Sparse Analysis Based on Generalized Gaussian Model for Spectrum Recovery with Compressed Sensing Theory

Imaging spectrometers can supply spatial data in abundant narrow and continuous wavelength bands. However, the huge data volume produced encounters the difficulty in storage and transmission. On the other hand, these hyperspectral data sets contain high redundancy, which offers an opportunity to reduce the number of spectral measurements and recover the full spectrum from limited samples without losing principal spectral information. This paper addresses the application of compressed sensing (CS) theory to hyperspectral data reconstruction. An important question involved is how to know a spectrum is sparse enough so that CS can be applied effectively. We provide a quantitative answer and develop a strategy to measure the degree of sparsity of a spectrum based on the generalized Gaussian distribution (GGD) model. The novelty includes the derivation of the sharpness of the GGD and how to estimate the sharpness of a spectral signal. The proposed strategy was tested using the spectral data from USGS database and an AVIRIS-HSI data set. The results demonstrate that it is important to introduce the sparsity measure, as CS offers a high reconstruction rate and low relative errors compared with the existing methods for sparse signals only.