Amelia Carolina Sparavigna

Image Segmentation Applied to the Analysis of Fabric Textures

Keywords: Image Processing, Image Segmentation, Texture Analysis, Textiles, Fabric Fault Detection

In this work, we are proposing the use of image segmentation to the analysis of the textures of fabrics, with the aim of applying this approach to a fabric fault detection based on image processing.

In spite of the presence of commercial systems, the research for further improvements of the methods for textile fault detection continues, as evidenced by the recent publications on this subject [START_REF] Jing | Fabric defect detection using Gabor filters and defect classification based on LBP and Tamura method[END_REF][START_REF] Schneider | A traverse inspection system for high precision visual on-loom fabric defect detection[END_REF][START_REF] Bhangale | An Introduction to Textile Defect Identification and Classification Using Wavelet Transform and Neural Networks[END_REF][START_REF] Li | The Real-Time Vision System for Fabric Defect Detection with Combined Approach[END_REF][START_REF] Huang | A Study on the Algorithm of Fault Information Automatic Detection for High-Precision Intelligent Instruments[END_REF][START_REF] Hu | Computer Technology for Textiles and Apparel[END_REF]. The reason for the necessity of further studies is motivated by the intrinsic difficulty of having an artificial vision of textiles; this difficulty consists in the fact that fabric faults are often very small and hardly detectable, having a visibility strongly dependent on illumination, reduced by the vibrations of the mounting devices [START_REF] Sari-Sarraf | Vision systems for on-loom fabric inspection[END_REF][START_REF] Campbell | Automatic vision inspection of woven textiles using a twostage defect detector[END_REF][START_REF] Yau | Optimization of the illumination beam size of an optical textile defect inspection system[END_REF][START_REF] Ribolzi | Real time fault detection on textiles using opto-electronic processing[END_REF][START_REF] Kasdan | Industrial application of diffraction pattern sampling[END_REF].

For the analysis of fabric textures, several statistical approaches had been used and developed [START_REF] Haralick | Textural features for image classification[END_REF][START_REF] Abouelela | A statistical approach for textile fault detection[END_REF], and also methods based on the Fourier analysis, on the Gabor filtering and on wavelets with adaptive bases [START_REF] Chan | Fabric defect detection by Fourier analysis[END_REF][START_REF] Arivazhagan | Fault segmentation in fabric images using Gabor wavelet transform[END_REF][START_REF] Kumar | Defect detection in textured materials using Gabor filters, Industry Applications[END_REF][START_REF] Jasper | Texture characterization and defect detection using adaptive wavelets[END_REF]. In addition, an approach based on an image processing, developed for the study of liquid crystals [START_REF] Montrucchio | A new image processing method for enhancing the detection sensitivity of smooth transitions in liquid crystals[END_REF], was given in [START_REF] Sparavigna | Performing Textile Fault Detection by Means of Texture Analysis[END_REF][START_REF] Sparavigna | Texture analysis for textile fault detection[END_REF][START_REF] Sparavigna | Diffractive optics for fabric fault detection[END_REF]. Recently we have also proposed the use of the GIMP Retinex filtering to enhance the visibility of defects [START_REF] Sparavigna | The GIMP Retinex Filter Applied to the Fabric Fault Detection[END_REF] (other applications of the Retinex filtering to microscopy, radiography and detection of vehicles in foggy images have been given in [START_REF] Sparavigna | GIMP Retinex for enhancing images from microscopes[END_REF][START_REF] Sparavigna | Effects of GIMP Retinex Filtering Evaluated by the Image Entropy[END_REF][START_REF] Marazzato | Retinex filtering of foggy images: generation of a bulk set with selection and ranking[END_REF][START_REF] Sparavigna | Evaluation of GIMP Retinex Filtering of Images by Means of the Shen++Max Shannon Entropy Finder[END_REF][START_REF] Sparavigna | An image processing approach based on Gnu Image Manipulation Program GIMP to the panoramic radiography[END_REF]). Here, we use another approach to the study of fabric textures, based on an image segmentation for evidencing and measuring the domains present in these textures. Before giving an example, let us discuss shortly the image segmentation.

Image Segmentation

In image processing, a segmentation is a method of partitioning an image into multiple sets of pixels, defined as super-pixels, in order to have a representation, which can be simpler than the original one or more useful to the following desired analyses [START_REF] Shapiro | Computer vision[END_REF]. For this reason, the segmentation of images is used in several applications; in particular, it is used in the medical image processing for stacking and comparing diagnostic results [START_REF] Pham | Current methods in medical image segmentation[END_REF][START_REF] Forghani | Parameter optimization of improved fuzzy c-means clustering algorithm for brain MR image segmentation[END_REF][START_REF] Vv | Wikipedia[END_REF].

A typical segmentation of an image is a method able to locate objects (the domains), which are present in the image frame, or the boundaries among the domains. Specifically, the segmentation is a process of assigning a label to every pixel in an image, such that the pixels having the same label share certain characteristics [START_REF] Vv | Wikipedia[END_REF]. As a consequence, the result of a segmentation is a set of "segments", or "superpixels", that are covering the whole image, or a set of contours, that is of "edges", extracted from the image. In this case, the segmentation gives the "edge detection". Several methods exist for segmentation, as we can appreciate from [START_REF] Vv | Wikipedia[END_REF].

Here we use a segmentation based on a thresholding of gray-scale images. By selecting a suitable threshold, the image is converted into a binary (black and white) image. In several cases, this is enough for evidencing the domains among the black or the white pixels. Details of the method of segmentation are given in [START_REF] Sparavigna | Image Segmentation Applied to the Study of Micrographs of Cellular Solids[END_REF]. In [START_REF] Sparavigna | A method for the segmentation of images based on thresholding and applied to vesicular textures[END_REF][START_REF] Sparavigna | Measuring the size of pores by the segmentation of images from scanning electron microscopy[END_REF][START_REF] Sparavigna | Measuring the particles in fly ash by means of an image segmentation[END_REF][START_REF] Sparavigna | Analysis of a natural honeycomb by means of an image segmentation[END_REF][START_REF] Sparavigna | Image segmentation applied to micrographs of microcellular plastics[END_REF][START_REF] Sparavigna | Measuring the size of tubules in phloem and xylem of plants[END_REF], we have demonstrated that this segmentation is suitable for the analysis of several textures, in particular, for those that we can define as "vesicular textures", where some vesicles or voids or empty areas are present. Actually, some "empty" areas can be present in the images of a fabric because of the void among the yarns. Therefore, we can segment the images by considering these voids as the super-pixels. The presence of a defect in the fabric can be evidenced by a change in the distribution of the super-pixels. Images are 600x600 pixels.

The two panels of Figure 1 are rendered into black and white images and segmented with the approach detailed in [START_REF] Sparavigna | Image Segmentation Applied to the Study of Micrographs of Cellular Solids[END_REF][START_REF] Sparavigna | A method for the segmentation of images based on thresholding and applied to vesicular textures[END_REF][START_REF] Sparavigna | Measuring the size of pores by the segmentation of images from scanning electron microscopy[END_REF][START_REF] Sparavigna | Measuring the particles in fly ash by means of an image segmentation[END_REF][START_REF] Sparavigna | Analysis of a natural honeycomb by means of an image segmentation[END_REF][START_REF] Sparavigna | Image segmentation applied to micrographs of microcellular plastics[END_REF][START_REF] Sparavigna | Measuring the size of tubules in phloem and xylem of plants[END_REF]. The result of the segmentation is shown in the Figure 2. Each black domain is evidenced in the segmentation by a different color tone (super-pixels). In the case of the "good" fabric, we use green tones; in the case of the "faulty" fabric, we use the red tones. Besides the maps, we can have the distributions of areas of the super-pixels. It is given in the Figure 3. Other examples Let us consider faulty textures discussed in [START_REF] Sparavigna | Performing Textile Fault Detection by Means of Texture Analysis[END_REF]. Faults in the structure of woven fabrics are deviations from the recurrence of a fundamental unit, and usually appear as subtle lines, dark or bright, in the image frame. The more frequently encountered defects are broken or missing picks. Dust, extraneous staples or oil spots can also be observed.

The first defect we are proposing is the mispick. This is a rather common defect, produced when a yarn is lacking or broken on the loom. It is a defect expanding on the surface fabric and involving several neighbor yarns. Mispicks are easy to find by eye inspection, such as dust and little oil spots. These defects are eye-inspected with back lighting. The same illumination system was used to record images in the Figures 5 and6 (left panels). In the Figure 5 we give a fabric and in the Figure 6 the same fabric with a mispick. In the middle, we see the corresponding binary image obtained by a threshold at the 111 grey-tone. On the right, the inverted image. The black domains represent areas of the fabric, which are transmitting more light.

Figure 6:

The same fabric of Figure 5 with a mispick (600x600 pixels, 600 pixels correspond to 40 mm).

In the middle, we find the binary image obtained by a threshold at 111 grey-tone. On the right, the inverted image. Here too the black domains represent areas that are transmitting more light.

The segmentation of the image on the right of Figure 5 gives us the "good" distribution that we can compare to the distribution coming from the segmentation of the right panel in the Figure 6. Figure 7 gives the maps resulting from the segmentation. On the left, the map is in green tones; on the right, in red tones. Again, the maps look quite different. The Figure 8 allows the comparison of the distributions. As previously observed for the image from Brodatz album, the distribution of areas of super-pixels obtained by means of an image segmentation allows to distinguish good and faulty textures.

