Edoardo Provenzi
email: provenzi.edoardo@gmail.com

Edwin H Land

John J Mccann

Formalizations of the Retinex model and its variants with variational principles and partial differential equations

Keywords: Retinex, partial differential equations, variational principles

introduced the Retinex model as a computational theory of color vision. However, Retinex details were described rather algorithmically than mathematically. Soon after the birth of Retinex, Horn proposed a drastic simplification of the model, which, however, had the merit of being mathematically explicit and rigorous. As a consequence, at least two different interpretations of Retinex appeared in the literature almost at the same time. This generated a lot of confusion about the name Retinex that persists even nowadays. The aim of this paper is to present an overview about the different interpretations and the corresponding mathematical formalizations of Retinex in terms of variational principles and partial differential equations.

Introduction

The most popular paper about the original Retinex formulation is. [START_REF] Land | Lightness and Retinex theory[END_REF] Retinex stands for 'Retina plus Cortex', which refers to the fact that the mechanisms underlying human color vision depend both on the retinal photoreceptors catches and on the cortex interpretation of these signals. The original Retinex is a computational model with the aim of finding a perceptual correlate of reflectance, called 'lightness' by Land, to be tested with psychophysical measurements.

Through a series of groundbreaking experiments, mostly performed with the famous 'Mondrian tableaux', Land and McCann proved that human perception of a surface's color is influenced by the spatial distribution of the surrounding surfaces, see Figure 1. As underlined by McCann in many papers and conference speeches, spatial locality of color perception is the central concept in the whole Retinex theory. Thus, at least in its original form, Retinex is not an algorithm for discarding illumination and recover the intrinsic reflectance of surfaces, as several authors claim in their papers even nowadays, but it is a computational model that uses the local information of the surround to recover color perception. In spite of their innovative and important experimental achievements, neither Land nor McCann 'carved their model into stone' through a rigorous mathematical formulation and this generated confusion, both about the role of the operations involved in the lightness computation, and in the essence of their model.

In this paper, we are not going to discuss Retinex as a vision model (see [START_REF] Mccann | The art and science of HDR imaging[END_REF] for more details about this), but the color enhancement algorithms for image processing that were inspired by it.

In particular, we will show that two major classes of Retinex that can be found in the literature: ratio-reset Retinex and Horn's Retinex. [START_REF] Horn | Determining lightness from an image[END_REF] We will underline how profound is the difference between these two interpretations thanks to variational principles and partial differential equations.

The variational models that will be discussed in this paper are the variational descriptions of existing models, it is thus necessary to introduce, in the following section, the basic formalization of the ratio-reset formula developed in. [START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF] This will help us fix the ideas and the notation about many concepts and notations that will be discussed in the following sections.

It is important to keep in mind that, since the Retinex theory attempts to estimate human color sensation, pre-and post-calibrations are required to map the device responses into human re-sponses. Here, since we will concentrate on image processing purposes, we will not take into account the calibrations procedures.

As a final remark: this paper is about mathematical formulations of Retinex models, their similarities and differences, not about their outputs, which strongly depend on their parameters.

Coherently with this, no Retinex output will be presented in this paper.

Land and McCann's ratio-threshold-reset Retinex model

As previously mentioned, the original Retinex model of Land and McCann 1 is based on the assumption that the HVS operates with three retinal-cortical systems, each processing independently the low, middle and high wavelengths of the visible electromagnetic spectrum. Every independent process forms a separate image determining a quantity that they called lightness and denoted with L. Land and McCann found a computational way to reproduce lightness for their Mondrian tableaux by introducing spatial comparisons among intensities, calculated over paths. The comparison is performed through a multiplicative chain of ratios, subjected to these non-linear operations:

Threshold mechanism: if the ratio does not differ from 1 more than a fixed threshold value, then it is set to be unitary; Reset mechanism: if the cumulated product of ratios overcomes the value 1 in a certain point of the path, then it is forced to 1, so that the computation restarts from it. In this way, this point becomes a local white reference, so that the reset mechanism is responsible for the white-patch behavior of Retinex.

Let us now present the mathematical formalization of the ratio-threshold-reset Retinex computation described in [START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF] as an interpretation of. [START_REF] Land | Lightness and Retinex theory[END_REF] Other possible interpretations exist in the literatures, see e.g. [START_REF] Mccann | The art and science of HDR imaging[END_REF] Given a discrete digital image function with normalized range, I : Ω ⊂ Z 2 → (0, 1], consider a collection of N oriented paths γ = {γ 1 , . . . , γ N } composed by ordered chains of pixels starting in y k and ending in x, k = 1, . . . , N . Let n k be the number of pixels traveled by the path γ k and let t k = 1, . . . , n k be its parameter, i.e.

γ k : {1, . . . , n k } → Ω ⊂ R 2 , γ k (1) = y k and γ k (n k ) = x.
Write, for simplicity, two subsequent pixels of the path as γ k (t k ) = y t k and γ k (t k + 1) = y t k +1 , for t k = 1, . . . , n k -1. Consider, in every fixed chromatic channel c ∈ {R, G, B}, their intensities I(y t k ), I(y t k +1 ) and then compute the ratio R t k = I(y t k +1 )

I(yt k ) with the initial condition R 0 = 1.
With this notation in mind, the value of lightness provided by the ratio-threshold-reset Retinex algorithm for a generic pixel x ∈ Ω, in every fixed chromatic channel c (that we avoid specifying for the same of a clearer notation), is given by:

L ε, γ (x) = 1 N N k=1 n k -1 t k =1 δ k (R t k ) (1)
where δ k : R + → R + , k = 1, . . . , N , are functions defined in this way: δ k (R 0 ) = 1 and, for

t k = 1, . . . , n k -1, δ k (R t k ) =                          R t k if 0 < R t k ≤ 1 -ε 1 if 1 -ε < R t k < 1 + ε R t k if 1 + ε ≤ R t k ≤ 1+ε t k -1 m k =0 δ k (Rm k ) 1 t k -1 m k =0 δ k (Rm k ) if R t k > 1+ε t k -1 m k =0 δ k (Rm k ) (2)
being ε > 0 a fixed threshold. The second option is the mathematical implementation of the threshold mechanism while the fourth implements the reset mechanism (and so the white patch behavior) of the algorithm.

It is useful to write the contribution of the single path γ k to L ε, γ (x) as:

L ε,γ k (x) = n k -1 t k =1 δ k (R t k ), (3)
so that formula (1) reduces simply to the average of these contributions, i.e. L ε, γ

(x) = 1 N N k=1 L ε,γ k (x).
Before passing to the analysis of the limit ε → 0, it is worthwhile specifying two issues. Firstly, an independent processing of the three chromatic channels is a correct assumption only for the very first stages of human vision. In fact, as soon as the visual signal arrives to ganglion cells, the output signals coming from the photoreceptors start to intertwine. [START_REF] Buchsbaum | Trichromacy, opponent colours coding and optimum colour information transmission in the retina[END_REF][START_REF] Ruderman | Statistics of cone responses to natural images: implications for visual coding[END_REF][START_REF] Provenzi | On the second order spatiochromatic structure of natural images[END_REF] Secondly, the ratio-threshold-reset algorithm described above is implemented in the so-called Milano Retinex algorithms 2 to perform color enhancement in digital images through a spatial color processing inspired by the Retinex principles.

The limit behavior ε → 0

The analytical formula to describe the ratio-threshold-reset Retinex algorithm just introduced allowed making predictions about the model. As explained in, [START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF] this can be done if the threshold mechanism is disregarded, or, equivalently, by considering the case ε → 0.

As ε → 0, the functions δ k become much simpler:

δ k (R t k ) =              R t k if 0 < R t k t k -1 m k =0 δ k (R m k ) ≤ 1 1 t k -1 m k =0 δ k (Rm k ) if R t k t k -1 m k =0 δ k (R m k ) > 1 (4)
hence, when ε → 0, δ k behaves either as the identity or the reset function.

In [START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF] it was proven that this implies the following formula:

L 0, γ (x) = 1 N N k=1 I(x) I(y H k ) , (5)
where y H k is the pixel with highest intensity traveled by γ k . From now on, we will refer to formula (5) as describing the 'ratio-reset Retinex algorithm'.

