Marcos Eduardo Valle
email: valle@ime.unicamp.br

Marco Aurélio Granero

Samuel Francisco

Santiago Velasco-Forero

Marco Aurélio Granero

Irregularity Index for Vector-Valued Morphological Operators

Keywords: mathematical morphology, vector-valued images, total order, irregularity issue, optimal transport

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Mathematical morphology (MM) is a nonlinear theory that uses geometric and topological concepts for image and signal processing [START_REF]Mathematical Morphology: From Theory to Applications[END_REF][START_REF] Soille | Morphological Image Analysis[END_REF]. The theory of mathematical morphology is usually defined on an algebraic structure called complete lattices, which is satisfactory for binary and gray-scale images [START_REF] Heijmans | Mathematical Morphology: A Modern Approach in Image Processing Based on Algebra and Geometry[END_REF][START_REF] Ronse | Why Mathematical Morphology Needs Complete Lattices[END_REF]. In the case of vector-valued images, vector spaces endowed with a total order are one of the most comfortable frameworks for the extension of morphological processing [START_REF] Angulo | Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis[END_REF][START_REF] Aptoula | A Comparative Study on Multivariate Mathematical Morphology[END_REF][START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF]. Approaches that have been recently formulated using total orderings include [START_REF] Lézoray | Complete lattice learning for multivariate mathematical morphology[END_REF][START_REF] Sangalli | Color Mathematical Morphology Using a Fuzzy Color-Based Supervised Ordering[END_REF][START_REF] Valle | Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary[END_REF][START_REF] Velasco-Forero | Mathematical Morphology for Vector Images Using Statistical Depth[END_REF][START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF][START_REF] Velasco-Forero | Random Projection Depth for Multivariate Mathematical Morphology[END_REF]. Despite their successful applications for color and hyperspectral image processing, Chevallier and Angulo showed that the information contained in a total order is too weak to reproduce the natural topology of the value space [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF]. As a consequence, morphological operators may introduce irregularities and aliasing on images. Our motivation is to formulate measures to study the irregularity implied by a morphological operator on vector-valued images. We believe this is the first work proposing a framework based on the Wasserstein metric to score the irregularity effect considering pairs of input/output images besides our conference paper [START_REF] Valle | Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric[END_REF]. Indeed, this paper extends [START_REF] Valle | Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric[END_REF] by presenting more efficient estimators for the irregularity measure using local windows and entropic regularized optimal transport methods [START_REF] Peyré | Computational optimal transport[END_REF][START_REF] Schmitzer | Stabilized sparse scaling algorithms for entropy regularized transport problems[END_REF]. This paper also presents extensive computational experiments to show the effectiveness of the proposed irregularity measures.

The paper is organized as follows: Section 2 reviews mathematical morphology concepts for vector-valued images and the difficulties produced by operators based on total orders. Sections 3 and 4 present our proposed irregularity measure. Additionally, results on natural images show the goodness of the proposed measure. The paper ends with Section 5 including conclusions and recommendations for future works.

Basic Concepts on Mathematical Morphology

Let us begin by presenting the basic concepts and the notations used in this paper. First, an image I corresponds to a mapping from a point set D to a value set V, that is, I : D → V. The set of all images from a domain D to V is denoted by V = V D . Throughout the paper, the set D is finite with N D = Card(D) points. Moreover, we assume D is included in a space E, where (E, +) is a group. Usually, we consider E = R 2 or E = Z 2 with the usual addition. We also assume the value set V is a complete lattice equipped with a metric d : V × V → [0, +∞). For simplicity, we consider the RGB color space endowed with the Euclidean distance throughout the paper.

Mathematical Morphology on Complete Lattices

Mathematical morphology (MM) is mainly concerned with image operators used to extract relevant geometric and topological information from an image [START_REF] Dougherty | Hands-on Morphological Image Processing[END_REF][START_REF] Heijmans | Mathematical Morphology: A Modern Approach in Image Processing Based on Algebra and Geometry[END_REF][START_REF]Mathematical Morphology: From Theory to Applications[END_REF][START_REF] Soille | Morphological Image Analysis[END_REF]. The two elementary operators of MM are dilations and erosions. Many other operators, such as opening, closing, and the morphological gradient, are obtained by combining the elementary morphological operators.

Complete lattices provide an appropriate mathematical background for defining the elementary morphological operators [START_REF] Heijmans | Mathematical Morphology: A Modern Approach in Image Processing Based on Algebra and Geometry[END_REF][START_REF] Ronse | Why Mathematical Morphology Needs Complete Lattices[END_REF]. A complete lattice L is a partially ordered set in which any subset X ⊂ L has both an infimum and a supremum [START_REF] Birkhoff | Lattice Theory[END_REF]. The infimum and the supremum of X are denoted by X and X, respectively.

The elementary morphological operators are those that commute with the supremum and the infimum operations in a complete lattice. When the value set V is a complete lattice, the operators δ S , ε S : V → V given by the following equations are respectively a dilation and an erosion:

I(p + s), (1)
where S ⊆ E is a finite set referred to as the structuring element (SE) [START_REF] Soille | Morphological Image Analysis[END_REF]. The images δ S (I) and ε S (I) are the dilation and the erosion of I by the structuring element S, respectively. Although there exist more general definitions, the elementary morphological operators given by ( 1) are widely used in practical situations. Combining dilations and erosions, we obtain many other morphological operators. In this paper, we focus on elementary operators defined by [START_REF] Angulo | Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis[END_REF]. We also consider openings γ S and closings φ S , which are obtained by the compositions γ S = δ S • ε S and φ S = ε S • δ S [START_REF] Soille | Morphological Image Analysis[END_REF].

Vector-Valued Mathematical Morphology

Let us now address morphological operators for vector-valued images. A vector-valued image is obtained by considering V ⊆ Rd , where R = R ∪ {-∞, +∞} and d ≥ 2. Vectorvalued dilations and erosions can be defined using (1) whenever the vector-valued set V is a complete lattice. However, there are several ordering schemes for vector-valued sets. Defining an appropriate ordering scheme is one of the main challenges of vector-valued MM. The following references address interesting directions of research on vector-valued MM [START_REF] Angulo | Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis[END_REF][START_REF] Aptoula | A Comparative Study on Multivariate Mathematical Morphology[END_REF][START_REF] Burgeth | An approach to color-morphology based on Einstein addition and Loewner order[END_REF][START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF][START_REF] Van De Gronde | Group-Invariant Colour Morphology Based on Frames[END_REF][START_REF] Lézoray | Complete lattice learning for multivariate mathematical morphology[END_REF][START_REF] Valle | Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary[END_REF][START_REF] Velasco-Forero | Mathematical Morphology for Vector Images Using Statistical Depth[END_REF][START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF][START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF].

As examples of ordering schemes on vector-valued sets, let us recall the marginal and the lexicographical orderings [START_REF] Aptoula | A Comparative Study on Multivariate Mathematical Morphology[END_REF]. The marginal ordering is defined as follows for all u

= (u 1 , . . . , u d ) ∈ V and v = (v 1 , . . . , v d ) ∈ V: u ≤ M v ⇐⇒ u i ≤ v i , ∀i = 1, . . . , d, (2)
where "≤" denotes the usual ordering on R. The marginal ordering is also known as the component-wise ordering or the Cartesian product ordering. The lexicographical ordering is defined as follows:

u ≤ L v ⇐⇒ ∃ i : u i ≤ v i and u j = v j , ∀j < i. (3)
In contrast to the marginal ordering, the lexicographical ordering is a total ordering. Thus, at least one of the inequalities u ≤ L v or v ≤ L u holds for any u, v ∈ V.

There are many other approaches to vector-valued mathematical morphology besides the marginal and lexicographical approaches. The following papers survey several interesting approaches [START_REF] Angulo | Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis[END_REF][START_REF] Aptoula | A Comparative Study on Multivariate Mathematical Morphology[END_REF][START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF]. For comparative purposes, in this paper, we consider two approaches based on reduced orderings [START_REF] Goutsias | Morphological Operators for Image Sequences[END_REF], and one approach based on the Loewner order [START_REF] Burgeth | An approach to color-morphology based on Einstein addition and Loewner order[END_REF].

A reduced ordering is defined by a surjective mapping ρ : V → L that maps the value set to a complete lattice as follows:

u ≤ ρ v ⇐⇒ ρ(u) ≤ L ρ(v). (4)
A reduced ordering is not a partial order but a preorder because it may fail to be antisymmetric; that is, the inequalities u ≤ ρ v and v ≤ ρ u do not necessarily imply u = v. Nevertheless, reduced orderings can be used for the development of effective vector-valued morphological operators [START_REF] Goutsias | Morphological Operators for Image Sequences[END_REF]. Precisely, combining (4) with look-up tables (LUTs), one can define computationally efficient vector-valued morphological operators taking advantage of gray-scale morphological operators [START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF]. Together with a LUT, a reduced ordering given by (4) becomes a total ordering on the set V (I) = {I(p) : p ∈ D} of values of an image I. Moreover, the surjective mapping ρ : V → L can be defined using machine learning techniques [START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF]. Accordingly, depending on how the mapping ρ is determined, a morphological approach can be classified as supervised or unsupervised. We consider a radial-basis function support vector machine (SVM) as a representative supervised vector-valued approach [START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF]. Also, the statistical projection depth function is considered as a representative example of an unsupervised reduced ordering-based approach [START_REF] Velasco-Forero | Random Projection Depth for Multivariate Mathematical Morphology[END_REF]. These two vector-valued morphological approaches are briefly revised in Appendix A.

Besides the marginal approach, we consider in this paper a second vector-valued morphological approach based on a partial order for comparison purposes. Specifically, we consider the approach to color-morphology based on the Loewner order proposed by Burgeth and Kleefeld [START_REF] Burgeth | An approach to color-morphology based on Einstein addition and Loewner order[END_REF]. The Loewner order is a partial order on the set of symmetric matrices. Formally, given symmetric matrices A and B, the Loewner order is defined by

A ≤ W B ⇐⇒ B -A is positive semidefinite. ( 5
)
The Loewner order has been successfully applied for the development of vector-valued morphological approaches [START_REF] Burgeth | A Unified Approach to the Processing of Hyperspectral Images[END_REF][START_REF] Burgeth | An approach to color-morphology based on Einstein addition and Loewner order[END_REF]. In particular, the Loewner order can be used as follows for defining morphological operators for color images. First, a color value is represented using hue (H), chroma (C), and luminance (L) instead of the traditional RGB representation [START_REF] Foley | Computer Graphics: Principles and Practice[END_REF][START_REF] Levkowitz | GLHS: A Generalized Lightness, Hue, and Saturation Color Model[END_REF][START_REF] Levkowitz | Color Theory and Modeling for Computer Graphics, Visualization, and Multimedia Applications[END_REF][START_REF] Gonzalez | Digital Image Processing[END_REF]. Geometrically, the considered HCL space corresponds to a bi-cone centered at zero, with tips at (0, 0, 1) and (0, 0, -1) and a circular base of radius one on the horizontal plane.

