Compressed Sensing Technology for Spectral Reconstruction Based on Maximum Entropy Criterion


 In order to choose the related sampling ratio in the information-poor and information-rich spectral fragments, this paper attempts to recover the spectral reflectance by compressed sensing technology based on maximum entropy criterion. The maximum entropy threshold method is the criterion that the optimal threshold is determined to segment the information content of spectral curves. The spectral reflectance in each sub-spectral fragment is reconstructed by compressed sensing. The wavelet orthogonal matrix performs a sparse representation of each segmented spectral curve. Undersampling spectral curve be collected by random gaussian measurement matrix. The orthogonal matching pursuit algorithm recovers sparse original signals from undersampling observed signals. In this paper, the four standard color blocks of Munsell and the spectral curves of five types of ground objects in the hyperspectral data set are used as the exper-imental objects. The reconstructed results are evaluated by spectral curve reconstruction, root mean square error and information entropy difference. The experimental results show that our approach improves the reconstruction accuracy of spectral reflectance effectively, compared with the traditional method.


Introduction
The re ectivity is a fundamental characteristic of object surface, which is not affected by the external environment, the illumination and the detector 1,2 . Spectral re ectance is associated with unique color information and rich texture information. Reconstruction of the spectral re ectance on every pixel can assist in the accurate reproduction of colors and object identi cation 3 . Spectral reconstruction technology is widely used in printed matter anti-counterfeiting, artwork restoration, construction industry and other elds.
There are some traditional strategies to reconstruct spectral re ectance [4][5][6][7] . Pratt and Mancill proposed pseudo-inverse method to reconstruct spectral re ectance 8 , the method can only have better re ectance reconstruction results when the number of camera response channels is large. Hardberg adopted principal eigenvector method which involves only the channel response that has a signi cant in uence on the reconstructed spectral re ectance. Murakami proposed the wiener estimation method is based on the least square tting algorithm to obtain the conversion matrix between the channel response value and the spectral re ectance, so that the difference between the reconstructed spectral re ectance and the original spectral re ectance is minimized 9 . Connah reconstructed the spectral re ectance by polynomial regression which employs different terms of higher-degree to determine the correlation of sampling points and the trend of interpolating points in the imaging response curve 10 .
Compressed sensing (CS) is the technology that the key idea is to adopt the sparsity of the signal to reduce the dimensionality of the signal with a far fewer samples than the traditional Nyquist-Shannon sampling theorem requires, and to restore the high-dimensional sparse vector by the optimization algorithm accurately [11][12][13] . Com-pressed sensing has been widely used in remote sensing imaging, medical imaging, dig-ital communications and other elds because of the characteristics of greatly reducing the measurement time and samples [14][15][16] .
In the early stage of our research group, compressed sensing has been applied to the spectral re ectance reconstruction, which achieved good reconstructed results 17 . Due to the smooth and non-smooth fragment in the spectral curve, the situation described as data rich but information poor will arise after random downsampling in the entire spectral fragment. It is necessary to adopt segmentation processing for the spectral fragment to improve the reconstructed accuracy in the non-smooth fragment and reconstructed e cency in the smooth fragment. The traditional uniform compressed sensing segmentation method which does not depend on the distribution of information in the spectral fragment may result in insu cient sampling in the non-smooth spectral frag-ment and excessive sampling in the smooth spectral fragment 18,19 . Aiming at the un-even distribution of information entropy in the spectral fragment, this paper proposes a compressed sensing technology for spectral reconstruction based on maximum entropy criterion.
First, we determine the optimal threshold by maximizing the entropy value of the spectral to nd the most reasonable segment length of the smooth spectral fragment and the non-smooth spectral fragment. The spectral fragment with a large information entropy value is allocated more sampling values, and the spectral fragment with a small information entropy value is allocated with fewer sampling values. Finally, the spectral re ectance in each sub-spectral fragment is reconstructed by compressed sensing to improve the reconstruction accuracy in the overall spectral curve.

