Abstract
A receptive field constitutes a region in the visual field where a visual cell or a visual operator responds to visual stimuli. This paper presents a theory for what types of receptive field profiles can be regarded as natural for an idealized vision system, given a set of structural requirements on the first stages of visual processing that reflect symmetry properties of the surrounding world. These symmetry properties include (i) covariance properties under scale changes, affine image deformations, and Galilean transformations of space–time as occur for real-world image data as well as specific requirements of (ii) temporal causality implying that the future cannot be accessed and (iii) a time-recursive updating mechanism of a limited temporal buffer of the past as is necessary for a genuine real-time system. Fundamental structural requirements are also imposed to ensure (iv) mutual consistency and a proper handling of internal representations at different spatial and temporal scales. It is shown how a set of families of idealized receptive field profiles can be derived by necessity regarding spatial, spatio-chromatic, and spatio-temporal receptive fields in terms of Gaussian kernels, Gaussian derivatives, or closely related operators. Such image filters have been successfully used as a basis for expressing a large number of visual operations in computer vision, regarding feature detection, feature classification, motion estimation, object recognition, spatio-temporal recognition, and shape estimation. Hence, the associated so-called scale-space theory constitutes a both theoretically well-founded and general framework for expressing visual operations. There are very close similarities between receptive field profiles predicted from this scale-space theory and receptive field profiles found by cell recordings in biological vision. Among the family of receptive field profiles derived by necessity from the assumptions, idealized models with very good qualitative agreement are obtained for (i) spatial on-center/off-surround and off-center/on-surround receptive fields in the fovea and the LGN, (ii) simple cells with spatial directional preference in V1, (iii) spatio-chromatic double-opponent neurons in V1, (iv) space–time separable spatio-temporal receptive fields in the LGN and V1, and (v) non-separable space–time tilted receptive fields in V1, all within the same unified theory. In addition, the paper presents a more general framework for relating and interpreting these receptive fields conceptually and possibly predicting new receptive field profiles as well as for pre-wiring covariance under scaling, affine, and Galilean transformations into the representations of visual stimuli. This paper describes the basic structure of the necessity results concerning receptive field profiles regarding the mathematical foundation of the theory and outlines how the proposed theory could be used in further studies and modelling of biological vision. It is also shown how receptive field responses can be interpreted physically, as the superposition of relative variations of surface structure and illumination variations, given a logarithmic brightness scale, and how receptive field measurements will be invariant under multiplicative illumination variations and exposure control mechanisms.
Highlights
When light reaches a visual sensor such as the retina, the information necessary to infer properties about the surrounding world is not contained in the measurement of image intensity at a single point, but from the relationships between intensity values at different points
2.3.3 Summary regarding intensity and illumination variations. This analysis shows that with image intensities parameterized on a logarithmic brightness scale and provided that the smoothing operation Ts has sufficient regularizing properties to make the computation of image derivatives well defined, receptive field responses in terms of spatial and spatio-temporal derivatives have a direct physical interpretation as the superposition of
Galilean covariant temporal derivative concept When considering temporal derivatives of spatio-temporal data computed for an object that moves with image velocity v = (v1, v2)T relative to the observer, it is natural to consider velocity-adapted temporal derivatives ∂talong the direction of motion according to
Summary
When light reaches a visual sensor such as the retina, the information necessary to infer properties about the surrounding world is not contained in the measurement of image intensity at a single point, but from the relationships between intensity values at different points. Despite the image measurements fundamentally being of an indirect nature, in terms of reflected light from external objects subject to unknown or uncontrolled illumination, this result shows how receptive field measurements can be related to inherent physical properties of objects in the environment This result provides a formal justification for using receptive field responses as a basis for visual processes, analogous to the way linear receptive fields in the fovea, LGN and V1 provide the basic input to higher visual areas in biological vision. Due to the formulation of the resulting receptive fields in terms of spatial and spatiotemporal derivatives of convolution kernels, it becomes feasible to analyze how receptive field responses can be related to properties of the environment using mathematical tools from differential geometry and thereby analyzing possibilities as well as constraints for visual perception
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