Another fabric, where we found a defect (missing pick), is proposed in the Figures 9 and10. In them, we see also the corresponding binary and inverted images. For both images, the threshold was chosen at the 177 grey-tone. The maps after segmentation are given in Figure 11, and in the Figure 12

Conclusion

As we have seen, the distributions are quite different. However, some points need to be investigated. among them, we have to find a method for the quantification of differences. This can be obtained by considering several images of the same textile, and investigating the corresponding set of distributions. Moreover, it is necessary to study the dependence of segmentation on the chosen threshold and also on physical conditions (illumination, contrast and noise). These seems being the main questions concerning the method, and that require further work. However, it is possible that during this work, other problems arise. In any case, the preliminary investigation here proposed is rather encouraging.

  Figure 1.

Figure 1 :

 1 Figure 1: A fabric from Brodatz Album (D103) on the left, and the simulation of a defect on the right.

Figure 2 :

 2 Figure 2: Segmentation of the images in Figure 1.

Figure 3 :

 3 Figure 3: Distributions of areas of super-pixels, in green for the good fabric and in red for the faulty fabric. Areas are spaced in intervals of 10 pixels. N means "number". There is a large number of superpixels containing about 200 pixels.

Figure 4 :

 4 Figure 4: The same as in the Figure 3, represented by means of logarithmic scales.

Figure 5 :

 5 Figure 5: Image of an almost good fabric on the left (600x600 pixels, 600 pixels correspond to 40 mm).

Figure 7 :

 7 Figure 7: Maps of the segmentation.

Figure 8 :

 8 Figure 8: Distributions of areas of super-pixels (in logarithmic scale), in green for the good fabric and in red for the faulty fabric. The faulty texture contains several large super-pixels.

  the distributions.

Figure 9 :

 9 Figure 9: Another fabric. Images 600x600 pixels.

Figure 10 :

 10 Figure 10: The missing pick. Images 600x600 pixels.

Figure 11 :

 11 Figure 11: Maps of segmentation.

Figure 12 :

 12 Figure 12: Distributions of areas of super-pixels (in logarithmic scale), in green for the fabric in Figure

Information about this Article

Published on Friday 10th November, 2017 at 18:53:27.

The full citation for this Article is:

Sparavigna, A.C. (2017). Image Segmentation Applied to the Analysis of Fabric Textures.

PHILICA Article number 1157.


Mathieu Carrière
email: mathieu.cariere@inria.fr

Steve Oudot
email: steve.oudot@inria.fr

Local Equivalence and Intrinsic Metrics Between Reeb Graphs *

Keywords: 1998 ACM Subject Classification F.2.2 [Nonnumerical Algorithms and Problems:]Geometrical Problems and Computations Keywords and phrases Reeb Graphs, Extended Persistence, Induced Metrics, Topological Data Analysis Digital Object Identifier 10.4230/LIPIcs.SoCG.2017.25

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudometric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough locally, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the intrinsic metrics induced by these distances, which turn out to be all globally equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them.

Introduction

In the context of shape analysis, the Reeb graph [START_REF] Reeb | Sur les points singuliers d'une forme de pfaff complètement intégrable ou d'une fonction numérique[END_REF] provides a meaningful summary of a topological space and a real-valued function defined on that space. Intuitively, it continuously collapses the connected components of the level sets of the function into single points, thus tracking the values of the functions at which the connected components merge or split. Reeb graphs have been widely used in computer graphics and visualization -see [START_REF] Biasotti | Reeb Graphs for Shape Analysis and Applications[END_REF] for a survey, and their discrete versions, including the so-called Mappers [START_REF] Singh | Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition[END_REF], have become emblematic tools of topological data analysis due to their success in applications [START_REF] Alagappan | From 5 to 13: Redefining the Positions in Basketball[END_REF][START_REF] Barra | 3D Shape Retrieval and Classification using Multiple Kernel Learning on Extended Reeb graphs[END_REF][START_REF] Lum | Extracting insights from the shape of complex data using topology[END_REF][START_REF] Nicolau | Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival[END_REF].

Finding relevant dissimilarity measures for comparing Reeb graphs has become an important question in the recent years. The quality of a dissimilarity measure is usually Our contributions. In the first part of the paper we show that the bottleneck distance can discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do, even though it is only a pseudo-metric globally. More precisely, we show that, given any constant K ∈ (0, 1/22], in a sufficiently small neighborhood of a given Reeb graph R f in the functional distortion distance (that is: for any Reeb graph R g such that d FD (R f , R g ) < c(f, K), where c(f, K) > 0 is a positive constant depending only on f and K), one has:

Kd FD (R f , R g ) ≤ d B (R f , R g ) ≤ 2d FD (R f , R g ).
(

) 1
The second inequality is already known [START_REF] Bauer | Measuring Distance between Reeb Graphs[END_REF], and it asserts that the bottleneck distance between Reeb graphs is stable. The first inequality is new, and it asserts that the bottleneck distance is discriminative locally, in fact just as discriminative as the other distances mentioned above. Equation (1) can be viewed as a local equivalence between metrics although not in the usual sense: firstly, all comparisons are anchored to a fixed Reeb graph R f , and secondly, the constants K and 2 are absolute.

The second part of the paper advocates the study of intrinsic metrics on the space of Reeb graphs, for the reasons mentioned above. As a first step, we propose to study the intrinsic metrics dGH , dFD , dI and dB induced respectively by d GH , d FD , d I and d B . While the first three are obviously globally equivalent because their originating metrics are, our second contribution is to show that the last one is also globally equivalent to the other three.

The paper concludes with a discussion and some directions for the study of the space of Reeb graphs as an intrinsic metric space.

Related work.

Interpolation between Reeb graphs is also the underlying idea of the edit distance recently proposed by Di Fabio and Landi [START_REF] Di Fabio | The Edit Distance for Reeb Graphs of Surfaces[END_REF]. The problem with this distance, in its current form at least, is that it restricts the interpolation to pairs of graphs lying in the same 25:3 homeomorphism class. By contrast, our class of admissible paths is defined with respect to the topology induced by the functional distortion distance, as such it allows interpolating between distinct homeomorphism classes.

Interpolation between Reeb graphs is also related to the study of inverse problems in topological data analysis. To our knowledge, the only result in this vein shows the differentiability of the operator sending point clouds to the persistence diagram of their distance function [START_REF] Gameiro | Continuation of Point Clouds via Persistence Diagrams[END_REF]. Our first contribution (1) sheds light on the operator's local injectivity properties over the class of Reeb graphs.

Background

Throughout the paper we work with singular homology with coefficients in the field Z 2 , which we omit in our notations for simplicity. In the following, "connected" stands for "path-connected", and "cc" stands for "connected component(s)". Given a map f : X → R and an interval I ⊆ R, we write X I f as a shorthand for the preimage f -1 (I), and we omit the subscript when the map is obvious from the context.

Morse-Type Functions

Definition 1. A continuous real-valued function f on a topological space X is of Morse type if: (i) there is a finite set Crit(f ) = {a 1 < ... < a n } ⊂ R, called the set of critical values, such
that over every open interval (a 0 = -∞, a 1 ), ..., (a i , a i+1 ), ..., (a n , a n+1 = +∞) there is a compact and locally connected space Y i and a homeomorphism µ i :

Y i × (a i , a i+1 ) → X (ai,ai+1) such that ∀i = 0, ..., n, f | X (a i ,a i+1 ) = π 2 • µ -1 i
, where π 2 is the projection onto the second factor; (ii) ∀i = 1, ..., n -1, µ i extends to a continuous function μi : Y i × [a i , a i+1 ] → X [ai,ai+1] ; similarly, µ 0 extends to μ0 : a1] and µ n extends to μn : +∞) ; (iii) Each levelset f -1 (t) has a finitely-generated homology.

Y 0 × (-∞, a 1 ] → X (-∞,
Y n × [a n , +∞) → X [an,
Let us point out that a Morse function is also of Morse type, and that its critical values remain critical in the definition above. Note that some of its regular values may be termed critical as well in this terminology, with no effect on the analysis.

Extended Persistence

Let f be a real-valued function on a topological space X. The family {X (-∞,α] } α∈R of sublevel sets of f defines a filtration, that is, it is nested w.r.t. inclusion: +∞) for all α ≤ β ∈ R. We can turn it into a filtration by reversing the order on the real line. Specifically, let R op = {x | x ∈ R}, ordered by x ≤ ỹ ⇔ x ≥ y. We index the family of superlevel sets by R op , so now we have a filtration:

X (-∞,α] ⊆ X (-∞,β] for all α ≤ β ∈ R. The family {X [α,+∞) } α∈R of superlevel sets of f is also nested but in the opposite direction: X [α,+∞) ⊇ X [β,
{X [ α,+∞) } α∈R op , with X [ α,+∞) ⊆ X [ β,+∞) for all α ≤ β ∈ R op .
Extended persistence connects the two filtrations at infinity as follows. First, replace each superlevel set X [ α,+∞) by the pair of spaces (X, X [ α,+∞) ) in the second filtration. This maintains the filtration property since we have (X, X [ α,+∞) ) ⊆ (X, X [ β,+∞) ) for all α ≤ β ∈ R op . Then, let R Ext = R∪{+∞}∪R op , where the order is completed by α < +∞ < β for all α ∈ R and β ∈ R op . This poset is isomorphic to (R, ≤).

