In order to verify the compressed sensing technology based on maximum entropy criterion, we first conducts experiments with four standard Munsell color blocks of red, green, blue and yellow. The evaluations are conducted by calculation of information entropy and root mean square error between the original and the reconstructed spectral reflectances.
Maximum Entropy Threshold Segmentation Method Of Spectral Reflectance
Presegmentation processing on the spectral curve was performed to calculate the information entropy on the cases of different segmentation thresholds. When the information 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 ti is selected in the same domain with i. Once ti is determined, the Spectral curve is divided into \({X_A}\)and \({X_B}\). \(X=\left( {{R_{400}},{R_{401}},{R_{402}}, \cdots ,{R_{700}}} \right)\). \({R_i}\) represents the spectral reflectance at wavelength i nm. \({X_A}=\left( {{R_{400}},{R_{401}},{R_{402}}, \cdots ,{R_{{t_i}}}} \right)\), \(A=[400,{t_i}]\).
AA shows the probability distribution of \({X_A}\). BB shows the probability distribution of \({X_B}\).
$$AA:\frac{{{P_{400}}}}{{{P_A}}},\frac{{{P_{401}}}}{{{P_A}}},\frac{{{P_{402}}}}{{{P_A}}}, \cdots ,\frac{{{P_{{t_i}}}}}{{{P_A}}}$$ (1)
$${P_A}=\sum\nolimits_{{i=400}}^{{{t_i}}} {{P_i}}$$ (2)
$$BB:\frac{{{P_{{t_i}+1}}}}{{{P_B}}},\frac{{{P_{{t_i}+2}}}}{{{P_B}}},\frac{{{P_{{t_i}+3}}}}{{{P_B}}}, \cdots ,\frac{{{P_{700}}}}{{{P_B}}}$$ (3)
\({P_i}\) represents the probability value of the spectral reflectance \({R_i}\) at the wavelength i nm.
The entropy related to the two probability densities of A and B are expressed by:
$$H\left( A \right)=  \sum\nolimits_{{i=400}}^{{{t_i}}} {\frac{{{P_i}}}{{{P_A}}}} \log 2\frac{{{P_i}}}{{{P_A}}}$$ (5)
$$H\left( B \right)=  \sum\nolimits_{{i={t_i}+1}}^{{700}} {\frac{{{P_i}}}{{{P_B}}}\log 2\frac{{{P_i}}}{{{P_B}}}} =  \sum\nolimits_{{i={t_i}+1}}^{{700}} {\frac{{{P_i}}}{{1  {P_A}}}\log 2\frac{{{P_i}}}{{1  {P_A}}}}$$ (6)
The maximum entropy of spectral fragments is calculated to determine the optimal threshold tO of the spectral curve:
$$HH\left( {{t_o}} \right)=\hbox{max} [H\left( A \right)+H\left( B \right)]$$ (7)
\(HH\left( {{t_o}} \right)\) indicates the maximum sum of the information entropy of the spectral segments divided by different thresholds ti. to 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 to is determined to find the most reasonable segment length of the smooth spectral fragment and the nonsmooth spectral fragment. The total fragment length of the spectral reflectance is \(N=301\). The total dimensionality reduction sample number is \(M=100\). There are two conditions to be met, \(M={M_a}+{M_b}\), \(N={N_a}+{N_b}\). Na is the length of the spectral fragment A, Nb is the length of the spectral fragment B. Ma is the downsampling size of the spectral fragment A, Mb is the downsampling size of the spectral fragment B.
The original signal is \(X=301 \times 1\) dimensional column vector, \(X={\left\{ {{R_{400}},{R_{401}},{R_{402}}, \cdots ,{R_{700}}} \right\}^T}\). By using a set of wavelet orthogonal matrix \(\varphi =\left\{ {{\varphi _1},{\varphi _2},{\varphi _3}, \cdots ,{\varphi _{301}}} \right\}\), 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 nonzero elements in the sparse coefficient α is \(K(K \leqslant N)\). The original signal X is undersampled by the measurement matrix \({\psi _{M \times N}}=\left\{ {{\psi _1},{\psi _2},{\psi _3}, \cdots ,{\psi _M}} \right\}\) for reduced dimensionality, namely:
$$Y=\psi X=\psi \varphi \alpha =C\alpha$$ (9)
Where C is the perception matrix. The signal Y is an Mdimensional 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). \(M \ll N\), X is derived from Y by an underdetermined equation. The norm \({l_1}\)is selected to solve the optimal solution\({\alpha ^\prime }\). The reconstruction for original signal adopts the orthogonal matching pursuit algorithm (OMP).
\(\alpha ^{\prime}=\arg \hbox{min} {\left\ \alpha \right\_1}\) s.t \(Y=C\alpha\) (10)
$$X^{\prime}=\varphi {\alpha ^\prime }$$ (11)
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\({X_A}={\left\{ {{R_{400}},{R_{401}},{R_{402}}, \cdots ,{R_t}} \right\}^T}\) is Ma. The signal \({X_A}\) is represented linearly by a set of orthogonal wavelet matrices \({\varphi _a}=\left\{ {{\varphi _1},{\varphi _2},{\varphi _3}, \cdots ,{\varphi _{Na}}} \right\}\). The sparsity coefficient is \({\alpha _{\text{1}}}\). The formula can be expressed as:
