Image Fusion by Compressive Sensing

We propose a new method of image fusion that utilizes the recently developed theory of compressive sensing. Compressive sensing indicates that a signal that is sparse in an appropriate set of basis vectors may be recovered almost exactly from a few samples via l(1)-minimization if the system matrix satisfies some conditions. These conditions are satisfied with high probability for Gaussian-like vectors. Since zero-mean image patches satisfy Gaussian statistics, they are suitable for compressive sensing. We create a dictionary that relates high resolution image patches from a panchromatic image to the corresponding filtered low resolution versions. We first propose two algorithms that directly use this dictionary and its low resolution version to construct the fused image. To reduce the computational cost of l(1)-minimization, we use Principal Component Analysis to identify the orthogonal "modes" of co-occurrence of the low and high resolution patches. Any pair of co-occurring high and low resolution patches with similar statistical properties to the patches in the dictionary is sparse with respect to the principal component bases. Given a patch from a low resolution multispectral band image, we use l(1)-minimization to find the sparse representation of the low resolution patch with respect to the sample-domain principal components. Compressive sensing suggests that this is the same sparse representation that a high resolution image would have with respect to the principal components. Hence the sparse representation is used to combine the high resolution principal components to produce the high resolution fused image. This method adds high-resolution detail to a low-resolution multispectral band image keeping the same relationship that exists between the high and low resolution versions of the panchromatic image. This reduces the spectral distortion of the fused images and produces results superior to standard fusion methods such as the Brovey transform and principal component analysis.