Alain Trémeau

Michele Gouiffes
email: gouiffes@sic.sp2mi.univ-poitiers.fr

Christophe Collewet

M Gouiffès

C Fernandez-Maloigne

) A Trémeau

Feature points tracking by using photometric model and colorimetric invariants

come

Introduction

Many computer vision algorithms rely on the accurate and robust matching of features between several frames through an image sequence, for example 3D reconstruction, video surveillance or visual servoing. The feature points have proved to be the only available feature in non-structured scenes, where no edges or lines can be extracted, such as in agricultural, spatial, medical or underwater applications [START_REF] Espiau | Robust features tracking for robotic applications : towards 2 1/2 D visual servoing with natural images[END_REF]. In addition, the whole of the points are not likely to be occluded at the same time. Nevertheless, using points is difficult since the only information available is the luminance (or the colour) of the neighboring pixels around the points to be tracked. In the seminal approaches, the luminance (or the colour) was assumed to be constant between successive frames, as in [START_REF] Lucas | An iterative image registration technique[END_REF][START_REF] Tomasi | Detection and tracking of point features[END_REF] for luminance images, or [START_REF] Heigl | Tracking points in sequences of color images[END_REF] for colour. Although some more robust approaches have been proposed thereafter, all of them are based on those same photometric constraints. These methods assume that the objets are lambertian and that no lighting changes occur during the wole sequence. However, in order to cope with bias and gain changes, a photometric normalization can be achieved [START_REF] Tommasini | Improving feature tracking with robust statistics[END_REF]. In a similar manner, Jin et al. [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF] have proposed to compensate for an affine photometric model in luminance images. In this work, the bias and the gain are estimated during the image sequence in order to improve the computation of the motion model.

Few works about colour points tracking have to be related. However, because of the additional information provided by the colour image, a larger number of points can be selected and tra-cked in colour image sequences. Althought two points have the same luminance, they can have different colours.

In this article, we propose several solutions to improve the robustness of points tracking according to specular highlights and illumination changes, by using colour attributes. Firstly, in the next section, we study the modeling of colour changes occuring during an image sequence. Secondly, the third section focuses on the use of colour invariants. The most efficient invariants do not compensate sufficiently well for specular highlights. Nevertheless, we improve the tracking process by jointly computing the colour invariants (which compensates for lighting changes) with the photometric model (which models the illumination changes). The fourth section will study the use of the photometric model and invariants in the tracking process.

These different techniques will be compared by considering several image sequences, which show some specular highlights variations and lighting changes.

Modeling of colour changes

The observed colour of a surface depends on several parameters such as the scene geometry (lighting, camera and objects locations) and the surface properties (roughness, reflectance). This physical phenomenon can be expressed by the use of reflection models, which are largely used in computer graphics, such as the Phong model [START_REF] Phong | Illumination for computer generated images[END_REF] and the Torrance-Sparrow one [START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF]. We consider a camera which moves and acquires an image sequence of a scene. The latter can also be in motion. We will call f f f and f f f the colour images acquired respectively at time k and k . Figure 1 shows the geometry of the scene which is used in the explanation of the photometric changes.

Colour model

Let be a point P of the object projected in f f f at point p of coordinates (x p , y p ). Owing to most reflection models [START_REF] Phong | Illumination for computer generated images[END_REF][START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF], the colour in p can be expressed by :

f f f (p) = C C C d d d (p) • a a a(p)m d (p) +C C C s s s (p)h f (p) +C C C a a a (1)
where a a a is related to the albedo and depends on the reflectance of the surface, C C C s s s is the colour of specular component, C C C d d d the colour of direct light and C C C a a a is the colour of the ambient light. Let us notice that « • » represents the Hadamard product. h f expresses the specular function in image f f f and refers to the scene geometry and to the surface properties. For example, with Torrance-Sparrow model [START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF], we have :

m d (p) = cos θ i (P) and h f (p) = exp -ρ(P) 2 2ς 2 / cos θ r (P) (2)
where θ i is the angle between the normal n in P and the lighting direction L L L. ρ is the angle between n and the bisecting line between the viewing and the lighting vectors V V V and L L L. ς expresses the roughness of the material. Thus, the specular component h f reaches its maximum value when ρ = 0. On the other hand, the higher ς is, the more quickly h f will vary in the image.

By using this colour model, we can deduce an analytic expression of colour changes between two frames.

Colour changes

Let us consider that a second image f f f is acquired after a relative motion between the scene and the camera. Then, point P is projected in p in image f f f . Furthermore, an illumination change can occur between f f f and f f f . Consequently, the colour f f f (p ) at time k can be expressed as a function of f f f at time k as follows :

f f f (p) = f f f (p) + ψ ψ ψ(p) (3)
In case of specular highlights variations, only h f (p) varies in equation ( 1) and we show that ψ ψ ψ = ψ ψ ψ 1 (p) with the following definition of ψ ψ ψ 1 (p) [START_REF] Gouiffès | Featue points tracking : robustness to specular highlights and lighting changes[END_REF] :

ψ ψ ψ 1 (p) = C C C s s s (p)(h f (p) -h f (p)) (4)
where h f (P) is the specular reflection component in image f f f . In case of both lighting changes and specular highlights occurrence ψ ψ ψ = ψ ψ ψ 2 (p) [START_REF] Gouiffès | Featue points tracking : robustness to specular highlights and lighting changes[END_REF] : The expressions (4) and ( 5) use a large number of parameters which are not easily computable without any knowledge about the properties of the materials or the geometry of the scene. Consequently, their use is difficult in computer vision applications. However, some simple local approximations are often used : in indexing [START_REF] Gros | Color illumination models for image matching and indexing[END_REF], tracking [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF] or optical flow [START_REF] Negahdaripour | Revised definition of optical flow : integration of radiometric and geometric cues for dynamic scene analysis[END_REF] contexts for example. The validity of such photometric models has been studied in [START_REF] Gouiffès | Featue points tracking : robustness to specular highlights and lighting changes[END_REF]. In the following section, we describe the photometric model which is used in this article.

ψ ψ ψ

Local approximation of colour changes

Let be W a small window of interest centered on the point to be tracked p. In computer vision, most local photometric models defined on W only assume a change of bias and/or gain of the camera (the affine model proposed by [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF] for example). Under this assumption, the two parameters of the model are constant at each pixel of the window W . However, according to (4) and ( 5), ψ is not a constant function since it depends on the vectors n, V V V , L L L, which can be different at each point of W , especially when the surface projected on W is non-planar. We assume that the function ψ can be correctly approximated by a continuous function in each point q, of coordinates (x, y), located in the window W . Thereby, this approximation can be expanded in Taylor series around the point to be tracked p. By neglecting the highest orders, we obtain the following photometric model :

f f f (q) f f f (q) + α α α(x -x p ) + β β β (y -y p ) + γ γ γ. ( 6
)
where γ γ γ = f f f (p) and α α α, β β β are the partial derivatives of f f f in directions x and y respectively. The model ( 6) is particularly well adapted for specular highlights occurrence, which is not correctly modeled by using an affine photometric model (biais and gain). It is valid locally, for surfaces which do not have any coarse discontinuity of curvature or roughness in W . Indeed, according to (4) and the expression of h f given by ( 2), ψ 1 depends on the angles and on the the roughness, which are supposed to vary smoothly in W .

The main limitation of this model appears when lighting changes occur as in [START_REF] