L Denis

F Tupin

X Rondeau

EXACT DISCRETE MINIMIZATION FOR TV+L0 IMAGE DECOMPOSITION MODELS

Keywords: denoising, discrete minimization, graphcuts, synthetic aperture radar

Penalized maximum likelihood denoising approaches seek a solution that fulfills a compromise between data fidelity and agreement with a prior model. Penalization terms are generally chosen to enforce smoothness of the solution and to reject noise. The design of a proper penalization term is a difficult task as it has to capture image variability. Image decomposition into two components of different nature, each given a different penalty, is a way to enrich the modeling. We consider the decomposition of an image into a component with bounded variations and a sparse component. The corresponding penalization is the sum of the total variation of the first component and the L0 pseudo-norm of the second component. The minimization problem is highly non-convex, but can still be globally minimized by a minimum s-t-cut computation on a graph. The decomposition model is applied to synthetic aperture radar image denoising.

INTRODUCTION

Image denoising has been considered in the seminal work of Rudin, Osher and Fatemi (ROF) [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] as the decomposition of an image into a component with bounded variations (BV), and a noise component. The noise is removed by minimizing the total variation (TV) of the BV component, subject to a data fidelity constraint. The popularity of TV-based denoising comes from its ability to preserve sharp edges in the denoised image (i.e., in the BV component).

It has long been noted that ROF model leads to "cartoonlike" images. This can be better understood by considering ROF in a Bayesian framework. Data fidelity expresses the likelihood while total variation models the prior on the denoised image. Piecewise constant images have low total variation, and are therefore given a high prior probability. TV prior therefore biases the solution towards piecewise constant (i.e., "cartoon") images. This is often considered a problem, sometimes referred to as a staircasing effect [START_REF] Buades | The staircasing effect in neighborhood filters and its solution[END_REF].

This work has been funded by DGA under contract REI 2008.34.0042.

Meyer proposed to take advantage of this effect to perform a decomposition of an image into its geometrical (socalled "structure") and textural components. To model the prior of textures as highly oscillating, he suggested the use of the G-norm [START_REF] Meyer | Oscillating patterns in image processing and nonlinear evolution equations: the fifteenth Dean Jacqueline B[END_REF]. Since that work, several image decomposition models have been studied in the literature [START_REF] Aujol | Structuretexture image decomposition: modeling, algorithms, and parameter selection[END_REF][START_REF] Yin | Image Cartoon-Texture Decomposition and Feature Selection Using the Total Variation Regularized L 1 Functional[END_REF][START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF][START_REF] Gilles | Image Decomposition: Theory, Numerical Schemes, and Performance Evaluation[END_REF]. In some approaches, the focus is put on the decomposition of a noiseless image into two parts. In others, the image is decomposed into two parts plus a noise component. We follow the latter approach in this paper.

We consider the restoration of an image formed by the sum of a piecewise constant component and by some isolated pixels. Such a model can for example be applied to synthetic aperture radar (SAR) images in which the scene can often be considered as a mix of homogeneous regions and point-like strong scatterers. The restoration of the two components can be obtained by minimizing the sum of TV and L0, as discussed in section 2. The minimization problem to handle is challenging: it is combinatorial due to the L0 term. The specific form of the energy (namely, the separability of the nonconvex part) can be exploited to design an algorithm for exact discrete minimization. The algorithm, described in section 3, is based on a minimum s-t-cut computation on a graph. We illustrate the method on numerical experiments and SAR data in section 4.

THE TV+L0 DECOMPOSITION MODEL

We describe the problem of the recovery of a scene u from an observed image v corrupted by noise. Images are considered sampled and quantized. The scene decomposition model is introduced in the following paragraph. Then, noise modeling is described. Finally, the TV+L0 minimization problem is stated.

Scene decomposition model

We consider scenes u that can be decomposed as the sum of two terms: u = u BV + u S , where u BV is a component with low total variation (i.e., close to piecewise constant), and u S is a sparse component (i.e., with all pixels but a few equal to zero).

The total variation of component u BV can be approximated 1 by TV(u BV ) = (k,l) w k,l |u BVk -u BVl |, where (k, l) denotes a clique (i.e., pair of neighboring pixels), and w k,l is a weight. The sparsity of component u S is defined by the value of the L0 pseudo-norm, i.e., the number of non-zero pixels of u S : L0(u S ) = k 1δ(u Sk ).

By considering that the two components u BV and u S are statistically independent, the joint prior distribution p(u BV , u S ) can be modeled in the framework of Markov random fields by:

-log p(u BV , u S ) = -log p(u BV ) -log p(u S ) = β BV TV(u BV ) + β S L0(u S ).
(1)

Image formation model

We assume that the likelihood distribution that relates the noisy image v to the scene u is separable, i.e., can be written as a product of the likelihood at each pixel k:

p(v|u) = k p(v k |u k ). (2)
We therefore consider no mixing (such as that caused by blurring) or noise correlation in our image formation model. If the noise is additive, the image v can be written as v = u BV + u S + n, with n a noise component. In the case of multiplicative noise, like speckle noise occurring in coherent imaging technique such as SAR, the image writes v = (u BV + u S ) × n. Note that for some noise distributions (e.g., Poisson noise, or Rice distribution), no such relation exist.

The likelihood distribution may depend on each term of the scene decomposition p(v k |u BVk , u Sk ), as will be discussed in section 4.

Energy minimization problem

Starting from an observed noisy image v, the components u BV and u S can be estimated in the maximum a posteriori sense by solving the following minimization problem:

(u BV , u S ) = arg min (uBV,uS) -log p(v|u BV , u S )-log p(u BV , u S ) = arg min (uBV,uS) D(v, u BV , u S ) + β BV TV(u BV ) + β S L0(u S ) (3)
Given an input image v with N pixels, minimization problem (3) has 2N unknowns. The objective function is noncontinuous and highly non-convex in variable u S (possibly also in variable u BV depending on D(•)) and is not separable in variable u BV (the TV term induces a coupling between values at different sites). It is thus very challenging to solve. 1 our choice of anistropic TV rather than isotropic TV will find its justification in the use of the discrete minimization technique described in section 3

We describe next section how the global minimum can still be found by exact discrete minimization via a combinatorial optimization technique based on a minimum s-t-cut computation.