F α = X (-∞,α] for α ∈ R, F +∞ = X ≡ (X, ∅) and F α = (X, X [ α,+∞) ) for α ∈ R op ,
where we have identified the space X with the pair of spaces (X, ∅) at infinity. The subfamily {F α } α∈R is the ordinary part of the filtration, while {F α} α∈R op is the relative part.

Applying the homology functor H * to this filtration gives the so-called extended persistence module V of f , which is a sequence of vector spaces connected by linear maps induced by the inclusions in the extended filtration. For functions of Morse type, the extended persistence module can be decomposed as a finite direct sum of half-open interval modules -see e.g. [START_REF] Chazal | The Structure and Stability of Persistence Modules[END_REF]:

V n k=1 I[b k , d k ), where each summand I[b k , d k ) is made of copies of the field of coefficients at every index α ∈ [b k , d k )
, and of copies of the zero space elsewhere, the maps between copies of the field being identities. Each summand represents the lifespan of a homological feature (cc, hole, void, etc.) within the filtration. More precisely, the birth time b k and death time d k of the feature are given by the endpoints of the interval. Then, a convenient way to represent the structure of the module is to plot each interval in the decomposition as a point in the extended plane, whose coordinates are given by the endpoints. Such a plot is called the extended persistence diagram of f , denoted Dg(f ). The distinction between ordinary and relative parts of the filtration allows us to classify the points in Dg(f ) as follows:

p = (x, y) is called an ordinary point if x, y ∈ R; p = (x, y) is called a relative point if x, y ∈ R op ; p = (x, y) is called an extended point if x ∈ R, y ∈ R op ;
Note that ordinary points lie strictly above the diagonal ∆ = {(x, x) | x ∈ R} and relative points lie strictly below ∆, while extended points can be located anywhere, including on ∆ (e.g. when a cc lies inside a single critical level, see Section 2.3). It is common to partition Dg(f ) according to this classification:

Dg(f ) = Ord(f ) Rel(f ) Ext + (f ) Ext -(f )
, where by convention Ext + (f ) includes the extended points located on the diagonal ∆.

Stability. An important property of extended persistence diagrams is to be stable in the so-called bottleneck distance d ∞ b . Given two persistence diagrams D, D , a partial matching between D and D is a subset Γ of D × D where for every p ∈ D there is at most one p ∈ D such that (p, p ) ∈ Γ, and conversely, for every p ∈ D there is at most one p ∈ D such that (p, p ) ∈ Γ. Furthermore, Γ must match points of the same type (ordinary, relative, extended) and of the same homological dimension only. The cost of Γ is: cost

(Γ) = max{max p∈D δ D (p), max p ∈D δ D (p )}, where δ D (p) = p -p ∞ if p is matched to some p ∈ D and δ D (p) = d ∞ (p, ∆) if p is unmatched -same for δ D (p ).

Definition 2. The bottleneck distance between two persistence diagrams D and

D is d B (D, D ) = inf Γ cost(Γ)
, where Γ ranges over all partial matchings between D and D .

Theorem 3 (Stability [14]). For any Morse-type functions

f, g : X → R, d B (Dg(f ), Dg(g)) ≤ f -g ∞ .
(2)

Reeb Graphs

Definition 4. Given a topological space X and a continuous function f : X → R, we define the equivalence relation ∼ f between points of X by x ∼ f y if and only if f (x) = f (y) and x, y belong to the same cc of f -1 (f (x)) = f -1 (f (y)). The Reeb graph R f (X) is the quotient space X/ ∼ f . As f is constant on equivalence classes, there is a well-defined induced map f : R f (X) → R.

Connection to extended persistence. If f is a function of Morse type, then the pair (X, f ) is an R-constructible space in the sense of [START_REF] Vin De Silva | Categorified Reeb Graphs[END_REF]. This ensures that the Reeb graph is a multigraph, whose nodes are in one-to-one correspondence with the cc of the critical level sets of f . In that case, there is a nice interpretation of Dg( f ) in terms of the structure of R f (X). We refer the reader to [START_REF] Bauer | Measuring Distance between Reeb Graphs[END_REF][START_REF] Cohen-Steiner | Extending persistence using Poincaré and Lefschetz duality[END_REF] and the references therein for a full description as well as formal definitions and statements. Orienting the Reeb graph vertically so f is the height function, we can see each cc of the graph as a trunk with multiple branches (some oriented upwards, others oriented downwards) and holes. Then, one has the following correspondences, where the vertical span of a feature is the span of its image by f :

The vertical spans of the trunks are given by the points in Ext + 0 ( f ); The vertical spans of the downward branches are given by the points in Ord 0 ( f ); The vertical spans of the upward branches are given by the points in Rel 1 ( f ); The vertical spans of the holes are given by the points in Ext - 1 ( f ). The rest of the diagram of f is empty. These correspondences provide a dictionary to read off the structure of the Reeb graph from the persistence diagram of the quotient map f . Note that it is a bag-of-features type of descriptor, taking an inventory of all the features together with their vertical spans, but leaving aside the actual layout of the features. As a consequence, it is an incomplete descriptor: two Reeb graphs with the same persistence diagram may not be isomorphic. See the two Reeb graphs in Figure 1 for instance.

Notation. Throughout the paper, we consider Reeb graphs coming from Morse-type functions, equipped with their induced maps. We denote by Reeb the space of such graphs. In the following, we have R f , R g ∈ Reeb, with induced maps f : R f → R with critical values {a 1 , ..., a n }, and g : R g → R with critical values {b 1 , ..., b m }. Note that we write f, g instead of f , g for convenience. We also assume without loss of generality (w.l.o.g.) that R f and R g are connected. If they are not connected, then our analysis can be applied component-wise.

Distances for Reeb graphs

Definition 5. The bottleneck distance between R f and R g is:

d B (R f , R g ) := d B (Dg(f ), Dg(g)).
(3) Definition 6. The functional distortion distance between R f and R g is:

d FD (R f , R g ) := inf φ,ψ max 1 2 D(φ, ψ), f -g • φ ∞ , f • ψ -g ∞ , (4)
where:

φ : R f → R g and ψ : R g → R f are continuous maps, D(φ, ψ) = sup {|d f (x, x ) -d g (y, y )| such that (x, y), (x , y ) ∈ C(φ, ψ)} , where: C(φ, ψ) = {(x, φ(x)) | x ∈ R f } ∪ {(ψ(y), y) | y ∈ R g }, d f (x, x ) = min π:x→x max t∈[0,1] f • π(t) -min t∈[0,1] f • π(t) , where π : [0, 1] → R f is a continuous path from x to x in R f (π(0) = x and π(1) = x ), d g (y, y ) = min π:y→y max t∈[0,1] g • π(t) -min t∈[0,1] g • π(t)
, where π : [0, 1] → R g is a continuous path from y to y in R g (π(0) = y and π(1) = y ).

Bauer et al. [START_REF] Bauer | Measuring Distance between Reeb Graphs[END_REF] related these distances as follows:

Theorem 7. The following inequality holds:

d B (R f , R g ) ≤ 3 d FD (R f , R g ).
S o C G 2 0 1 7

25:6

Local Equivalence and Intrinsic Metrics Between Reeb Graphs This result can be improved using the end of Section 3.4 of [START_REF] Bjerkevik | Stability of Higher Dimensional Interval Decomposable Persistence Modules[END_REF], then noting that level set diagrams and extended diagrams are essentially the same [START_REF] Carlsson | Zigzag Persistent Homology and Real-valued Functions[END_REF], and finally Lemma 9 of [START_REF] Bauer | Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs[END_REF]:

Ext + 0 Ord + 0 Rel - 1 Ext - 1 R f R g Dg(f ) = Dg(g)
Theorem 8. The following inequality holds: d B (R f , R g ) ≤ 2 d FD (R f , R g ).
We emphasize that, even though Theorem 8 allows us to improve on the constants of our main result -see Theorem 9, the reduction from 3 Theorem 8 is not fundamental for our analysis and proofs.

d FD (R f , R g ) in Theorem 7 to 2 d FD (R f , R g ) in
Since the bottleneck distance is only a pseudo-metric -see Figure 1, the inequality given by Theorem 8 cannot be turned into an equivalence result. However, for any pair of Reeb graphs R f , R g that have the same extended persistence diagram Dg(f ) = Dg(g), and that are at positive functional distortion distance from each other, every continuous path in d FD from R f to R g will perturb the points of Dg(f ) and eventually drive them back to their initial position, suggesting first that d B is locally equivalent to d FD -see Theorem 9 in Section 3, but also that, even though d B (R f , R g ) = 0, the intrinsic metric dB (R f , R g ) induced by d B is positive -see Theorem 17 in Section 4.

Local Equivalence

Let a f = min 1≤i≤n a i+1 -a i > 0 and a g = min 1≤j≤m b j+1 -b j > 0. In this section, we show the following local equivalence theorem:

Theorem 9. Let K ∈ (0, 1/22]. If d FD (R f , R g ) ≤ max{a f , a g }/(8(1 + 22K))
, then:

Kd FD (R f , R g ) ≤ d B (R f , R g ) ≤ 2d FD (R f , R g ).
Note that the notion of locality used here is slightly different from the usual one. On the one hand, the equivalence does not hold for any arbitrary pair of Reeb graphs inside a neighborhood of some fixed Reeb graph, but rather for any pair involving the fixed graph. On the other hand, the constants in the equivalence are independent of the pair of Reeb graphs considered. The upper bound on d B (R f , R g ) is given by Theorem 8 and always holds. The aim of this section is to prove the lower bound.

Convention:

We assume w.l.o.g. that max{a f , a g } = a f , and we let ε

= d FD (R f , R g ).

Proof of Theorem 9

Let K ∈ (0, 1/22]. The proof proceeds by contradiction. Assuming

d B (R f , R g ) < Kε, where ε = d FD (R f , R g ) < a f /(8(1 + 22K
)), we progressively transform R g into some other Reeb graph R g (Definition 12) that satisfies both d FD (R g , R g ) < ε (Proposition 14) and d FD (R f , R g ) = 0 (Proposition 15). The contradiction follows from the triangle inequality.

Graph Transformation

The graph transformation is defined as the composition of the simplification operator from [START_REF] Bauer | Measuring Distance between Reeb Graphs[END_REF] and the Merge operator 1 from [START_REF] Carrière | Structure and Stability of the 1-Dimensional Mapper[END_REF]. We refer the reader to these articles for the precise definitions. Below we merely recall their main properties. Given a set S ⊆ X and a scalar α > 0, we recall that S α = {x ∈ X | d(x, S) ≤ α} denotes the α-offset of S.

Lemma 10 (Theorem 7.3 and following remark in [START_REF] Bauer | Measuring Distance between Reeb Graphs (v2)[END_REF]). Given α > 0, the simplification operator S α :

Reeb → Reeb takes any Reeb graph R h to R h = S α (R h ) such that Dg(h ) ∩ ∆ α/2 = ∅ and d B (R h , R h ) ≤ 2 d FD (R h , R h ) ≤ 4α.
Lemma 11 (Theorem 2.5 and Lemma 4.3 in [START_REF] Carrière | Structure and Stability of the 1-Dimensional Mapper[END_REF]). Given a ≤ b, the merge operator Merge a,b : Reeb → Reeb takes any Reeb graph R h to R h = Merge a,b (R h ) such that Dg(h ) is obtained from Dg(h) through the following snapping principle (see Figure 2 for an illustration):

(x, y) ∈ Dg(h) → (x , y ) ∈ Dg(h ) where x = x if x / ∈ [a, b]

Properties of the transformed graph

Let R f , R g ∈ Reeb such that d B (R f , R g ) < Kε where ε = d FD (R f , R g ) < a f /(8(1 + 22K)). Letting R g = F Kε (R g ), we want to show both that d FD (R g , R g ) < 22Kε < ε and d FD (R f , R g ) =
0, which will lead to a contradiction as mentioned previously.

Let B ∞ (•, •) denote balls in the ∞ -norm.

1 Strictly speaking, the output of our Merge is the Reeb graph of the output of the Merge from [START_REF] Carrière | Structure and Stability of the 1-Dimensional Mapper[END_REF].

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Local Equivalence and Intrinsic Metrics Between Reeb Graphs

a i a i+1 a i+1 -9α a i+1 + 9α a i -9α a i + 9α S 2α Merge 9α a i-1 a i-1 -9α a i-1 + 9α Figure 3 Illustration of Fα.
Lemma 13. Let R h = S 2Kε (R g ). Under the above assumptions, one has

Dg(h) ⊆ τ ∈Dg(f ) B ∞ (τ, 9Kε).
( 5)

Proof. Since d B (R f , R g ) < Kε, we have Dg(g) ⊆ τ ∈Dg(f ) B ∞ (τ, Kε) ∪ ∆ Kε . Since R h = S 2Kε (R g ), it
follows from Lemma 10 that d B (Dg(h), Dg(g)) ≤ 8Kε. Moreover, since every persistence pair in Dg(g) ∩ ∆ Kε is removed by S 2Kε , it results that:

Dg(h) ⊆ τ ∈Dg(g)\∆ Kε B ∞ (τ, 8Kε) ⊆ τ ∈Dg(f ) B ∞ (τ, 9Kε). Now we bound d FD (R g , R g ). Recall that, given an arbitrary Reeb graph R h , with critical values Crit(h) = {c 1 , ..., c p }, if C is a cc of h -1 (I),

Proposition 14. Under the same assumptions as above, one has d

FD (R g , R g ) < 22K . Proof. Let R h = S 2Kε (R g ). We have d FD (R g , R g ) ≤ d FD (R g , R h ) + d FD (R h , R g )
by the triangle inequality. It suffices therefore to bound both d FD (R g , R h ) and d FD (R h , R g ). By Lemma 10, we have d FD (R h , R g ) < 4Kε. Now, recall from (5) that the points of the extended persistence diagram of R h are included in τ ∈Dg(f ) B ∞ (τ, 9Kε). Moreover, since R g = Merge 9Kε (R h ), R g and R h are composed of the same number of arcs in each [a i + 9Kε, a i+1 -9Kε]. Hence, we can define explicit continuous maps φ : R h → R g and ψ : R g → R h as depicted in Figure 4. More precisely, since R h and R g are composed of the same number of arcs in each [a i + 9Kε, a i+1 -9Kε], we only need to specify φ and ψ inside each interval (a i -9Kε, a i + 9Kε) and then ensure that the piecewise-defined maps are assembled consistently. Since the critical values of R h are within distance less that 9Kε of the critical values of f , there exist two levels a i -9Kε < α i ≤ β i < a i + 9Kε such that R h is only composed of arcs in (a i -9Kε, α i ] and [β i , a i + 9Kε) for each i (dashed lines in Figure 4). For any cc C of h -1 ((a i -9Kε, a i + 9Kε)), the map φ sends all points of C ∩ h -1 ([α i , β i ]) to the corresponding critical point y C created by the Merge in R g , and it extends the arcs of

C ∩ h -1 ((a i -9Kε, α i ]) (resp. C ∩ h -1 ([β i , a i + 9Kε))) into arcs of (g ) -1 ([a i -9Kε, a i ]) (resp. (g ) -1 ([a i , a i + 9Kε])).
In return, the map ψ sends the critical point y C to an arbitrary point of C. Then, since the Merge operation preserves connected components, for each arc A of (g ) -1 ((a i -9Kε, a i + 9Kε)) connected to y C , there is at least one corresponding path A in R h whose endpoint in h -1 (a i -9Kε) or h -1 (a i + 9Kε)

φ : R h → R g ψ : R g → R h 9K 9K a i a i α i α i β i R h R g R h R g R h R g R h R g Figure 4
The effects of φ and ψ around a specific critical value ai of f . Segments are matched according to their colors (up to reparameterization).

matches with the one of A (see the colors in Figure 4). Hence ψ sends A to A and the piecewise-defined maps are assembled consistently.

Let us bound the three terms in the max{• • • } in (4) with this choice of maps φ, ψ:

We first bound g -h • ψ ∞ . Let x ∈ R g . Either g (x) ∈ i∈{1,...,n-1} [a i + 9Kε, a i+1 - 9Kε],
and in this case we have g (x) = h(ψ(x)) by definition of ψ; or, there is i 0 ∈ {1, ..., n} such that g (x) ∈ (a i0 -9Kε, a i0 + 9Kε) and then h(ψ(x)) ∈ (a i0 -9Kε,

a i0 + 9Kε). In both cases |g (x) -h • ψ(x)| < 18Kε. Hence, g -h • ψ ∞ < 18Kε.
Since the previous proof is symmetric in h and g , one also has h -g • φ ∞ < 18Kε. We now bound D(φ, ψ). Let (x, φ(x)), (ψ(y), y) ∈ C(φ, ψ) (the cases (x, φ(x)), (x , φ(x )) and (ψ(y), y), (ψ(y ), y ) are similar). Let π g : [0, 1] → R g be a continuous path from φ(x) to y which achieves d g (φ(x), y).

Assume h(x) ∈ i∈{1,...,n-1} [a i + 9Kε, a i+1 -9Kε]. Then one has ψ • φ(x) = x. Hence, π h := ψ • π g is a valid path from x to ψ(y). Moreover, since g -h • ψ ∞ < 18Kε, it follows that max im(h • π h ) < max im(g • π g ) + 18Kε, min im(h • π h ) > min im(g • π g ) -18Kε. ( 6
)
S o C G 2 0 1 7

25:10 Local Equivalence and Intrinsic Metrics Between Reeb Graphs

Hence, one has

d h (x, ψ(y)) ≤ max im(h • π h ) -min im(h • π h ) < d g (φ(x), y) + 36Kε, -d h (x, ψ(y)) ≥ min im(h • π h ) -max im(h • π h ) > -d g (φ(x), y) -36Kε.
This shows that |d h (x, ψ(y)) -d g (φ(x), y)| < 36Kε. Assume that there is i 0 ∈ {1, ..., n} such that h(x) ∈ (a i0 -9Kε, a i0 + 9Kε). Then, by definition of φ, ψ, we have g (φ(x)) ∈ (a i0 -9Kε, a i0 +9Kε), and, since φ and ψ preserve connected components, there is a path π h : [0, 1] → R h from x to ψ • φ(x) within the interval (a i0 -9Kε, a i0 + 9Kε), which itself is included in the interior of the offset im(g • π g ) 18Kε . Let now π h be the concatenation of π h with ψ • π g , which goes from x to ψ(y). Since g -h • ψ < 18Kε, it follows that im(h

• ψ • π g ) ⊆ int im(g • π g ) 18Kε ,
and since im(h

• π h ) = im(h • π h ) ∪ im(h • ψ • π g ) by concatenation, one finally has im(h • π h ) ⊆ int im(g • π g ) 18Kε .
Hence, the inequalities of ( 6) hold, implying that |d h (x, ψ(y)) -d g (φ(x), y)| < 36Kε. Since these inequalities hold for any couples (x, φ(x)) and (ψ(y), y), we deduce that D(φ, ψ) ≤ 36Kε.

