From Gradient Magnitude to Level-Set Geometry: A 2nd-Order Aligned Regularizer for 1st-Order Image Priors

Natural images exhibit coherent spatial organization of local variations, particularly in gradient orientations along contours and smoothly varying structures. Image corruption can disrupt this organization, causing neighboring gradient orientations to fluctuate irregularly even when individual variations remain plausible. In image denoising, 1st-order regularizers control local variation but do not explicitly constrain how these variations are organized across space. This raises a question: can spatial organization be regularized without introducing a preference that conflicts with the original 1st-order prior? In this paper, we show that it can be done using the first variation of a 1st-order prior. This defines a spatial response on the level-set geometry whose squared norm, the 2nd-order Aligned Regularizer (SOAR), penalizes how the normal profile and the level-set curvature vary across the image. In a local tangential Fourier picture, SOAR adds a 4th-order spectral response on top of the 2nd-order spectral response of the parent prior, acting as an additional low-pass filter on geometric fluctuations while preserving the parent prior's stationary points. For denoising, we combine image-adaptive weighting of the 1st-order regularizer with selective SOAR activation. The resulting single-image zero-shot denoising method consistently improves over the 1st-order baseline and achieves the highest PSNR on real-world benchmarks.