EXACT DISCRETE MINIMIZATION BY GRAPH-CUTS

We show in the following paragraphs how to solve minimization problem (3) exactly among discrete images u BV and u S .

At each of their N pixels, these images take values respectively in the discrete ordered sets

{α 1 , α 2 , • • • , α mBV } and {γ 1 , γ 2 , • • • , γ mS }.

Problem reformulation

We begin by considering minimization problem (3) for u BV fixed. We write u ⋆ S (u BV ) the minimizer of one such restricted problem:

u ⋆ S (u BV ) = arg min uS D(v, u BV , u S ) + β S L0(u S ) (4)
It comes from the separability of the data fidelity and L0 terms (see our hypotheses discussed in paragraph 2.2) that the computation of u ⋆ S (u BV ) requires to solve N mono-dimensional minimization problems: We can now reformulate minimization problem (3) into an equivalent problem involving only u BV :

     u Sk ⋆ (u BVk ) = u Sk † if -log p(v k |u BVk , u Sk † ) + β S < -log p(v k |u BVk , u Sk = 0), u Sk ⋆ (u BVk ) = 0 otherwise; (5
arg min uBV D(v, u BV , u ⋆ S (u BV )) + β S L0(u ⋆ S (u BV )) (i) +β BV TV(u BV ) (ii) (6)
The term (i) is of the form k f k (u BVk ) (i.e., it is separable), and (ii) is a convex term of the form (k,l) g kl (u BVk , u BVl ) (i.e., involving only first order cliques). Solving problem [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF] gives u ⋆ BV . The pair (u ⋆ BV , u ⋆ S (u ⋆ BV )) obtained is then a global minimizer of problem (3). We describe in next paragraph how problem (6) can be stated equivalently as a minimum s-t-cut computation on a graph.

Graph representation

Minimization problems of the form of problem [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF], with a non-convex but separable discrete potential f k and a con- vex discrete potential over cliques g kl can be exactly solved by computing a minimum s-t-cut on a graph with N × m BV nodes [START_REF] Ishikawa | Exact optimization for Markov random fields with convex priors[END_REF][START_REF] Darbon | Image restoration with discrete constrained Total Variation part II: Levelable functions, convex priors and non-convex cases[END_REF]. We give here a quick overview of how to restate such a minimization problem as a min-cut computation.

We begin by giving a brief description of Ishikawa's graph construction [START_REF] Ishikawa | Exact optimization for Markov random fields with convex priors[END_REF]. As depicted in figure 1, the graph is a superimposition of layers of nodes. Each layer has N nodes, one for each image pixel, and pairs of arcs connect neighboring nodes (the cliques of the underlying pixel grid shown in Fig. 1(a)). The m BV different layers represent each possible level of the discrete set {α 1 , α 2 , • • • , α mBV }. Two special nodes (terminals) are added: a source s and a sink t. Each node is also connected to its two counterparts in the next layer below and above. The first and last layers are connected respectively to the source and sink. Finally, a capacity is set to each arc, as depicted in Fig. 1(c). The node corresponding to pixel k, located on the i-th layer, represents the value u BVk = α i . It is connected to its neighbors from the same layer with horizontal arcs with capacity = β BV • (α i+1α i ). The vertical arc going downstream (from source to sink) is given the weight = f k (α i ). The vertical arc going upstream is given infinite cost = ∞.

With this construction, cuts with finite cost are in bijection with the set {α 1 , α 2 , • • • , α mBV } N of all possible discrete images u BV (see [START_REF] Ishikawa | Exact optimization for Markov random fields with convex priors[END_REF]). The cost of each cut equals the value of the energy minimized in [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF] for the corresponding image u BV . Thus, the minimum cut gives the global minimizer u ⋆ BV of problem [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF], from which we deduce u ⋆ S (u ⋆ BV ) to solve the initial image decomposition problem (3).

RESULTS

We illustrate in the following paragraphs the application of the discrete minimization algorithm to decompose a SAR image into homogeneous regions and strong scatterers with the TV+L0 model.

Model for SAR images

High and very high resolution SAR scenes are generally a mix of rather homogenous regions and point-like strong scatterers. These scatterers have strong contrast (amplitudes several orders of magnitude higher than that of the homogeneous background). Man-made structures (buildings, bridges, pylons, . . . ) produce such strong echoes.

The observed SAR amplitude image v is corrupted by speckle noise generated by the coherent summation of echoes coming from different scatterers. In a homogeneous region, noise follows Rayleigh distribution (Gamma distribution when considering intensity rather than amplitude). When a strong scatterer is present in a resolution cell, noise distribution is better described by a Rice distribution [START_REF] Goodman | Speckle phenomena in optics: theory and applications[END_REF] (see Fig. 2):

-log p(v k |u BVk , u Sk ) = v k 2 + u Sk 2 2u BVk 2 + 2 log u BVk -log I 0 v k • u Sk u BVk 2 , (7)
with I 0 modified Bessel function of the first kind. Note that the amplitude u BVk must be strictly positive and that u Sk is either equal to zero (absence of strong scatterer) or strictly positive. In the absence of a strong scatterer, equation [START_REF] Gilles | Image Decomposition: Theory, Numerical Schemes, and Performance Evaluation[END_REF] simplifies to the widely used Rayleigh model.

Energy minimization

Rice neg-log-likelihood defined in equation ( 7) is quasiconvex (but non-convex) in each variable. For a given value of u BV , it reaches its unique minimum at u * S (u BV ) (red curve on Fig. 2). At each pixel k, and for each value {α 1 , α 2 , • • • , α mBV }, the corresponding value of u Sk * (u BVk ) can be computed efficiently by bisection. The number of quantization levels m S of the sparse component can be chosen large to account for the high dynamic range of strong scatterers with very little overhead. The bounded variations component has very low dynamic range and choosing m BV ≈ 100 is often enough in practice. The positivity constraints (∀k, u BVk > 0 and u Sk ≥ 0) are straightforwardly enforced by considering discrete levels that satisfy the constraints. The tuning of parameters β BV and β S is done by hand by first setting a suitable β BV value with β S = 0, and then choosing a value of β S that gives a satisfying number of strong scatterers.