Then, each coordinate of the bi-cone is mapped onto a symmetric 2 × 2 matrix A such that

- 1 √ 2 I ≤ W A ≤ W 1 √ 2 I,
where I denotes the identity matrix. Finally, the isomorphism between colors and symmetric 2 × 2 matrices allows defining the so-called Loewner approach for color morphology1 .

The False Colors Problem Versus the Irregularity Issue

One problem on vector-valued MM is the creation of "false colors" or, more generally, false values [START_REF] Serra | The "False Colour" Problem[END_REF]. A morphological operator ψ : V → V introduces false values whenever there are values in ψ(I) which do not belong to the original image I. Formally, let 2 V denote the power set of V and let V : V → 2 V be the mapping given by

V (I) = {I(p) : p ∈ D}, ∀I ∈ V. (6)
A morphological operator ψ introduces false colors if the set difference V (ψ(I)) \ V (I) is not empty. The abnormal false values can be a problem in some applications such as when dealing with satellite data [START_REF] Serra | The "False Colour" Problem[END_REF]. Using a partial order such as the marginal or the Loewner orders, the dilation and the erosion given by (1) usually yield false colors. A total order, such as the lexicographical order and a reduced ordering combined with a look-up table, circumvents the problem of false values [START_REF] Serra | The "False Colour" Problem[END_REF]. Using a total order, the supremum and the infimum of a finite set are elements of the set, i.e., they coincide with the maximum and minimum operations, respectively. As a consequence, if D is finite, the elementary morphological operators given by ( 1) only contain values of the input image I.

On the downside, a total order can be irregular in a metric space. According to Chevallier and Angulo, the irregularities follow because the topology induced by a total order may not reproduce the topology of a metric space [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF]. Specifically, let the value set V be a totally ordered set as well as a metric space, with metric d : V × V → [0, +∞). Chevallier and Angulo showed that there exist u, v, w ∈ V such that u ≤ v ≤ w but d(u, w) < d(u, v) under mild conditions with respect to the connectivity of V. In words, although u is closer to w than to v, the inequalities u ≤ v ≤ w suggest w is farther from u than v. Since the morphological operators are defined using the extrema operators, they do not take the metric of V into account.

A visual interpretation of the irregularity is shown in Figure 1, which is very similar to an example provided in [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF]. Figure 1a) shows an image with three RGB colors, namely u = (0, 0, 0), v = (0, 0, 1), and w = (1/255, 0, 0). The toy image I is obtained by replacing pure black values u by w with probability 0.3 from an image of size 32 × 64 with two stripes of colors blue and black. The dilations J L = δ L S (I) and J M = δ M S (I) by a cross structuring element S obtained using the lexicographical RGB and the marginal ordering schemes are also depicted in Figure 1. Visually, u and w are black colors while v is a pure blue. Using the Euclidean distance, we obtain d(u, v) = 1 and d(u, w) = 0.005. These distances agree with our color perception. However, using the lexicographical ordering, we obtain u ≤ L v ≤ L w. As a consequence, the following happens when we compute the dilation δ L S (I) = J L using the lexicographical ordering: the blue v advances over the black u but it is overlaid by the black w, resulting in the irregularities shown in Figure 1b). In contrast, the dilated image depicted in Figure 1c) obtained using the marginal ordering does not present any visual irregularity.

Although we know that the irregularity results from a divergence between the topologies induced by the metric and the total order, no consensual measure agrees with our visual perception. A measure for the irregularity can help to choose an appropriate ordering scheme for vector-valued mathematical morphology. The following section proposes a measure of irregularity using the Wasserstein metric.

Measuring the Irregularity

In this section, we present a measure for the irregularity, referred to as the global irregularity index. Although we are interested in measuring the irregularity implied by a total ordering, we will not assume V is totally ordered. Indeed, the proposed irregularity measure is well defined whenever D is finite and V is a metric space. For simplicity, however, the value set V corresponds to the RGB color space equipped with the Euclidean distance in the following examples and computational experiments.

The Wasserstein Metric and the Generalized Sum of Pixel-wise Distances

The global irregularity index is given by the relative gap between the Wasserstein metric and a generalized sum of pixel-wise distances. Let us begin by reviewing the generalized sum of pixel-wise distances.

Consider an image operator ψ : V → V. Given an input image I ∈ V, let J = ψ(I) denote the output of the image operator. The generalized sum of pixel-wise distances of I and J is an operator

D p : V × V → [0, +∞) given by D p (I, J) = x∈D d p I(x), J(x) 1 p , p ≥ 1. (7)
The generalized sum of pixel-wise distances is one of the simplest measures that considers the metric d and the pixel locations. However, D p is usually not properly scaled, possibly because its dimension is the same as the metric d. For example, the images shown in Figure 1 yield the values D 1 (I, J L ) = 34.12 and D 1 (I, J M ) = 66.05. Note that the inequality D 1 (I, J L ) < D 1 (I, J M ) holds true although J L is more irregular than J M . Hence, the generalized sum of pixel-wise distances is not an appropriate measure for the irregularity.

Let us now review the Wasserstein metric, also known as the Earth mover's distance or the Kantorovich-Rubinstein distance in some contexts [START_REF] Rubner | Earth mover's distance as a metric for image retrieval[END_REF][START_REF] Villani | Optimal Transport, Grundlehren der mathematischen Wissenschaften[END_REF]. The Wasserstein metric, named after the Russian mathematician Leonid Vaseršteȋn, has been previously used by Rubner et al. for content-based image retrieval [START_REF] Rubner | Earth mover's distance as a metric for image retrieval[END_REF]. The Wasserstein metric is formulated as a transport problem and can measure distances between probability distributions [START_REF] Villani | Optimal Transport, Grundlehren der mathematischen Wissenschaften[END_REF].

The objective of a transport problem is to minimize the cost to deliver items from n factories to m shops [START_REF] Peyré | Computational optimal transport[END_REF]. In our context, the transport problem minimizes the cost to transform the input image I into the output image J. The cost is defined using the metric on the value set V. Precisely, let V (I) = {v 1 , . . . , v n } and V (J) = {u 1 , . . . , u m } be the sets of color values of I and J, respectively. Also, let

f i = Card({x : I(x) = v i }) and g j = Card({x : J(x) = u j }), (8)
denote respectively the number of pixels of value v i in the image I and the number of occurrences of the value u j in J, for i = 1, . . . , n and j = 1, . . . , m. Given p ≥ 1, the cost to transform a value v i of I into a value u j of J is defined by

c ij = d p (v i , u j ), ∀i = 1, . . . , n, ∀j = 1, . . . , m. (9)
The Wasserstein metric, denoted by

W p : V × V → [0, ∞) for p ≥ 1, is given by W p (I, J) =   n i=1 m j=1 c ij x ij   1 p , p ≥ 1, (10)
where x ij solves the linear programming problem

                         minimize n i=1 m j=1 c ij x ij subject to m j=1 x ij = f i , ∀i = 1, . . . , n, n i=1 x ij = g j , ∀j = 1, . . . , m, x ij ≥ 0, ∀i = 1, . . . , n, ∀j = 1, . . . , m. (11)
The Wasserstein metric is the p-th root of the minimal cost to transform I into J. In the transport problem [START_REF] Goutsias | Morphological Operators for Image Sequences[END_REF], the variable x ij represents the (optimal) number of pixels with value v i of I transformed to pixels with value u j of J. Moreover, the solution of ( 11), which can be arranged in a matrix X ∈ R n×m , is an optimal transport plan. An optimal transport plan is a cyclically monotone plan in the sense that the cost n i=1 m j=1 c ij x ij cannot be improved by changing the number of pixels with value v i transformed to pixels with value u j [START_REF] Villani | Optimal Transport, Grundlehren der mathematischen Wissenschaften[END_REF]. For the images shown in Figure 1, we obtain W 1 (I, J L ) = 6.18 and W 1 (I, J M ) = 65.94. Note that the inequality W 1 (I, J L ) < W 1 (I, J M ) holds despite J L being more irregular than J M . Like the generalized sum of pixel-wise distances, the Wasserstein metric is not appropriate for measuring the irregularity.

The Global Irregularity Index

Although both the generalized sum of pixel-wise distances and the Wasserstein metric are, per se, not appropriate to evaluate the irregularity, we advocate that they can be combined to yield a useful measure.

First of all, note that the generalized sum of pixel-wise distances satisfies

D p (I, J) =   n i=1 m j=1 c ij y ij   1 p , p ≥ 1, (12)
where

y ij = Card ({x : I(x) = v i and J(x) = u j , x ∈ D}) , (13)
for all i = 1, . . . , n and j = 1, . . . , m. Moreover, it is not hard to see that y ij ≥ 0 and the identities

m j=1 y ij = f i and n i=1 y ij = g j , (14)
where f i and g j are given by [START_REF] Fatras | Learning with minibatch Wasserstein : asymptotic and gradient properties[END_REF], hold for all i = 1, . . . , n and j = 1, . . . , m. Therefore, the generalized sum of pixel-wise distances also measures the cost of transforming I into J. Because W p is the minimal cost, the inequality W p (I, J) ≤ D p (I, J) holds for any I and J = ψ(I). The y ij 's given by ( 13), which can be arranged in a matrix Y ∈ R n×m , is called the operator's plan. The operator's plan is probably not an optimal transport plan. Indeed, one usually can reduce the cost n i=1 m j=1 c ij y ij by rerouting the number of pixels with value v i in I transformed to pixels with value u j in J. In some sense, the difference D p (I, J) -W p (I, J) measures the cycles in the operator's plan that can be reduced. Example 1 below illustrates the relationship between cycles and the irregularities introduced by a morphological operator using the toy image shown in Figure 1.