$${X_A}={\varphi _a}{\alpha _1}$$ (12)
Where \({\alpha _{\text{1}}}\) is also means linear projection of signal \({X_A}\) on \({\varphi _a}\) domain, the number of nonzero elements in sparse coefficient is \({K_a}\left( {{K_a} \leqslant {N_a}} \right)\). The signal \({X_A}\) is linearly projected by a random Gaussian measurement matrix for dimensionality reduction. The measurement matrix \({\psi _{Ma \times Na}}=\left\{ {{\psi _1},{\psi _2},{\psi _3}, \cdots ,{\psi _{Ma}}} \right\}\)is not related with the wavelet basis, then the compressed sensing problem can be formulated as follows:
$${Y_A}={\psi _{Ma \times Na}}{X_A}={\psi _{Ma \times Na}}{\varphi _a}{\alpha _1}={C_1}{\alpha _1}$$ (13)
Where \({C_{\text{1}}}\) is the perception matrix. \({Y_A}\)is an Mdimensional column vector. After the linear projection of the signal \({X_A}\)on the measurement matrix \({\psi _{Ma \times Na}}\), the undersampling measured value \({Y_A}\) is obtained, \({M_a} \ll {N_a}\). The orthogonal matching pursuit algorithm is selected for recovering the sparsest solution of underdetermined system.
\({\alpha _1}^{\prime }=\arg \hbox{min} {\left\ \alpha \right\_1}\) s.t \({Y_A}={C_1}{\alpha _1}\) (14)
$${X_A}^{\prime }={\varphi _a}{\alpha _1}^{\prime }={\left\{ {{R_{400}}^{\prime },{R_{401}}^{\prime },{R_{402}}^{\prime }, \cdots ,{R_t}^{\prime }} \right\}^T}$$ (15)
The CSbased reconstruction of the spectral fragment A is obtained.
Spectral fragment B is reconstruct by CS. The sampling size of signal \({X_B}={\left\{ {{R_{t+1}},{R_{t+2}},{R_{t+3}}, \cdots ,{R_{700}}} \right\}^T}\) is Mb. The spectral signal \({X_B}\) is represented linearly by a set of orthogonal wavelet matrices \({\varphi _b}=\left\{ {{\varphi _{184}},{\varphi _{185}},{\varphi _{186}}, \cdots ,{\varphi _{301}}} \right\}\). The sparsity coefficient is \({\alpha _{\text{2}}}\). The formula can be expressed as:
$${X_B}={\varphi _b}{\alpha _2}$$ (16)
Where \({\alpha _{\text{2}}}\) is also means linear projection of signal \({X_B}\) on \({\varphi _b}\) domain, the number of nonzero elements in sparse coefficient is \({K_b}\left( {{K_b} \leqslant {N_b}} \right)\). The signal \({X_B}\) is linearly projected by a random gaussian measurement matrix for dimensionality reduction. the measurement matrix \({\psi _{Mb \times Nb}}=\left\{ {{\psi _1},{\psi _2},{\psi _3}, \cdots ,{\psi _{Mb}}} \right\}\)is not related to the wavelet basis, then the compressed sensing problem can be formulated as follows:
$${Y_B}={\psi _{Mb \times Nb}}{X_B}={\psi _{Mb \times Nb}}{\varphi _b}{\alpha _2}={C_2}{\alpha _2}$$ (17)
Where \({C_{\text{2}}}\) is the perception matrix. \({Y_B}\) is an Mdimensional column vector. After the linear projection of the signal \({X_B}\)on the measurement matrix \({\psi _2}\), the undersampling measured value \({Y_B}\) is obtained, \({M_b} \ll {N_b}\). The orthogonal matching pursuit algorithm is used to solve the underdetermined equations.
\({\alpha _{\text{2}}}^{\prime }=\arg \hbox{min} {\left\ \alpha \right\_1}\) s.t \({Y_B}={C_2}{\alpha _2}\) (18)
$${X_B}^{\prime }={\varphi _b}{\alpha _2}^{\prime }={\left\{ {{R_{t+1}}^{\prime },{R_{t+2}}^{\prime },{R_{t+3}}^{\prime }, \cdots ,{R_{700}}^{\prime }} \right\}^T}$$ (19)
The CSbased reconstruction of the spectral fragment B is obtained .
Integrating the reconstructed spectral reflectance \({X_A}^{\prime }\)and \({X_B}^{\prime }\), the result is \({X^\prime }={\left\{ {{R_{400}}^{\prime },{R_{401}}^{\prime },{R_{402}}^{\prime }, \cdots ,{R_{700}}^{\prime }} \right\}^T}\).
Table 1 shows the maximum entropy threshold segmentation results of the four standard color blocks (red, green, blue and yellow). The optimal thresholds are to=639nm, 536nm, 461nm, 583nm for red, green, blue and yellow blocks respectively.
Table 1
The maximum entropy threshold segmentation results.

Red

Green

Blue

Yellow

to

639

536

461

583

A

[400:639]

[400:536]

[400:461]

[400:583]

B

[640:700]

[537:700]

[462:700]

[584:700]

Taking the yellow spectral curve for example. The optimal threshold to of the yellow spectral curve is 583nm. The spectral curve is divided into fragments A=[400nm583nm], B=[584nm700nm]. The total fragment length of the spectral reflectance 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.
N a =183, Nb=118, Ma=60, Mb=40.
The reconstructed spectral reflectance for yellow is shown in Fig. 1(d).