Commutativity of spatiochromatic covariance matrices in natural image statistics

Yiye Jang , Jérémie Bigot (1) , Edoardo Provenzi (1)

1 IMB - Institut de Mathématiques de Bordeaux

Statistic of natural images is a growing field of research both in vision and image processing. On the vision research side, fine statistical details about object distribution in real-world scenes help understanding the human visual system behavior. On the image processing side, by using the information gathered from statistics of natural scenes, we can obtain reliable priors and insights that can be used in many models. In has been rigorously proven in [15] that, if second order stationarity and commutativity of spatiochromatic covariance matrices hold true for natural scenes, then the codification of spatial and chromatic information by the human visual system can be separated through a tensor product. Spatial features are coded via local and oriented Fourier basis elements, while color features are coded via a triad given by an achromatic channel followed by two color opponent channels. In this paper, we will show that, while stationarity is guaranteed, commutativity is not. However, we shall see that commutativity of spatiochromatic covariance matrices can be approached if the database of images used to model visual scenes is modified accordingly to a suitable transformation that describes the response of retinal photoreceptors to light absorption: the Michaelis-Menten formula. A thorough investigation of the effects of a parameter of this formula will be performed and its influence on commutativity of covariance matrices will be detailed.

Domaines: Statistiques [math.ST]; Traitement du signal et de l'image [eess.SP]

Yiye Jang, Jérémie Bigot, Edoardo Provenzi. Commutativity of spatiochromatic covariance matrices in natural image statistics. Mathematics in Engineering, 2020. ⟨hal-02141503v2⟩