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Alias Geometry and Density Compensation in Perturbed Fourier Sampling — arXiv:2611.02379

Source

arXiv (en)

Original

https://arxiv.org/abs/2611.02379

License (as archived)

CC0-1.0

Archived

2026-11-03T11:35:42.812Z

SHA-256 of the archived text

e3cb42c62f69b98d9503cf108a5d301ff34b5aba10c1fee767f4b680fb3df84a

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Alias Geometry and Density Compensation in Perturbed Fourier Sampling — arXiv:2611.02379 — arXiv (en), https://arxiv.org/abs/2611.02379 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bdd4dcd438b2 (Metraipolis archive dc_bdd4dcd438b2, sha256 e3cb42c62f69b98d9503cf108a5d301ff34b5aba10c1fee767f4b680fb3df84a)

@misc{dcbdd4dcd438b2,
  title = {Alias Geometry and Density Compensation in Perturbed Fourier Sampling — arXiv:2611.02379},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.02379},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bdd4dcd438b2; sha256 e3cb42c62f69b98d9503cf108a5d301ff34b5aba10c1fee767f4b680fb3df84a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.02379. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

On a uniform grid with equal weights, Fourier modes that remain distinct on the grid are exactly orthogonal. When the nodes are perturbed, density compensation readjusts the weights. We ask how much of the resulting error reweighting can remove, measuring it by the Frobenius norm of the Gram defect, whose square is proportional to the mean-square error of density-compensated adjoint reconstruction. The answer is governed by aliasing. The grid cannot distinguish certain frequency differences, so a weight correction must treat each such alias class as a single number, while a node displacement affects every frequency difference in the class separately. To first order, reweighting removes the class average but not the spread. The irreducible error is a sum of covariance matrices, one for each class of colliding frequency differences. If no differences collide, every small node perturbation can be repaired exactly with positive weights. With collisions, a repair that works to first order can fail at second order. At the two-dimensional DFT, row and column shears are each exactly repairable, but their combination is not, and we compute the obstruction. In one dimension, oversampling protects a band of low-frequency node motions, and under random jitter optimal reweighting asymptotically removes a fraction $1-\kappa^2(2-\kappa)^2$ of the expected squared first-order error at oversampling ratio $\kappa\in[1,2]$. At critical sampling, node and weight perturbations are orthogonal: no reweighting helps, Voronoi weights can never reduce the error, and for the one-dimensional DFT the remaining error is a log-Vandermonde energy with the circular $H^{1/2}$ energy as its continuum limit.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.NA, nuovi lavori del giorno).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2611.02379 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T11:05:07.218Z dalla pagina https://arxiv.org/list/math.NA/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).