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CLONE: A 3DGS-Based Closed-Loop Differentiable Optimization Framework for Single-Image Normal Estimation

We propose CLONE, a 3DGS-based Closed-Loop differentiable Optimization framework for single-image Normal Estimation. The core idea is to construct an "image-geometry-image" consistency loop that unifies and jointly constrains the limitations of both paradigms: the reliance on explicit supervision without cross-domain geometric constraints in discriminative methods, and the absence of stable differentiable optimization pathways in generative methods despite strong generative priors. Specifically, we first employ 3D Gaussian Splatting to explicitly parameterize the scene and derive continuous and differentiable surface normals via covariance eigen-decomposition, providing an analytical gradient pathway for geometric modeling. We then introduce a differentiable illumination model with a learnable light modulation kernel to establish a continuous mapping between surface normals and image radiance, enabling reprojection errors to directly supervise the underlying 3D geometry. Furthermore, to compensate for the limited local detail expressiveness of Gaussian representations, we design a one-step deterministic diffusion-inspired refinement network, which enhances local geometric details while preserving end-to-end differentiability. A cross-domain gating fusion mechanism is introduced to coordinate global geometric consistency and local detail reconstruction. Finally, all components are jointly optimized under a unified reprojection objective, forming a closed-loop and stable gradient propagation pathway. This enables effective constraint of the multi-solution space and improved geometric consistency without requiring ground-truth normal supervision.

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