Notice that he presence of paths makes the ratio-reset Retinex a local algorithm, where locality is intrinsically represented by the geometry of paths used. However, when n k → |Ω|, the ratioreset Retinex loses its local properties and reduces, see, [START_REF] Provenzi | Mathematical definition and analysis of the retinex algorithm[END_REF] to the global diagonal von Kries model. [START_REF] Kries | Chromatic adaptation[END_REF] On the other hand, n k and N contribute to the creation of noise when their value is small.

Finally, it is important to underline that, since intensity values are normalized, 0 < I(y H k ) ≤ 1

for every k = 1, . . . , N and then N k=1 1

I(y H k ) ≥ N . It follows that L(x) ≥ I(x)
for every pixel i and this proves that an image filtered with the ratio-reset Retinex is always brighter or equal to the original one. This shows an important limitation of this algorithm: an over-exposed picture can only be worsened by the application of the ratio-reset Retinex used as a color corrector.

From paths to pixel sprays: RSR and related algorithms

The information obtained thanks to the mathematical formulation of Retinex has important consequences on the structure of P x (Ω), the set of paths embedded in the image domain Ω and ending in the point x. After formula (5), on this set it is natural to define the following equivalence relation:

given γ, η ∈ P x (Ω), γ ∼ η ⇔ max y∈γ * {I(y)} = max y∈η * {I(y)} (6)
where γ * and η * are the co-domains of the paths, i.e. the collections of pixels traveled by γ and η, respectively.

Paths belonging to different equivalence classes give different contributions to the lightness computation, while every path in a given equivalence class gives rise to the same value of L 0,γ k (x).

It follows immediately that P x (Ω) contains redundant paths and that the correct set of paths to consider is given by the quotient set P x (Ω)/ ∼, whose elements are the equivalence classes of paths with respect to the equivalence relation defined in (6).

In each equivalence class one can choose a single representative path to compute L 0,γ k (x), in particular, the most efficient one is the two-points path whose co-domain is simply given by {y H k , x}. Moreover, by a mathematical point of view, paths are topological manifolds of dimension 1 embedded in the image, which is a topological manifold of dimension 2, so paths do not really scan local neighborhoods of a pixel, rather particular directions in these neighborhoods. This directional extraction of information can lead to halos or artifacts in the filtered image, see e.g. [START_REF] Frankle | Method and apparatus for lightness imaging[END_REF] These considerations led the authors of 10 to consider 2-dimensional objects such as areas instead of 1-dimensional paths to analyze image locality for an efficient color correction. Roughly speaking, their idea is to implement spatial locality by selecting a fraction of pixels from these areas with a density sample that changes according to a given function of their distance with respect to the target pixel x. Each function generates a different kind of pixel selection around x, leading to different kind of 'sprays', as the one depicted in Figure 2, each of which shows different local filtering properties. The new implementation of the ratio-reset Retinex that follows this idea is called RSR for 'Random Sprays Retinex'.

In RSR the role of a path γ k traveling n k pixels and ending in the target x is played by S k (x), a spray with n k pixels centered in x. Actually, once the number of points per spray is chosen, there is no need to vary it with k, hence, from now on, we will write n instead of n k to denote the number of pixels per spray. The ratio-reset operation along a path is substituted by the search of the pixel with highest intensity in the whole spray. The functional expression of formula ( 5) to compute the lightness remains exactly the same in both algorithms, so the ratio-reset Retinex and RSR share the same intrinsic properties.

In [START_REF] Banić | Light random sprays retinex: Exploiting the noisy illumination estimation[END_REF][START_REF] Banić | Smart light random memory sprays retinex: a fast retinex implementation for high-quality brightness adjustment and color correction[END_REF] two techniques have been proposed to reduce noise generation also decreasing the computational time of RSR.

Let us start by describing the strategy described in [START_REF] Banić | Light random sprays retinex: Exploiting the noisy illumination estimation[END_REF]

C * k 1 ,k 2 (x) = (C k 1 (x) * k 2 )(x).
The output image O of LRSR is calculated via the formula:

O(x) = I(x) C * k 1 ,k 2 (x) ∀x ∈ Ω, (7)
the size of kernels k 1 and k 2 used in [START_REF] Banić | Light random sprays retinex: Exploiting the noisy illumination estimation[END_REF] 25 × 25.

In, [START_REF] Banić | Smart light random memory sprays retinex: a fast retinex implementation for high-quality brightness adjustment and color correction[END_REF] the same authors attacked the problem of computational complexity reduction with a technique called 'Smart Light Random Memory Sprays Retinex' (SLRMSR). The basic concept behind SLRMSR is that of 'spray memory', which consists in creating a single spray that will be gradually modified while browsing the image by modifying just one pixel of S(x) each time we move from one pixel to another, by randomly selecting a pixel that lies in the neighborhood of the last pixel. By iterating this procedure, the original spray is gradually and smoothly modified.

Thanks to this idea, the computational complexity passes from O(nN |Ω|) to O(n|Ω|) because now one spray is used. In, [START_REF] Chauvin | SLMRACE: a noise-free race implementation with reduced computational time[END_REF] these techniques have been extended from RSR to RACE.

In [START_REF] Provenzi | Spatially variant white patch and gray world method for color image enhancement driven by local contrast[END_REF] the spray technique was used to fuse RSR with ACE, [START_REF] Rizzi | A new algorithm for unsupervised global and local color correction[END_REF] another perceptually-inspired color correction algorithm that makes use of the gray-world hypothesis. [START_REF] Buchsbaum | A spatial processor model for object colour perception[END_REF] The hybrid algorithm is called RACE and it is able to color correct both under and over exposed images.

A more recent proposal to fuse white-patch (WP) and gray-world (GW) features in a single algorithm is that presented in [START_REF] Kolås | Spatio-temporal retinex-inspired envelope with stochastic sampling: A framework for spatial color algorithms[END_REF] and called STRESS for Spatio-Temporal Retinex-like Envelope with Stochastic Sampling. As Retinex, STRESS computes, for each pixel, the local white reference, but also the black reference in each chromatic channel. This is done through calculating the maximum and minimum envelope functions, denoted as E max (x) and E min (x), respectively.

The computational steps needed to get the envelope functions are the following: fix a pixel x ∈ Ω and N random sprays 10 S k (x), k = 1, . . . , N , centered in x, then compute M k = max

x∈S k (x) I(x), m k = min x∈S k (x)
I(x) and the following objects:

r k (x) = M k (x) -m k (x), v k (x) =          1 2 if M k (x) = m k (x) I(x)-m k (x) M k (x)-m k (x)
otherwise, thus, essentially, the value of v k (x) is either the middle gray if x lies in a homogeneous area, or it is the linearly stretched value of I(x) with respect to the interval [m k (x), M k (x)]. By denoting with r(x) and v(x) the average values of r k (x) and v k (x) over the N sprays, i.e.

r(x) = 1 N N k=1 r k (x), v(x) = 1 N N k=1 v k (x),
the authors can finally define the two envelope functions E min (x) as follows:

E min (x) = I(x) -v(x)r(x), E max (x) = I(x) + (1 -v(x)r(x)) = E min + r(x). (8)
The final output of STRESS is the stretched value of I(x) on the interval [E min , E max ], namely:

STRESS(I(x)) = I(x) -E min E max -E min . (9)
STRESS, as RACE, is capable of handling both under and over exposed images, but it is affected by the same noise problems of RSR. In section 3 we will see a variational extension of the technique used to compute the envelope functions based on the total variation. This will allow avoiding the noise problems related to the random spray technique.

Finally, let us mention some related works. In, [START_REF] Gianini | Qbrix: a quantile-based approach to retinex[END_REF] the RSR sampling technique has been studied from a probabilistic point of view, resulting in the algorithm QBRIX, in, [START_REF] Gianini | A retinex model based on absorbing markov chains[END_REF] further comparisons among Retinex models are discussed, in 20 a noise-free, exact mapping of RSR into a probabilistic framework is presented, in [START_REF] Gianini | A fuzzy set approach to retinex spray sampling[END_REF] fuzzy variants of RSR are taken into account and, finally, in [START_REF] Lecca | Grass: A gradient-based random sampling scheme for milano retinex[END_REF] a random spatial sampling method halfway through path-based and 2D-based is discussed.