In order to reduce the impact of the metric d on the value set V and the impact on the choice of the parameter p ≥ 1, we propose to measure the irregularity using the mapping

Φ g p : V × V → [0, 1]
given by the relative gap between D p and W p . Precisely, given images I, J ∈ V, we define the global irregularity index by means of the equation

Φ g p (I, J) = D p (I, J) -W p (I, J) D p (I, J) , if D p (I, J) = 0, (15)
and Φ g p (I, J) = 0 if D p (I, J) = 0. Note that the larger the gap between W p (I, J) and D p (I, J), the larger the global irregularity index. Equivalently, we have

Φ g p (I, J) =    0, if D p (I, J) = 0, 1 - W p (I, J) D p (I, J) , otherwise. ( 16
)
The irregularity index is symmetric and bounded, that is, Φ p (I, J) = Φ p (J, I) and 0 ≤ Φ p (I, J) ≤ 1. Moreover, because D p and W p have the same units and similar magnitudes, Φ g p (I, J) is a dimensionless quantity. The more irregular is J = ψ(I), the larger the value of Φ g p (I, J) is expected to be. For example, the irregularity index of the dilated images J L and J M shown in Figure 1b) and 1c) are Φ g 1 (I, J L ) = 81.9% and Φ g 1 (I, J M ) = 0.17%, respectively. The following examples explores further the global irregularity index using toy images.

Because u = (0, 0, 0), v = (0, 0, 1), and w = (1/255, 0, 0), we have

D 1 (I, J L ) = 18d(u, v) + 541d(u, w) + 14d(v, w) = 34.12, (18)
W 1 (I, J L ) = 4d(u, v) + 555d(u, w) = 6.18, (19)
and

D 1 (I, J L ) -W 1 (I, J L ) = 14 d(u, v) + d(v, w) -d(u, w) = 27.95, (20)
From [START_REF] Pitié | Advances in colour transfer[END_REF], we conclude that the cost of the operator's plan can be reduced by replacing 14 times the route u → v → w by u → w or, equivalently, by avoiding the cycle (u, v), (v, w), (w, u). Moreover, note that the route u → v → w reflects the lexicographical inequalities u ≤ L v ≤ L w. Thus, the difference D 1 (I, J L ) -W 1 (I, J L ) is proportional to the number of pixels with irregular values (14 in this example) as well as to the distance between the pixel values (in this example, the amount

d(u, v) + d(v, w) -d(u, w)). The global irregularity index is obtained by dividing the difference D 1 (I, J L ) -W 1 (I, J L ) by D 1 (I, J L ), that is, Φ g 1 (I, J L ) = D 1 (I, J L ) -W 1 (I, J L ) D 1 (I, J L ) = 27

Dilated images

Fig. 2 Top: Global irregularity index by the probability of replacing a pure black pixel value u = (0, 0, 0) by the black w = (0.005, 0, 0) in an image similar to Figure 1a). Bottom: Lexicographical dilation of images obtained using π = 0.0, 0.25, 0.5, 0.75, and 1.0, respectively.

Example 2 Figure 2 shows the percentage of the global irregularity index computed on dilated versions of toy images similar to the one provided by Chevallier and Angulo [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF]. Precisely, we first construct an image of size 64 × 32 with two stripes of the same width but with the pure colors black u = (0, 0, 0) and blue v = (0, 0, 1). Then, a pure black pixel value is replaced by the black w = (1/255, 0, 0) with probability π ∈ [0, 1]. The resulting image I is dilated by a cross structuring element using both the lexicographical and the marginal ordering schemes, yielding the images J L and J M , respectively. The simulation has been repeated 500 times for each probability π ∈ [0, 1]. Figure 2 shows the average of the global irregularity index by the probability π. For a better interpretation of this plot, dilated samples of the images generated with probabilities π = 0.0, 0.25, 0.50, 0.75, and 1 are shown at the bottom of Figure 2. Note that the irregularity index for the lexicographical RGB ordering increases until close to π = 0.3 and then decreases. Furthermore, the irregularity indexes agree with the visual irregularity provided in the sample images at the bottom of Figure 2. In contrast, the marginal approach yields small irregularity indexes. This toy example confirms our expectation that the more irregular is the image J = ψ(I), the larger is the global irregularity index Φ g 1 (I, J). Fig. 3 Illustrative examples of the global irregularity index Φ g 1 computed for several tiny color images using different morphological operators.

Computational Experiments with Tiny Color Images

Let us now provide some computational experiments with tiny color images 2 . Precisely, we compute the global irregularity index Φ g 1 of morphological operators applied on one hundred color images from the CIFAR10 dataset. The measures have been computed using erosion, dilation, opening, and closing by a 3 × 3 square structuring element. Specifically, we have computed twenty-five dilations, erosions, openings, and closings on different images from the CIFAR10 dataset.

We considered three approaches based on total orderings. The first is the RGB lexicographical order in which the colors are ranked sequentially according to the red, green, and blue channels. The other two approaches are based on reduced orderings [START_REF] Velasco-Forero | Vector Ordering and Multispectral Morphological Image Processing[END_REF]. Specifically, we considered the supervised reduced ordering based on an SVM with radial basis function kernel [START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF]. In this computational experiment, we trained the SVM to distinguish the central object on an image from the rest. The last approach uses an unsupervised reduced ordering based on the random projection depth, which aims to discriminate between background and foreground pixels [START_REF] Velasco-Forero | Random Projection Depth for Multivariate Mathematical Morphology[END_REF]. For comparison, we also include two approaches based on partial orderings. Namely, the marginal approach and the approach based on Loewner order [START_REF] Burgeth | An approach to color-morphology based on Einstein addition and Loewner order[END_REF]. Because these two approaches are not based on total orders, their output images are not expected to be very irregular. Global Irregularity Index (%)

Fig. 4 The boxplot of the global irregularity index of dilations, erosions, openings, and closings for 100 color images of size 32 × 32 using five different morphological approaches.

erators defined using the marginal, Loewner, lexicographical RGB, supervised SVM-based, and projection depth approaches, respectively. We provide the global irregularity index below the images produced by the morphological operators. As expected, the marginal and the Loewner approaches yielded global irregularity indexes smaller than the lexicographical, SVM-based, and projection depth approaches. The supervised SVM-based approach yielded the most irregular dilated image. The irregularity index of 5.38% produced by projection depth's dilation of the cat image is an outlier of the global irregularity index produced by this unsupervised morphological approach. Indeed, the median of the global irregularity index produced by the morphological operators based on the projection depth is 45.22%. The eroded image depicted in the last column of Figure 3 is a typical outcome of the projection depth approach. The median of the irregularity indexes produced by the marginal, Loewner, lexicographical, and the SVM-based approaches are 2.56%, 2.81%, 6%, and 22.98%, respectively. The truck image's openings and the car image's closings provide typical examples of the global irregularity index produced by the marginal, Loewner, and lexicographical approaches. The opening of the car image produced by the projection depth approach yielded the largest global irregularity index of this experiment. The boxplot shown in Figure 4 summarizes the outcome of this computational experiment. As expected, the morphological operators based on the marginal and the Loewner partial orderings resulted in minor global irregularity indexes. In contrast, the morphological operators based on the lexicographical RGB, SVM-based, and the projection depth yielded the largest global irregularity indexes. Accordingly, the morphological approaches can be ranked increasingly using the global irregularity index as follows: marginal, Loewner, lexicographical RGB, supervised SVM-based, and projection depth. We confirmed the ranking of the morphological approaches statistically using the Wilcoxon signed-rank test with a confidence level of 99%. Concluding, the morphological operators based on total orderings yielded more significant global irregularity indexes than the morphological operators obtained using partial orderings.

Finally, we would like to emphasize that neither the generalized sum of pixel-wise distances nor the Wasserstein metric are appropriate for measuring the irregularity of morphological operators. For example, the generalized sum of pixel-wise distances and the Wasserstein metric between the original and the eroded frog image produced by the marginal approach are 182.31 and 180.61, respectively. Similarly, the same measures between the original and the eroded frog image produced by the projection depth approach are 135.44 and 74.13. Although the projection depth approach yielded a smaller generalized sum of pixelwise distances and Wasserstein metric values, the former outcome is a visually more irregular image than the image produced by the marginal approach. Conversely, a more irregular image does not necessarily result in a smaller generalized sum of pixel-wise distances or a smaller Wasserstein metric value. Indeed, the generalized sum of pixel-wise distances and the Wasserstein metric between the original truck and its opening by the marginal approach are 31.05 and 30.18, respectively. The same measures applied to the original truck and its closing by the projection depth approach are 53.96 and 37.17. However, the projection depth approach visually yielded a more irregular truck image than the opening by the marginal approach.

Computing the Irregularity Index

Despite its mathematical formulation, computing the global irregularity index is not an easy task for natural images. Precisely, the irregularity index requires solving a linear programming problem with mn variables, where m and n are the numbers of distinct pixel values of the images I and J, respectively. As a consequence, the complexity for computing the global irregularity index analytically grows as O m 3 n 3 log(mn) [START_REF] Pitié | Advances in colour transfer[END_REF]. In practical situations, the dimension of the linear programming problem [START_REF] Goutsias | Morphological Operators for Image Sequences[END_REF] is enormous, making it impossible to compute the global irregularity index analytically in real-time. We propose to compute the Wasserstein metric and the generalized sum of pixel-wise distances locally and aggregate the values into a single index to circumvent this computational burden.

The Local Irregularity Index

Let {W 1 , W 2 , . . . , W k } be a partition of the image domain D, that is, W 1 , W 2 , . . . , W k are non-overlapping local windows such that D = ∪ k i=1 W i . Also, let W p (I, J|W ) denote the Wasserstein metric computed restricting the images I and J to the local window W . The local irregularity index is defined by the following equation for all I, J ∈ V:

Φ l p (I, J) = 1 - k =1 W p (I, J|W ) p 1/p D p (I, J) . ( 22
)
The size of the local windows W 1 , . . . , W k plays an important role in the local irregularity index. On the one hand, it is impractical to compute the irregularity index using large local windows. On the other hand, the local irregularity index approaches zero as the size of the local windows decreases. Thus, the size of the local windows is a trade-off between computational cost and accuracy of the local irregularity index.

