A geometry-based deep equilibrium model for image restoration under multiplicative Gamma noise

Multiplicative Gamma noise is a signal-dependent degradation in coherent imaging; synthetic aperture radar (SAR) despeckling is its most prominent real-world instance. Existing diffusion denoisers parameterize their forward process by abstract signal-to-noise schedules rather than by the physical look number $L$, so different deployment scenarios typically require separately trained models, and transfer from synthetic Gamma training to real SAR remains challenging without clean ground truth. We introduce $γ$-Bridge, a look-parametric bridge whose schedule $L(t)$ connects the noisy observation at $L_{obs}$ to the clean limit through exact multiplicative Gamma marginals. Its closed-form Gamma--Lévy reverse posterior admits both stochastic and deterministic processes, while observation conditioning and a two-step consistency loss stabilize multi-step inference in the low-SNR single-look regime. Because bridge time directly represents $L$, one conditioned network can smart-start from any admissible input look and stop at a target look number. These two orthogonal controls enable zero-shot restoration over the full admissible grid after training only at $L_{obs} = 1$ on natural images with synthetic Gamma corruption. Combined with a homogeneous-patch look estimator, $γ$-Bridge processes data from six spaceborne and airborne SAR sensors without sensor-specific fine-tuning, achieving leading results on standard synthetic benchmarks while providing physically interpretable input and output controls absent from prior denoisers. Codes are released \href{https://github.com/Teriri1999/GammaBridge}{here}.

Image restoration models are typically trained with a fixed set of capabilities. When new restoration requirements emerge, existing solutions usually train additional models or jointly retrain the original model with both new and historical data. Instead of designing another restoration backbone, we investigate how a trained restorer can continually acquire new capabilities without forgetting those learned previously. We propose RestoreMore, a continual capability-expansion framework that preserves the pretrained restoration model as a frozen capability anchor and learns residual expansion modules for newly arriving degradations. RestoreMore introduces a capability-oriented bi-level routing mechanism at multiple feature stages. The first routing level identifies restoration capabilities relevant to the current input, while the second selects and combines a sparse set of complementary degradation experts. This design enables newly introduced tasks to selectively reuse historical restoration knowledge and progressively enriches the expert bank available for subsequent restoration tasks. Extensive experiments on a wide range of restoration benchmarks demonstrate that RestoreMore consistently acquires new restoration abilities while preserving and improving previously learned capabilities.

Diffusion models represent the state-of-the-art for solving inverse problems such as image restoration tasks. Diffusion-based inverse solvers incorporate a likelihood term to guide prior sampling, generating data consistent with the posterior distribution. However, due to the intractability of the likelihood, most methods rely on isotropic Gaussian approximations, which can push estimates off the data manifold and produce inconsistent, poor reconstructions. We propose Equivariance Regularized (EquiReg) diffusion, a general plug-in framework that improves posterior sampling by penalizing trajectories that deviate from the data manifold. EquiReg formalizes manifold-preferential equivariant functions that exhibit low equivariance error for on-manifold samples and high error for off-manifold ones, thereby guiding sampling toward symmetry-preserving regions of the solution space. We highlight that such functions naturally emerge when training non-equivariant models with augmentation or on data with symmetries. EquiReg's largest gains are under reduced sampling and measurement consistency steps, where many methods suffer severe quality degradation. By regularizing trajectories toward the manifold, EquiReg implicitly accelerates convergence and enables high-quality reconstructions. EquiReg consistently improves performance in linear and nonlinear image restoration tasks and solving partial differential equations. Our code is available at https://github.com/Anima-Lab/EquiReg