A variational framework for the ratio-reset Retinex

The similarities between the ACE formula [START_REF] Rizzi | A new algorithm for unsupervised global and local color correction[END_REF] and the gradient descent equations for histogram equalization obtained in, [START_REF] Sapiro | Histogram modification via differential equations[END_REF] led to the discovery of a variational interpretation of ACE in the paper. [START_REF] Bertalmío | Perceptual color correction through variational techniques[END_REF] The framework were further extended in [START_REF] Palma-Amestoy | A perceptually inspired variational framework for color enhancement[END_REF] and, finally, in [START_REF] Bertalmío | Issues about the retinex theory and contrast enhancement[END_REF] a variational framework for (an anti-symmetric version of the) ratio-reset Retinex has been discussed. In order to understand how this is possible, let us come back to the lightness formula (5). An algorithmic analysis of these models has been provided in. [START_REF] Ferradans | An algorithmic analysis of variational models for perceptual local contrast enhancement[END_REF] Interestingly, this variational setting can be also extended to consider cognitive phenomena, as it has been shown in. [START_REF] Gronchi | A variational model for context-driven effects in perception and cognition[END_REF] Land proposed a further Retinex mechanism, the scaling, [START_REF] Land | The Retinex theory of color vision[END_REF] implemented via a strictly increasing compressive (concave) function f : (0, 1] → (0, 1] such that f (r) ≥ r for all r ∈ (0, 1] applied to the ratio r = I(x) I(y H k ) , so that the Retinex lightness formula becomes:

L 0, γ,f (x) = 1 N N k=1 f I(x) I(y H k ) . ( 10
)
The reset mechanism of Retinex and the scaling operation can be merged: in fact, we can extend f to (0, +∞) preserving its continuity by defining

f (r) =          f (r) if r ∈ (0, 1] 1 if r ∈ [1, +∞).
It is clear that applying this new scaling function f to the ratios I(x)/I(y), with x fixed and y that varies in Ω, jointly implements the scaling and the reset mechanism.

Now we have all the elements to introduce the continuous version of the Retinex algorithm presented in [START_REF] Bertalmío | Issues about the retinex theory and contrast enhancement[END_REF] under the name 'Kernel-Based Retinex', or KBR for short. Given x ∈ Ω, let Y w,x be the random variable modeling the selection of a pixel in the neighborhood of x according to the density w(x, y).

The output L KBR w (x) of the KBR algorithm at the pixel x is defined as the conditional expectation of f I(x) I(Yw,x) with respect to the distribution w of pixels around x, i.e.

L KBR w (x) = E Yw,x f I(x) I(Y w,x ) . ( 11
)
This formula is used independently for each color channel and can be written more explicitly as

L KBR w (x) = {y∈Ω:I(y)≥I(x)} w(x, y) f I(x) I(y) + {y∈Ω:I(y)<I(x)} w(x, y). ( 12
)
All the basic properties of the ratio-reset Retinex are faithfully implemented in (12): KBR is founded on the propagation of a two-pixel ratio comparison between the fixed target x and the generic pixel y that runs across the image; these comparisons are then subjected to the reset and scaling performed by f and, finally, locally averaged with weight w, in order to produce the value of L KBR w (x).

To study the action of KBR of pixel intensities, it is useful to rewrite (12) introducing the functions

sign + (ξ) :=                1 if ξ > 0, 1 2 if ξ = 0, 0 if ξ < 0, sign -(ξ) = 1 -sign + (ξ),
so that eq. ( 12) can be re-written as

L KBR w (x) =
Thanks to eq. ( 13) we can verify that KBR always increases brightness as the ratio-reset Retinex implementation. In fact, since f (r) ≥ r for all r ∈ (0, 1], then f I(x) I(y) ≥ I(x) I(y) ≥ I(x), so

L KBR w (x) ≥
where aKBR stands for anti-symmetrized KBR.

In [START_REF] Bertalmío | Issues about the retinex theory and contrast enhancement[END_REF] it was proven that the right-hand side of the previous equation can be interpreted as the minimization of the energy functional given by:

C f w (I) =
Minimizing C f w (I) corresponds to maximizing the contrast in a local (due to the presence of the weight w) and non linear way (due to the ratio and to the presence of f ). In fact, since the functional C f w (I) is defined via a quotient, its argmin will balance the minimization of the numerator and maximization of the denominator, thus the minimal and the maximal intensity will be spread apart and this will generate a contrast enhancement. This explained in a quantitative and qualitative way how and why the somewhat involved ratio-reset mechanism of Retinex allows for a unilateral contrast enhancement, always directed towards the highest intensity.

KBR, ACE, RACE and STRESS corrected this unilateral behavior. In 30 the spatially-based variational framework was translated into a wavelet-based setting.

4 Retinex: a 'melody' that everyone plays differently

In image processing it is hard to find a model whose name has been interpreted in so many different ways as 'Retinex'. In this section, we present a synthetic description of the evolution of the Retinex interpretation.

Path-wise Retinex share a local WP nature and mostly differ from each other by the path geometry used to explore spatial locality: Land and McCann used piecewise linear paths in. [START_REF] Land | Lightness and Retinex theory[END_REF] In [START_REF] Cooper | Analysis and extensions of the Frankle-McCann Retinex algorithm[END_REF][START_REF] Marini | A computational approach to color adaptation effects[END_REF] and 33 those paths were substituted by double spirals, Brownian paths and traces of a specialized swarm of termites, respectively.

Center/surround Retinex are local GW algorithms originated from, [START_REF] Land | An alternative technique for the computation of the designator in the retinex theory of color vision[END_REF] where Land noticed that he could reproduce Mach bands originated by a spinning white square on a black background by using a different Retinex formulation. Precisely, for every image point, the intensity of the center

x is replaced by the ratio between I(x) and the average value of the surround, sampled with a density that decays as the inverse of the square distance from the center. Writing with L CS this 'center/surround lightness', we have: L CS (x) = I(x)/ < {I(y), y ∈ Surround} >, where < • > represents the average operator. Comparing this last formula with (5), it can be seen that there is a fundamental difference between this formulation and the ratio-reset one: there the ratio is performed over the pixel with highest intensity, while in this formulation it is implemented over the mean value of the surround. In practice, this last formulation can be seen as a gray-world method to remove the illuminant component of the image. [START_REF] Buchsbaum | A spatial processor model for object colour perception[END_REF] In 1997, Johbson, Rahman and Woodell [START_REF] Jobson | Properties and performance of a center/surround Retinex[END_REF] re-elaborated Land's idea presented in: [START_REF] Land | An alternative technique for the computation of the designator in the retinex theory of color vision[END_REF] they worked with logarithmic data, approximating the average of the surround by convolving the image func-tion I with a normalized kernel function F , usually a Gaussian. If we use again, for simplicity, the symbol L CS , we can write this model as follows: L CS (x) = log(I(x)) -log((F * I)(x)), ∀x ∈ Ω.

Multilevel Retinex algorithms were pioneered by Frankle and McCann in [START_REF] Frankle | Method and apparatus for lightness imaging[END_REF] and further refined in. [START_REF] Funt | Retinex in MATLAB[END_REF] In these works a multilevel version of the ratio-reset local WP Retinex is presented, the authors abandon paths and consider a computation that takes into account all pixels. The input image is progressively sub-sampled averaging a number of pixel that grows as increasing powers of 2. On each sub-sample level a ratio-reset computation (without threshold) is operated a certain number of times, from the coarser sub-sample level to the finest one. Because of the sub-sampling, as we go far from the target pixel, we do not consider actual pixel values, but average values of macroareas of increasing size. A rigorous mathematical formulation of these multilevel algorithms is still lacking. If we apply a threshold operator δ T defined as follows:

δ T (s) =          s if |s| > T 0
∆(log S c (x)) = δ T (∆(log I c (x))), (18)
whose solution allows to recover the logarithmic reflectance log S c (x). It is clear that Horn's method is based on quite restrictive hypotheses: smoothness of illumination (violated by scenes with deep shadows, for instance) and a Mondrian-like world (violated each time edges are not sharp).

Mathematical formalizations of Horn's interpretation

Besides the variational framework described in section 3, in the literature there exist alternative variational models of Retinex-like algorithms and also formalizations based on partial differential equations (PDE). The aim of this section is not to give an exhaustive list, rather to discuss the main features of the most famous alternative mathematical formalizations of Retinex-like algorithms present in the literature.

