A Convex Formulation for Hyperspectral Image Superresolution via Subspace-Based Regularization

Hyperspectral remote sensing images (HSIs) usually have high spectral resolution and low spatial resolution. Conversely, multispectral images (MSIs) usually have low spectral and high spatial resolutions. The problem of inferring images that combine the high spectral and high spatial resolutions of HSIs and MSIs, respectively, is a data fusion problem that has been the focus of recent active research due to the increasing availability of HSIs and MSIs retrieved from the same geographical area. We formulate this problem as the minimization of a convex objective function containing two quadratic data-fitting terms and an edge-preserving regularizer. The data-fitting terms account for blur, different resolutions, and additive noise. The regularizer, a form of vector total variation, promotes piecewise-smooth solutions with discontinuities aligned across the hyperspectral bands. The downsampling operator accounting for the different spatial resolutions, the nonquadratic and nonsmooth nature of the regularizer, and the very large size of the HSI to be estimated lead to a hard optimization problem. We deal with these difficulties by exploiting the fact that HSIs generally “live” in a low-dimensional subspace and by tailoring the split augmented Lagrangian shrinkage algorithm (SALSA), which is an instance of the alternating direction method of multipliers (ADMM), to this optimization problem, by means of a convenient variable splitting. The spatial blur and the spectral linear operators linked, respectively, with the HSI and MSI acquisition processes are also estimated, and we obtain an effective algorithm that outperforms the state of the art, as illustrated in a series of experiments with simulated and real-life data.

I. Introduction

IMAGES are an efficient way to describe and store visual information about our world. This paper will deal with a special kind of image, the so-called spectral images. A spectral image or data cube is a set of 2-D images, also termed bands, representing the reflectance or radiance of a scene in different parts of the electromagnetic (EM) spectrum. They find applications in the fields of remote sensing (agriculture, mineralogy, etc.), astronomy, and biomedicine, for example [1]. Our focus will be on the remote sensing field, where spectral images are typically generated from airborne or spaceborne sensors.