Remark 1 We would like to point out that we previously used the geometric mean to aggregate the quotient of W p (I, J|W ) by D p (I, J|W ) [START_REF] Valle | Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric[END_REF]. However, the geometric mean does not scale well for moderate and large size images because 0 < W p (I,J|W )

D p (I,J|W ) < 1 implies k =1 W p (I,J|W )
D p (I,J|W ) → 0 as the number k of local windows increases. In this paper, the local irregularity index is given by ( 22) which, besides scaling well to moderate and large scale images, is closely related to the minibatch Wasserstein metric [START_REF] Fatras | Learning with minibatch Wasserstein : asymptotic and gradient properties[END_REF].

The local irregularity index provides a lower bound to the global irregularity index, i.e., the inequality Φ l p (I, J) ≤ Φ g p (I, J) holds true for all I ∈ V and J = ψ(I). Precisely, let

f i = Card {x ∈ W : I(x) = v i } and g j = Card {x ∈ W : J(x) = u j } , ( 23
)
denote respectively the number of pixels of I with value v i and the number of pixels of J with value u j , both restricted to the local window W , for = 1, . . . , k, i = 1, . . . , n, and j = 1, . . . , m. Note that f i = 0 and g j = 0 if the images I and J have no pixels with values v i and u j in the windows W , respectively. Moreover, since {W : = 1, . . . , k} is a partition of the images domain D, the identities

k =1 f i = f i and k =1 g j = g j , (24)
where f i and g j are given by [START_REF] Fatras | Learning with minibatch Wasserstein : asymptotic and gradient properties[END_REF], hold for all i = 1, . . . , n and j = 1, . . . , m. Although in practice we compute W p (I, J|W ) using only the pixel values of I and J in the local window W , the restricted Wasserstein metric satisfies

W p (I, J|W ) p = n i=1 m j=1 c ij x ij , (25)
where x ij solves the linear programming problem

           minimize n i=1 m j=1 c ij x ij subject to m j=1 x ij = f i , n i=1
x ij = g j , and x ij ≥ 0, ∀i, ∀j.

Now, define the non-negative variables

z ij = k =1
x ij , ∀i = 1, . . . , n and ∀j = 1, . . . , m.

From ( 24), we conclude that

m j=1 z ij = m j=1 k =1 x ij = k =1 f i = f i and n i=1 z ij = k =1 g j = g j . ( 28
)
Thus, the variables z ij satisfy the constraints of the linear programming problem [START_REF] Goutsias | Morphological Operators for Image Sequences[END_REF]. Because W p (I, J) p is the minimum value of [START_REF] Goutsias | Morphological Operators for Image Sequences[END_REF], from [START_REF] Sangalli | Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings[END_REF] we conclude that Local Irregularity Index (%)

W p (I, J) p ≤ n i=1 m j=1 c ij z ij = k =1 n i=1 m j=1 c ij x ij = k =1 W p (I, J|W ) p , (29)
Fig. 5 The boxplot of the local irregularity index with local windows of size 16 × 16 obtained from dilations, erosions, openings, and closings for 100 natural color images using five different morphological approaches.

which results in the desired inequality

Φ g p (I, J) = 1 - W p (I, J) p D p (I, J) ≥ 1 - k =1 W p (I, J|W ) p 1 p D p (I, J) = Φ l p (I, J), (30)
for all I ∈ V and J = ψ(I).

Computational Experiments with Natural Color Images

In analogy to the previous section, let us perform some computational experiments with natural color images. However, instead of computing the global irregularity index on tiny color images from the CIFAR10 dataset, we compute the local irregularity index Φ l 1 of morphological operators applied on one hundred colors images from the Berkeley segmentation dataset 3 . The local irregularity index is computed using local windows of size 16×16 for dilations, erosions, openings, and closings by a 9 × 9 square structuring element. The boxplot shown in Figure 5 summarizes the outcome of this computational experiment.

Note from Figure 5 that the approaches based on partial orderings, namely the marginal and the Loewner approaches, yielded the minor local irregularity indexes. In contrast, the approaches based on total orderings produced the largest irregularity indexes. Using the Wilcoxson signed-rank test with a confidence level at 99%, we rank the local irregularity indexes increasingly by the marginal, Loewner, lexicographical RGB, supervised SVM-based, and the projection depth approaches. and the Loewner approaches produced the smallest local irregularity indexes among the dilated images shown in Figure 6. In contrast, the SVM-based and the projection depth approaches yielded the largest local irregularity indexes. We would like to point out that the value of 2.31% is an outlier corresponding to the smallest local irregularity index produced by the projection depth approach on the one hundred natural color images considered in this computational experiment. Similarly, Figure 7 depicts an original color image and the corresponding eroded images. The SVM-based and the projection depth approaches again yielded the most significant local irregularity indexes. Furthermore, in contrast to the outlier irregularity value attained by the dilated image shown in Figure 6, the eroded image in Figure 7 is a typical image produced by the projection depth approach in this experiment.

Other typical images produced by the five morphological approaches are shown in Figures 8 and9.

We would like to conclude this subsection by calling the reader's attention to the qualitative similarity between the boxplots shown in Figures 4 and5 ularized methods aim to solve the convex optimization problem

                         minimize n i=1 m j=1 c ij x ij + n i=1 m j=1 x ij (log(x ij ) -1) subject to m j=1 x ij = f i /N D , ∀i = 1, . . . , n, n i=1 x ij = g j /N D , ∀j = 1, . . . , m, x ij ≥ 0, ∀i = 1, . . . , n, ∀j = 1, . . . , m. (31)
where > 0 is a regularization term and N D denotes the number of pixels of the processed images. Note that, for obtaining a meaningful entropic regularization, the variables metric is then approximated by

x ij must satisfy n i=1 m j=1 x ij = (1/N D ) n i=1 f i = (1/N D ) m j=1 g j = 1. The Wasserstein
W r p (I, J) =   N D n i=1 m j=1 c ij x ij   1/p , p ≥ 1, (32)
where x ij solves [START_REF] Valle | Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary[END_REF]. Note that the approximation W r p is computed without the regularization term. Since both [START_REF] Goutsias | Morphological Operators for Image Sequences[END_REF] and [START_REF] Valle | Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary[END_REF] have equivalent constraints (they differ only because of the term N D ), the inequality W p (I, J) ≤ W r p (I, J) holds for all image I and J = ψ(I). As a consequence, the irregularity index computed by an entropic regularized method also yields a lower bound for the analytical irregularity index.

As pointed out by Peyré and Cuturi [START_REF] Peyré | Computational optimal transport[END_REF], the objective in ( 31) is an -strongly convex function. Therefore, the regularized optimization problem has a unique optimal solution. Furthermore, the optimal solution can be computed using the Sinkhorn method, which uses only matrix-vector products and, thus, it is suited to be executed in GPU. Also, the unique solution of [START_REF] Valle | Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary[END_REF] converges to the analytical solution as tends to zero. On the downside, the convergence of the Sinkhorn method slows as the entropic term approaches zeros. For small , the regularized optimization problem defined by ( 31) can be solved using stabilized versions of the Sinkhorn method [START_REF] Schmitzer | Stabilized sparse scaling algorithms for entropy regularized transport problems[END_REF]. The Sinkhorn method and its stabilized versions are implemented in optimal transport libraries such as the Python Optimal Transport4 (POT) and Julia's Optimal Transport 5 .

Figure 10 compares the local irregularity index computed using an entropic regularized optimal transport method with the analytical Wasserstein metric. Precisely, the scatter plot in Figure 10 shows the local irregularity index computed on the considered subset of the Berkeley segmentation dataset. Similarly to the previous experiments, we considered dilations (δ S ), erosions (ε S ), openings (γ S ), and closings (φ S ) by a 9 × 9 square structuring element S. We distinguished the marginal, Loewner, lexicographical, SVM-based, and projection depth approaches using the superindexes M , W , L, S, and P , respectively. The horizontal and the vertical axes in the scatter plot shown in Figure 10 correspond to the local irregularity index obtained using the analytical and the stabilized Sinkhorn method, respectively. In this computational experiment, we have used the regularization term = 10 -3 . Figure 10 also shows the line y = x. The closer a point is to the line y = x, the better is the approximation obtained by the stabilized Sinkhorn method to the analytical solution. Notably, the coefficient of determination obtained from the 500 points is R 2 = 0.98. Thus, the stabilized Sinkhorn method yielded good estimations for the analytical local irregularity index using local windows of size 16 × 16. Furthermore, computing the local irregularity index by the stabilized Sinkhorn method is much faster than using the analytical Wasser- stein metric. Thus, we encourage computing the local irregularity index using the Sinkhorn method or its variations. We would like to point out that other methods, such as sliced approximations of the optimal transport problem [START_REF] Pitié | Advances in colour transfer[END_REF][START_REF] Pitié | N-dimensional probability density function transfer and its application to colour transfer[END_REF], can also be considered for estimating the irregularity index.

Additional Remarks on the Local Irregularity Index

Let us finish this section by briefly addressing the sensibility of the local irregularity index with respect to the parameter p. Let us also address the role of the sizes of the local windows and the structuring element on the local irregularity index.

First, analogous to Figure 10, Figure 11 compares the local irregularity index obtained using p = 1 and p = 2. In Figure 11, a point above the line y = x means that the local irregularity index for p = 1 is smaller than for p = 2, that is, Φ l 1 (I, J) < Φ l 2 (I, J). Conversely, the inequality Φ l 1 (I, J) > Φ l 2 (I, J) yields a point below the line y = x. Although the local irregularity index is predominantly larger for p = 2 than for p = 1, they are similar for values greater than 5%. Indeed, the coefficient of determination increases from 0.77 for the 500 images to 0.89 restricting both Φ l 1 (I, J) > 5% and Φ l 2 (I, J) > 5%. The less irregular the morphological operator is, the more significant is the parameter p to the irregularity index. Indeed, although the local irregularity index with p = 2 ranked the approaches based on the total orders well, it could not distinguish the irregularity introduced by the morphological operators based on the marginal and the Loewner orders.

