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.

翻译：PCA可用于旋转不变特征，通过其$p_{ab}=E[(x_i-E[x_a])(x_b-E[x_b])]$协方差矩阵用椭球近似形状来描述形状，并允许利用其幂迹等旋转不变量。然而，真实形状通常复杂得多，因此提出将其扩展到例如$p_{abc}=E[(x_a-E[x_a])(x_b-E[x_b])(x_c-E[x_c])]$的三阶或更高阶张量来描述中心矩，或多项式乘以高斯函数以实现任意高精度的可解码形状描述符及其类似旋转不变量。其实际应用包括旋转不变特征以包含模旋转的形状，例如用于分子形状描述符，或用于2D图像/3D扫描中直至旋转的对象识别（或许也适用于3D场景理解），或作为形状相似性度量，允许对模旋转的物体进行廉价比较，避免代价高昂的旋转优化。