Methods
In order to verify the compressed sensing technology based on maximum entropy criterion, we rst conducts experiments with four standard Munsell color blocks of red, green, blue and yellow. The evaluations are conducted by calculation of information en-tropy and root mean square error between the original and the reconstructed spectral re ectances.

Maximum Entropy Threshold Segmentation Method Of Spectral Re ectance
Pre-segmentation processing on the spectral curve was performed to calculate the information entropy on the cases of different segmentation thresholds. When the in-formation entropy reaches the maximum, it is determined as the optimal threshold. The original spectral domain is divided into two spectral fragments. The symbol A denotes the spectral fragment where the spectral wavelength is smaller than the threshold. The symbol B denotes the spectral fragment where the spectral wavelength is bigger than the threshold.
The spectral wavelength i ranges from 400nm to 700nm . The threshold t i is selected in the same domain with i. Once t i is determined, the Spectral curve is divided into X A and X B . X = R 400 , R 401 , R 402 , ⋯, R 700 . R i represents the spectral re ectance at wavelength i nm.
AA shows the probability distribution of X A . BB shows the probability distribution of X B .

AA
P A + P B = 1 (4) P i represents the probability value of the spectral re ectance R i at the wavelength i nm.
The entropy related to the two probability densities of A and B are expressed by: The maximum entropy of spectral fragments is calculated to determine the optimal threshold t O of the spectral curve: HH t o indicates the maximum sum of the information entropy of the spectral segments divided by different thresholds t i . t o is the optimal threshold for dividing the smooth spectral segment and the nonsmooth spectral segment. The next step of the experiment is to reconstruct segmented spectral by compressed sensing.

