The influence of natural image transformations on receptive field responses is crucial for modelling visual operations in computer vision and biological vision. In this regard, covariance properties with respect to geometric image transformations in the earliest layers of the visual hierarchy are essential for expressing robust image operations, and for formulating invariant visual operations at higher levels. This paper defines and proves a set of joint covariance properties for spatio-temporal receptive fields in terms of spatio-temporal derivative operators applied to spatio-temporally smoothed image data under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations. Specifically, the derived relations show how the parameters of the receptive fields need to be transformed, in order to match the output from spatio-temporal receptive fields under composed spatio-temporal image transformations. For this purpose, we also fundamentally extend the notion of scale-normalized derivatives to affine-normalized derivatives, that are computed based on spatial smoothing with affine Gaussian kernels, and analyze the covariance properties of the resulting affine-normalized derivatives for the affine group as well as for important subgroups thereof. We conclude with a geometric analysis, showing how the derived joint covariance properties make it possible to relate or match spatio-temporal receptive field responses, when observing, possibly moving, local surface patches from different views, under locally linearized perspective or projective transformations, as well as when observing different instances of spatio-temporal events, that may occur either faster or slower between different views of similar spatio-temporal events.

翻译：自然图像变换对感受野响应的影响对于计算机视觉和生物视觉中视觉操作的建模至关重要。在这方面，视觉层次最早层中关于几何图像变换的协变性质对于表达鲁棒的图像操作以及在更高层次上构建不变视觉操作至关重要。本文定义并证明了一组时空感受野的联合协变性质，这些性质涉及在空间缩放变换、空间仿射变换、伽利略变换和时间缩放变换的组合作用下，应用于时空平滑图像数据的时空导数算子。具体而言，推导出的关系展示了感受野参数需要如何变换，才能匹配在组合时空图像变换下时空感受野的输出。为此，我们还将尺度归一化导数的概念根本性地扩展到仿射归一化导数，这些导数基于使用仿射高斯核进行空间平滑计算，并分析了所得仿射归一化导数对于仿射群及其重要子群的协变性质。最后，我们通过几何分析表明，当从不同视角观察可能运动的局部表面块（在局部线性化的透视或投影变换下），以及当观察时空事件的不同实例（这些实例在相似时空事件的不同视角间可能发生得更快或更慢）时，所推导的联合协变性质如何使得关联或匹配时空感受野响应成为可能。