The first authors to embed a Retinex-like algorithm in a variational framework were Kimmel and colleagues in. [START_REF] Kimmel | A variational framework for retinex[END_REF] They did not considered the ratio-threshold-reset Retinex, but Horn's inter- Avoiding the subscript c, the functional proposed in, [START_REF] Kimmel | A variational framework for retinex[END_REF] with the notations of this paper, can be expressed as follows:

E α,β (log L) = x∈Ω [|∇ log L(x)| 2 + α(log L(x) -log I(x)) 2 + β|∇(log L(x) -log I(x))| 2 ] (19)
with the constraints log L(x) ≥ log I(x), because the reflectance S(x) is always between 0 and 1, and the boundary condition ∇ log L, n = 0 on ∂Ω, i.e. log L orthogonal to the normal n to the boundary ∂Ω of Ω.

The first term of the functional forces spatial smoothness on the illumination L. The authors chose that particular analytical form because the Euler-Lagrange equation associated to

x∈Ω |∇ log L(x)| 2 is the Laplace PDE ∆ log L = 0, whose steepest descent solution is equivalent to a Gaussian smoothing. The second penalty term forces a proximity between log L and log I, so that their difference log S, the logarithmic reflectance, tends to 0, i.e. the reflectance R tends to 1, or white. The authors declare that the principal objective of this term is to regularize the problem, so that it is better conditioned in view of a numerical solution and they set the constant α to a very small value not to force too much log L towards log I. The third term represents a Bayesian penalty, which forces reflectance gradients to be smooth. The authors declared to have introduced it to force R to be visually pleasing, without abrupt variations.

         -∆L(x) = F (x) x ∈ Ω ∂L(x) ∂ n = 0 x ∈ ∂Ω,
where F is a suitable scalar field, see [START_REF] Morel | A pde formalization of retinex theory[END_REF] page 2830.

Let us now consider the algorithm STRESS. We recall that the basic information needed by STRESS is given by the two envelope functions E min and E max which, in the original formulation, are computed through the same random spray technique of RSR. [START_REF] Provenzi | Random spray retinex: A new retinex implementa-tion to investigate the local properties of the model[END_REF] To avoid the typical noise problems related to this technique, in, [START_REF] Simone | Spatio-temporal retinex-like envelope with total variation[END_REF] the authors proposed to compute the envelope functions via the minimization of a functional based on the total variation, instead of using the random spray technique. For this reason the corresponding algorithm is called STRETV and corresponds to the minimization of the following functional for E (in this case E denotes the envelope and not the energy functional):

x∈Ω |∇E(x)| + λ 2 |E(x) -I(x)| 2 (20)
subjected to E(x) ≥ I(x) to compute E max and to E(x) ≤ I(x) for E min .

The minimization of the first (total variation) term, assures the spatial smoothness of the envelope functions, the second term is a fidelity term used not to depart too much from the original image values. The authors declare that the coefficient λ must be 1 for good results. The authors do not specify if they consider a spatial kernel to localize their computation or not.

The last variational formalization that we discuss here is that presented in [START_REF] Lecca | Energy-driven path search for termite retinex[END_REF] relative to the termite Retinex. [START_REF] Simone | Termite retinex: a new implementation based on a colony of intelligent agents[END_REF][START_REF] Simone | On edge-aware path-based color spatial sampling for retinex: from termite retinex to light energy-driven termite retinex[END_REF] Here an energy functional is taken into account to determine the geometry of the paths used by Retinex. Fixed a pixel x ∈ Ω, the authors search for the path γ : [0, 1] → Ω, γ(0) = x, the minimizes the energy functional defined by:

E(γ) = 1 0 1 1 + (D 2 -x -γ(s) 2 ) ∇I(γ(s)) 2 + θ(γ(s)) ds, ( 21
)
where D is the diagonal of Ω and 1 is introduced to avoid singularities in the case the denominator is 0. The paths that minimize E(γ) are those which balance the fact to remain as close to x as possible and, simultaneously, to explore image areas with high values of the gradient. Both features maximize the denominator of the first term. If x lies in an area with a high density of edges, γ will not go too far from x, instead, if x lies in a rather homogeneous area, γ will be forced to explore the image points far away from x to find the important edge information. θ(γ(s)), the so-called 'poison term', is set to zero at the beginning, and it increases each time a path has been traveled, to prevent from exploring the same image area all the time. Once a set of N path has been selected, the intensity I(x) of the pixel x in each separate chromatic channel is modified with the Retinex formula (5).

Conclusions

In the past fifteen years, variational methods have been used to formalize color correction algorithms. This permitted to point out similarities and differences among several models that were difficult to detect just looking at their direct equations. In this paper we have described, in a selfcontained way, both the direct and the variational versions of several color enhancement algorithms inspired by the seminal Retinex theory of color vision. A particular emphasis has been put in highlighting the very different variational formulations of the original Retinex of Land and McCann and those referring to Horn's interpretation, which are often misleadingly mixed in the literature.

When the variational formulation of the ratio-reset Retinex is compared with Horn's a clear difference is underlined, both in the functional terms and in the argument of the variational energies. A more complete treatise about this subject can be found in. [START_REF] Provenzi | Computational Color Science: Variational Retinex-like Methods[END_REF]

Fig 1

 1 Fig 1 Digital (personal) reproduction of a typical Mondrian tableau used by Land and McCann for their color matching experiments.

Fig 2

 2 Fig 2 An example of (normalized) random spray. Normalization refers to the fact that the horizontal and vertical axes have been normalized between 0 and 1.

  and called 'Light Random Sprays Retinex' (LRSR) devised to avoid noise formation when a small value of n and/or N is used. Consider an arbitrary input image I and apply RSR to it, obtaining the image L, the ratio C = I L is called intensity change image. In LRSR, the noise is reduced through a convolution with a kernel function k. This can be done after the computation of C, obtaining C k = (C * k)(x), ∀x ∈ Ω, or before it, i.e. applying the blurring on I and L, obtaining C k (x) = (I * k)(x) (L * k)(x) , ∀x ∈ Ω. By combining the two approaches, i.e filtering before and after calculation with two (possibly identical) kernels k 1 and k 2 , respectively, one can define the new intensity change matrix:

  y∈Ω w(x, y) f I(x) I(y) sign + (I(y) -I(x)) + y∈Ω w(x, y) sign -(I(y) -I(x)).

  y∈Ω w(x, y) I(x) sign + (I(y) -I(x)) + y∈Ω w(x, y) sign -(I(y) -I(x))(14)moreover, being I(x) ≤ 1, we can writeL KBR w (x) ≥ y∈Ω w(x, y) I(x) sign + (I(y) -I(x)) + y∈Ω w(x, y) I(x) sign -(I(y) -I(x)) = I(x) y∈Ω w(x, y) sign + (I(y) -I(x)) + sign -(I(y) -I(x)) = I(x) y∈Ω w(x, y) = I(x),(15)having used the fact that the kernel is normalized. As in the ratio-reset formulation, this property implies that over-exposed pictures could not be enhanced with Retinex unless we use a post-processing stage and that further iterations of Retinex keep on increasing the intensity until a white image is reached. This equation of KBR does not correspond to the minimization of an energy functional. However, let us consider the sum of the function f I(x) I(y) sign + (I(y) -I(x)) and of its the antisymmetrized version on the region {x ∈ Ω : I(y) ≤ I(x)}, i.e. sign + (I(y) -I(x)) -y∈Ω w(x, y) f I(y) I(x) sign -(I(y) -I(x))

  , y)f min(I(x), I(y)) max(I(x), I(y)).

