Because of the variabilities of real-world image structures under the natural image transformations that arise when observing similar objects or spatio-temporal events under different viewing conditions, the receptive field responses computed in the earliest layers of the visual hierarchy may be strongly influenced by such geometric image transformations. One way of handling this variability is by basing the vision system on covariant receptive field families, which expand the receptive field shapes over the degrees of freedom in the image transformations. This paper addresses the problem of deriving relationships between spatial and spatio-temporal receptive field responses obtained for different values of the shape parameters in the resulting multi-parameter families of receptive fields. For this purpose, we derive both (i) infinitesimal relationships, roughly corresponding to a combination of notions from semi-groups and Lie groups, as well as (ii) macroscopic cascade smoothing properties, which describe how receptive field responses at coarser spatial and temporal scales can be computed by applying smaller support incremental filters to the output from corresponding receptive fields at finer spatial and temporal scales, structurally related to the notion of Lie algebras, although with directional preferences. The presented results provide (i) a deeper understanding of the relationships between spatial and spatio-temporal receptive field responses for different values of the filter parameters, which can be used for both (ii) designing more efficient schemes for computing receptive field responses over populations of multi-parameter families of receptive fields, as well as (iii)~formulating idealized theoretical models of the computations of simple cells in biological vision.

翻译：由于真实世界图像结构在自然图像变换下的可变性——当在不同观测条件下观察相似物体或时空事件时，这些变换会产生——视觉层级最早层计算得到的感受野响应可能受到此类几何图像变换的强烈影响。处理这种可变性的一种方法是基于协变感受野族构建视觉系统，这些感受野族在图像变换的自由度上扩展感受野形状。本文旨在推导在所得多参数感受野族中，针对不同形状参数值获得的空域与时空感受野响应之间的关系。为此，我们推导了（i）无穷小关系（大致对应于半群与李群概念的结合），以及（ii）宏观级联平滑特性（描述了如何通过将较小支撑域的增量滤波器应用于更精细时空尺度上对应感受野的输出，来计算更粗糙时空尺度上的感受野响应，其结构与李代数概念相关，但具有方向偏好）。所提出的结果提供了（i）对滤波器参数取不同值时空域与时空感受野响应之间关系的更深入理解，这可用于（ii）设计更高效的计算多参数感受野族总体响应的方案，以及（iii）构建生物视觉中简单细胞计算过程的理想化理论模型。