Let us now turn our attention to the size of the local windows and the structuring element. We consider square structuring elements of size 3 × 3, 5 × 5, . . . , 25 × 25 and local windows of size 4 × 4, 8 × 8, 16 × 16, and 32 × 32. Figure 12 shows the median of the local irregularity index by the size of the structuring element obtained by considering four different local windows. The area between the first and third quartiles is also included in Figure 12. Note from Figure 12 that the local irregularity index decreases with the size of the structuring element independently of the local windows. Moreover, there is no clear relationship between the size of the local windows and the structuring element. In particular, the local window does not need to be larger nor smaller than the structuring element. Also, note that the local irregularity index increases with the size of the local windows. Despite the noticeable differences between the median of the local irregularity index, the filled area between the first and third quartiles overlaps for local windows of size 16 × 16 and 32 × 32 for some approaches. Accordingly, the local irregularity index should converge to the global irregularity index as the size of the local windows increases. Finally, although we face a trade-off between computational cost and approximating the global irregularity index, fixed local windows can be confidently used for comparing the irregularity of different morphological operators or approaches.

Concluding Remarks

Despite the rigorous formulation of the irregularity issue provided by Chevallier and Angulo [START_REF] Chevallier | The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology[END_REF], as far as we know, there was no effective measure for this problem. This paper proposed two measures for the irregularity of morphological operators: the global and the local irregularity indexes, denoted by Φ g p and Φ l p respectively. The global irregularity index is given by the relative gap between the Wasserstein metric W p and the generalized sum of pixel-wise distances D p . The computational experiments detailed in Section 3 confirmed good values for the irregularity of tiny color images. However, in practical situations, the number of distinct pixel values makes it impossible to compute the global irregularity index. We proposed computing the irregularity index on several non-overlapping local windows to circumvent this computational drawback, resulting in the local irregularity index. Furthermore, the computational cost can be reduced significantly by replacing the analytical Wasserstein metric with entropic regularized approximations of the optimal transport problem, such as the one produced by the Sinkhorn or the stabilized Sinkhorn methods [START_REF] Peyré | Computational optimal transport[END_REF][START_REF] Schmitzer | Stabilized sparse scaling algorithms for entropy regularized transport problems[END_REF]. The computational experiments detailed in Section 3 confirmed the practical application of the local irregularity index as a measure of the irregularity in natural color images. Indeed, besides the illustrative examples provided in Figures 6789, we provided a statistical analysis with one hundred natural images from the Berkeley segmentation dataset using four different morphological operators obtained from five vector-valued approaches.

Finally, we would like to point out that the irregularity index can be used in the future to evaluate the performance of morphological operators. The irregularity index can also be used for the design of efficient morphological operators. For example, it can be used as the objective function for the design of vector-valued morphological operators based on uncertain reduced orderings [START_REF] Sangalli | Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings[END_REF].

where σ > 0 is a parameter. Moreover, α 1 , . . . , α K and β 1 , . . . , β M solve the quadratic optimization problem

                           maximize K i=1 α i + M j=1 β j - 1 2 K i,l=1 α i α l κ(f i , f l ) - 1 2 M j,l=1 β j β l κ(b j , b l ) + 1 2 K i=1 M j=1 α i β j κ(f i , b j ) subject to K i=1 α i - M j=1 β j = 0, 0 ≤ α i , β j ≤ C, (35)
where the parameter C > 0 controls the trade-off between the classification error and the margin of separation between background and foreground values [START_REF] Haykin | Neural Networks and Learning Machines, 3rd edn[END_REF][START_REF] Schölkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF].

In an unsupervised morphological approach, the mapping ρ : V → L is determined using a set of unlabeled values. The statistical depth projection-based approach, for example, determines the mapping ρ based on "anomalies" with respect to a background composed of the majority of pixel values of an image [START_REF] Velasco-Forero | Random Projection Depth for Multivariate Mathematical Morphology[END_REF]. Formally, suppose V ⊂ R d and L ⊂ R. Given a training sample represented by a matrix X = [x 1 , . . . , x n ] ∈ R d×n , the projection depth function ρ * P : R d → R is defined by

ρ * P (x) = sup u∈S d-1 |u T x -MED(u T X)| MAD(u T X) , ∀x ∈ R d , (36)
where S d-1 = {x ∈ R d : ||x|| 2 = 1}, MED : R n → R is the median operator, and MAD : R n → R is the median absolute deviation from the median operator. Recall that the median absolute deviation from the median is given by

MAD(t) = MED(|t -1 n MED(t)|), (37)
where 1 n ∈ R n denotes the vector of ones and the absolute value | • | is computed in a component-wise manner. In practice, we compute the depth projection function by replacing the supremum with the maximum on a finite set of elements in the hypersphere S d-1 . Formally, the function ρ P : R d → R given by

ρ P (x) = max u∈U |u T x -MED(u T X)| MAD(u T X) , ∀x ∈ R d , (38)
where U = {u 1 , u 2 , . . . , u k } ⊂ S d-1 , is taken as an approximation of the theoretical depth projection function ρ * P . The projection depth morphological approach is defined by ranking the vector-values according to the mapping ρ P : R d → R given by (38) together with a look-up table.

δ

  S (I)(p) = s∈S p-s∈D I(p -s) and ε S (I)(p) = s∈S p+s∈D

Fig. 1

 1 Fig. 1 Illustrative example of the irregularity issue. Image with three colors and its corresponding dilation by a cross structuring element using the RGB lexicographical and marginal orderings.

Example 1

 1 Consider the toy image I and the dilated image J L = δ L S (I) shown in Figure1. The operator's and the optimal transport plans are arranged respectively in the following matrices:

  of g 1 (I, J L ) Average of g 1 (I, J M )

Figure 3

 3 depicts some original images, the outcome of a morphological operator, and the corresponding global irregularity index. The images in the first column correspond to the original color images. The following columns present the output of morphological op-

Figures 6 -Fig. 6

 66 Fig. 6 Original color image and the corresponding dilation by a 9 × 9 square produced by the marginal, the Loewner, lexicographical, SVM-based, and projection depth approaches. The local irregularity index, obtained using local windows of size 16 × 16, is given together with the dilated images.

Fig. 8

 8 Fig. 8 Original color image and the corresponding opening by a 9 × 9 square produced by the marginal, the Loewner, lexicographical, SVM-based, and projection depth approaches. The local irregularity index, obtained using local windows of size 16 × 16, is given together with the opening images.

Fig. 9

 9 Fig. 9 Original color image and the corresponding opening by a 9 × 9 square produced by the marginal, the Loewner, lexicographical, SVM-based, and projection depth approaches. The local irregularity index, obtained using local windows of size 16 × 16, is given together with the opening images.

Fig. 10

 10 Fig.10Scatter plot of the local irregularity index computed using the analytical Wasserstein metric and the stabilized Sinkhorn method with regularization term = 10 -3 , both with local windows of size 16 × 16.

Fig. 11

 11 Fig.11Scatter plot of the local irregularity index computed using the parameters p = 1 and p = 2.

Fig. 12

 12 Fig. 12 Local irregularity index by the size of the structuring element. The first and third quartiles delimit the filled area.

The source-codes for the Loewner morphological approach can be found at https://www.math. tu-cottbus.de/INSTITUT/lsnmwr/kleefeld/SourcePRL/.

The Julia's source-code for the global irregularity index is available at https://github.com/ mevalle/Irregularity-Index.

The Julia's source-code for the local irregularity index is available at https://github.com/ mevalle/Irregularity-Index.

https://optimaltransport.github.io/

https://github.com/JuliaOptimalTransport/OptimalTransport.jl

This work was supported in part by the São Paulo Research Foundation (FAPESP) under grant no 2019/02278-2.

Original image

Marginal: 0.47% Loewner: 0.77% Lexicographical: 0.95% SVM-based: 5.34% Projection Depth: 18.11% the global and local irregularity represents well the visual irregularity introduced by the morphological operators. Moreover, although the local irregularity index yields a lower bound for the global irregularity index, we advocate its use for measuring the irregularity of vectorvalued morphological operators.

Entropic regularized methods for approximating the Wasserstein metric

Besides partitioning the domain of a vector-valued image into local windows, we can further reduce the computational cost for computing the irregularity index by considering entropic regularized methods for approximating the analytical Wasserstein metric. In a few words, entropic regularized methods like the Sinkhorn method and its stabilized version add an entropic penalty to the original transport problem. In mathematical terms, the entropic reg-

Appendix A Supervised and Unsupervised Morphological Approaches

Let us briefly review the supervised and unsupervised vector-valued morphological approaches, whose details can be found in [START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF][START_REF] Velasco-Forero | Random Projection Depth for Multivariate Mathematical Morphology[END_REF].

In a supervised ordering, the surjective mapping ρ : V → L is defined using a set F ⊂ V of foreground values and a set B ⊂ V of background values such that F ∩ B = ∅. Given the sets F and B, the mapping ρ is expected to satisfy the inequality ρ(f ) > ρ(b) for f ∈ F and b ∈ B. Considering V ⊂ R d and L ⊂ R, the decision function of a SVM can be used to accomplish this goal [START_REF] Schölkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF][START_REF] Vapnik | Statistical Learning Theory[END_REF][START_REF] Velasco-Forero | Supervised Ordering in Rp: Application to Morphological Processing of Hyperspectral Images[END_REF]. Precisely, consider sets F = {f 1 , . . . , f K } ⊂ R d and B = {b 1 , . . . , b M } ⊂ R d of foreground and background values, respectively. An SVMbased morphological approach is obtained by considering the mapping ρ S : R d → R given by

where κ : R d × R d → R is a Mercer kernel [START_REF] Schölkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF]. For example, Gaussian radial basis function kernel is given by


Abdou Rachid Thiam
email: thiam@ens.fr

Vincent Faugeras

Olivier Duclos

Didier Bazile

Impact of cyclization and methylation on peptide penetration through droplet interface bilayers