Thus, d FD (R h , R g ) < 4Kε and d FD (R h , R g ) ≤ 18Kε, so d FD (R g , R g ) < 22Kε as desired.
Now we show that R g is isomorphic to R f (i.e. it lies at functional distortion distance 0).

Proposition 15. Under the same assumptions as above, one has d

FD (R f , R g ) = 0.
Proof. First, recall from ( 5) that the points of the extended persistence diagram of R h are included in τ ∈Dg(f ) B ∞ (τ, 9Kε). Since R g = Merge 9Kε (R h ), it follows from Lemma 11 that Crit(g ) ⊆ Crit(f ). Hence, both R g and R f are composed of arcs in each (a i , a i+1 ). Now, we show that, for each i, the number of arcs of (g ) -1 ((a i , a i+1 )) and f -1 ((a i , a i+1 )) are the same. By the triangle inequality and Proposition 14, we have:

d FD (R f , R g ) ≤ d FD (R f , R g ) + d FD (R g , R g ) < (1 + 22K)ε. ( 7
)
Let φ : R f → R g and ψ : R g → R f be optimal continuous maps that achieve d FD (R f , R g ). Let i ∈ {1, ..., n -1}. Assume that there are more arcs of f -1 ((a i , a i+1 )) than arcs of (g ) -1 ((a i , a i+1 )). For every arc

A of f -1 ((a i , a i+1 )), let x A ∈ A such that f (x A ) = ā = 1 2 (a i + a i+1
). First, note that φ(x A ) must belong to an arc of (g ) -1 ((a i , a i+1 )). Indeed, since

f -g • φ ∞ < (1 + 22Kε), one has g (φ(x A )) ∈ (ā -(1 + 22K)ε, ā + (1 + 22K)ε) ⊆ (a i , a i+1
). Then, according to the pigeonhole principle, there exist x A , x A such that φ(x A ) and φ(x A ) belong to the same arc of (g ) -1 ((a i , a i+1 )).

Since x A and x A do not belong to the same arc, we have

d f (x A , x A ) > a f /2. Now, since f -g • φ ∞ < (1 + 22K)ε and φ(x A ), φ(x A ) belong to the same arc of (g ) -1 ((a i , a i+1 )), we also have d g (φ(x A ), φ(x A )) < 2(1 + 22K)ε (see Figure 5). Hence, D(φ, ψ) ≥ |d f (x A , x A ) -d g (φ(x A ), φ(x A ))| > a f /2 -2(1 + 22K)ε, which is greater than 2(1 + 22K)ε because ε < a f /(8(1 + 22K)). Thus, d FD (R f , R g ) > (1 + 22K)ε,
which leads to a contradiction with [START_REF] Biasotti | Reeb Graphs for Shape Analysis and Applications[END_REF]. This means that there cannot be more arcs in f -1 ((a i , a i+1 )) than in (g ) -1 ((a i , a i+1 )). Since the proof is symmetric in f and g , the numbers of arcs in (g ) -1 ((a i , a i+1 )) and in f -1 ((a i , a i+1 )) are actually the same.

Finally, we show that the attaching maps of these arcs are also the same. In this particular graph setting, this is equivalent to showing that corresponding arcs in R f and R g have the same endpoints. Let a i be a critical value. Let A - f,i and A + f,i (resp. A - g ,i and A + g ,i ) be the sets of arcs in f -1 ((a i-1 , a i )) and f -1 ((a i , a i+1 )) (resp. (g ) -1 ((a i-1 , a i )) and (g ) -1 ((a i , a i+1 ))).

φ(x A ) a i+1 a i x A R f R g 2(1 + 22K) φ(x A ) a i+1 a i x A f (x A ) = f (x A ) = ā Figure 5
Any path between xA and x A must contain the red segments, and the blue segment is a particular path between φ(xA) and φ(x A ).

a i x y φ(x) φ(y) 2(1 + 22K) Rf Rg
Figure 6 Any path from x to y must go through an entire arc, hence d f (x, y) ≥ a f . On the contrary, there exists a direct path between φ(x) and φ(y), hence d g (φ(x), φ(y)) < 2(1 + 22K)ε.

Morevover, we let ζ i

f and ξ i f (resp. ζ i g and ξ i g ) be the corresponding attaching maps that send arcs to their endpoints in f -1 (a i ) (resp. (g ) -1 (a i )). Let A, B ∈ A - f,i . We define an equivalence relation ∼ f,i between A and B by:

A ∼ f,i B iff ζ i f (A) = ζ i f (B), i.
e. the endpoints of the arcs in the critical slice f -1 (a i ) are the same. Similarly, C, D ∈ A + f,i are equivalent if and only if ξ i f (C) = ξ i f (D). One can define ∼ g ,i in the same way. To show that the attaching maps of R f and R g are the same, we need to find a bijection b between the arcs of R f and R g such that A ∼ f,i B ⇔ b(A) ∼ g ,i b(B) for each i.

We will now define b then check that it satisfies the condition. Recall from (7) that

d FD (R f , R g ) < (1 + 22K)ε. Hence there exists a continuous map φ : R f → R g such that f -g • φ ∞ < (1 + 22K)ε.
This map induces a bijection b between the arcs of R f and R g . Indeed, given an arc

A ∈ A - f,i , let x ∈ A such that f (x) = ā = 1 2 (a i-1 + a i ). We define b(A) as the arc of A - g,i that contains φ(x). The map b is well-defined since g •φ(x) ∈ [ā -(1 + 22K)ε, ā + (1 + 22K)ε] ⊆ (a i-1 , a i ), hence φ(x) must belong to an arc of (g ) -1 ((a i-1 , a i )). Let us show that b(A) ∼ g ,i b(B) ⇒ A ∼ f,i B. Assume there exist A, B ∈ A - f,i (the treatment of A, B ∈ A + f,i is similar) such that A ∼ f,i B and b(A) ∼ g ,i b(B). Let x = ζ i f (A) and y = ζ i f (B). Then we have d f (x, y) ≥ a f while d g (φ(x), φ(y)) < 2(1 + 22K)ε (see Figure 6). Hence |d f (x, y) -d g (φ(x), φ(y))| > a f -2(1 + 22K)ε > 2(1 + 22K)ε, so d FD (R f , R g ) > (1 + 22K)ε,
which leads to a contradiction with [START_REF] Biasotti | Reeb Graphs for Shape Analysis and Applications[END_REF]. The same argument applies to show that

A ∼ f,i B ⇒ b(A) ∼ g ,i b(B).

Induced Intrinsic Metrics

In this section we leverage the local equivalence given by Theorem The following result is, in our view, the starting point for the study of intrinsic metrics over the space of Reeb graphs. It comes as a consequence of the (local or global) equivalences between d B and d FD stated in Theorems 8 and 9. The intuition is that integrating two locally equivalent metrics along the same path using sufficiently small integration steps yields the same total length up to a constant factor, hence the global equivalence between the induced intrinsic metrics2 . Theorem 17. dB and dFD are globally equivalent. Specifically, for any R f , R g ∈ Reeb,

dFD (R f , R g )/22 ≤ dB (R f , R g ) ≤ 2 dFD (R f , R g ). ( 8
)
Proof. We first show that dB (R f , R g ) ≤ 2 dFD (R f , R g ). Let γ be an admissible path and let Σ = {t 0 , ..., t n } be a partition of [0, 1]. Then, by Theorem 8,

n-1 i=0 d FD (γ(t i ), γ(t i+1 )) ≥ 1 2 n-1 i=0 d B (γ(t i ), γ(t i+1 )).
Since this is true for any partition Σ of any finite size n, it follows that

L d FD (γ) ≥ 1 2 L d B (γ) ≥ 1 2 dB (R f , R g ).
Again, this inequality holds for any admissible path γ, so dB

(R f , R g ) ≤ 2 dFD (R f , R g ). We now show that dFD (R f , R g )/22 ≤ dB (R f , R g ).
(R f , R g ) ≤ L d FD (γ) ≤ 22 L d B (γ).
Again, this inequality is true for any admissible path γ, so dFD

(R f , R g ) ≤ 22 dB (R f , R g ).
Theorem 17 implies in particular that dB is a true metric on Reeb graphs, as opposed to d B which is only a pseudo-metric. Moreover, the simplification operator defined in Section 3.1.1 makes it possible to continuously deform any Reeb graph into a trivial segment-shaped graph then into the empty graph. This shows that Reeb is path-connected in d FD . Since the length of such continuous deformations is finite if the Reeb graph is finite, dFD and dB are finite metrics. Finally, the global equivalence of dFD and dB yields the following: Remark. Note that the first inequality in (8) and, consequently, Corollary 18, are wrong if one defines the admissible paths for dB to be the whole class of maps [0, 1] → Reeb that are continuous in d B -hence our convention. For instance, let us consider the two Reeb graphs R f and R g of Figure 1 such that Dg(f ) = Dg(g), and let us define γ : [0, 1] → Reeb by γ(t) = R f if t ∈ [0, 1/2) and γ(t) = R g if t ∈ [1/2, 1]. Then γ is continuous in d B while it is not in d FD at 1/2 since d FD (R f , R g ) > 0. In this case, dB (R f , R g ) ≤ L d B (γ) = 0 < dFD (R f , R g ).

