Commutativity of spatiochromatic covariance matrices in natural image statistics

Yiye Jiang; Jérémie Bigot; Edoardo Provenzi; Yiye Jiang; Jérémie Bigot; Edoardo Provenzi

doi:10.3934/mine.2020016

Mathematics in Engineering

2020, Volume 2, Issue 2: 313-339. doi: 10.3934/mine.2020016

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Research article

Commutativity of spatiochromatic covariance matrices in natural image statistics

Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France

Received: 18 April 2019 Accepted: 23 December 2019 Published: 17 February 2020

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 ^[16] 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.
- statistics of natural images,
- spatiochromatic covariance,
- exponential decay of the covariance,
- Michaelis-Menten transformation,
- joint diagonalization
Citation: Yiye Jiang, Jérémie Bigot, Edoardo Provenzi. Commutativity of spatiochromatic covariance matrices in natural image statistics[J]. Mathematics in Engineering, 2020, 2(2): 313-339. doi: 10.3934/mine.2020016

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Abstract

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 ^[16] 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.

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