  Based on this idea, Marini, Rizzi and De Carli[START_REF] Marini | A computational approach to color adaptation effects[END_REF] constructed a local WP multilevel version of Brownian path Retinex that reduced the amount of noise in the output images. A different multilevel proposal has been pointed out by Johbson, Rahman and Woodell in: 37 they introduced a certain number S of scales where performing the convolutions with normalized Gaussian functions F s , s = 1, . . . , S. Each scale is associated to a suitable weight w s , which gives more importance to finer scales than to coarser ones. Finally there are WP Retinex versions based on solving a Poisson equation. They rely on a work of Horn, 3 in which he remarkably pointed out, for the first time, the need for a spatially isotropic two-dimensional version of Retinex. Horn considered, as Land, only Mondrian tableaux illuminated by a smoothly varying light. However, differently from Land, he explicitly tackled the ill-posed problem of inverting the equation I c (x) = S c (x)L c (x), c ∈ {R, G, B}, with respect to S c (x), the reflectance of the point x, knowing only the image intensity I c (x) and not the illumination L c (x). If we pass to logarithmic values, i.e. log I c (x) = log S c (x) + log L c or, equivalently, log S c (x) = log I c (x) -log L c and we apply a differential operator D to both sides, then D(log L c (x)) will be small but finite everywhere, while D(log S c (x)) will be different from zero only if x is close to sharp edges.

  elsewhere, for all s ∈ R and if the threshold T > 0 is small enough, then we obtain D(log S c (x)) = δ T (D(log I c (x))). Horn insisted on the choice of the Laplacian for D instead of the gradient, arguing that first order derivatives are one-dimensional, while the second order derivatives involved in the Laplacian are isotropic and thus more suited for the topology of an image. By substituting D with the Laplacian operator ∆, the last formula becomes a Poisson equation:

  pretation. In fact, they started from the logarithmic equation log I c (x) = log S c (x) + log L c (x), c ∈ {R, G, B} and tried to solve it with respect to log L c (x) by imposing the hypothesis of smoothness on the illuminant part of the logarithmic image. Once obtained an estimation of the illumination, they could infer the reflectance information S c (x). This one then undergoes suitable transformations and gives an illuminant-invariant version of the original image.It is important to underline a fundamental difference between this variational technique and the one presented in the previous sections: here contrast enhancement of the original image log I c (x) is obtained by decreasing the contrast of the illuminant image log L c (x). In fact, log I c (x) is measured by the camera and so it is a fixed data, log L c (x) is estimated by using a smoothness prior, thus the estimated reflectance log S c (x) = log I c (x) -log L c (x), or S c (x) = I c (x)/L c (x) is forced to have a stronger contrast than the original image data. Instead, the variational principles previously discussed act directly on the contrast of the original image, without taking into account the separation between reflectance and illuminant and related approximations and priors.

  Morel, Petro and Sbert[START_REF] Morel | A pde formalization of retinex theory[END_REF] analyzed the Retinex model[START_REF] Land | The retinex[END_REF] without the reset mechanism. They showed that, if the Retinex paths are interpreted as symmetric random walks, then Retinex is equivalent to the following Neumann problem for a Poisson equation:

List of Figures

An example of (normalized) random spray. Normalization refers to the fact that the horizontal and vertical axes have been normalized between 0 and 1.


Lucien Oulahbib
email: lucien.oulahbib@free.fr

HOW THE LEGACY OF LENINISM STILL STRUCTURES THE INTELLECTUAL DEBATE IN FRANCE

Keywords: Lenin, Althusser, Bataille, Blanchot, Derrida, Foucault, Deleuze, Nietzsche

It should be interesting to study here the moment when Louis Althusser, just before the "sixty-eight" period, poses the doctrinal return to Lenin as a precondition towards "theoretical anti-humanism". This concept opens the way to ideas such as the "death of Man", the ideological underpinning of transhumanism, borderless, and transgender movements. It is necessary, therefore, to draw the theoretical path of this return, as happened to the detriment of this other model of theoretical explanation of history's grammar, such as "social interaction" (which thinks the morphology of complex societies as a dynamic movement dialectically generated by the permanence of the so-called "social conflict"), and that it is possible institutional resolution, at least in a democratic regime.

The French filiation is very different, even opposed in some ways to the German one, operating from Adorno to Honneth via Habermas. These critical and interactionist positions are different from the hypercritical negativist ones, like those belonging to the first. The French concept of "domination" 4 , for instance, is perceived by Honneth and Habermas as "behaviorist" 5 style in contrast with "social interaction" 6 (as the root of "social conflict" in that it establishes in permanence new practices favourable to the greatest number of people, since these one fight for this and not just stays in a passive way, as suggested by the French direction. According to the German filiation, the social struggle is reminiscent -in addition to the question of the permanence of structural problems in order to build a society, as Leo Strauss states (Strauss 1988, 39) -the Simmelian definition of the permanent conflict as a vector of all social forms; without forgotten that of Machiavelli's (conceived in the same line as Aristotle when it comes to speak about the very notion of politeia or belongings), when he states in his Speeches on the first decade of Titus-Livius that "the disunion of the Senate and the people has made the Roman Republic powerful and free" (Machiavelli 1980, 44).

The althuserian filiation to Lenin can be seen as a return to the frame genuine of 17 October (Althusser,Thesis 7), observed already in 1963 within the PCF (French Communists Party) elite, for example in the French theorist journal Pensée, where Roger Garaudy is accused by Althusser as fomenting a "humanist deviation from Marxism" 7 . This is also established in the context opened by the Khrushchev Report (1956), exclusively charging Stalin, as well as the creation of the Berlin Wall (1961) and the beginning of the so-called Maoist cultural revolution (1966), without to forget the strengthening of Castroism and Sovietism (or "scientific socialism") in Africa (in Algeria, for example, with Ben Bella and Boumediene) and finally the Eurocommunism of the PCI (the so call aggiornamento, in 1972). A whole environment, which Garaudy also relates it in a note to the PCF's PO, in 1965, stating that Althusser and his filiation refuse any "historical compromise" embodied precisely in the "aggiornamento" of Eurocommunism (also affecting the PCE (Spanish), animated by Semprun and criticized by Garaudy, in his 1965's Note).

Lenin was "a proletarian leader", Althusser said (Althusser 1982, 15), confided this way during the meeting with the Executive Committee of the Third International (June 19, 1920), according to Frossard's report:

There are profound differences between us on the conception of proletarian dictatorship. It is the most advanced fraction of the proletariat that becomes the state itself, against the bourgeoisie and against the least advanced part of the working class, including the reformists whom we treat as bourgeois ». (Verdier 1981, 36-37) This debate is a permanent one. It is, for example, one of the keys to understand the split of 1920 within the French Section of the Workers International (SFIO), between the Blum-style legitimists and the pro-Bolshevik followers of the acceptance of the "21 conditions" dictated by the so-called "Third International", in the sense that it is not a question of opposing the notion of "dictatorship of the proletariat", as one might think, but of opposing the very idea of taking power. Is it a question of perfecting a morphological continuity -as defined by historical materialism, namely that the bourgeois stage should be over-passed, being unable in itself to overcome the objective contradictions of class clashes? Or is it a question of just taking power in order to replace just like that the "bourgeois class" (as Claude Lefort explains so well in Complications: Communism and the Dilemmas of Democracy, 1998 8 ), by the so-called "proletarian class" (the Lenin "professional revolutionaries", the "metaphysical race" in Nietzschean way)? But for what purpose, if the structural conditions are not reached to do so (as the Bolsheviks have well seen), and force the return to the market via the NEP, a return that Lenin slowly admitted to having to prolong? This is what Leon Blum deduced from it, as Robert Verdier reports in his book on the Congress of Tours 9 .

The problem of "replacement" remains relevant today for the up to date adepts of the French filiation, who are seeking to constitute a kind of "metaphysical race". The notion of "race" coupled with that of "strength" is fundamental to Nietzsche (on which Foucault relies in the exergue). He writes this way: (...) our desire, our very will of knowledge, is a symptom of monstrous decadence... We aspire, on the contrary, to what strong races, strong natures want -understanding is the end of something (…)". (Nietzsche 1977, 167) The notion of "race" next to that of "strong" must then be read in a political way. Heidegger sees also its metaphysical side:

Nor is the Will of Power biologically conceived, whereas it is much more ontologically conceived, the Nietzschean notion of race has no biological significance, but metaphysical significance. (Ibidem,247) And what is it to become an element of the "strong race", in the metaphysical and political sense, concrete, practically, if not a god human becoming a "professional revolutionary" or the passage from "theory to practice", as for Althusser when he tells, in his Memoirs, how he "governed" Ulm?