Decomposition results

We illustrate the method on a real SAR image (Fig. 3.a: south of Toulouse city, France, c CNES/DGA). Fig. 3 compares the results obtained when denoising with a total variation minimization approach and with the TV+L0 decomposition approach. The TV approach corresponds to setting β S to infinity. Strong scatterers are either suppressed or distorted (reduced amplitude, spreading) as they do not fit to the prior model (result of TV denoising shown Fig. 3.b). TV+L0 de-composition produces much smoother regions in the BV component (Fig. 3.c left) and sharp targets in the sparse component (Fig. 3.c right).

CONCLUSION

The proposed decomposition provides a better model to images such as in SAR imaging (both in terms of scene prior and model). The challenging highly non-convex can be exactly solved by computation of a min-cut. Though the number levels can be limited, the graphbased minimization algorithm is limited its use by the of the memory representation of the graph: N × m BV nodes and about 3N × m BV pairs of arcs.

  ) with u Sk † = arg min uS k -log p(v k |u BVk , u Sk ) either known in closed form, or numerically computed in O(m S ) operations in the worst case.

Fig. 1 .

 1 Fig. 1. Graph construction for discrete minimization.

Fig. 2 .

 2 Fig. 2. Surface representation of Rice neg-log-likelihood (surface drawn for v k = 1).

Fig. 3 .

 3 Fig. 3. SAR image denoising: comparison of TV denoising and TV+L0 decomposition (SAR image c CNES/DGA).


Ngoc-Lam L Phung
email: ngoclam250@yahoo.com

Véronique Favier
email: veronique.favier@ensam.eu

Nicolas Ranc
email: nicolas.ranc@ensam.eu

Fréderic Vales
email: frederic.vales@ensam.eu

Haël Murghrabi
email: hael.mughrabi@ww.uni-erlangen.de

F Valès

H Mughrabi

Very high cycle fatigue of copper: Evolution, morphology and locations of surface slip markings

Keywords:

come

Introduction

There is currently a growing demand to investigate the very high cycle fatigue (VHCF) regime (higher than 10 7 cycles) to improve the reliability of design calculations and to extend the service lifetimes of some components such as engines. The shape of the S-N curve (plot of stress versus number of cycles to failure) varies with the material [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF]. The hardest materials were found to exhibit a gradual decrease of the fatigue resistance with increasing number of cycles, whereas the softest alloys were found to present a plateau from 10 7 up to 10 9 -10 10 cycles [START_REF] Wang | Gigacycle fatigue of ferrous alloys[END_REF]. Titanium and tantalum [START_REF] Papakyriacou | Influence of loading frequency on high cycle fatigue properties of b.c.c. and h.c.p. metals[END_REF] and aluminium [START_REF] Stanzl-Tschegg | Fatigue and fatigue crack growth of aluminium alloys at very high numbers of cycles[END_REF] exhibit a decrease of the fatigue resistance at a constant rate from 10 4 up to 10 8 cycles. The S-N curve for pure copper displays two slopes: the decrease of the fatigue resistance is stronger from 10 4 up to 10 7 cycles than from 10 7 up to 10 9 cycles [START_REF] Awatani | A study of the fatigue limit of copper[END_REF][START_REF] Lukas | High cycles fatigue life of metals[END_REF][START_REF] Mughrabi | Fatigue mechanisms in the ultrahigh cycle regime[END_REF][START_REF] Mughrabi | On multi-stage fatigue life diagrams and the relevant life controlling mechanisms in ultrahigh-cycle fatigue[END_REF]. All these results demonstrated that failure could occur beyond 10 7 cycles. However, the reason why such discrepancy as far as the VHCF response with regard to material is concerned is not clearly understood. Mughrabi [START_REF] Mughrabi | On multi-stage fatigue life diagrams and the relevant life controlling mechanisms in ultrahigh-cycle fatigue[END_REF] suggested distinguishing two classes of metallic materials when they are loaded in the VHCF regime: type I materials are pure ductile single-phase metals containing no extrinsic internal defects; type II materials contain internal defects such as precipitates or non-metallic inclusions. Most current studies have focused on the VHCF behavior of type II materials and revealed that, in the transition from high cycle fatigue (HCF) to VHCF, the origins of fatigue failure changed from surface to interior ''fish-eye'' fracture for instance at nonmetallic inclusions [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF][START_REF] Mughrabi | Fatigue mechanisms in the ultrahigh cycle regime[END_REF]. For type I materials, only few investigations mainly on copper [START_REF] Hessler | Near-threshold behaviour of polycrystalline copper[END_REF][START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF][START_REF] Weidner | Fatigue damage in copper polycrystals subjected to ultrahigh-cycle fatigue below the PSB threshold[END_REF][START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF] are available regarding the shape of the fatigue life curve and the damage evolution [START_REF] Hessler | Near-threshold behaviour of polycrystalline copper[END_REF][START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF][START_REF] Weidner | Fatigue damage in copper polycrystals subjected to ultrahigh-cycle fatigue below the PSB threshold[END_REF][START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF][START_REF] Krupp | The significance of microstructural barriers during fatigue of a duplex steel in the high-and veryhigh-cycle-fatigue (HCF/VHCF) regime[END_REF][START_REF] Höppel | Influence of grain size and precipitation state on the fatigue lives and deformation mechanisms of CP aluminium and AA6082 in the VHCF-regime[END_REF]. Hessler et al. [START_REF] Hessler | Near-threshold behaviour of polycrystalline copper[END_REF] and Weidner et al. [START_REF] Weidner | Fatigue damage in copper polycrystals subjected to ultrahigh-cycle fatigue below the PSB threshold[END_REF] showed that cyclic strain localization in VHCF below the PSB threshold occurs in some form of PSBs. Despite the fact that there seems to be no possibility for the formation of a pronounced PSB structure at very low stress amplitudes, Mughrabi's model suggests that cracks initiate at the surface owing to the accumulation of very small but irreversible plastic deformation over very large number cycles [START_REF] Mughrabi | On the life-controlling microstructural fatigue mechanisms in ductile metals and alloys in the gigacycle regime[END_REF]. In this work, the morphology and the location of slip markings observed after fatigue testing in the VHCF regime is clarified for pure copper. To perform experiments up to a very high number of cycles in a reasonable time, ultrasonic equipment at a testing frequency of 20 kHz was used. Three types of persistent slip markings are classified for the first time in terms of the stress amplitude ranges at which they appear predominantly. Slip markings morphology observed at 20 kHz and frequencies lower than 100 Hz are compared. The relationship between the appearance of persistent slip markings and failure is discussed. emax 2. Material and experimental procedure