Discussion

In this article, we proved that the bottleneck distance, even though it is only a pseudometric on Reeb graphs, can actually discriminate a Reeb graph from the other Reeb graphs in a small enough neighborhood, as efficiently as the other metrics do. This theoretical result legitimates the use of the bottleneck distance to discriminate between Reeb graphs in applications. It also motivates the study of intrinsic metrics, which can potentially shed new light on the structure of the space of Reeb graphs and open the door to new applications where interpolation plays a key part. This work has raised numerous questions, some of which we plan to investigate in the upcoming months: Can the lower bound be improved? We believe that ε/22 is not optimal. Specifically, a more careful analysis of the simplification operator should allow us to derive a tighter upper bound than the one in Lemma 10, and improve the current lower bound on d B .

S o C G 2 0 1 7

25:14

Local Equivalence and Intrinsic Metrics Between Reeb Graphs

Do shortest paths exist in Reeb?

The existence of shortest paths achieving dB is an important question since a positive answer would enable us to define and study the intrinsic curvature of Reeb. Moreover, characterizing and computing these shortest paths would be useful for interpolating between Reeb graphs. The existence of shortest paths is guaranteed if the space is complete and locally compact. Note that Reeb is not complete, as shown by the counter-example of Figure 7. Hence, we plan to restrict the focus to the subspace of Reeb graphs having at most N features with height at most H, for fixed but arbitrary N, H > 0. We believe this subspace is complete and locally compact, like its counterpart in the space of persistence diagrams [START_REF] Blumberg | Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces[END_REF]. Is Reeb an Alexandrov space? Provided shortest paths exist in Reeb (or in some subspace thereof), we plan to determine whether the intrinsic curvature is bounded, either from above or from below. This is interesting because barycenters in metric spaces with bounded curvature enjoy many useful properties [START_REF] Ohta | Barycenters in Alexandrov spaces of curvature bounded below[END_REF], and they can be approximated effectively [START_REF] Ohta | Gradient flows on Wasserstein spaces over compact Alexandrov spaces[END_REF]. Can the local equivalence be extended to general metric spaces? We have reasons to believe that our local equivalence result can be used to prove similar results for more general classes of metric spaces than Reeb graphs. If true, this would shed new light on inverse problems in persistence theory.

Finally, define the extended S o C G 2 0 1 7 25: 4 Local

 74 Equivalence and Intrinsic Metrics Between Reeb Graphsfiltration of f over R Ext by:

Figure 1

 1 Figure 1 Example of two different Reeb graphs R f and Rg that have the same extended persistence diagram Dg(f ) = Dg(g). These graphs are at bottleneck distance 0 from each other, while their functional distortion distance is positive.

Figure 2

 2 Figure 2 Left: effect of Merge a,b on a Reeb graph R h . Right: Effect on its persistence diagram.

Definition 12 .

 12 Let R f be a fixed Reeb graph with critical values {a 1 , • • • , a n }. Given α > 0, the full transformation F α : Reeb → Reeb is defined as F α = Merge 9α • S 2α, where Merge 9α = Merge an-9α, an+9α • ... • Merge a1-9α, a1+9α . See Figure3for an illustration.

  where I is an open interval such that ∃c i , c i+1 s.t. I ⊆ (c i , c i+1 ), then C is a topological arc, i.e. homeomorphic to an open interval.

7 25: 12 LocalDefinition 16 .

 71216 [START_REF] Blumberg | Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces[END_REF] to derive a global equivalence between the intrinsic metrics dB and dFD induced by d B and d FD . Note that we already know dFD to be equivalent to dGH and dI since d FD is equivalent to d GH and d I . ToS o C G 2 0 1 Equivalence and Intrinsic Metrics Between Reeb Graphsthe best of our knowledge, the question whether d FD , d I or d GH is intrinsic on the space of Reeb graphs has not been settled, although d GH itself is known to be intrinsic on the larger space of compact metric spaces -see e.g.[START_REF] Ivanov | The Gromov-Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic[END_REF].Convention. In the following, whatever the metric d : Reeb × Reeb → R + under consideration, we define the class of admissible paths in Reeb to be those maps γ : [0, 1] → Reeb that are continuous in d FD . This makes sense when d is either d FD itself, d GH , or d I , all of which are equivalent to d FD and therefore have the same continuous maps γ : [0, 1] → Reeb. In the case d = d B our convention means restricting the class of admissible paths to a strict subset of the maps γ : [0, 1] → Reeb that are continuous in d (by Theorem 8), which is required by some of our following claims. Let d : Reeb × Reeb → R + be a metric on Reeb. Let R f , R g ∈ Reeb, and γ : [0, 1] → Reeb be an admissible path such that γ(0) = R f and γ(1) = R g . The length of γ induced by d is defined as L d (γ) = sup n,Σ n-1 i=0 d(γ(t i ), γ(t i+1 )) where n ranges over N and Σ ranges over all partitions 0 = t 0 ≤ t 1 ≤ ... ≤ t n = 1 of [0, 1]. The intrinsic metric induced by d, denoted d, is defined by d(R f , R g ) = inf γ L d (γ) where γ ranges over all admissible paths γ : [0, 1] → Reeb such that γ(0) = R f and γ(1) = R g .

Corollary 18 .

 18 The metrics dFD and dB induce the same topology on Reeb, which is a refinement of the ones induced by d FD or d B .

  Let γ be an admissible path and Σ = {t 0 , ..., t n } a partition of [0, 1]. We claim that there is a refinement of Σ (i.e. a partition Σ = {t 0 , ..., t m } ⊇ Σ for some m ≥ n) such that d FD (γ(t j ), γ(t j+1 )) < max{c t j , c t j+1 }/16 for all j ∈ {0, ..., m -1}, where c t > 0 denotes the minimal distance between consecutive critical values of γ(t). Indeed, since γ is continuous in d FD , for any t ∈ [0, 1] there existsδ t > 0 such that d FD (γ(t), γ(t )) < c t /16 for all t ∈ [0, 1] with |t -t | < δ t . Consider the open cover {(max{0, t -δ t /2}, min{1, t + δ t /2})} t∈[0,1] of [0, 1]. Since [0, 1]is compact, there exists a finite subcover containing all the intervals (t i -δ ti /2, t i + δ ti /2) for t i ∈ Σ. Assume w.l.o.g. that this subcover is minimal (if it is not, then reduce the δ ti as much as needed). Let then Σ = {t 0 , ..., t m } ⊇ Σ be the partition of[0,[START_REF] Agarwal | Computing the Gromov-Hausdorff Distance for Metric Trees[END_REF] given by the midpoints of the intervals in this subcover, sorted by increasing order. Since the subcover is minimal, we have t j+1 -t j < (δ t j + δ t j+1 )/2 < max{δ t j , δ t j+1 } hence d FD (γ(t j ), γ(t j+1 )) < max{c t j , c t j+1 }/16 for each j ∈ {0, m -1}. It follows that

	n-1	m-1
	d FD (γ(t i ), γ(t i+1 )) ≤	d FD (γ(t j ), γ(t j+1 )) by the triangle inequality since Σ ⊇ Σ
	i=0	j=0
		m-1
	≤ 22	d B (γ(t j ), γ(t j+1 )) by Theorem 9 with K = 1/22
		j=0
	≤ 22 L d B (γ).

Since this is true for any partition Σ of any finite size n, it follows that dFD

  Figure 7 A sequence of Reeb graphs that is Cauchy but that does not converge in Reeb because the number of critical values goes to +∞. Indeed, each Rn has n + 2 critical values.

				• • •
	R 1	R 2	R 3	R 4

Provided the induced metrics are defined using the same class of admissible paths, hence our convention.

This work was partially supported by ERC Grant Agreement No. 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions) and was carried out while the second author was visiting the ICERM at Brown University.


Arthur Levy

Steven Le Corre

Arnaud Poitou
email: arnaud.poitou]@ec-nantes.fr.

Eric Soccard
email: eric.soccard@eads.net.

Ultrasonic Welding of Thermoplastic Composites, Modeling and Simulation of the Process

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INTRODUCTION

This work aims at modeling an original welding process for composite material with thermoplastic matrix. Triangular bulges, called "energy directors" are molded with matrix only, on a width of two centimeters on the border of one of the plate to be welded. The two plates are then positioned in order to cover each other on the width of the energy directors. The process then consists in applying an ultrasonic (20 KHz) sinusoidal compression stress between the two plates as shown in figure 1(a). The strain concentrates at the (a) Assembly overview.

(b) Melting and flowing of an energy director.

Figure 1: Ultrasonic welding of two plates interface, in the energy directors, that melt because of the viscous dissipation [START_REF] Grewell | Plastics and Composites Welding Handbook[END_REF]. The triangles then progressively flow on the whole interface and perform welding of the two composite planes (see figure 1(b)). Even if no fiber crosses the interface, the cover is wide enough to transmit the stress which allows the process to be used for large scale assemble.

