# Hyperspectral Data: Efficient and Secure Transmission
## 2 An Efficient and Secure Transmission Framework
### An Effective Implementation of the Proposed Framework
#### 2.1.1 Reversible Invisible Watermarking Scheme for Hyperspectral Images
\(N_{i}=lengthOf(w_{i})\)
7. repeat
8. \(b_{w}=w_{i}[idx]\);
9. By using \(G\) along with \(K\), selects \(x_{(0)}\), \(x_{(1)}\), \(x_{(2)}\), \(x_{(3)}\) from \(HI_{(i)}\);
10. \(S_{P}=\{x_{(0)},\ x_{(1)},\ x_{(2)},\ x_{(3)}\}\);
11. \(S_{P}^{\text{E}}=estimate(S_{P})\);
12. \(D=|x_

#### 2.1.2 Multiband Lossless Compression of Hyperspectral Images

The predictive-based multiband lossless compression for hyperspectral images (LMBHI) [11] algorithm exploits the inter-band correlation (i.e., the correlation among the neighboring pixels of contiguous bands) as well as the intra-band correlations (i.e., the correlations among the neighboring pixels of the same band), by using a predictive coding model. Each pixel of the input hyperspectral image \(HI\) is predicted by using one of the following predictive structures: the 2-D linearized median predictor (2-D LMP) [15] and the 3-D multiband linear predictor (3D-MBLP). 2-D LMP considers only the intra-band correlation and it is used only for the pixels of the first band for which there are no previous reference bands. 3D-MBLP exploits the intra-band and the inter-band correlations instead and is used to predict the pixels of all the bands except for the first one. Once the prediction step is performed the prediction error, \(e\), is modelled and coded. In particular, \(e\) is obtained by subtracting the value of the prediction of the current pixel from its effective value. The 3D-MBLP predictor uses a three-dimensional prediction context which is defined by considering two parameters: \(B\) and \(N\), where \(B\) indicates the number of the previous reference bands and \(N\) indicates the number of pixels that will be included in the prediction context that are in the current band and in the previous \(B\) reference bands. In order to permit an efficient and relative indexing of the pixels that form the prediction context of the 3D-MBLP, an enumeration, \(E\), is defined. We denote with \(I_{i,j}\) the \(i\)th pixel (according to the enumeration \(E\)) of the \(j\)th band. In addition, we suppose that the current band is the \(k\)th band. In this manner, \(I_{0,j}\) is referred to the pixel that has the same spatial coordinates of the current pixel (denoted as \(I_{0,k}\)), in the \(j\)th band. The 3D-MBLP predictor is based on the least squares optimization technique and the prediction is computed, as in Equation (1). \[\widehat{I_{0,k}}=\sum_{i=1}^{B}\alpha_{i}\times I_{0,k-i} \tag{1}\]

It is important to point out that the coefficients \(\alpha_{0}=[\alpha_{1},\,\ldots,\,\alpha_{B}]^{T}\) are chosen to minimize the energy of the prediction error, as in Equation (2). \[P=\sum_{i=1}^{N}\left(I_{i,k}-\widehat{I_{i,j}}\right)^{2} \tag{2}\]

It should be observable that Equation (2) can be rewritten (by using the matrix notation), as outlined in the equation

\[P=(C\alpha-X)^{T}. (C\alpha-X),\text{ where }C=\left[ \begin{array}{ccc}I_{1,k-1}& \cdots&I_{1,k-B}\\ \vdots&\ddots&\vdots\\ I_{N,k-1}&\ldots&I_{N,k-B}\end{array} \right]\text{ and }X=[I_{1,k},\,\ldots,\,I_{N,k}]^{T}.\]

Subsequently, by computing the derivate of \(P\) and by setting it to zero, the optimal coefficients can be obtained

\[(C^{T}C)\alpha_{0}=(C^{T}X) \tag{3}\]

Once the coefficients \(\alpha_{0}\), which solve the linear system of Equation (3), are computed, the prediction \(\widehat{I_{0,k}}\), of the current pixel \(I_{0,k}\), can be calculated. ## 3 Experimental Results

To validate the proposed framework, we have experimentally tested our proposed algorithms on two datasets, which are composed by airborne visible/infrared imaging spectrometer (AVIRIS)hyperspectral images. Such data is obtained by AVIRIS hyperspectral sensors (NASA Jet Propulsion Laboratory (JPL) [16]), which measure the spectrum in the wavelengths ranging from 380 to 2500 nm (subdivided into 224 spectral bands). ### Description of the Test Datasets

**Dataset 1. ** The first dataset we used in the testing phase is composed by five AVIRIS hyperspectral images provided by the JPL (Jet Propulsion Laboratory) of NASA.