is a sparse representation of the spectrum map; represents the updated sparse basis, W ij is a weight matrix used to ensure that the Euclidean distance of the sparse representation in the reconstructed map is minimized; J(s χ ) represents sparsity regularization, G(s χ ) represents weighted smoothing regularization.

10. The method for compressing and mapping time-varying spectrum maps based on dictionary learning and smoothing regularization according to claim 9, characterized in that: Iteratively update the sparse representation of the spectrum map until convergence, and optimize to obtain a fixed sparse representation The spectrum map x is: in, represents the updated sparse basis of the spectrum map, is a sparse representation of the three-dimensional space χ.

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