The main problem when modeling and simulating such a process comes from the existence of two time scales, which would induce very fine time steps and therefore huge computation times. This process has been subject for few models. First we can mention Benatar [START_REF] Benatar | Ultrasonic welding of advanced thermoplastic composites[END_REF][START_REF] Grewell | Plastics and Composites Welding Handbook[END_REF] who described the process as the succession of five phenomena physical phenomena. Some authors like Tolunay et al. [START_REF] Tolunay | Heating and bonding mechanisms in ultrasonic welding of thermoplastics[END_REF] Suresh et al. [START_REF] Suresh | Modeling of temperature distribution in ultrasonic welding of thermoplastics for various joint designs[END_REF] or Wang et al. [START_REF] Wang | Fem investigation of the temperature field of energy director during ultrasonic welding of peek composites[END_REF] described the mechanical dissipation in a linear harmonic viscoelastic framework. Nevertheless, their models were limited to the thermal aspects only. We propose here a more general framework, which also enables to take into account the flowing of polymer.

MODELING OF THE PROCESS

General Thermo -Mechanical problem

Let us consider a single energy director as described in figure 2. The objective is to obtain a thermo-mechanical formulation able to describe the flow and heating of the polymer, due to the mechanical loading and the internal dissipation caused by vibrations. Mechanical problem. Due to the high frequency of the sonotrode and the rather long time of the process, using a visco-elastic model for the polymer seems necessary. As a first approach, a Maxwell model in small displacement framework is used here :

λ σ + σ = 2η ε ( 1
)
where λ is the relaxation time, η the viscosity and ε the strain tensor. Neglecting the inertia terms and the volumic force, the equilibrium equation can be written as :

div(σ -pI) = 0 ( 2
)
where σ is the extra stress tensor and p the pressure so that σ -pI is the stress tensor.

The composites plates are supposed to be perfectly rigid compared to the energy director. The displacement of the tip of the director is then supposed to be null, whereas the displacement of the upper part can be split into a "micro-chronological" sinusoidal displacement a.sin(ωt) due to the short period of the ultrasound, and a "macro-chronological" displacement u d due to the squeezing of the director during the process. The whole mechanical problem can finally be written as follow:

   λ σ + σ = 2.η.ε on Ω div(σ -pI) = 0 on Ω div(u) = 0 on Ω and   u = u d (t) + asin(ωt) (Γ sup ) u = 0 (Γ 0 ) (σ -pI).n = 0 (Γ lat ) (3)
Thermal problem. The director is supposed to be insulated. A fraction α of the viscous part of the mechanical energy σ : εvis is supposed to be dissipated in the director during the process. Considering the maxwell constitutive law (1), we can assume that εvis = 1 2η σ . The thermal problem can then be written as:

ρc θ = k∆θ + α 2η σ : σ k.grad(θ ).n = 0 on (Γ lat ∪ Γ 0 ∪ Γ sup ) ( 4
)
where θ is the temperature.

Dimensionless quantities

In order to deal with comparable quantities, we will transform the initial problem into a dimensionless problem. We proceed by using characteristic magnitude of the process: Viscosity : As set before, we choose a characteristic viscosity η 0 = 10 7 Pa.s that is equivalent to the newtonian viscosity at temperature θ = θ g (where θ g = 143˚C is the glassy temperature).

Length : The characteristic length used in this work is e = 300 µm which is the initial height of the director.

Temperature : A temperature θ will be transformed into a dimensionless temperature θ * by applying the following formula:

θ * = θ -θ amb θ re f (5)
where θ amb is the ambient temperature and θ re f = θ mθ amb with θ m the melting temperature. This allows to have an dimensionless temperature θ * that varies around [0, 1]. Time : The time λ 0 = 1 s which is the time of the process is used as characteristic time.

Variables : We finally introduce dimensionless variables which are the following stared variables defined by:

           σ = η 0 λ 0 σ * u = eu * ε = ε * ε = grad * s u * θ = θ re f θ * + θ re f (6)
Operators : In the same way, dimensionless space operators are introduced:

grad ≡ 1 e grad * , div ≡ 1 e div * and ∆ ≡ 1 e 2 ∆ * (7)
Dimensionless time derivation : One can notice that there are two time scales in the process: a short time scale of about 50 micro-seconds corresponding to the ultrasonic period that will be refered to as the micro-chronological time scale ; and a long time scale of about 1 second corresponding to the process time that will be refered to as the macro-chronological time scale.

Dimensionless time scales t * and τ * are then introduced such as t * = t λ 0 and τ * = ωt. t * being the macro-chronological time variable and τ * the micro-chronological one. Then, one can define the time scale factor :

ξ = t * τ * = 1 ωλ 0 ∼ 10 -5 (8)
This good scales separation will reasonably allow us to use a technique of homogenization in time to model the process [START_REF] Guennouni | Sur une méthode de calcul de structures soumises à des chargements cycliques: l'homogénéisation en temps[END_REF]. To proceed, each variable φ of the problem is written as a function of t * and τ * , the time derivative of φ being written as:

φ = dφ dt = 1 λ 0 ∂ φ ∂t * + 1 λ 0 1 ξ ∂ φ ∂ τ * (9)
2.2 Homogenization.

Dimensionless resulting problems

Dimensionless equations corresponding to systems ( 3) and ( 4) can be deduced 1 :

                 Λξ l ∂ σ ∂t + Λξ l 1 ξ ∂ σ ∂ τ + σ = 2.Nξ n . ∂ ε ∂t + 2.Nξ n . 1 ξ ∂ ε ∂ τ div(σ -pI) = 0 div(v) = 0   u = ξ p U d (t) + Rξ r sin(τ) (Γ sup ) u = 0 (Γ 0 ) (σ -pI).n = 0 (Γ lat ) (10)
∂ θ ∂t + 1 ξ ∂ θ ∂ τ = Aξ a ∆θ + Bξ b 1 Nξ n σ : σ grad(θ ).n = 0 (Γ lat ∪ Γ 0 ∪ Γ sup ) (11)
where, for the sake of legibility, the stared notation has immediately been dropped. All the capital letters introduced are of order 0 in ξ . Process and material order of magnitude found in the literature [2, 1, 7, 4, 9] allowed to determine the exponents of the equations :

           |v d | ∼ 300 µm/s k ∼ 0.24W /m/K ρ ∼ 1300 Kg/m 3 c ∼ 2000 J/Kg/K |a| ∼ 20 µm            U d ξ p = u d e ∼ 2.10 -4 3.10 -4 ∼ 0, 6 ⇒ p = 0 Rξ r = a e ∼ 2.10 -5 3.10 -4 ∼ 0, 06 ⇒ r = 0 Aξ a = kλ 0 ρce 2 ∼ 0.8 2.10 6 (3.10 -4 ) 2 ∼ 3 ⇒ a = 0 Bξ b = αη 0 ρcθ re f λ 0 ∼ 10 7
2.10 6 .2.10 2 .1 ∼ 5.10 -2 ⇒ b = 0 (12) Concerning Λξ l and Nξ n we consider a temperature above the glassy temperature T g . Using the work of Nicodeau [START_REF] Nicodeau | Modelisation du Soudage en Continu des composites a Matrice Thermoplastique[END_REF], around T = 410˚C, which is widely above the melting temperature (T m = 334˚C), λ = 0.25 s, and around T g extrapolating the arrhenius law, 1 note that the constitutive law has been divided by η 0 λ 0 and the thermal equation by ρcθ re f λ = 230 s. Therefore we can set λ λ 0 ∼ ξ 0 , ie l = 0. Proceeding the same way for the viscosity, we get n = 0.

Finally, the constitutive equation and boundary condition of the mechanical problem [START_REF] Sethian | Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry[END_REF] become:

Λ ∂ σ ∂t + Λ 1 ξ ∂ σ ∂ τ + σ = 2.N. ∂ ε ∂t + 2.N. 1 ξ ∂ ε ∂ τ u = U d (t) + Rsin(τ) (Γ sup ) (13)
and the thermal constitutive equation:

∂ θ ∂t + 1 ξ ∂ θ ∂ τ = A∆θ + B N σ : σ (14)

Asymptotic expansion

Asymptotic expansion in time consists in writing each unknown as an expansion in power of ξ :

φ = φ 0 + φ 1 ξ + φ 2 ξ 2 + ... ( 15
)
where φ stands for σ , p, u, ε or θ . Each φ i is supposed to be periodic in τ as in Guennouni or Boutin & Wong [START_REF] Guennouni | Sur une méthode de calcul de structures soumises à des chargements cycliques: l'homogénéisation en temps[END_REF][START_REF] Boutin | Study of thermosensitive heterogeneous media via space-time homogenisation[END_REF]. Later on, formulation of the mechanical problem in terms of velocity instead of displacement will be simpler. In our time derivation framework the velocity is defined as :

v = ∂ u ∂t + 1 ξ ∂ u ∂ τ
, which means that the velocity expansion starts at order -1 with:

v -1 = ∂ u 0 ∂ τ v i = ∂ u i ∂t + ∂ u i+1 ∂ τ ∀i ≥ 0 (16)
We also expand the strain rate tensor as:

   D -1 = grad s v -1 = ∂ ε 0 ∂ τ D i = grad s v -1 = ∂ ε i ∂t + ∂ ε i+1 ∂ τ ∀i ≥ 0 (17)

Identification on the mechanical problem.

