Hyperspectral Image Fusion with Implicit Regularization

Abstract

Hyperspectral (HS) satellite images contain rich spectral information but generally suffer from low spatial resolution. In contrast, multispectral (MS) images offer higher spatial resolution but only a limited number of spectral bands. The aim of HS–MS fusion is to combine these complementary data sources to produce an image with high spatial and spectral resolution. The resulting fused images are valuable for tasks such as scene interpretation, material identification, and environmental monitoring.

The fusion task can be formulated as a linear inverse problem, where the forward model is determined by the optics of the HS and MS cameras. However, this problem is ill-posed, and proper regularization is necessary for accurate reconstruction. Several regularization methods have been explored, including Tikhonov and total variation regularization, sparsity-based priors in transform domains, and low-rank and subspace constraints. More recently, it has been shown that smoothing operators (originally designed for image denoising) can act as implicit regularizers within iterative reconstruction algorithms. This implicit approach offers considerable flexibility, avoiding the need to handcraft explicit penalty terms, and has demonstrated strong empirical performance across many imaging problems. However, repeated smoothing can render the iterations unstable or even divergent, and the absence of rigorous convergence guarantees limits the reliability of such methods.

This work develops a class of image smoothers that provide strong regularization while remaining analytically tractable. The core idea is to use kernel-based denoisers that exploit both spatial and spectral correlations inherent in hyperspectral data. In particular, we introduce a Cascaded Kernel Denoiser (CasKD), which combines two kernel denoisers: the first captures inter-band (spectral) correlations, while the second enhances intra-band (spatial) smoothness. Although cascading denoisers do not guarantee improved performance, the complementary roles of these two operators lead to consistently superior fusion quality in our setting.

We incorporate CasKD into two standard Plug-and-Play (PnP) frameworks, PnP–Proximal Gradient Descent and PnP–Half Quadratic Splitting. Although prior PnP results have shown that certain self-adjoint smoothing operators can be interpreted as proximal operators of convex functions, these results do not apply here because CasKD is not self-adjoint. As a result, the resulting reconstruction process cannot be viewed as minimizing a well-defined objective function, and we cannot use existing convergence guarantees. We instead analyze convergence by viewing the algorithm as a fixed-point iteration. In this setup, the convergence problem reduces to understanding the fixed-point properties of an affine operator derived from CasKD and the forward model. A key technical challenge is that the optimization variables are matrices, which requires us to work with operators acting between matrix spaces. We show that, for a suitable choice of parameters (such as the gradient step size), this operator is contractive. This implies global linear convergence of the iterates to a unique fixed point, thereby ensuring algorithmic stability and consistent reconstruction performance.

Extensive experiments on benchmark datasets show that our framework achieves strong and consistent fusion performance across datasets and initialization conditions. It outperforms classical methods and remains competitive with recent techniques, with particularly notable gains at high noise levels. We further analyze the reconstruction operator using the power method and verify that it remains contractive across parameter settings. In addition to the theoretical convergence guarantee, we observe in practice that the PSNR typically stabilizes within only 10–15 iterations. Overall, the proposed method is stable, reliable, and delivers consistently strong performance in practical fusion scenarios.