In this paper, we present a discussion about perceptual-based color correction of digital images in the framework of variational techniques. We propose a novel image functional whose minimization produces a perceptually inspired color enhanced version of the original. The variational formulation permits a more flexible local control of contrast adjustment and attachment to data. We show that a numerical implementation of the gradient descent technique applied to this energy functional coincides with the equation of automatic color enhancement (ACE), a particular perceptual-based model of color enhancement. Moreover, we prove that a numerical approximation of the Euler-Lagrange equation reduces the computational complexity of ACE from theta(N2) to theta(N log N), where N is the total number of pixels in the image.

In order to investigate the local filtering behavior of the Retinex model, we propose a new implementation in which paths are replaced by 2-D pixel sprays, hence the name "random spray Retinex." A peculiar feature of this implementation is the way its parameters can be controlled to perform spatial investigation. The parameters' tuning is accomplished by an unsupervised method based on quantitative measures. This procedure has been validated via user panel tests. Furthermore, the spray approach has faster performances than the path-wise one. Tests and results are presented and discussed.

We present a detailed mathematical analysis of the original Retinex algorithm due to Land and McCann [J. Opt. Soc. Am. 61, 1 (1071)]. To this end, we propose an analytic formula that describes the algorithm behavior. More than one Retinex version (e.g., with and without threshold) is examined. The behavior of Retinex varying the number of paths is predicted, and its recursive iterations are mathematically analyzed using the formula. The mathematical setting presented serves as a common ground for the various Retinex implementations. Its validity is confirmed by the tests on images that we have performed.

Basic phenomenology of human color vision has been widely taken as an inspiration to devise explicit color correction algorithms. The behavior of these models in terms of significative image features (such as, e.g., contrast and dispersion) can be difficult to characterize. To cope with this, we propose to use a variational formulation of color contrast enhancement that is inspired by the basic phenomenology of color perception. In particular, we devise a set of basic requirements to be fulfilled by an energy to be considered as 'perceptually inspired', showing that there is an explicit class of functionals satisfying all of them. We single out three explicit functionals that we consider of basic interest, showing similarities and differences with existing models. The minima of such functionals is computed using a gradient descent approach. We also present a general methodology to reduce the computational cost of the algorithms under analysis from O(N2) to O(N logN), being N the number of pixels of the input image.

We present an interpretation of Land's Retinex theory that we show to be consistent with the original formulation. The proposed model relies on the computation of the expectation value of a suitable random variable weighted with a kernel function, thus the name Kernel-Based Retinex (KBR) for the corresponding algorithm. KBR shares the same intrinsic characteristics of the original Retinex: it can reduce the effect of a color cast and enhance details in lowkey images but, since it can only increase pixel intensities, it is not able to enhance over-exposed pictures. Comparing the analytical structure of KBR with that of a recent variational model of color image enhancement, we are able to perform an analysis of the action of KBR on contrast, showing the need to anti-symmetrize its equation in order to produce a two-sided contrast modification, able to enhance both under and over-exposed pictures. The anti-symmetrized KBR equations show clear correspondences with other existing color correction models, in particular ACE, whose relationship with Retinex has always been difficult to clarify. Finally, from an image processing point of view, we mention that both KBR and its antisymmetric version are free from the chromatic noise due to the use of paths in the original Retinex implementation and that they can be suitably approximated in order to reduce their computational complex-M. Bertalmío · V. Caselles · E. Provenzi ( )

Tone Mapping is the problem of compressing the range of a High-Dynamic Range image so that it can be displayed in a Low-Dynamic Range screen, without losing or introducing novel details: The final image should produce in the observer a sensation as close as possible to the perception produced by the real-world scene. We propose a tone mapping operator with two stages. The first stage is a global method that implements visual adaptation, based on experiments on human perception, in particular we point out the importance of cone saturation. The second stage performs local contrast enhancement, based on a variational model inspired by color vision phenomenology. We evaluate this method with a metric validated by psychophysical experiments and, in terms of this metric, our method compares very well with the state of the art.

Starting from the revolutionary Retinex by Land and McCann, several further perceptually inspired color correction models have been developed with different aims, e.g. reproduction of color sensation, robust features recognition, enhancement of color images. Such models have a differential, spatially-variant and non-linear nature and they can coarsely be distinguished between white-patch (WP) and gray-world (GW) algorithms. In this paper we show that the combination of a pure WP algorithm (Random Spray Retinex (RSR) )and an essentially GW one (Automatic Color Equalization (ACE)) leads to a more robust and better performing model (RACE). The choice of RSR and ACE follows from the recent identification of a unified spatially-variant approach for both algorithms. Mathematically, the originally distinct non-linear and differential mechanisms of RSR and ACE have been fused using the spray technique and local average operations. The investigation of RACE allowed us to put in evidence a common drawback of differential models: corruption of uniform image areas. To overcome this intrinsic defect, we devised a local and global contrast-based and image-driven regulation mechanism that has a general applicability to perceptually inspired color correction algorithms. Tests, comparisons and discussions are presented.

In this paper, we propose a variational formulation for histogram transfer of two or more color images. We study an energy functional composed by three terms: one tends to approach the cumulative histograms of the transformed images, the other two tend to maintain the colors and geometry of the original images. By minimizing this energy, we obtain an algorithm that balances equalization and the conservation of features of the original images. As a result, they evolve while approaching an intermediate histogram between them. This intermediate histogram does not need to be specified in advance, but it is a natural result of the model. Finally, we provide experiments showing that the proposed method compares well with the state of the art.