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Bioavailability represents the yield of a drug that reaches the systemic circulation. It is a crucial parameter for the design of oral drugs. Unfortunately, improvements in drugs specificity and efficiency often come with a loss of intestinal absorption, leading to poor oral bioavailability. [START_REF] Macarron | Impact of High-Throughput Screening in Biomedical Research[END_REF][START_REF] Gard | Applied Pharmacology[END_REF] As such, activity and bioavailability both challenge the development of treatments, especially those for protein-related diseases which require high specificity. [START_REF] Kola | Can the Pharmaceutical Industry Reduce Attrition Rates?[END_REF] By combining the protein-protein blocking capabilities of biologics with the bioavailability of small molecules, some peptides potentially provide a solution to this problem. [START_REF] Morrison | Constrained Peptides' Time to Shine?[END_REF][START_REF] Martin-Loeches | A New Antibiotic Class in the Pipeline[END_REF] A growing family of peptides called cell-penetrating peptides (CPP) emerged as a promising family to chaperone active macromolecules through cell membranes, such as oligonucleotides, proteins, and medical imaging agents to target sites. [START_REF] Patel | Cell Penetrating Peptides: Intracellular Pathways and Pharmaceutical Perspectives[END_REF] Such an asset makes CCPs potent and versatile candidates for disease treatment. [START_REF] Chugh | Cell-Penetrating Peptides: Nanocarrier for Macromolecule Delivery in Living Cells[END_REF][START_REF] Zhang | Cell-Penetrating Peptide Mediates Intracellular Membrane Passage of Human Papillomavirus L2 Protein to Trigger Retrograde Trafficking[END_REF] However, CPPs often exhibit high proteolytic degradation sensitivity and poor endosomal membrane escape ability. [START_REF] Shibata | Improvement of Biological Activity and Proteolytic Stability of Peptides by Coupling with a Cyclic Peptide[END_REF][START_REF] Verdurmen | A Quantitative Comparison of Cytosolic Delivery via Different Protein Uptake Systems[END_REF] Yet, exceptions exist, such as CPPs with constrained patterns, e.g., cycles, staples, or turns. 11 These constraints result in a loss of entropy and the presence of hidden structures in the molecular backbone. Thus, cyclization, for example, likely offers resistance to proteolytic degradations [START_REF] Shibata | Improvement of Biological Activity and Proteolytic Stability of Peptides by Coupling with a Cyclic Peptide[END_REF] , might enhance cell membrane permeability, and, thereupon, represent potent candidates for oral delivery. [START_REF] Gentilucci | Chemical Modifications Designed to Improve Peptide Stability: Incorporation of Non-Natural Amino Acids, Pseudo-Peptide Bonds, and Cyclization[END_REF][START_REF] Liu | Membrane Permeable Cyclic Peptidyl Inhibitors against Human Peptidylprolyl Isomer Ase Pin1[END_REF][START_REF] Bock | Getting in Shape: Controlling Peptide Bioactivity and Bioavailability Using Conformational Constraints[END_REF] In particular, the constrained structure of hydrophobic cyclic peptides likely offer a higher bilayer membrane permeability. 11,[START_REF] Jobin | The Role of Tryptophans on the Cellular Uptake and Membrane Interaction of Arginine-Rich Cell Penetrating Peptides[END_REF][START_REF] Hoang | Connecting Hydrophobic Surfaces in Cyclic Peptides Increases Membrane Permeability[END_REF] This is the case of Cyclosporin-A which is a cyclopeptide found in fungi that exhibits a high bioavailability and bioactivity. [START_REF] Tjia | Cyclosporin Metabolism by the Gastrointestinal Mucosa[END_REF][START_REF] Borel | Biological Effects of Cyclosporin A: A New Antilymphocytic Agent[END_REF] The cyclic nature and multiple backbone N-methylations of Cyclosporin-A has been found to be the key features of its bioavailability. [START_REF] Räder | Improving Oral Bioavailability of Cyclic Peptides by N-Methylation[END_REF][START_REF] Hill | Constraining Cyclic Peptides to Mimic Protein Structure Motifs[END_REF] N-methylation can affect conformation and hydrogen bonding potentials. In this way, it may modulate the bioavailability and bioactivity of peptides. [START_REF] Rezai | Testing the Conformational Hypothesis of Passive Membrane Permeability Using Synthetic Cyclic Peptide Diastereomers[END_REF] Accordingly, multiple pieces of evidence showed a correlation between backbone methylation and membrane permeability. [START_REF] Chatterjee | N-Methylation of Peptides: A New Perspective in Medicinal Chemistry[END_REF][START_REF] Doedens | Multiple N -Methylation of MT-II Backbone Amide Bonds Leads to Melanocortin Receptor Subtype HMC1R Selectivity: Pharmacological and Conformational Studies[END_REF][START_REF] Chatterjee | Multiple N-Methylation by a Designed Approach Enhances Receptor Selectivity[END_REF] Considering structural features that can induce constrained patterns in molecular backbones, somatostatin-inspired peptides were designed to identify minimum sequences required for an efficient cell permeation. Notably, few hexa-and nona-peptides derived from somatostatin showed significant intestinal permeability and oral bioavailability. [START_REF] Liu | Membrane Permeable Cyclic Peptidyl Inhibitors against Human Peptidylprolyl Isomer Ase Pin1[END_REF][START_REF] Biron | Improving Oral Bioavailability of Peptides by Multiple N-Methylation: Somatostatin Analogues[END_REF] With the benefits offered by these peptides, several methods have been set up for the earlystage assessment of their activity and bioavailability. They are often based on cell culture methodologies coupled with microscopy studies. [START_REF] Ter-Avetisyan | Cell Entry of Arginine-Rich Peptides Is Independent of Endocytosis[END_REF][START_REF] Patel | Cell-Penetrating Peptide Sequence and Modification Dependent Uptake and Subcellular Distribution of Green Florescent Protein in Different Cell Lines[END_REF] Unfortunately, such methods are not suitable for the pharmaceutical industry because of frequent reproducibility issues and inability to scale them up to high throughput screening. Therefore, most permeability studies rely on Caco-2 cell models and PAMPA systems. Caco-2 cells are small intestine cells put in culture to mimic the intestinal barrier. This method cumulates contributions from both passive and active permeabilities, and diffusion through cell junctions, which cannot be decorrelated.

PAMPA is based on a filter plate made of pores, infused with a micrometer thick solution of phospholipids in oil, and placed between two aqueous compartments. [START_REF] Wang | Rational Design and Synthesis of an Orally Bioavailable Peptide Guided by NMR Amide Temperature Coefficients[END_REF][START_REF] Rezai | Conformational Flexibility, Internal Hydrogen Bonding, and Passive Membrane Permeability: Successful in Silico Prediction of the Relative Permeabilities of Cyclic Peptides[END_REF][START_REF] Ghasemy | Fine-Tuning the Physicochemical Properties of Peptide-Based Blood-Brain Barrier Shuttles[END_REF][START_REF] Morimoto | A Parallel Permeability Assay of Peptides across Artificial Membranes and Cell Monolayers Using a Fluorogenic Reaction[END_REF] Despite the presence of phospholipids, the water-oil-water film is distinct from phospholipid bilayer membranes.

Model membranes are a suitable tool to assess the passive membrane permeability of peptides, potentially translatable to on-chip high throughput systems. Unilamellar vesicles particularly enable to assess the ability of polycationic and amphipathic peptides to cross cell membranes and to depict permeation mechanisms. [START_REF] Bárány-Wallje | A Critical Reassessment of Penetratin Translocation across Lipid Membranes[END_REF][START_REF] Persson | Vesicle Size-Dependent Translocation of Penetratin Analogs across Lipid Membranes[END_REF][START_REF] Swiecicki | The Efficacies of Cell-Penetrating Peptides in Accumulating in Large Unilamellar Vesicles Depend on Their Ability to Form Inverted Micelles[END_REF][START_REF] Wheaten | Translocation of Cationic Amphipathic Peptides across the Membranes of Pure Phospholipid Giant Vesicles[END_REF] Droplet interface bilayers (DIBs) are droplet-based method which represent an interesting alternative for identifying peptidescargo systems suitable for membrane transport. [START_REF] Lein | Protein Transport across Membranes: Comparison between Lysine and Guanidinium-Rich Carriers[END_REF][START_REF] Li | Peptide-Mediated Membrane Transport of 12369-12374[END_REF][START_REF] Huang | Direct Quantitation of Peptide-Mediated Protein Transport across a Droplet-Interface Bilayer[END_REF] Indeed, DIBs are versatile, made rapidly with controlled membrane composition, and they offer a straightforward visualisation and quantification of permeation. [START_REF] Lee | Static and Dynamic Permeability Assay for Hydrophilic Small Molecules Using a Planar Droplet Interface Bilayer[END_REF][START_REF] Czekalska | Passive and Parallel Microfluidic Formation of Droplet Interface Bilayers (DIBs) for Measurement of Leakage of Small Molecules through Artificial Phospholipid Membranes[END_REF][START_REF] Faugeras | Membrane Determinants for the Passive Translocation of Analytes through Droplet Interface Bilayers[END_REF][START_REF] Strutt | UV-DIB: Label-Free Permeability Determination Using Droplet Interface Bilayers[END_REF] Here, we used DIBs to study the passive permeation of a range of hydrophobic peptides derived from somatostatin, a somatotropin release-inhibiting factor. The impact of cyclization and backbone methylation on the resulted permeability was studied. Cyclization improved the permeation of the tested peptides. Methylation also improved permeation but not systematically. By exploring the physical chemistry of the interaction of the peptides with the DIB bilayer, we discussed possible hypotheses for the increased permeability of the peptides.

Results and discussion

Droplet pairing and permeability determination. With a millifluidic device, we generated separately donor (filled with peptides) and acceptor water droplets in an oil phase containing phospholipids. Binary droplets decorated with a monolayer of phospholipids were then brought together under the microscope (Figure 1.a). When the droplets were approached to each other, they spontaneously adhered and formed a droplet interface bilayer (DIB). [START_REF] Huang | Direct Quantitation of Peptide-Mediated Protein Transport across a Droplet-Interface Bilayer[END_REF][START_REF] Poulin | Adhesion of Water Droplets in Organic Solvent[END_REF] The bilayer was fully formed within a minute, depending on the solvent and phospholipids used. [START_REF] Thiam | From Stability to Permeability of Adhesive Emulsion Bilayers[END_REF] We used glyceryl trioctanoate (TO) for the oil phase and DOPE for phospholipids unless stated otherwise. We previously showed that the passive permeability of soluble fluorophores to such a DIB bilayer is similar to that of common membranes. [START_REF] Faugeras | Membrane Determinants for the Passive Translocation of Analytes through Droplet Interface Bilayers[END_REF][START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF] We used a buffer solution containing 50 mM HEPES, 120 mM KAcetate, and 1 mM MgCl 2 in Milli-Q water (pH 7.4, 293 mOsm, I = 173 mM) with an addition of DMSO (5 % v/v), enabling a complete dissolution of the peptides at the concentration of 5 µM, used for all experiments.