Material and specimens

Oxygen-free high-conductivity commercially pure (99.95%) copper was tested. The polycrystalline material was hot rolled and supplied by Griset company. The specimens were extracted from the center of a 14 mm thick plate (Fig. 1). The mean grain size was about 30 lm. The microstructure contains about 0.3 volume fraction of annealed twins. The texture is quasi-isotropic. Fatigue tests were performed using a piezoelectric fatigue machine designed by Bathias and Paris [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF]. Cylindrical hourglass shaped specimens were fatigued to get the S-N curve while flat hourglass shaped specimen were designed to carry out surface observations after interrupted fatigue tests. The specimens were designed to run in longitudinal vibration resonance with the piezoelectric machine at 20 kHz. The diameter in the center of the cylindrical specimens was 3 mm. The cross section in the center of the flat specimens was 2 mm thick and 3 mm wide. The dimensions of both specimen types are indicated on Fig. 2. After machining, the specimens were heat treated at 250 °C for 60 min to relieve the residual stress without change of microstructure. After thus thermal treatment, the tensile strength of the material is about 230 MPa. Then, all specimens were mechanically polished and soldered into a screw to attach them to the ultrasonic transducer. Finally, they were electrolytically polished to remove all hardened layers on the specimen surface. As a result, before testing, the specimens were mirror polished and without residual stresses. For all calculations, the Young's modulus was taken equal to 130 GPa [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF].

Experimental procedure

The piezoelectric transducer of the ultrasonic fatigue machine converts an electronic signal into a mechanical displacement. This displacement is amplified via a horn and transmitted to the specimen. It was calibrated at the beginning of each series of tests with a laser extensometer. The displacement was found to be perfectly sinusoidal in the course of time. Assuming pure linear elastic behavior, the strain and stress distributions along the specimen were calculated using a one-dimensional approach (the mechanical variables depend only on x coordinate) in forced vibration regime. The strain was then measured by means of a small strain gage stuck on the specimen surface center to check the calibration. The strain R ratio defined as e min was -1. In the following, the stress amplitude rather than the strain amplitude is used to define the fatigue test. In hour-glass shaped specimens, the strain and stress amplitudes strongly decrease towards the specimen ends. They are 97% of maximum at places 1.25 mm and 1.5 mm apart from the center for the cylindrical and flat specimens, respectively (Fig. 2). Additional 3D elastic finite element simulations in forced vibration regime, carried out on the hour-glass shaped flat specimen, showed that the stress amplitude at the border of the smallest cross section is slightly greater (rv5%) than the stress amplitude at the center of the cross section.

To get the S-N curve for a number of cycles higher than 10 6 , hour-glass shaped cylindrical specimens were continuously fatigued up to failure at various stress amplitudes. Failure occurred when a crack had grown large enough to decrease the natural frequency of the system below the standard operating range (19.5-20.5 kHz) leading to a machine stop. The specimens were cooled with compressed air to avoid self-heating due to intrinsic dissipation. After the fatigue test, they were broken in liquid nitrogen for fracture surface examination.

In addition, fatigue loadings performed on the hour-glass shaped flat specimens were interrupted repeatedly in order to perform surface studies by optical microscopy, scanning electron microscopy (SEM), electron back scattering diffraction (EBSD) technique and atomic force microscopy (AFM) after different numbers of cycles. The stress amplitude range was chosen much smaller than for the S-N curve to observe the early traces of plasticity on the surface specimen. In this stress amplitude range, the specimen heating was found lower than 20 °C so the specimens were not cooled. The morphology of the surface patterns was observed by means of a Hitachi S3600 FEG-SEM. EBSD analyses clarify the places of the plastic markings with regard to the polycrystalline microstructure. AFM was operating directly to the surface specimen (no replica) to keep the details of the roughness. The AFM Vecco nanoscope V was used in tapping mode in ambient air and room temperature. The AFM has a V-shaped Si 3 N 4 tip with a spring constant of about 20 N/m of the cantilever beam. The side angle of the tip is 17 nm and the radius of its curvature 8 nm.

Results

S-N curve

The S-N curve of the studied copper obtained by ultrasonic fatigue machine at 20 kHz is shown in Fig. 3 [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF][START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF]. Their results revealed a ''fatigue limit at 1 x 10 10 cycles'' also called ''fatigue life threshold'' equal to 92.2 MPa.

