Normalized cross-correlation is the reference approach to carry out template matching on images. When it is computed in Fourier space, it can handle efficiently template translations but it cannot do so with template rotations. Including rotations requires sampling the whole space of rotations, repeating the computation of the correlation each time. This article develops an alternative mathematical theory to handle efficiently, at the same time, rotations and translations. Our proposal has a reduced computational complexity because it does not require to repeatedly sample the space of rotations. To do so, we integrate the information relative to all rotated versions of the template into a unique symmetric tensor template -which is computed only once per template-. Afterward, we demonstrate that the correlation between the image to be processed with the independent tensor components of the tensorial template contains enough information to recover template instance positions and rotations. Our proposed method has the potential to speed up conventional template matching computations by a factor of several magnitude orders for the case of 3D images.

翻译：摘要：归一化互相关是图像模板匹配的基准方法。该方法在傅里叶空间中计算时，虽能高效处理模板平移，却无法应对模板旋转。引入旋转需对整个旋转空间进行采样并重复计算相关性。本文提出一种可同时高效处理旋转与平移的替代数学理论。本方案无需重复采样旋转空间，从而显著降低计算复杂度。具体而言，我们将模板所有旋转版本的相关信息整合为唯一的对称张量模板（每个模板仅需计算一次），继而证明：待处理图像与该张量模板的独立张量分量之间的相关性，足以恢复模板实例的位置与旋转参数。对于三维图像场景，本方法有望将传统模板匹配的计算速度提升数个数量级。