Hence the fact that it is not surprising to observe an Althusser reconnecting with Lenin (who knows his way around) and thus with the idea of the a priori separation between "revolutionaries" and "reactionaries", which Althusser summarizes in his testamentary book (The Future Lasts Long...) when he analyses the function of the "Red Guards":

The Marxist-Leninist Cahiers 10 , after a difficult start, were selling very well. I had given them, from the first issue, dedicated to the Cultural Revolution, which had just broken out, an unsigned article (whose authenticity I recognize here, after Rancière,) where I implemented a simple and false theory based on the principle: there are three forms of class struggle, the economic, the political and the ideological. They, therefore, require three separate organizations to conduct it. We know two of them: the union and the Party. The Chinese have just invented the third one: The Red Guards. It was a little simple, but it was pleasing (…).

The note published by Roger Garaudy (supra) in June 1965 on Althusser thus sheds light on the latter's strict affiliation with Lenin (Thesis 7) in the name of a "scientific" approach distinct from that advocated by the first, accused -according to him -of "humanist deviation from Marxism". In this note, Garaudy observes that Althusser's relationship with "science" is very rigid and simplistic, referring to various discussions with Italians and Chinese. In fact, the "science" of which Althusser speaks is only framed by the Leninist-Marxist standard, as Garaudy indicates in his note.

It turns out that in the 1960s and 1970s, to get out of phenomenology (as Foucault points out 11 ) was somehow strategic, especially in the midst of the Maoist cultural revolution which, to resume, claimed to be not only Lenin who died too soon, but also the purest and hard Leninist traces (following Marx and Engels) in the entire history of Western thought; a "break" between "idealists and materialists" (whereas they are only "moments" of the conceptual movement, as Hegel has repeatedly indicated). Why indeed Althusser relies this way on Lenin if not to find a theoretical model capable of selfgeneration, emancipating himself from the relationship both with reality and with the inter-subjectivity of scientists. Thus, Althusser relies in his lecture both on Lenin's Materialism and Empirical Criticism, as has been said on Dietzgen also, who keeps to distance "the philosophers" treated as "graduate minions".

However (let's start with Derrida), it is precisely this "margin" (as in the title of Derrida's book) that holds the lines, in the sense that it allows the latter to rely on Althusser (an Althusser who considers Derrida a "giant" (Foucault 1994: 170, 174). This one indeed followed not only Althusser's return to Lenin, which practised philosophy politically (Althusser 1982: 18, 19) but tried, with Tel Quel's team (Hourmant 1996: 112-128) (including the Maoists Philippe Sollers, Julia Kristeva, Alain Badiou, etc.) to go even further than Lenin. This is what he states in Positions (Derrida 1972: 86-89). Derrida considers that absolute negativity must go so far as to deny this need for accountability. He even sees this lack of negativity in Lenin (Ibided, 86). Derrida, therefore, claims not only to follow Lenin's "strategy" but to amplify it, by "reworking" the "rules". In what sense? That of the targeted negation for itself, which would extend to the whole of reality and to the very depths of the intimacy of language, even in its intimate and secret traces:

So the destruction of the discourse is not a simple neutralization of erasure. It multiplies words, precipitates them against each other, engulfs them also in an endless and bottomless substitution whose only rule is the sovereign affirmation of the meaningless game. Not the reserve or the withdrawal, the infinite murmur of a white word erasing the traces of the classical discourse but a kind of potlatch of signs, burning, consuming, wasting words in the gay affirmation of death: a sacrifice and a challenge. Note 1: Play is nothing but an open and unreserved challenge to what opposes play" (Note on the margins of this unpublished Theory of Religion that Bataille planned to call "Dying with laughter and laughing with death. (Derrida 1967, 403) This "position" is strategic. Because, at the same time, it is destroyed, it makes possible to be replaced by one which destroys it (as Lenin has shown); for instance, in the same movement, Derrida "deconstructs" but escapes the ruins of this absolute negativity, and becomes, at the same time, the positive reference framework for any sketch of negativity while saying the opposite, of course. It's not too far of Lenin's way:

The unity (coincidence, identity, equivalence) of opposites is conditional, temporary, transitory, relative. The struggle between mutually exclusive opposites is absolute, as is the development and movement." (Lenin 1973, 344).

What is preserved or denied as positive is therefore not a given determination, set aside, suspended or elevated in a synthesis, as in Hegel or Marx, but only the need to bind oneself to the opposite, it is the moment of "unity" in order to struggle then with it, to exclude it absolutely; it is precisely this struggle for life in a pure state (criticized by Horkheimer as a bourgeois moment, techniques of science, positivism reducing reason to instrumental logic [START_REF] Horkheimer | Éclipse de la raison[END_REF], which is also defended by Nietzsche. What is then preserved is only the "moment of the connection": it serves only to trigger the movement in view of self-movement, since Lenin says it is the "negativity which is the internal pulsation of selfmovement and life" (Lenin 1973, 135). And this is done in a concrete way. In Lenin, every concrete being is placed as a potential void if its mass contains a direction or density that does not correspond to the desired measure. This is indeed absolute subjectivism ("above the Begriff is the Idea" 12 ), which tactically avoids "pure negation" because Lenin wants something real, sovereign power (but not as Derrida wants just spiritual power). Lenin wants to be the living principle that decides the appropriate reality. Without the sacred garment, however, which Stalin hastened to put on. Derrida will thus undermine this claim to extend negativity by extending her domain towards language (take a look on John R.Searle/Derrida controversy about Austin's legacy 13 and the letter against Derrida's Honorary Degree in The Times (London) (Saturday, May 9, 1992 14 ), towards also reason and social order in itself, and not this or that kind of rationality and social order. Lenin, then. But not Stalin. While the last continues this first (Ellenstein 1984: 162, 171, 189), as Bataille pointed out (Bataille 1949: 183, 186, 198-199, 222):

The resolute disregard for individual interest, thought, convenience and personal rights was from the very beginning the result of the Bolshevik revolution. In this respect, Stalin's politics accuse the features of Lenin's but do not innovate it. The "Bolshevik firmness" is opposed to "rotten liberalism". (...) In truth, marvellous mental chaos results from the action of Bolshevism in the world, and from the passivity, the moral non-existence, that it has encountered. But history may be the only one likely to put an end to it, by some military decision. We can only propose to look for the nature of this action, which disturbs the established order before our eyes, much more deeply than Hitler could have done. (...) If we are exhausted, only terror and exaltation allow us to escape the slackness. Without a violent stimulus, Russia could not get back on track. (...) It is cruel to desire the extension of a regime based on a secret police, the gagging of thought and many concentration camps. But there would be no Soviet camps in the world if a huge movement of human masses had not responded to an urgent need.

The same framework can be seen in Maurice Blanchot:

I would like to quote this text by Alexander Blok, the great poet of the Twelve, whom the October Revolution, however, frightened: "The Bolsheviks do not prevent us from writing verses, but they prevent us from feeling like a master; the one who carries within himself the pole of his inspiration, his creation and holds the rhythm is a master. (Blanchot 1971, 113-114) And Blanchot, on whom Derrida (in Parages, Spectres de Marx) relies first and foremost, also writes this about "negation":

He who stays with negation cannot use it. Who belongs to him, in this belonging can no longer leave him, because he belongs to the neutrality of absence where he is already no longer himself. This situation is, perhaps, despair, not what Kierkegaard calls "disease to death", but this disease where dying does not lead to death, where one no longer hopes in death, where death is no longer to come but is that who no longer comes" (Blanchot 1955, 125).

It is also in this sense that Gilles Deleuze uses the term "differential". In its meaning, order, unity, self, or integrative support, exists only as "variation" (Deleuze 1968, 224) (roaming, random) of the permanent drift, the only accepted constant, the only "Idea" (in the sense of Lenin, supra) whose contortions then create "varieties" (of human being, for instance, such as various "queer" with "care values" in Foucault's system, which is very near from Deleuze matrix, as Foucault often said). And these only exist for themselves in so far as they express the un(limited) in(defined)finite aspect of the mother matrix, that of permanent drift without ending, even the Dead, since we must (sollen) be already dead (according to Blanchot). This is also observed by Lyotard:

Where do you make your criticism from? Don't you see that criticizing is still knowing, knowing better? That the critical relationship is still inscribed in the sphere of knowledge, of "awareness" and therefore of empowerment? We must drift out of criticism. Much more: the drift is by itself the end of criticism. (Lyotard 1972, 15) The drift is carried out by acting on it, by putting it in condition, such as the wave which needs an obstacle to refract because it is unable to do so internally since there is no impetus or external-internal differentiation in Deleuze. There is no orientation, no hierarchy, no priority, no better or worse, there is only one thing, only one, the drift, knowing that there are no more shores, except that of dissolving -the supreme "Idea". For example, it is a question of posing at the beginning of any emotional series that exists an idea acting as a "pure element of potentiality" (Deleuze 1968, 227) without past or future, such as a matrix with frames without inside or outside, except the instant, and which possess the "first coefficient or first derivative", which will then determine "the other derivatives and consequently all the terms of the series resulting from the same operations" (Ibidem). This means, moreover, that it is not pleasure but the random triggering of desire in what it brings as a plane, form, click of clicks in (de)finite and repeated allowing the unlimited drift of the dissolved self in the generalized hallucination or the "I cracked" and the "dissolved self" (Deleuze 1968: 223, 332). This is why Deleuze prefers Artaud to Carroll. What matters is, above all, the drift alone, and not what can trigger it as a primary element, as Lyotard said anyway (supra).