Fracture surface observations (Fig. 4) showed that the crack initiated at the specimen surface as commonly observed for type I materials [START_REF] Mughrabi | On multi-stage fatigue life diagrams and the relevant life controlling mechanisms in ultrahigh-cycle fatigue[END_REF][START_REF] Höppel | Influence of grain size and precipitation state on the fatigue lives and deformation mechanisms of CP aluminium and AA6082 in the VHCF-regime[END_REF][START_REF] Cartensen | Very high cycle fatigue of thin walled tubes made from austenitic stainless steel[END_REF]. SEM investigations of the fracture surface and of the surface grains close to the fracture surface reveal that the roughness at the crack initiation zone is very low with regard to the rest of the fracture surface. Crack initiates at a surface grain boundary with one side marked with slip bands while the other side was free of them. Similar results were found by Provan and Zhai [START_REF] Provan | Fatigue crack initiation and stage I propagation in polycrystalline materials I: Micromechanisms[END_REF], Cretigny and Saxena [START_REF] Cretegny | AFM characterization of the evolution of surface deformation during fatigue in polycrystalline copper[END_REF] and Mughrabi et al. [START_REF] Mughrabi | Fatigue crack initiation by cyclic slip irreversibilities in high-cycle fatigue[END_REF] for pure copper loaded in the LCF regime. It is well-known that crack initiation is intergranular in the LCF regime [START_REF] Provan | Fatigue crack initiation and stage I propagation in polycrystalline materials I: Micromechanisms[END_REF][START_REF] Cretegny | AFM characterization of the evolution of surface deformation during fatigue in polycrystalline copper[END_REF][START_REF] Mughrabi | Fatigue crack initiation by cyclic slip irreversibilities in high-cycle fatigue[END_REF][START_REF] Kim | Crack nucleation and stage I propagation in high strain fatigue-I. Microscopic and interferometric observations[END_REF][START_REF] Figueroa | Crack initiation mechanisms in copper polycrystals cycled under strain amplitudes and in step tests[END_REF]. The fatigue cracks initiate frequently along the large-angle grain boundaries. The HCF regime is the PSB dominated regime. It was commonly admitted that cracks initiate at the interface between the matrix and the PSB in polycrystals as it is in single crystals [START_REF] Mughrabi | Fatigue crack initiation by cyclic slip irreversibilities in high-cycle fatigue[END_REF][START_REF] Ma | Overview of fatigue behaviour in copper single crystals-II population, size distribution and growth kinetics of stage I cracks for tests at constant amplitude[END_REF]. However many experimental observations showed that PSBs are not only responsible for slip band cracking in PSB, but also induce intergranular cracks [START_REF] Mughrabi | Fatigue crack initiation by cyclic slip irreversibilities in high-cycle fatigue[END_REF][START_REF] Neumann | Crack initiation at grain boundaries in FCC materials -strength of metals and alloys[END_REF][START_REF] Pineau | Fatigue intergranulaire, dans Joints de grain et plasticité cristalline, sous la direction de Louisette Priester[END_REF]. The latter is promoted by the presence of large-angle grain boundary and the decrease of the stacking fault energy [START_REF] Zhang | low-cycle fatigue cracking mechanisms in fcc crystalline materials[END_REF]. In the VHCF regime, Stanzl-Tschegg and Schönbauer [START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF] mentioned that the grain boundaries are the preferred crack nucleation and propagation sites for pure copper loaded at 19 kHz in agreement with the present results.

Slip markings morphology and location

Observation of different types of slip markings

Slip markings on the copper specimen were investigated using SEM and AFM. Three types of markings were distinguished. They are labeled as type I, type II and type III in the following sections (Fig. 5). All the slip markings were shown to be persistent, namely when the surface was electropolished after fatigue, they reappeared at the same sites on the specimen surface when fatigue was resumed. All persistent slip markings are oriented at more or less 45° from the loading direction showing that they are related to the activation of the easiest crystallographic slip systems. They also displayed different features exposed in the following sections.

Characterization of type I persistent slip markings

Figs. 5a and6 show examples of SEM micrographs of type I persistent slip markings. The latter are long, straight and cross the grains. They gather in family of around ten parallel slip markings. Their roughness is more marked at the grain boundary than in the grain center suggesting that the slip markings initiated at the grain boundary. Fig. 7a illustrates AFM image of type I persistent slip markings. Fig. 7b exhibits the roughness along the white line perpendicular to the persistent slip markings. The peaks are clearly associated with persistent slip markings while the surface is very smooth between them. The height of the persistent slip markings reached 600 nm. The average height deduced from many similar profiles was found about 500 nm. The width of persistent slip markings varies from 0.5 to 3 lm and its average is 1.4 lm. The distance between slip markings increases with increasing grain size. It is for example about 5-7 lm for the mean grain size of 30 lm (Figs. 6 and7).

Characterization of type II persistent slip markings

Figs. 5b and8a show examples of SEM micrographs of type II persistent slip markings. As type I persistent slip markings, they are long and straight but they are isolated and do not cross the grains. EBSD images demonstrate that they are located very near and along grain boundaries and most of the time along and at twin boundaries (Fig. 8b). The affected twin boundaries are oriented at about 45° from the loading direction. The average height of type II is close to type I persistent slip markings height but the maximum height is higher and can reach 1-2 lm depending on the stress amplitude and the number of cycles (Fig. 9) (see Section 3.4).

Characterization of type III persistent slip markings

The early slip markings of type III are invisible under optical microscopy. SEM observations revealed the presence of clusters of very fine slip markings formed inside the grains and near the grain boundaries (Fig. 5c andd). Two slip marking directions were usually observed. The first one is more marked and occupies the majority of the cluster surface. It is very probably associated with the slip band of the primary slip system. The second is less marked and is called secondary slip band (Fig. 10). Fig. 11 shows an AFM image of type III persistent slip markings at a stress amplitude of 50 MPa after 2 x 10 8 cycles. Their roughness is less than 50 nm. They are located close to large-angle grain boundaries, twin boundaries and preferentially triple junctions.

Relation between stress amplitude and early persistent slip markings

The appearance of the different types of slip markings depends essentially on stress amplitude. At high stress amplitudes (close to or higher than 92.2 MPa, the fatigue strength at 1 x 10 10 cycles), the majority of the early slip markings is of type I (Fig. 12a). The amount of type I persistent slip markings decreases with decreasing stress amplitudes and type II persistent slip markings become more and more numerous. At stress amplitude of 65 MPa, the early persistent slip markings are of type II (Fig. 12b). At very low stress amplitudes such as 44 MPa, type III persistent slip markings are predominant and type I persistent slip markings are very rarely observed (Fig. 12c).