Compressed Sensing
The maximum entropy threshold segmentation method is that the optimal threshold t o is determined to nd the most reasonable segment length of the smooth spectral fragment and the non-smooth spectral fragment. The total fragment length of the spectral re ectance is N = 301. The total dimensionality reduction sample number is M = 100. There are two conditions to be met, M = M a + M b , The original signal is X = 301 × 1 dimensional column vector, X = R 400 , R 401 , R 402 , ⋯, R 700 T . By using a set of wavelet orthogonal matrix φ = φ 1 , φ 2 , φ 3 , ⋯, φ 301 , the sparse representation of the original signal will be obtained.
Where α is the linear projection of the original signal X on the wavelet orthogonal basis φ. The number of non-zero elements in the sparse coe cient α is K(K ⩽ N). The original signal X is undersampled by the measurement matrix ψ M × N = ψ 1 , ψ 2 , ψ 3 , ⋯, ψ M for reduced dimensionality, namely: The signal Y is an M-dimensional column vector. After the linear projection of the signal X on the measurement matrix,the undersampling measured value Y is obtained. The measurement matrix selected in this experiment is gaussian random matrix. The gaussian random matrix obey standard normal distribution with mean 0 and variance 1. Gaussian random measurement matrix complies with the restricted isometry property (RIP). M ≪ N, X is derived from Y by an underdetermined equation. The norm l 1 is selected to solve the optimal solutionα ′ . The reconstruction for original signal adopts the orthogonal matching pursuit algorithm (OMP).
On the promise that the total sampling number remains unchanged, different sampling numbers are distributed on the two spectral fragments A and B. The spectral fragment with the large information entropy value is allocated more sampling number, and the spectral fragment with the small information entropy value is allocated fewer sample number. The reconstruction of compressed sensing is performed on the two spectral fragments A and B respectively.
Spectral fragment A is reconstructed by CS. The sampling size of signal The signal X A is represented linearly by a set of orthogonal wavelet matrices φ a = φ 1 , φ 2 , φ 3 , ⋯, φ Na . The sparsity coe cient is α 1 . The formula can be expressed as: Where α 1 is also means linear projection of signal X A on φ a domain, the number of non-zero elements in sparse coe cient is K a K a ⩽ N a . The signal X A is linearly projected by a random Gaussian measurement matrix for dimensionality reduction. The measurement matrix ψ Ma × Na = ψ 1 , ψ 2 , ψ 3 , ⋯, ψ Ma is not related with the wavelet basis, then the compressed sensing problem can be formulated as follows: Y A = ψ Ma × Na X A = ψ Ma × Na φ a α 1 = C 1 α 1 (13) Where C 1 is the perception matrix. Y A is an M-dimensional column vector. After the linear projection of the signal X A on the measurement matrix ψ Ma × Na , the undersampling measured value Y A is obtained, M a ≪ N a . The orthogonal matching pursuit algorithm is selected for recovering the sparsest solution of under-determined system.
The CS-based reconstruction of the spectral fragment A is obtained.
Spectral fragment B is reconstruct by CS. The sampling size of signal X B = R t + 1 , R t + 2 , R t + 3 , ⋯, R 700 T is M b . The spectral signal X B is represented linearly by a set of orthogonal wavelet matrices φ b = φ 184 , φ 185 , φ 186 , ⋯, φ 301 . The sparsity coe cient is α 2 . The formula can be expressed as: (16) Where α 2 is also means linear projection of signal X B on φ b domain, the number of non-zero elements in sparse coe cient is K b K b ⩽ N b . The signal X B is linearly projected by a random gaussian measurement matrix for dimensionality reduction. the measurement matrix ψ Mb × Nb = ψ 1 , ψ 2 , ψ 3 , ⋯, ψ Mb is not related to the wavelet basis, then the compressed sensing problem can be formulated as follows: Where C 2 is the perception matrix. Y B is an M-dimensional column vector. After the linear projection of the signal X B on the measurement matrix ψ 2 , the undersampling measured value Y B is obtained, The orthogonal matching pursuit algorithm is used to solve the under-determined equations.
The CS-based reconstruction of the spectral fragment B is obtained .
Integrating the reconstructed spectral re ectance X A ′ and X B ′ , the result is X ′ = R 400 ′ , R 401 ′ , R 402 ′ , ⋯, R 700 ′ T . Table 1 shows the maximum entropy threshold segmentation results of the four standard color blocks (red, green, blue and yellow). The optimal thresholds are t o =639nm, 536nm, 461nm, 583nm for red, green, blue and yellow blocks respectively. Taking the yellow spectral curve for example. The optimal threshold t o of the yellow spectral curve is 583nm. The spectral curve is divided into fragments A=[400nm-583nm], B=[584nm-700nm]. The total fragment length of the spectral re ectance is N = 301. The total dimensionality reduction sample value is M = 100. Subject to the constrains M a > M b , 100 = M a + M b , 301 = N a + N b , we yeild the following parameters after segmentation.
The reconstructed spectral re ectance for yellow is shown in Fig. 1(d).

Results
The spectal reconstruction for the standard color blocks The spectral curves of the four standard Munsell color blocks (red, green, blue and yellow) are reconstructed with different method. Figure 1 shows the reconstructed spectral curve with the traditional approaches and our approach. Solid red line is the original spectral curve. Dot solid blue line, solid purple line, dashed-dotted black line and solid green line are the spectral curves reconstructed by compressed sensing, polynomial regression, pseudo-inverse and compressed sensing technology based on maximum entropy criterion (MECCS) respectively. It can be clearly observed that the two spectral curves which are reconstructed by polynomial regression and pseudo-inverse both deviate from the original curve. The solid green curve is closer to the original spectral curve.
Root mean square error (RMSE) can be used to discuss the deviation between the original spectral re ectance and the reconstructed spectral re ectance. Original spectral re ectance is X = R 400 , R 401 , R 402 , ⋯, R 700 T , the reconstructed spectral re ectance is. (20) Table 2 shows RMSEs for the aboved methods which reconstruct spectral re ectance of four standard color blocks. RMSE1 reveals the difference between the original spectral data and the reconstructed spectral data with the polynomial regression. RMSE2 reveals the difference between the original spectral data and the reconstructed spectral data with the pseudo-inverse method. RMSE3 reveals the difference between the original spectral data and the reconstructed spectral data with the compressed sensing method. RMSE4 reveals the difference between the original spectral data and the reconstructed spectral data with compressive sensing based on maximum entropy criterion. It is clearly that RMSE4 has the smallest value, i.e., the deviation between the original spectral re ectance data and the reconstructed spectral re ectance data is smallest. Information entropy is used to evaluate the uncertainty of information quantity. The greater the uncertainty of the information is, the greater the information entropy is. The smaller the difference in information entropy is, the higher the stability of the two information is. (25) Table 3 shows the information entropy between the original and reconstructed curves with different methods. Δ 2−1 is the difference between the original spectral data and the reconstructed spectral data with the polynomial regression, Δ 3−1 refers to the difference between the original spectral data and the reconstructed spectral data with the pseudo-inverse, Δ 4−1 is the difference between the original spectral data and the reconstructed spectral data with the compressed sensing, Δ 5−1 refers to the difference between the original spectral data and the reconstructed spectral data with the compressive sensing based on maximum entropy criterion. It is clear that Δ 5−1 is 2-6 times less than Δ 4−1 , Δ 5−1 is 2-10 times less than Δ 2−1 and Δ 3−1 . The experiment result shows that our method is better than other methods. The performance that our method reconstructs the spectral re ectance is excellent.