Thus, in a Sado-Maso experience (also highly prized by Foucault) at the borders of a strong opiate intake, apparently necessary to anaesthetize oneself, especially when it comes to seams and burns:

What is this masochist doing? He seems to imitate the horse, Equus Eroticus, but that's not it. The horse, and the master trainer, the mistress, are no more images of mother or father. It is a completely different question, an animal becoming essential to masochism, a question of forces. The masochist presents it this way: training axiom -destroy instinctive forces to replace them with transmitted forces (...). The renunciation of external pleasure, or its delay, its infinite distance, on the contrary, testifies to a conquered state where desire no longer lacks anything, fills itself with itself and builds its field of immanence. Pleasure is the affection of a person or a subject, it is the only way for a person to "find his way around" in the process of desire that overflows him; pleasures, even the most artificial, are reterritorializations. But precisely, is it necessary to meet again? (Deleuze & Guattari 1980: 191-194) This is typically the very economy of what can be called "neo-Leninism". Because we do not see in this name that this process would not enter into general equivalence with the rest, with what is transmitted at the same time by the blow: a way of being in the world, of being submissive, ordered, to go and haunt the Château of masters in "negation" (especially if they boast of being from the 1807's vintage version of Blanchot), selling themselves as voluntary victims who believe they also command the process. Since they are already consenting, to what? To integrate the becoming "red guard" (as Blacks Blocs) in the Althusser way. How? By wishing to "libidinally" address boots and then gradually "destroy instinctive forces to replace them with transmitted forces" which, in the long run, make increasingly illusory to find one's way around when it comes to politically fight these boots that we so psychophysiologically desire (as Blacks Blocs). Thus, for Foucault, Deleuze's book The Anti-Oedipus is "the book" par excellence of the destruction of the self, reduced to a conjunctural and conjectural cluster of polarization of intensities. It is considered as a book of ethics, the "only book of ethics written for a long time" -Foucault wrote. Do thus see that the Bolsheviks correspond perfectly to it, putting in shape this "new barbarity" of which Blanchot speaks, adds Bataille, their "marvellous mental chaos" (Bataille 1949, 186) and their "resolute disregard for self-interest, though, convenience and personal rights" (Ibidem, 183)?

In the end, it is a question of destroying oneself in order to prevent the body from using its power to organize itself in the world. This is the artist's last work: Brutus, who can become Orpheus if he knows how to handle the dagger of negation reduced to itself: He who recognizes for his essential task the effective action within history, cannot prefer artistic action. Art acts badly and acts little. It is clear that if Marx had followed his childhood dreams and written the most beautiful novels in the world, he would have enchanted the world, but would not have shaken it. We must, therefore, write Le Capital and not War and Peace. We must not paint Caesar's murder, we must be Brutus. These connections, these comparisons will seem absurd to the viewers. But, as soon as art is measured against action, immediate and urgent action can only prove it wrong, and art can only prove itself wrong. (Blanchot 1955, 284) This is therefore where the Blanchotien-Battaillian (and therefore Foucaldo-Deleuziano-Lyotardo-Derridian) "communism" is located. Thus, a Brutus -the Blanchotian artist -is in the search of an "end" that would have "the heaviness of a new beginning" (L'espace littéraire, 330) like a knife wound which, unceasingly falls down and repeats itself. Since that which kills something unreal, unimportant, simple, anonymous real material of inspiration in order to exhale, to suck in endlessly, like a voracious spectrum whose image perceived in the eyes of the Eurydice is the only possible sceptre:

The basis of failure is the restarting of the experiment, not the fact that it is not successful. Everything always starts all over again -yes, once again, again, again, again. (Blanchot 1955: 331-332) Or, how to scatter its melodies and other members in order, Blanchot said, to "repudiate also the principle of which the god is only the support and try to leave the circle where, since always, under his guard as under the guard of humanism, we remain locked in the fascination of unity" (Blanchot 1971, 85).

It would also seem that, when Bataille uses the notions of "acephalus" i.e. "not knowing" (as Nietzsche said, supra) and when Blanchot talks about Eurydice and Brutus, it seems that Bataille and Blanchot were trying to make France, after Russia, the permanent acephalon.

More generally, what appears in the French intelligentsia under fascist and Leninist influence is a kind of psychological and aesthetic preparation (well seen by Walter Benjamin (Hollier 1979, 586) when he observes Bataille and his group) to flight forward within the destruction of internal and external reason by justifying it by the anti-capitalist struggle. For Bataille et Blanchot, it is, therefore, necessary to have the strength to say no to the man of work, to his reason and needs, and, at the same time, yes, to force whatever comes forth.

Look now at Foucault. Let us start, for example, from Foucault's fascination for Pierre Rivière's speech, who explains, calmly, without a wrinkle or a word underlined, not what he has done but what he has already written in the heart of reality, the a posteriori speech only reflecting on it to erase it:

We have here, however, a phenomenon of which I do not see any equivalents in the history of both crime and discourse: that is, a crime accompanied by such a strong and strange discourse that the crime ends up no longer existing, escaping, by the very fact of this discourse which is held on it by the person who committed it. (Foucault 1994, T. III, 98) To imitate this, Foucault uses Borges' work, whose presence occupies the book Les mots et les choses (1966) from the first line of the Pre/face (before/face):

This book was born in a text by Borges. (...). This text quotes "a certain Chinese encyclopedia" where it is written that "animals are divided into: a) belonging to the Emperor, b) embalmed, c) tamed, d) suckling pigs, e) sirens, f) fabulous, g) dogs in freedom, h) included in this classification, i) which agitate like madmen, j) countless, k) drawn with a very fine brush of camel hair, l) and caetera, m) which break the crust, n) which from afar seem to be flies. In the wonder of this taxonomy, what we reach with a leap, what, thanks to the apologist, is indicated to us as the exotic charm of another thought, is the limit of ours: the naked impossibility of thinking that. (...) This text by Borges made me laugh for a long time, not without certain unease and difficult to overcome. (Foucault 1966, 7) Let us also observe that this text, which so amazes Foucault and which "made him laugh for a long time" is nevertheless, in the beginning, that of a Borges who, it seems, works as a literary writer and not as a sociologist or historian of the human sciences. In other words, this writer, Borges, seeks to bring together, even to the absurd (in the pataphysical sense of an Ionesco), the distance between words and things in various syntactic spins. This implies that "the naked impossibility of thinking" is not a naked impossibility if one thinks of its possibility in the context of poetics. What is surprising, however, is Foucault's astonishment.

As if he was wondering, at heart, whether it would be possible or not to think of this "naked impossibility" in a dressed, disguised way.

Indeed, if there is "the naked impossibility of thinking", in the scientific framework where the book Les mots et les choses is located, it is possible to dress this nudity, there, with a trompe l'oeil scientifichere History -precisely to make the absurd real and the real absurd. Under these conditions, the result achieved allows the construction of a custom-made covering that gives the appearance of rigorous research but which has been made only to shape things (in Foucault's words) and not in the real history of the sciences that he addresses in his book.

The habit, therefore, not only acts as a monk but, above all, becomes the only form of reality. Like these ghost ships which, seen from the distance, seem to be inhabited, there is even a man at the helm (inside, in fact), only one skeleton (that of a Borges-style classification) who remains the last work on the canvas of meaning. Thus, in Les mots et les choses Foucault seems like taking the word Borges: building a classification that, for example, artificially opposes the categories of representation and meaning or who so meticulously perceives, in the presence of an "organization" (Foucault 1966, 243) of science, a place in which "death prowls around" (Ibidem, p. 395). In short, Foucault transvestite in Borges will have to create a story, a fiction, in order to put it as a framework for "atopy", "aphasia" (Ibidem, 10). What does this mean? Foucault, although he like Pierre Rivière (supra), writes beforehand what he will accomplish (like Lenin).

