发表机构
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出高斯映射变分模型用于图像去噪，通过几何分析保持边缘和角点，并采用Anderson加速的majorization-minimization算法求解，实验验证了其边界保持和噪声去除性能。
AI 中文摘要
我们提出了一种用于图像去噪的高斯映射变分（GMV）模型，该模型度量缩放图像图切平面投影仪的空间变化。我们建立了正则化器在相应高斯映射下的等价表示，并利用包括管状坐标和Frenet框架在内的微分几何工具，分析其在一般$C^2$和分段$C^2$边界上的行为。由此得到的估计提供了边缘和角点对比度保持特性。为了求解所提出的模型，我们引入了一个涉及单位法向量场和标量幅度场的双线性分解，并开发了一种Anderson加速的majorization-minimization算法。法向场子问题允许显式的逐点majorization-minimization更新，并与Anderson加速相结合。对于$L^1$和$L^2$数据保真项，我们建立了迭代的充分下降性和有界性，并证明了生成的序列收敛到惩罚模型的临界点。在合成图像和自然图像上的数值实验证明了所提出模型的边界保持能力及其在去除高斯噪声和脉冲噪声方面的竞争性能。
英文摘要
We propose a Gauss-map variation (GMV) model for image denoising that measures the spatial variation of the tangent-plane projectors of the scaled image graph. We establish an equivalent representation of the regularizer in terms of the corresponding Gauss map and, using differential geometric tools including tubular coordinates and the Frenet frame, analyze its behavior across general $C^2$ and piecewise $C^2$ boundaries. The resulting estimates provide edge- and corner-contrast preservation properties. To solve the proposed model, we introduce a bilinear decomposition involving a unit normal field and a scalar magnitude field and develop an Anderson-accelerated majorization--minimization algorithm. The normal field subproblem admits an explicit pointwise majorization--minimization update, which is combined with an Anderson acceleration. For both $L^1$ and $L^2$ data fidelity terms, we establish sufficient decrease and boundedness of the iterates and prove that the generated sequence converges to a critical point of the penalized model. Numerical experiments on synthetic and natural images demonstrate the boundary preserving capability of the proposed model and its competitive performance in removing Gaussian and impulsive noise.