The numbers of cycles needed to form the early persistent slip markings and the type of persistent slip markings as a function of the stress amplitude are shown in Fig. 13. At a given stress amplitude, the test was regularly interrupted to observe the slip markings on the specimen surface. In Fig. 13, the experimental data indicate the last observation before slip markings appeared and the next observation when the early slip markings are first detected using SEM. From 62 MPa to 34 MPa, the stress amplitude varies linearly with the logarithm of the number of cycles necessary to form the early slip markings (dashed line in Fig. 13). No slip marking was observed in the region below the dashed line. Above it, persistent slip markings are present at the surface of specimen. Again, our results are in very good agreement with the experimental results obtained by Stanzl-Tschegg et al. [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF][START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF]. All kinds of persistent slip markings (I, II and III) have been reported before [e.g. [START_REF] Krupp | The significance of microstructural barriers during fatigue of a duplex steel in the high-and veryhigh-cycle-fatigue (HCF/VHCF) regime[END_REF][START_REF] Mughrabi | On the life-controlling microstructural fatigue mechanisms in ductile metals and alloys in the gigacycle regime[END_REF]. However, here these three types of early persistent slip markings are classified for the first time in terms of the stress amplitude ranges at which they appear predominantly: the range of stress amplitude higher than 65 MPa is associated with the type I, the range [45-65 MPa] is associated with the type II and the range of stress amplitude lower than 45 MPa with the type III. In particular, it is shown that the persistent slip markings of types II and III are dominant in the VHCF regimes of rather low stress amplitudes.

The S-N curve and the curve for the appearance of the early slip markings are compared in Fig. 14. Up to 10 8 cycles, at a specified number of cycles, the stress amplitude required to break the specimen is twice larger than the stress amplitude needed to form the early slip markings on the specimen surface.

Evolution of the slip markings morphology and spatial distribution over cycles

When the early type I persistent slip markings appeared on the surface specimen, 5% of grains exhibit them. With increasing the numbers of cycles, more parallel slip bands formed inside the grain and the height of the persistent slip markings rises. Concerning the type II persistent slip markings, the majority is formed on twin boundaries. The early type II persistent slip markings are very few, only one slip marking in rv5000 lm 2 . The initial persistent slip markings become longer, larger and higher during increasing cycles. AFM measurements showed that the height of type II persistent slip markings can reach 1 lm (Fig. 9a) and sometimes 2 lm. So, type II persistent slip markings can be much higher than type I persistent slip markings. Fig. 15 shows the evolution of a type II persistent slip marking at stress amplitude of 55 MPa after 10 7 cycles, 10 8 cycles and 2 x 10 8 cycles. Note that the persistent slip marking grows only in the left grain. In addition, some new persistent slip markings appeared 2-3 grains further. Concerning the persistent slip markings of type III, when the early slip markings appeared, their quantity is very small as for type II persistent slip markings. With increasing the number of cycles, the clusters of type III persistent slip markings expand, namely new persistent slip markings are observed beside the first ones. Persistent slip markings with other directions are also observed. Fig. 16 shows clusters exhibiting very fine and more pronounced type III persistent slip markings along a twin boundary (Fig. 16a) and at rv30° from a grain boundary (Fig. 16b). In the latter case, the more pronounced persistent slip markings are spaced out 5-10 lm as type I persistent slip markings. A zoom (Fig. 16c) reveals that the persistent slip markings are slightly curved, in contrast to the early type III persistent slip markings. In addition, persistent slip markings having another orientation than the main orientation were detected. These observations suggest that the slip markings type III changes into slip markings type II or type I during increasing cycles due to plastic strain localization mechanisms.

Discussion

Morphology and location of persistent slip markings

Slip markings of type I correspond to the so-called Persistent Slip Bands (PSBs) since their morphology (height and width) and their spatial distribution crossing the grains. They are commonly observed in low and high cycle fatigue regime (see e.g. [START_REF] Cretegny | AFM characterization of the evolution of surface deformation during fatigue in polycrystalline copper[END_REF][START_REF] Mughrabi | Fatigue crack initiation by cyclic slip irreversibilities in high-cycle fatigue[END_REF][START_REF] Polak | Fatigue damage in polycrystalline copper below the fatigue limit[END_REF] for polycrystalline copper). PSBs are usually associated with the ladder-like dislocation structure under the slip marking [START_REF] Mughrabi | Dislocations and properties of real materials[END_REF][START_REF] Laufer | Dislocation structures in fatigued copper single crystals[END_REF].

