Adaptive Radial Projection on Fourier Magnitude Spectrum for Document Image Skew Estimation

This paper proposes an algorithm for estimating the similarity transformation, namely translation, scale, and rotation, between two images with subpixel accuracy. Image registration is a fundamental technique for aligning images acquired under different viewpoints and imaging conditions, and a representative approach based on maximizing discrete cross-correlation is the Fourier--Mellin registration. However, the Fourier--Mellin approach often fails to achieve sufficient alignment accuracy when subpixel-level estimation is required. The proposed method integrates (i) scale-and-rotation estimation from the Fourier magnitude spectrum in a log-polar representation and (ii) maximization of phase-only correlation based on the auxiliary function method. This integration enables a two-stage estimation procedure: it first estimates scale and rotation without being affected by translation, and then estimates translation with subpixel precision in the spatial domain using the corrected image pair. A simulation experiment on image pairs subjected to random similarity transformations demonstrates that the proposed method reduces estimation errors in scale, rotation, and translation compared with Fourier--Mellin-based registration methods using discrete cross-correlation.

In this article, we analyze and propose a Python implementation of the method "Pith Estimation on Rough Log End images using Local Fourier Spectrum Analysis", by Rudolf Schraml and Andreas Uhl. The algorithm is tested over two datasets.

Compact representations that support geometrically meaningful comparisons are valuable for learning from probability distributions. We study optimal, free-weight $M$-atom quantization of one-dimensional probability measures in quadratic Wasserstein distance and extend it to higher dimensions via slicing. Using $L$ shared projections yields a sketch with $O(LM)$ comparison time. Our main result is an exact decomposition of squared distance error into residual and partition-mismatch terms, yielding computable certificates for arbitrary centroid sketches. For optimal sketches, a finite $(2+δ)$ moment for some $δ>0$ guarantees $O(M^{-1})$ error in the finite-direction sliced distance, while equal-mass sketches can converge more slowly. Each slice preserves its mean, while storing residual energies enables covariance recovery from sufficiently many generic directions. Experiments on document, point-cloud, and flow-cytometry benchmarks show that optimal sketches substantially reduce storage while preserving the classification performance of uncompressed sliced distances. They also improve generalization from few labelled subjects and support end-to-end learning as compact pooling layers.