AutoLumNet: Monotone Optimal Transport for Single-Shot Exposure Correction

Single-shot exposure correction aims to map an arbitrarily degraded image---whether under-exposed, over-exposed, or a spatial mixture of both---to a well-exposed output from a single capture. We present AutoLumNet, a framework that decomposes this task into a global monotone tone curve and a bounded local residual, making the global component the locus of formal guarantees. The tone curve is parameterized as the normalized cumulative integral of a strictly positive density, ensuring strict monotonicity by construction rather than by penalty. We prove that this parameterization (i)~preserves the pairwise luminance ordering of all pixels and all spatial extrema unconditionally, and (ii)~is dense in the space of valid tone corrections, containing the one-dimensional optimal-transport map from the input to any target luminance distribution. A differentiable sorted-sample Wasserstein-2 objective drives the learned curve toward the OT optimum during training. Spatially varying effects that the global map provably cannot address---local shading, chrominance shifts, and clipped-region restoration---are handled by a bounded residual decoder with dual-branch convex fusion, for which we provide an explicit sufficient condition for local order preservation. Experiments on five benchmarks (MSEC, SICE, LCDP, LOL-v1, LOL-v2-real) show that AutoLumNet achieves state-of-the-art PSNR and SSIM across both under- and over-exposure regimes at 11.2\,ms per frame, and generalizes zero-shot to pure low-light benchmarks without retraining. To our knowledge, AutoLumNet is the first exposure-correction method to unite structural monotonicity, optimal-transport optimality, and bounded local adaptivity within a single trainable architecture. Code is available at https://github.com/kraihan/Autolumnet.

Orthogonal Polynomial Approximation for Matrix Log Normalization in Global Covariance Pooling

Global Covariance Pooling (GCP) improves deep networks by capturing second-order feature statistics, and is especially effective for fine-grained recognition. Because covariance matrices live on the Symmetric Positive Definite (SPD) manifold, a normalization step is required before the Euclidean classifier. The faithful choice is the matrix logarithm (MLN-COV), which maps the SPD manifold to its tangent space; in practice it was abandoned in favour of the matrix square root because its eigendecomposition-based gradient is numerically unstable. We show that this instability is an artifact of computing the logarithm spectrally, not of the logarithm itself. Approximating the logarithm with finite polynomials in the covariance matrix removes the eigendecomposition from both passes: every operation becomes a General Matrix Multiplication (GEMM), the gradient stays bounded on the spectral support of the pre-normalized covariance, and the unstable 1/(lambda_i-lambda_j) term never appears. The key ingredient is a mean-eigenvalue pre-normalization that centres the spectrum near 1, away from the singularity of log, with a scalar post-compensation that returns the singular part of log(A) in closed form. Our recommended normalizer is a degree-8 Chebyshev expansion evaluated by a three-term matrix recurrence, with a matching reverse recurrence for the backward pass; Legendre, Laguerre, Taylor and Pade expansions are studied as controls that isolate the roles of the basis and of the target function. On three fine-grained benchmarks and ImageNet-1k the decomposition-free logarithm is both faster and more accurate than the spectral logarithm and than the square-root approximations it replaces, and at matched basis and degree the log target beats the square-root target, confirming that the gain comes from the faithful Riemannian map rather than from a better polynomial family.