Persistent slip markings of type II and III are localized at or close to grain and twin boundaries. These observations reveal the key role of grain/twin boundaries acting as stress concentration factors and promoting microplasticity. More precisely, the high majority of the early type II persistent slip markings is located at twin boundaries while triple junctions are preferred sites for type III persistent slip markings. Polak and Vasek [START_REF] Polak | Fatigue damage in polycrystalline copper below the fatigue limit[END_REF] observed PSBs similar to type II persistent slip markings at strain amplitudes lower than the ''conventional fatigue limit'' (De/2 = 5 x 10 -5 [START_REF] Mughrabi | Cyclic deformation of face-centred cubic polycrystals: a comparison with observations on single crystals. Deformation of polycrystals: mechanisms and microstructures[END_REF]) in polycrystalline copper. In the same way, Neumann and Tönnessen [START_REF] Neumann | Crack initiation at grain boundaries in FCC materials -strength of metals and alloys[END_REF] and Peralta et al. [START_REF] Peralta | Fatigue fracture at copper bicrystal interfaces: fractography[END_REF] have observed PSBs along twin boundaries oriented at around 45° from the loading direction but not inside the grains at low stress amplitudes. Llanes and Laird [START_REF] Llanes | The role of annealing twin boundaries in the cyclic deformation of f.c.c. materials[END_REF] showed that twin boundaries promote very early slip and PSB formation. Recently, Man et al. [START_REF] Man | Study of cyclic strain localization and fatigue crack initiation using FIB technique[END_REF] demonstrated the presence of a ladder-like dislocation structure under a slip marking located along a twin boundary after HCF tests in fatigued nickel specimens. The presence of slip bands along twin boundaries was not so much studied in literature. However, more investigations [START_REF] Zhang | low-cycle fatigue cracking mechanisms in fcc crystalline materials[END_REF][START_REF] Neumann | Crack initiation at grain boundaries in FCC materials -strength of metals and alloys[END_REF][START_REF] Pineau | Fatigue intergranulaire, dans Joints de grain et plasticité cristalline, sous la direction de Louisette Priester[END_REF][START_REF] Thompson | The origin of fatigue fracture in copper[END_REF] demonstrated the key role of the large-angle grain and twin boundaries on crack initiation. In LCF and HCF, the numerous type I persistent slip markings or equivalently PSBs hide the type II persistent slip markings explaining why few literature results mentioned slip bands along grain and twin boundaries. Besides, the literature results quoted above [START_REF] Kim | Crack nucleation and stage I propagation in high strain fatigue-I. Microscopic and interferometric observations[END_REF][START_REF] Neumann | Crack initiation at grain boundaries in FCC materials -strength of metals and alloys[END_REF][START_REF] Pineau | Fatigue intergranulaire, dans Joints de grain et plasticité cristalline, sous la direction de Louisette Priester[END_REF][START_REF] Polak | Fatigue damage in polycrystalline copper below the fatigue limit[END_REF][START_REF] Llanes | The role of annealing twin boundaries in the cyclic deformation of f.c.c. materials[END_REF][START_REF] Man | Study of cyclic strain localization and fatigue crack initiation using FIB technique[END_REF] were obtained after symmetrical strain-controlled fatigue tests operated at frequencies below 100 Hz, i.e. much lower than the present frequency (20 kHz). To our knowledge, the key role of twin boundaries on microplasticity initiation was never highlighted for ultrasonic fatigue tests. Comparisons between our results obtained at 20 kHz and literature results obtained at low frequencies [START_REF] Kim | Crack nucleation and stage I propagation in high strain fatigue-I. Microscopic and interferometric observations[END_REF][START_REF] Neumann | Crack initiation at grain boundaries in FCC materials -strength of metals and alloys[END_REF][START_REF] Pineau | Fatigue intergranulaire, dans Joints de grain et plasticité cristalline, sous la direction de Louisette Priester[END_REF][START_REF] Polak | Fatigue damage in polycrystalline copper below the fatigue limit[END_REF][START_REF] Llanes | The role of annealing twin boundaries in the cyclic deformation of f.c.c. materials[END_REF][START_REF] Man | Study of cyclic strain localization and fatigue crack initiation using FIB technique[END_REF] show that, in both cases, (i) the length, the height, the width and the location of slip markings of type I and II are similar and (ii) the relative amount of the type II slip marking with regard to type I slip band increases with decreasing stress or strain amplitudes. These comparisons suggest that similar mechanisms are responsible for formation of PSBs despite the plastic strain amplitude is much lower and the number of cycles much higher in the VHCF characterized by 20 kHz fatigue tests than in the HCF characterized by low frequency tests. In view of the large height of type II persistent slip markings, it is reasonable to suggest that the type II persistent slip markings could promote intergranular crack initiation at twin boundaries, consistently with the observations. It is worth noticing that Stanzl-Tschegg et al. [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF][START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF] and Weidner et al. [START_REF] Weidner | Fatigue damage in copper polycrystals subjected to ultrahigh-cycle fatigue below the PSB threshold[END_REF] discussed the origin of the discrepancy between the PSB thresholds determined by conventional and by ultrasonic fatigue machines. The PSB threshold stress and the stress fatigue limit, measured in ultrasonic tests, are systematically higher than the values found at conventional frequencies. It is attributed to different ramp loading conditions, a strain rate sensitivity effect and/or a time effect related to impurities and diffusion processes.

The morphology of type III persistent slip markings is different from the two others. The type III persistent slip markings are very fine and spread into clusters of closely-spaced parallel bands. This type of persistent slip markings in ductile single-phase metals is rarely mentioned in literature. Stanzl-Tschegg et al. [START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF] observed parallel 14 nm-height slip bands at triple junctions after 1.05 x 10 10 cycles at 45 MPa under fatigue at 20 kHz in similar copper as the present one. Basinski et al. [START_REF] Basinski | Early stages of fatigue in copper single crystals[END_REF] detected very fine, closely-spaced parallel slip bands at low frequency tests on copper single-crystal but after very few number if cycles (lower than 10 and typically one quarter and one half cycles). More pronounced slip markings very rapidly appears to the detriment of the others, very similarly to what we observed in Fig. 15b. Basinski et al. [START_REF] Basinski | Early stages of fatigue in copper single crystals[END_REF] concluded that the deformation associated with the fine slip bands is very homogeneous and probably took place in larger regions without any markings because of recoverable (reversible) slip. Besides, type III persistent slip markings morphology is not so far from the morphology of the so-called shear bands found in underaged alloys [START_REF] Risbet | Some comments about fatigue crack initiation in relation to cyclic slip irreversibility[END_REF]. The shear bands are tens of microns in height and spaced about 0.5-2 lm. Generally speaking, literature reviews [START_REF] Zhang | low-cycle fatigue cracking mechanisms in fcc crystalline materials[END_REF][START_REF] Risbet | Some comments about fatigue crack initiation in relation to cyclic slip irreversibility[END_REF] concluded that the distance, the height and the width of the slip markings decrease when planar slip is promoted.

Thus, the type III persistent slip markings very likely result from predominant planar slip of one or two active slip systems. The dislocation interactions between these two systems induce some slip irreversibility leading to markings on the specimen surface. Dislocation pile-ups at grain boundaries produce stress concentrations and promote cross slips. As a consequence, dislocation structures form inducing plastic strain localization. This mechanism is consistent with the fact that the type III persistent slip markings change into the type I or II persistent slip markings during increasing cycles. At higher stress amplitudes, the plastic strain localization mechanisms take place much rapidly leading to more pronounced type I persistent slip markings and type II at lower number of cycles.

