1. A method for image reconstruction using a compressed sensing framework for a hyperspectral video system, the method comprising: constructing, for each of a plurality of sub-data cubes, a N×N representation matrix ψ, each of said sub-data cubes containing a 2D array of pixels and collectively comprising a first hyperspectral image data cube constructed from a first set of different spectral planes of a hyperspectral image captured using a reference sensor, where N is a total number of bands in said reference sensor, wherein constructing each representation matrix comprises: creating a matrix of random numbers; deriving a set of natural basis vectors in a spectral direction; selecting a first few of said natural basis vectors to restrict said representation matrix to a space of interest; replacing a first few columns of said random number matrix with said selected first few natural basis vectors; and performing Gram-Schmidt ortho-normalization to obtain said representation matrix; receiving a M×N sampling matrix Φ comprising a non-square matrix filled with 1's at locations corresponding to the peak wavelengths of filters in a multi-filter grid of a target hyperspectral camera and with remaining elements filled with 0's, where M a number of bands in said target camera such that M<<N; and using said constructed representation matrices and said sampling matrix to reconstruct a full signal ƒ* of a hyperspectral image captured by said target hyperspectral camera.