The Spectal Reconstruction For Hyperspectral Image
The above experimental results con rm the feasibility of our proposed method. In this part of the work, we use the hyperspectral data from Pavia university to extend the application of our method. The size of the data cube is 340*610*103. The wavelength range is 430nm-860nm and the original channel number is 115. The roof image with 56*56 size in the 30th channel is selected as the area of interest to complete reconstruction by compressed sensing and compressed sensing based on maximum entropy criterion, as shown in Fig. 2. The experimental results are shown in Fig. 3. Figure 3(a) is original image and the spectral curve; Fig. 3 (b) is image and curve reconstructed by CS; Fig. 3 (c) is image and curve reconstructed by MECCS. The difference on intensity of image pixel can be seen. Our method improves the spectral reconstruction. Table 4 shows PSNR and image entropy reconstructed by CS and MECCS method. We can observe that the result of MECCS is better than CS'. The image entropy reconstructed by MECCS is closer to the original image entropy, that is to say, these two images are more similar. The Spectal Reconstruction For The Objects On The Ground We further reconstruct the spectral re ectance curves of the different ground objects in the hyperspectral data such as trees, shadows, meadows, bricks, metal. The evaluations are conducted by calculating the information entropy of spectral curve and RMSE between the original and the reconstructed spectral.
In Fig. 4, solid blue line is the original spectral curve. Solid black line and dashed-dotted red line are spectral curve reconstructed by CS and MECCS respectively. It is clear that the red curves are closer to the blue curves. Table 5 shows RMSEs of CS and MECCS method. RMSE1 represents the difference between the original spectral data and the reconstructed spectral data with the compressed sensing. RMSE2 represents the difference between the original spectral data and the reconstructed spectral data with compressive sensing based on maximum entropy criterion. It is clear that RMSE2 is 2-4 times less than RMSE1.  Table 6 shows information entropy of ground objects. Δ 2−1 represents the difference between the original spectral data and the spectral data reconstructed by CS, Δ 3−1 represents the difference between the original spectral data and the reconstructed spectral data with MECCS. Δ 3−1 is 2-5 times less than Δ 2−1 .

Conclusion
A compressed sensing technology based on the maximum entropy criterion is pro-posed to reconstruct the spectral curve in this paper. Maximum entropy is the criterion for spectral signal segmentation. The wavelet base is a sparse matrix, and the gaussian random measurement matrix reduces the dimensionality of the spectral re ectance signal. The orthogonal matching pursuit algorithm solves the optimization problem to recover original spectral re ectance precisely. Hyperspectral cube of Pavia university and the four standard Munsell color blocks as the object are used to verify the excellent performance of the experiment. Compared with other methods, the reconstructed spectral curve with the CS based on maximum entropy is closer to the original spectral curve, Root mean square error and information entropy difference are both given to evaluate our method quantitatively.

Declarations
Data availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request Competing interests (mandatory) The authors declare no competing nancial interests.  The 30th channel image and the extracted region of interest.