[论文] Graded convolutional networks: scale invariance without the scale and ...

论文概要

研究领域: CV
作者: Tanush Shaska
发布时间: 2026-11-06
arXiv: 2611.09878

中文摘要

图像的高斯喷射（Gaussian jet）是一个分次向量空间：以 λ 膨胀会将 k 阶导数乘以 λ^{-k}。局部分次卷积恰好是齐次微分算子，局部特征是膨胀不变的当且仅当它通过尺度归一化jet分解，或无尺度时通过分次加权球面的收缩（retraction）分解。在未知高斯模糊下，收缩仅改变尺度，而尺度归一化jet会获得任何径向归一化都无法消除的分次因子。在零jet处，收缩能分离径向归一化和连续分类器无法分离的图像族。基于收缩构建的网络对不变特征具有通用性。在MNIST上，这种网络在两像素模糊下保持88.4%准确率，而使用尺度归一化导数的同一网络仅为66.2%；在Lindeberg的高斯导数网络中，收缩将该模糊下的准确率从91.5%提高到95.1%。

原文摘要

The Gaussian jet of an image is a graded vector space: a dilation by \(λ\) multiplies derivatives of order \(k\) by \(λ^{-k}\). Local graded convolutions are exactly homogeneous differential operators, and a local feature is dilation-invariant if and only if it factors through the scale-normalized jet or, without the scale, through the retraction onto the weighted sphere of the grading. Under unknown Gaussian blur the retraction changes only by a change of scale, whereas the scale-normalized jet acquires a grading factor that no radial normalization removes. At the zero jet the retraction separates image families that radial normalizations and continuous classifiers cannot. Networks built on the retraction are universal for invariant features. On MNIST such a network keeps 88.4% accuracy under a...

自动采集于 2026-11-10

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