The buffer osmolarity of ~ 300 mOsm was slightly reduced by the addition of DMSO but maintained in a physiological range. Peptides at 5 µM barely modulated the buffer osmolarity. Thus, osmotic fluxes between donor and acceptor compartments are nonexistent.

For the studied peptides, the possible partitioning to the oil phase was largely negligible in front of their membrane permeation rate. The concentration of the fluorescent peptide in the acceptor droplet was thus obtained from the following mass conservation:

where is the concentration and V is the volume; " " denoting for acceptor and " " for donor compartment;

is the concentration of the solute in the donor droplet at t 0 . Since the concentration is proportional to the fluorescence intensity F, the mass conservation can be written simply as: which yields:

Here, the contribution of bleaching is irrelevant since we consider the ratio of the fluorescence between the acceptor and donor droplets that would be similarly bleached. We defined the translocation parameter at 30 minutes as:

which quantifies the permeability of the bilayer to the considered analyte ( Cyclization significantly increases permeation rate. We investigated the impact of cyclization for a range of peptides derived from somatostatin (Figure 1.c; L stands for linear, C for cyclic, m for methylated). C1, the cyclic counterpart of L1 was found to be much more permeable (Figure 2.a-c). Indeed, permeability was improved by about one order of magnitude by cyclization, from 1.4.10 -7 cm.s -1 to 1,0.10 -6 cm.s -1 . Yet, both peptides have similar chemical properties (MW, hydrophilicity). Importantly, we attached a specific terminal group, acetamide for both N-and C-terminus, to prevent their ionization (Figure S2.a).

Thus, the net charges of peptides L1 and C1 were equal and the difference in permeation should then result from the cyclization of the peptide.

We also compared the cyclic peptide C1 with well-established linear peptides, the polycationic TAT, and the amphipathic Penetratin (Pen). Likewise, the linear peptides were much less permeable compared with the cyclic C1, with respectively 5.3.10 -8 cm.s -1 for Pen, 1.4.10 -8 cm.s -1 for TAT, and 1.8.10 -7 cm.s -1 for C1 (Figure S3.a-d). Yet, these discrepancies may not be solely ascribed to the cyclic nature of C1, as the peptides have different net charges, 0, 8, and 9 respectively for C1, Pen, and TAT. Indeed, charge is a determinant parameter for leaving the water environment and to cross the bilayer.

Our findings are consistent with several studies showing that cyclic peptides are better permeants than their linear counterparts. [START_REF] Dougherty | Understanding Cell Penetration of Cyclic Peptides[END_REF][START_REF] Nischan | Covalent Attachment of Cyclic TAT Peptides to GFP Results in Protein Delivery into Live Cells with Immediate Bioavailability[END_REF] However, whether cyclization systematically improves permeability may depend on the amino acid sequence. [START_REF] Kwon | Quantitative Comparison of the Relative Cell Permeability of Cyclic and Linear Peptides[END_REF] In any case, designing peptides to increase their lipophilicity and reduce interaction with solvents will improve their permeation. [START_REF] Wang | Exploring Experimental and Computational Markers of Cyclic Peptides: Charting Islands of Permeability[END_REF] The contribution of cyclization in this picture is likely related to constraints in the molecular structure of the peptide, e.g., through hiding H-bond in the backbone and exposing hydrophobic group. [START_REF] Wang | Rational Design and Synthesis of an Orally Bioavailable Peptide Guided by NMR Amide Temperature Coefficients[END_REF] Indeed, such constraint would increase the apparent lipophilicity, reduce interactions with the solvent and, hence, improve permeation rates. Impact of backbone N-methylation on permeation. We investigated the impact of amide group methylations on permeation rate for the same group of peptides. To do so, we designed a methylated counterpart of C1 peptide, C1m (Figure 1.c), with two methylated amide groups on its two first leucine residues. Compared to C1, C1m showed a significant increase in permeation (Figure 3.a-b). Permeability increased from 1.0.10 -6 cm.s -1 for C1 to 3.7.10 -6 cm.s -1 for C1m.

We decided to change the nature of the cargo dye, by using a FITC-labelled analog, to test whether our findings hold (Figure S3.b,c; (f) stands for FITC-labelled). The impact of methylation on permeability for C1 was then studied by repeating the permeation assay.

First, we found the permeation rate to differ from TAMRA. Indeed, for the FITC cargo, we measured reduced permeability values, from 1.0.10 -6 cm.s -1 to 1,77.10 -7 cm.s -1 and 3.7.10 -6

cm.s -1 to 1,43.10 -6 cm.s -1 , respectively for C1(f) and C1m(f). These variations indicated that the nature of the cargo modulated permeability. Here, the slowdown in permeation may be ascribed to the net charge difference between FITC, having two additional charges, and the zwitterionic TAMRA at our physiological-working pH. Second and foremost, C1m(f) still had a much higher permeability than C1(f), by almost an order of magnitude (Figure S3.b, c).

Thus, we found that methylation improved the permeability of the cyclic peptide C1 with two different cargos.

We also studied whether methylation improves the permeability of the corresponding linear L1. Like C1m, L1m features two methylated groups on the same residues (two first leucine residues). As between L1 and C1, L1m had a lower permeation rate than C1m, i.e. 3.1.10 -8 cm.s -1 to 3.7.10 -6 cm.s -1 , a two orders magnitude difference (Figure 3.c). While C1m had a higher permeability than C1, it is interesting to note that the permeability for L1m was significantly reduced compared to L1. This trend could be due to the reduction of donor Hbond for methylated L1m, which would reduce the possibilities of intramolecular H-bonding constraining the peptide. In any case, these findings support that the relevance of methylation in improving permeability is likely more prominent to cyclic peptides. C2m, i.e., from 5.6.10 -6 cm.s -1 and 2.4.10 -6 cm.s -1 , but increased by methylation between C3 and C3m, 2.6.10 -6 cm.s -1 and 3.7.10 -6 cm.s -1 (Figure 4.b). Thus, for C1 and C3, and not C2, methylation improved permeability.

Methylation may modulate both peptide lipophilicity and interaction with the solvent. When amides are methylated, peptide interaction with the solvent is decreased through the reduction of H-bonds. In this way, methylation might confer higher intramolecular constraints. [START_REF] Wang | Rational Design and Synthesis of an Orally Bioavailable Peptide Guided by NMR Amide Temperature Coefficients[END_REF] These constraints may reduce conformational possibilities offering interactions with water molecules and, thereupon, improve permeability. This might have been the case for C1m and C3m. Yet, the position of the methylation would be crucial as it could also prevent intramolecular H-bonding. In such an eventuality, the nature of the resulting constraint may not be favorable for membrane permeation. This might have occurred with

C2m that showed reduced permeability compared to C2. In conclusion, the methylation of cyclic peptides can improve the passive permeability but not systematically, and the methylation position might be detrimental to the improvement of permeability. The DIB assay represents an accurate model for permeability assessment. The permeability of the studied peptides was assayed with the industrial gold-standard methods PAMPA and Caco-2 (Figure 5.a). [START_REF] Wang | Exploring Experimental and Computational Markers of Cyclic Peptides: Charting Islands of Permeability[END_REF] However, the results with these methods are often discrepant. [START_REF] Wang | Rational Design and Synthesis of an Orally Bioavailable Peptide Guided by NMR Amide Temperature Coefficients[END_REF] In the PAMPA approach, a phospholipid bilayer is not generated and does not reproduce properly a phospholipid bilayer. The Caco-2 approach cumulates multiple permeabilities: through cell junctions, passive permeability across the plasma membrane, and permeability via endocytic pathways. Thus, it may not be surprising that these methods become discrepant for specific conditions. Also, these methods would not yield the absolute bilayer permeability, as model bilayers such as DIBs would enable. Therefore, we decided to compare the permeability values of the cyclic peptides between the three approaches.

The permeability values of C1, C1m, C2, C2m, C3, and C3m were evaluated via PAMPA and Caco-2 cells by Wang et al. (2015). [START_REF] Wang | Exploring Experimental and Computational Markers of Cyclic Peptides: Charting Islands of Permeability[END_REF] Considering the impact of the fluorescent cargo reporter we attached to the peptides, we renormalized all results and analyzed the relative permeability of the peptides for each assay ( Permeability dependence on membrane recruitment. We found that the peptides were recruited to the DIB interfaces (Figure 3.a, 4.a). Since their translocation requires contact with the membrane, we asked whether the interfacial signal of the peptides could be a proxy for permeability.

We defined an interfacial recruitment parameter, R m and R b , respectively quantifying the signal of the peptides on the monolayer and the bilayer (Figure 6.a). Methylation systematically increased the interfacial recruitment R m and R b of the cyclic peptides (Figure

6.b,c).

To know whether the peptides were inserted between phospholipids, we conducted the assay with a lipid composition offering a better tightening of the phospholipid bilayer. [START_REF] Faugeras | Membrane Determinants for the Passive Translocation of Analytes through Droplet Interface Bilayers[END_REF][START_REF] Caillon | Triacylglycerols Sequester Monotopic Membrane Proteins to Lipid Droplets[END_REF] Namely, we used a phospholipid composition made of DOPE:DOPC 1:1. With this new composition, the peptides were almost totally excluded from the interface (Figure S4.a,b).

Therefore, the phospholipid bilayer packing level correlates with the capacity of the peptides to insert into the membrane. Methylated peptides have thus a higher tendency to insert.

Interestingly, the decrease in the peptide recruitment level to the bilayer was accompanied by a reduction in permeability for C1m, from 3.7.10 -6 cm.s -1 for DOPE and 1.5.10 -6 cm.s -1 for DOPE:DOPC 1:1 (Figure S4.b). At least for this peptide, the insertion capacity of the peptides seems to correlate with permeability.