Relationship between persistent slip markings appearance and fracture

Results in VHCF demonstrate that when the stress amplitude decreases, the number of cycles to failure and the number of cycles needed to form persistent slip markings increase and no limits are observed. In a previous work [START_REF] Blanche | Dissipation assessments during dynamic very high cycle fatigue[END_REF], a dissipated energy was evidenced at all applied stress ranges, although very small at the lowest stress amplitudes. In addition, the dissipated energy rate was found to always raise during cycles. This result is consistent with the gradual increase of the number of persistent slip markings observed on the specimen surface. The presence of increasing dissipation over cycles indicates that no stabilized cyclic state was reached during the VHCF fatigue tests on copper. In other words, the results show that the fatigue strength of copper (about 93 MPa at 10 10 cycles) cannot be considered as the fatigue limit from a thermodynamic standpoint, considering that, ideally, the fatigue limit is associated with no change of dissipated and stored energy from one cycle to another. At a given stress amplitude, it is not possible to observe the appearance of the early slip markings and the fracture during the number of cycles reached during experiments. Moreover, the number of cycles for the evolution of slip markings and possibly crack propagation are higher than 10 12 cycles. These results are in good agreement with [START_REF] Hessler | Near-threshold behaviour of polycrystalline copper[END_REF][START_REF] Stanzl-Tschegg | Life-time and cyclic slip of copper in the VHCF-regime[END_REF][START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF]. Stanzl-Tschegg et al. [START_REF] Stanzl-Tchegg | Mechanisms of strain localization, crack initiation and fracture of polycrystalline copper in the VHCF regime[END_REF] suggested that numerous small cracks do not likely lead to failure below 90 MPa because the cyclic stress intensities remain lower than the stress intensity threshold necessary for crack growth.

Conclusions

The surfaces of specimens of commercially pure polycrystalline copper were investigated after 20 kHz fatigue tests in the VHCF regime. The main results are:

• Three types of slip markings were classified for the first time in terms of their morphology, their spatial distribution in the polycrystalline material and the stress amplitude ranges at which they appear predominantly. • For stress amplitudes ranging from 45 MPa to 65 MPa, straight and long early persistent slip markings of type II are produced and located at twin boundaries oriented at about 45° from the loading direction. They are sometimes found in the HCF regime. • For stress amplitudes below 45 MPa, clusters of fine and short early persistent slip markings of type III are produced and mainly located at triple junctions. They are rarely observed in HCF. • The type III persistent slip markings are suggested to change into the type I or type II due to plastic strain localization mechanisms. • Similar mechanisms are responsible for formation of persistent slip markings SBs at 20 kHz and frequencies lower than 100 Hz but do not take place in the same stress amplitude -number of cycles range.

• Results in VHCF demonstrated that when the stress amplitude decreases, the number of cycle to failure and the number of cycle needed to form persistent slip markings increases, and no limits are observed. • The mechanical state responsible for the location of the three SM types will be investigated in another paper.

Fig. 1 .

 1 Fig. 1. Polycrystalline pure copper microstructure: cross section (a) in rolling direction and (b) perpendicular to the rolling direction.

Fig. 2 .

 2 Fig. 2. Ultrasonic fatigue specimens and stress distributions along the specimen axis x (a) cylindrical hour glass shaped specimen and (b) flat hour-glass shaped specimen.

Fig. 3 .

 3 Fig. 3. S-N curve of polycrystalline pure copper obtained from ultrasonic fatigue tests at 20 kHz.

Fig. 4 .

 4 Fig. 4. Fracture specimen at stress amplitude of 105 MPa; N = 3.1 x 10 7 cycles -(a) fracture surface: the crack initiation zone is indicated by the arrow, (b) zoom on the crack initiation zone, (c) specimen surface normal to the fracture, left side, zone free of slip bands, and (d) specimen surface normal to the fracture surface, right side, zone marked with slip bands.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Type I persistent slip markings at stress amplitude of 89 MPa; N = 10 6 cycles -the loading direction is horizontal.

Fig. 8 .

 8 Fig. 8. Type II persistent slip markings (indicated by arrows) at stress amplitude of 57.5 MPa -N = 10 7 cycles -(a) SEM image and (b) EBSD analyze at the same zone showing the initiation of type II slip markings SMs at twin boundaries -the loading direction is horizontal.

Fig. 9 .

 9 Fig. 9. Surface roughness of a type II persistent slip marking at stress amplitude of 57.5 MPa -N = 10 7 cycles -(a) AFM image and (b) surface profiles along the white lines numbered 1 and 2.

Fig. 10 .

 10 Fig. 10. Type III persistent slip markings at stress amplitude of 48 MPa -N = 10 8 cycles oriented along two directions.

Fig. 11 .

 11 Fig. 11. Surface roughness of type III persistent slip markings at stress amplitude of 50 MPa -N =2 x 10 8 -(a) AFM image, and (b) surface profile along the white line.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. SEM images of slip band formed at (a) stress amplitude of 90 MPa -N =5 x 10 5 cycles, (b) stress amplitude of 65 MPa -N = 10 6 cycles and (c) stress amplitude of 44 MPa -N = 4.5 x 10 8 cycles.

Fig. 14 .

 14 Fig. 14. Stress amplitude-number of cycles for fracture (open and full circles) and for forming the early slip markings (open and full squares).

Fig. 15 .

 15 Fig. 15. Evolution of one type II persistent slip marking with numbers of cycles at stress amplitude of 55 MPa -(a) after 10 7 cycles, (b) after 10 8 cycles and (c) after 2 x 10 8 cycles.

Fig. 16 .

 16 Fig. 16. Plastic strain localization in the cluster of type III persistent slip markings indicated by the arrows -(a) at a twin boundary like type II persistent slip markings at stress amplitude of 48 MPa -N = 2 x 10 8 cycles, (b) close to a grain boundary but pointed at the grain center like type I persistent slip markings at stress amplitude of 44 MPa -N = 4.5 x 10 8 cycles, and (c) similar as (b) but at another place and at higher magnification at which slip markings with another orientation were observed at the top on the micrograph.

Acknowledgements

We would like to acknowledge the Agence Nationale de la Recherche France ANR-09-BLAN-0025-01 for the funding that enabled this work to be carried out and Camille Pissinier and Fabrice Détrez for their help in AFM measurements.