Once injected in the previous problem, identification of different order of ξ leads to several sets of equations. First, we can notice that the equilibrium equation, incompressibility constrain and two boundary conditions can be identified trivially at every order i of ξ . We get:

         div(σ i -p i I) = 0 div(u i ) = 0 u i = 0 (Γ 0 ) (σ i -p i I).n = 0 (Γ lat ) or          div(σ i -p i I) = 0 div(v i ) = 0 v i = 0 (Γ 0 ) (σ i -p i I).n = 0 (Γ lat ) (18)
The displacement boundary condition on Γ sup given in the system (13) only appears at order 0 as :

u i = 0 ∀i > 0 (Γ sup ) u 0 = u d (t) + Rsin(τ) (Γ sup ) or      v -1 = ∂ u 0 ∂ τ = Rcos(τ) (Γ sup ) v 0 = ∂ u 1 ∂ τ + ∂ u 0 ∂t = ∂ u d ∂t (Γ sup ) v i = 0 ∀i > 0 (Γ sup ) (19)
Then, the constitutive law ( 13) is identified at the first orders:

order ξ -1 : Λ ∂ σ 0 ∂ τ + = 2N ∂ ε 0 ∂ τ (20) order ξ 0 : Λ ∂ σ 0 ∂t + Λ ∂ σ 1 ∂ τ + σ 0 = 2N. ∂ ε 0 ∂t + 2.N. ∂ ε 1 ∂ τ (21)
Introducing the short time average operator < • >= 1 |κ| κ (•)dτ where κ is an ultrasonic period, periodicity of σ i implies < ∂ σ i ∂ τ >= 0. We obtain from previous equation:

Λ ∂ < σ 0 > ∂t < σ 0 >= 2N. ∂ < ε 0 > ∂t (22)
Finally, combining constitutive laws (20) and ( 22) , equilibrium, incompressibility and boundary conditions ( 18) and ( 19), we can write two mechanical problems.

Micro-mechanical problem : deals with short time τ:

     Λ ∂ σ 0 ∂ τ = 2N ∂ ε 0 ∂ τ = 2ND(v -1 ) div(σ 0 -p 0 I) = 0 div(v -1 ) = 0   v -1 = R.sin(τ) (Γ sup ) v -1 = 0 (Γ 0 ) σ 0 .n = 0 (Γ lat ) (23)
We get an hypo-elastic problem which is equivalent to an elastic problem in the small displacement framework. Indeed, for short time solicitation, the maxwell model is mainly elastic. This problem describes a stress fluctuation of order ξ 0 that is linked to a velocity of order ξ -1 . The boundary condition on Γ sup is a micro-chronological harmonic condition only.

Macro-mechanical problem : deals with long time t:

     Λ ∂ <σ 0 > ∂t + < σ 0 >= 2N < D 0 > div(< σ 0 > -< p 0 > I) = 0 div(< v 0 >) = 0   < v 0 >= v d (t) (Γ sup ) < v 0 >= 0 (Γ 0 ) < σ 0 > .n = 0 (Γ lat ) (24)
It describes the slow mechanical variation as a visco-elastic maxwell flow. The stress average is of order ξ 0 as well but is linked to a velocity of order ξ 0 . The boundary condition on Γ sup is the macro-chronological velocity condition only.

Identification of the thermal problem

Concerning the thermal problem, the boundary condition of isolation ( 11) is trivially identified at every order of ξ as:

∀i ≥ 0 grad(θ i ).n = 0 (Γ lat ∪ Γ 0 ∪ Γ sup ) (25)
The constitutive equation ( 14) can be identified at order ξ -1 as ∂ θ 0 ∂ τ = 0, which implies that < θ 0 >= θ 0 (t), and at order ξ 0 as:

∂ θ 0 ∂t + ∂ θ 1 ∂ τ = A∆θ 0 + B N σ 0 : σ 0 ( 26
)
This allows to define the temperature variation as the superposition of a short time fluctuation θ 1 one order smaller than the long time variation θ 0 (t), which is completely defined by averaging the previous equation:

∂ θ 0 ∂t = A∆θ 0 + B N < σ 0 : σ 0 > (27)
Finally, the homogenization method allowed to split the initial multi time scale thermomechanical problem into three mono time scale coupled problems : the mechanical microchronological elastic problem that induces a source term in the thermal diffusion problem and a therm-dependent macro-chronological problem that gives the geometry changes.

NUMERICAL PROCEDURE

Simplification of the problem

Although the macro-chronological should strictly be visco-elastic, it was set as a purely viscous problem. Indeed, the macroscopic time of the process is rather long compared to the characteristic Maxwell time so that the steady state is suposed to be rapidly reached. Of course this assumption will be valid for rather high temperatures (close to θ m ). Nevertheless, for a better deformed shape representativeness, a strain rate dependent viscosity was used, therefore implying a non-linear flow problem.

Moreover, though rigorously valid above T g the homogenized equation were used over the whole temperature range, but it was checked that th epresent formulations could be extended assuming that n = -1 and l = -1 in equation [START_REF] Sethian | Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry[END_REF].

Resolution Scheme

The systems are solved using a classical Galerkin FEM. For each time step the three systems of equations are solved successively. Since coupling between the three systems is rather weak, they are solved iteratively until the residual of the three variational forms are simultaneously smaller than a prescribed tolerance. To solve each problem (which can be non-linear) a Newton-Raphson method was used.

The resolution is performed on a fixed mesh where the free surface is described by a levelset field. This is done using the C++ Library X-FEM. At each time step, once the three physical problems are solved, the levelset is propagated using a Hamilton-Jacobi method [START_REF] Sethian | Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry[END_REF]. Then, using an operator splitting method [START_REF] Smolianski | Finite-element / level-set / operator-splitting (felsos) approach for computing two-uid unsteady ows with free moving interfaces[END_REF], the temperature field is convected with the macro-chronological solution, using an SUPG method. The mesh is made of unstructured triangles over the whole domain, which is reduced to a half energy director, a part of the lower and upper plates and the void between directors. The initial levelset, that defines the free surface is shown on figure 3. Note that the singularity at the tip of the director has been smoothed by a bevel.

Material Data

The material parameters used in the simulations were chosen from the literature. Elasticity : For the elastic micro-chronological system, the material was assumed to be isotropic with temperature dependent Young modulus E. In a first approximation based on [START_REF] Cogswell | Thermoplastic aromatic polymer composites[END_REF][START_REF] Nicodeau | Modelisation du Soudage en Continu des composites a Matrice Thermoplastique[END_REF][START_REF] Lamethe | Etude de l'adhesion de composites semi-cristallins ; Application a la mise en oeuvre par soudure[END_REF]we propose a simple linear dependency:

E = 3.7e9 -1e7.T (˚C) Pa (28)
Viscosity : In order to take into account the shear-thining effect the viscosity was set as a Carreau law where the newtonian viscosity has an arrhenius type form [START_REF] Nicodeau | Modelisation du Soudage en Continu des composites a Matrice Thermoplastique[END_REF] :

η = η 0 (T ) 1 + (λ careau D eq ) a m-1 a η 0 = A.exp( E a R.T ) (29)
where a = 0.7, m = 0.54, E a = 74400 and A = 5, 6e -3 Thermal parameters : Using data provided by [START_REF] Ageorges | Characteristics of resistance welding of lap shear coupons : Part i : Heat transfer[END_REF][START_REF] Cogswell | Thermoplastic aromatic polymer composites[END_REF][START_REF] Lamethe | Etude de l'adhesion de composites semi-cristallins ; Application a la mise en oeuvre par soudure[END_REF], we propose simple approximation for ρc and k :

ρc = 1.3 10 6 + 5 10 3 T (˚C) J/m 3 /K k = 0.24W /m/K (30)

RESULTS AND DISCUSSION

Despite the presentation of material data, we solved the dimensionless problems and therefore present dimensionless quantities in the following.

Figure 5 shows the evolution of the levelset negative part, which is the material domain. We clearly observe a flow of polymer that begins at the tip of the director. As the loading is pursued the flowing zone increases and fills the gap between the two plates. This is in agreement with the deformed shape of the micrography 4, made on a sample obtained with a stopped experiment. It would correspond to the simulated shape at time t = 1.1 s. Next step would be to obtain better controlled interrupted experiments. We can also notice that a local refinement of the mesh around the levelset iso-zero would be necessary in order to describe the free surface more precisely.

This shape is explained by the temperature fields of figure 6 associated with the thermo-dependent mechanical parameters. It clearly shows that the tip of the director heats first as found in the literature [START_REF] Wang | Fem investigation of the temperature field of energy director during ultrasonic welding of peek composites[END_REF][START_REF] Benatar | Ultrasonic welding of advanced thermoplastic composites[END_REF]. Despite those good first results, at this stage the temperature only reaches θ * = 0.4 which seems rather low according to the expected welding at θ * = 1.

This shows the need for finer instrumented experiments and material parameter identification. They are be the subject of ongoing works that will tend to validate and adjust the presented model.

Figure 2 :

 2 Figure 2: Geometry of the initial energy director.

Figure 3 :

 3 Figure 3: Initial Level Set. The iso-zero is in black.

Figure 4 :

 4 Figure 4: Micrography of a stopped experiment.

Figure 5 :Figure 6 :

 56 Figure 5: Evolution of the LevelSet. Negative part only.