Duality Principle for Image Regularization and Perceptual Color Correction Models

Abstract

Abstract. In this paper, we show that the anisotropic nonlocal total variation involved in the image regularization model of Gilboa and Osher [15] as well as in the perceptual color correction model of Bertalmío et al.[4] possesses a dual formulation. We then obtain novel formulations of their solutions, which provide new insights on these models. In particular, we show that the model of Bertalmío et al. can be split into two steps: first, it performs global color constancy, then local contrast enhancement. We also extend these two channel-wise variational models in a vectorial way by extending the anisotropic nonlocal total variation to vector-valued functions.

A Geometric Model of Brightness Perception and Its Application to Color Images Correction

“…We show that the model induced by the optimal covariant derivative yields a local contrast enhancement model that enhances more the dark than the bright areas of the image unless we consider the limit value α = (+∞, +∞, +∞) that makes the model enhance both areas in a similar extent. This result is coherent with the results in sections 4-5, where we show that the connection 1-form encodes Weber's law for α = (0, 0, 0) and Weber-Fechner's law for some α ∈ (R + * ) 3 . This is a desirable property when dealing with under-exposed images.…”

“…The study of the flatness property of the covariant derivatives and the construction of the moving frames in which the connection 1-forms of the flat ones vanish enables to get an insight about the behavior of the model. Indeed, the variational model induced by a flat covariant derivative can be written, up to a moving frame change, as the vectorial extension of the channel-wise model of Bertalmío et al [5] that we proposed in [3]. This allows a much faster implementation of the algorithm as there is no need to compute the parallel transport map.…”

“…In this paper, we extend this approach in the sense that we consider more Lie group representations acting on a three dimensional space and by considering an optimal covariant derivative associated to each group representation. Indeed, besides the group D + * 3 aforementioned, we consider group representations of SO (2) and SO(3) on R 3 , and the corresponding optimal covariant derivatives are the ones constructed in Batard and Sochen [2] and that are image dependent. This point is developed in the sections 4 and 5.…”

“…where G is the orthonormal moving frame in which the connection 1-form vanishes. The variational model in (77) is the Euclidean restriction of the model (31), and that we studied in [3]. Then, the expression of the solutions of the model (77) in the discrete case is…”

“…The case δ/κ ∈ {0, ∞}. For δ/κ = ∞, the connection 1-form (51) vanishes in the standard frame (e 1 , e 2 , e 3 ), meaning that the variational model (31) is nothing but the Euclidean model we introduced in [3], and which is the vectorial extension of the model (29).…”