To further verify our conclusion, we plotted all the permeability values for each peptide as a function of the bilayer recruitment R b, and regardless of the membrane composition. We observed that permeability tends to increase with R b for both C1/C1m and C3/C3m pairs (Figure 6.d); we noticed an almost linear dependence between the peptide recruitment to the phospholipid bilayer with permeability (Figure 6.d). For these peptides, the recruitment to the bilayer seemed determinant to permeation. Their membrane recruitment was likely mediated by the hydrophobic residues in the peptides (L, Y) [START_REF] Campelo | The Hydrophobic Insertion Mechanism of Membrane Curvature Generation by Proteins[END_REF][START_REF] Bird | Biophysical Determinants for Cellular Uptake of Hydrocarbon-Stapled Peptide Helices[END_REF] and the methylation at the Leucine could improve their orientation and interaction with the membrane. For instance, such a membrane recruitment-based mechanism promotes the translocation of amphipathic peptides. [START_REF] Jobin | The Role of Tryptophans on the Cellular Uptake and Membrane Interaction of Arginine-Rich Cell Penetrating Peptides[END_REF][START_REF] Gehan | Penetratin Translocation Mechanism through Asymmetric Droplet Interface Bilayers[END_REF] However, C2/C2m showed inconsistent behaviors (Figure 6.d), as its translocation rate was not reflected by its interfacial recruitment (except if C2m was considered alone). This observation indicates that the capacity of a peptide to insert into the bilayer membrane represents only one parameter controlling permeability. Other parameters might become more prominent depending on the nature of the peptide, as for C2. Thus, although the peptide insertion in the bilayer could be a proxy for permeability, the insertion level does not systematically correlate with the penetrating ability of the peptide. This will be especially true for peptides having a high intrinsic translocation capacity, e.g., with high flexibility allowing them to fit both hydrophilic and hydrophobic environment.

Conclusion

Poor intestinal permeability is a key limitation to the development of bioavailable drugs suitable for oral delivery. Therefore, there is a crucial need for assays that can enable robust and high-throughput measurement of cell permeability. Current gold-standard systems are PAMPA wells and Caco-2 cells. [START_REF] Nigsch | In Vitro Models for Processes Involved in Intestinal Absorption[END_REF] However, the permeability of many compounds assessed with these systems was discrepant. [START_REF] Hallifax | Clearance-Dependent Underprediction of in Vivo Intrinsic Clearance from Human Hepatocytes: Comparison with Permeabilities from Artificial Membrane (PAMPA) Assay, in Silico and Caco-2 Assay, for 65 Drugs[END_REF] In this study, we used a biomimetic DIB assay to investigate the permeability of various molecules tested by these methods and highlight determinant features that had not been revealed. We found that DIB assay can efficiently discriminate peptides based on their structure when they have similar amino acid content.

Indeed, tiny structural modifications such as amide methylation or amino acids chirality inversion induced significant changes in peptide permeation through bilayers. The DIB assays allowed us to quantify permeability variations in three orders of magnitude, with intermediate performance between PAMPA and Caco-2. Altogether, the DIB assay appears as a solid alternative to PAMPA to study cell permeability and is highly complementary to Caco-2. It demonstrates the ability to consistently distinguish the passive permeation capacity of peptides designed with tiny structural differences. Translating this assay into high throughput will make it a promising tool for the pharmaceutical industry.

Figure 1 .

 1 b). During permeation, the volume of the droplets and the patch area were almost constant (Figure S1.a). Unless mentioned otherwise, the fluorescent probe used to study the peptides' translocation and distribution was the fluorescein-derivative TAMRA, attached to the lysine residue of the peptide (Figure S2.a,b).

Figure 1 :

 1 Figure 1: Passive permeability DIB assay with hydrophobic peptides. a) Left: Schematic of the proposed DIB permeability assay. Two sets of W/O droplets, empty and peptide-filled (respectively acceptor and donor droplets), are produced with a millifluidic device and dropped on a treated glass slide for monitoring under a confocal microscope. Right: Microscopic pictures showing droplet paring and DIB parameters. Scale bars, 100 µm. b) Example of permeation kinetics of peptide L1 through glyceryl trioctanoate DOPE DIB; introduction of permeation parameter p 30 . c) Sequence of the studied peptides, specific methylations are highlighted in grey.

Figure 2 :

 2 Figure 2: Comparison of the linear and cyclic peptides L1 and C1. a) Confocal microscopy pictures of DIB permeation assay for L1 and C1 peptides over time. The same signal enhancement correction was applied to every single image to be able to detect significative signals of translocation for both peptides (image brightness is enhanced). Corresponding L1 and C1 fluorescence profiles are provided. Scale bars, 100 µm. b) Permeation kinetics for peptides L1 and C1. c) Calculated permeability for L1 and C1 peptides (defined as permeation parameter p 30 ) (mean value, error bars represent the SD; **** indicates p < 0.0001).

Figure 3 :

 3 Figure 3: Comparison of cyclic peptides C1 and C1m. a) Confocal microscopy pictures of DIB permeation assay for C1 and C1m peptides over time. Corresponding C1 and C1m fluorescence profiles are provided. Scale bars, 100 µm. b) c) Calculated permeability for C1, C1m, L1 and L1m peptides (defined as permeation parameter p 30 ) (mean value, error bars represent the SD; **** indicates p < 0.0001).

Figure 4 :

 4 Figure 4: Comparison of cyclic peptides C1, C1m, C2, C2m, C3, and C3m. a) Confocal microscopy pictures of DIB permeation assay for C2, C2m, C3, and C3m peptides over time. Corresponding fluorescence profiles are provided. Scale bars, 100 µm. b) Calculated permeability for C1, C1m, C2, C2m, C3 and C3m peptides (defined as permeation parameter p 30 ) (mean value, error bars represent the SD; **** indicates p < 0.0001).

Figure 5 .

 5 b). Overall, we found that permeability values from the DIB assay were intermediate between PAMPA and Caco-2 assays. Yet, while the permeability value for C3m relative to C3 shows a decrease of permeability in the case of PAMPA (0.5 times), it is increased for the DIB (1.4 times) and Caco-2 (13 times) assays. Moreover, methylated peptides showed close permeability patterns for the DIB and Caco-2 assays when PAMPA significantly underestimates the value for C2m and C3m relative to C1m (Figure5.b). These comparisons reveal that the DIB assay is an encouraging approach to evaluate the passive permeability of peptides, given that Caco-2 and PAMPA might misevaluate permeability. This will be especially the case when peptides can cross cell membranes through several mechanisms. In such a case, Caco-2 might overestimate the passive permeability, e.g., because of active transports or para-and transcellular diffusion, while PAMPA might underestimate the permeation efficiency because of the thick layer of oil between the two phospholipid monolayers. Collectively, our results suggest that the DIB assay reflects better than PAMPA the passive permeability of peptides across bilayers and it is a good compromise between PAMPA and Caco-2.

Figure 5 :

 5 Figure 5: a) Schematic illustration of common in vitro assays used to assess membrane permeability,

Figure 6 :

 6 Figure 6: a) Schematic representation of DIB and the associated method to obtain fluorescence profile and peptide recruitment R m and R b . Scale bars, 100 µm. b) Monolayer donor (mD) and bilayer (b) recruitment values for peptides C1 and C1m. c) Bilayer recruitment values R b for all three peptide couples (C1, C1m, C2, C2m, C3, C3m) d) Permeability value for each DIB assays as a function of

Material and Method

Products. All phospholipids (Avanti Polar Lipids), glyceryl trioctanoate (Sigma-Aldrich), fluorescein sodium salt (VWR Chemicals), HEPES (Sigma-Aldrich), potassium acetate KCH 3 COO (Sigma-Aldrich) and magnesium chloride MgCl 2 (Sigma-Aldrich) are used without further purification. Penetratin and TAT peptides were purchased at Anaspec. Linear peptides L1, L1m and cyclic peptides C2, C2m, C3, C3m were synthesized at Peptides & Elephants GmbH. Other cyclic peptides were purchased by Peptides International (USA).

Structure and purity were respectively checked by mass spectroscopy and HPLC (purity > 90 %).

Unless mentioned otherwise, peptides are first aliquoted down to 100 µM in a mixture of HKM buffer and DMSO (50/50% v/v) suitable for the complete dissolution of peptide. Peptide samples are then diluted down to 5 µM for W/O droplets fabrication and assays.

Preparation of the Oil Phase. Phospholipids used for the oil phase are conditioned in chloroform. We first evaporate chloroform under a stream of argon; the dried lipids are subsequently re-solubilized to the desired concentration in glyceryl trioctanoate (TO). Unless stated otherwise, a lipid concentration of 0.1 wt% is used for all experiments, this value is above the critical concentration for every studied phospholipid composition.

Millifluidic setup to fabricate monodisperse W/O droplets.

Experiments are performed with a 0.1 wt. % lipids in TO solution and the following HKM buffer recipe: 50 mM HEPES, 120 mM KCH 3 COO, and 1 mM MgCl 2 in Milli-Q water (at pH 7.4). For our assays, it is necessary to produce two types of droplets: acceptor droplets filled with buffer and donor droplets filled with a 5 μM solution of peptides in buffer. To do so, we use two parallel circuits Permeability Assay. All permeability assays are conducted at room temperature T lab = 21°C under a laser scanning microscope (LSM 800, Carl Zeiss, Oberkochen, Germany). Two droplets, one acceptor and one donor are selected. We then bring the droplets into contact enabling the adhesion of monolayers and the spontaneous formation of a bilayer. Just after DIB formation, we follow the passage of the analyte through the interface by taking pictures of the droplets every minute. The permeation is studied over 30 minutes (Figure 1 We finally define the translocation parameter at 30 minutes: p 30 as:

Where and are respectively the measured peptide concentration at t = 30 min and the initial peptide concentration in the acceptor droplet, is the patch area and is the duration of the experiment (here 30 minutes). The translocation ratio is weighted with both and to consider variation in droplet morphology and divided by to be homogeneous to a permeability. Each permeability value was determined by its mean and standard deviation.

We performed a minimum of 10 measurements for each peptide or lipid composition. The Measurement of monolayer and bilayer peptide recruitment. From confocal microscopy picture of DIBs, we can study the interfacial behaviour of peptides. The fluorescent intensity profile is obtained via mean segment profile integration (25 px lateral integration) for each droplet interface bilayers (Fig. 6a). We then introduce a peptide recruitment factor calculated from the mean fluorescence value of droplet bulk and the maximum fluorescence intensity on bilayer and/or monolayers. The peptide recruitment factor is defined as:

Where , and respectivly refer to the bilayer, the bulk and the monolayer of donor/acceptor droplet fluorescence intensity (a.u.). Peptide recruitment was considered at the end of the permeability assay, after 30 minutes, letting enough time for peptide to reach interfacial equilibrium.

Quantification and statistical analysis.

Supporting information

Peptide details and additional data (control experiments, FITC-labeled peptides permeation behavior, data treatment) are provided.