Indeed, the Foucauldian discourse seizes (as a Nietzschean member of the "metaphysical race") upon the "fundamental codes of a culture" (Foucault 1966, 11) in order not to explain but to emerge as this culture itself (Ibidem, 396-397):

Thus, the last man is both older and younger than the death of God; since he has killed God, it is he himself who must answer for his own finiteness; but since it is in the death of God that he speaks, thinks and exists, his murder himself is doomed to die; new, the same gods are already swelling the future Ocean; man will disappear. More than the death of God, -or rather in the wake of that death and according to a deep correlation with it, what Nietzsche's thought announces is the end of his murderer; it is the bursting of man's face in laughter, and the return of masks; it is the dispersion of the deep flow of time by which he felt carried and of which he suspected the pressure in the very being of things; it is the identity of the Return of the Same and the absolute dispersion of man.

Foucault thus floats like these "new gods" who "already inflate the future Ocean" by accomplishing this so-called "absolute dispersion of man" previously thought of by Althusser (who also quotes Foucault on this point in his testament book, L'avenir dure longtemps.

Let us also look at this other extract 15 , which will allow us to see how Foucault produces some too obvious traces of his History of Madness:

I was told that madness did not exist, whereas the problem was absolutely the opposite: it was a question of how madness, under the different definitions that could be given to it at one time, could be integrated into an institutional field that constituted it as a mental illness with a certain place alongside other diseases.

Thus, the question was how madness "could be integrated into an institutional field that constituted it as a mental illness". It was, therefore, according to Foucault, "an institutional field" that "constituted" it as "mental illness".

In other words, without this "institutional field", madness would not be constituted as a mental illness. What would madness have been constituted as? Has "madness" thus constituted itself? Could madness think of itself? Foucault said: "I was told that madness did not exist, whereas the problem was absolutely the opposite". If the problem is the other way around, could it mean that what exists as madness is not madness? If the problem is "absolutely opposed" to the fact that madness does not exist, therefore something exists, whose word is not madness in the sense of "mental illness". Or does Foucault presuppose that what is called "madness" is not, in fact, as such, so beyond the "constitutional field" that constituted it?

This is what he will say next since he will advance, and rather in retrospect, that for him it is rather a status, but he has not studied its medical aspect. Nevertheless, he pronounces himself on it both implicitly and quite explicitly by presupposing that the institution, the word, constitutes the thing, madness, as a mental illness whereas it is, for Foucault, quite another thing.

In short, under the pretext, in fact, that certain medical practices may have hastily categorized certain people as "crazy" when they were rather weakened, out of phase, etc., it does not follow that we must deny the morphological problems of personality dislocation beyond the word that claims to subsume the thing.

Unless an a priori arbitrary posit that this dislocation is as valid as a deepening of its unity and, even better, is preferable because it can thus perceive multiplicity as "acephalic" (Bataille) and thus avoid bourgeois accumulation, a premise of the liberal order is a question of drying up at its source. Now, this domination is what Foucault wants to fight against, at least if we want to eradicate the "General who is in us", according to Deleuze's words in L'anti-Oedipe II (Deleuze & Guattari 1980, 36).

At least that is what Foucault believes, whereas the consequences are frightening, including for others, since in the horizon of this "end of man" it is not certain that the "alternatives" envisaged by him will best suit this "end":

It is often said that the definitions of the Islamic government are imprecise. On the contrary, they seemed very familiarly clear to me, but I must say, rather uncomfortable. These are the basic formulas of democracy, bourgeois or revolutionary, I said; we have not stopped repeating them since the 18th century, and you know what they have led to. But I was immediately answered: The Koran had stated them long before your philosophers and if the Christian and industrial West have lost their meaning, Islam will be able to preserve their value and effectiveness 16 .

Thus, what would not be very "reassuring" does not come from the much more totalitarian than communitarian essence of this so-called "Islamic government" which claims, through "Islam", to find "meaning", but from the fact that "the definitions of the Islamic government" are "the basic formulas of democracy, bourgeois or revolutionary (...) we have not stopped repeating them since the 18th century, and you know what they have led to".

What does this mean? Where did they "lead"? Undoubtedly towards the most "confinement", "order", in the official version of the Foucauldian word. While in Iran everyone is immersed in the joy of rediscovering the "meaning".

Thus, this "destruction" resulting from Leninism and (vulgar) Nietzscheism and produced by the French filiation is, in fact, a kind of new totalitarianism, i.e. it is theoretically rectified in the sense that it is no longer a question of killing a race or a class but everything that in the human race would be similar to self-development, the supposed source of evil, and whose ("theoretical") killing would be a proof of the work to be done to rise to the rank of the superhuman killer of God.

In this case, it is a question of an unlimited absolutism which nevertheless wants to be authentic, originating in its theoretical revolt, whereas in practice it is a question of the renewed yoke of this new "metaphysical race" resulting from the "professional revolutionaries" (appointed by the State as "organic intellectuals" such as Althusser at Ulm street in Paris, inside the famous Latin Quarter etc. etc.) for whom it is no longer a question of biology but of applying the "class struggle in theory". The important thing is to know for which "class" this "struggle" is being carried out.

All depends on which step we have come. Like Blanchot said, Brutus killing Caesar, Orpheus killing Eurydice; is it better to write War and Peace, as Tolstoï? And this until a repetition without difference: Republic, language, thought, sex, everything must wander, error, (t)error. That of an empty, emptied, lifeless, bloodless, and high-speed "differrance" (or permanent voluntary random) while remaining in a tremendous way, waiting maybe the new God (the very death of Man and his Freedom Right) in secret excitement. The postmodern deconstructivist way of life seems opening up, more and more, mind and arms, at the pre-democratic, tyranny regime, while hypocritically denouncing it on the media network mainstream.

Leninism, in fact, never ends to "move away" from us only if we refuse this; who "we"? processes that lead to domination (...), then there is no chance that we can explain how continuous local struggles must be able to consolidate into institutional power. This is a problem from which Axel Honneth has energetically identified the ins and outs" (Habermas 1985, 340). 5. "Foucault's argument, finally, contains certain elements that, coming from the opposite side, carry a raw behaviorism. Yet it is precisely they who will abound in the direction of a fundamentally mechanistic conception (…) The behaviour of human beings and, in particular, the physical manifestations of

  NOTES 1. "Communication Presented to the French Society of Philosophy on 24-2-1968 and Reproduced with the Approval of its President, Mr Jean Wahl", Small Maspero Collection (PCM), 1975 (2nd edition: 15,000 to 25,000 copies). 2. Ibidem (PCM 1975, 75-77, Theses 1 to 7). For example, Thesis 7, where Althusser not only explicitly refers to Materialism and Empirical Criticism but considers Lenin's work (1908, with the subtitle "Critical note on a reactionary philosophy." Paris: Éditions Sociales, 1948) as containing "all" the theses counted on this subject (Idem, 76): "7 -The Marxist-Leninist revolution in philosophy consists in refusing the idealistic conception of philosophy (philosophy as "interpretation of the world") which, as it always does, denies that philosophy expresses a class position, and in adopting in philosophy the position of proletarian class, which is materialist, thus establishing a new practice of philosophy, materialist and revolutionary, causing the effects of class division in theory. All these theses are contained, either explicitly or implicitly in Materialism and Empirical Criticism. I have done nothing but begin to explain them". See also Louis Althusser, Etienne Balibar in Lire le Capital, editions FM, Small Collection Maspero, 1968, Book II, p. 80: "in the expression 'historical materialism', 'materialism' means nothing other than science, and the expression is strictly synonymous with 'science of history'". 3. "Interview with Jean-Louis Houdebine and Guy Scarpetta", published in Promesse, no. 30-31 (automne et hiver), 1971 (reprinted in 1972 as a collection -Paris: Midnight Editions), in particular pp. 86-88 (see also below for an analysis of this explicit affiliation to Althusser and Lenin). 4. "If, like Foucault, we only tolerate as a model the one provided (...) by the