提出高阶张量描述形状，实现旋转不变的精确匹配。
Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation
- 用三阶及以上中心矩张量替代传统PCA，捕捉复杂形状特征。
- 通过幂次迹等旋转不变量，实现任意精度的形状描述。
- 适合分子结构比对、2D/3D物体识别与快速旋转不变相似性计算。
PCA可用于生成旋转不变特征，通过协方差矩阵 $p_{ab}=E[(x_i-E[x_a])(x_b-E[x_b])]$ 近似形状为椭球体，进而导出如幂次迹等旋转不变量。然而真实形状通常更复杂，本文提出将其扩展至 $p_{abc}=E[(x_a-E[x_a])(x_b-E[x_b])(x_c-E[x_c])]$ 等三阶或更高阶张量，用于描述中心矩，或结合多项式与高斯函数以实现可解码的任意高精度形状描述符及其旋转不变量。该方法可应用于分子形状描述、二维图像/三维扫描中模旋转的物体识别，以及无需昂贵旋转优化即可快速比较物体间相似性的旋转不变度量。
原文摘要 · Abstract (English)
PCA can be used for rotation invariant features, describing a shape with its $p_{ab}=E[(x_i-E[x_a])(x_b-E[x_b])]$ covariance matrix approximating shape by ellipsoid, allowing for rotation invariants like its traces of powers. However, real shapes are usually much more complicated, hence there is proposed its extension to e.g. $p_{abc}=E[(x_a-E[x_a])(x_b-E[x_b])(x_c-E[x_c])]$ order-3 or higher tensors describing central moments, or polynomial times Gaussian allowing decodable shape descriptors of arbitrarily high accuracy, and their analogous rotation invariants. Its practical applications could be rotation-invariant features to include shape modulo rotation e.g. for molecular shape descriptors, or for up to rotation object recognition in 2D images/3D scans maybe also for 3D scene understanding, or shape similarity metric allowing inexpensive comparison of objects modulo rotation avoiding costly optimization over rotations.
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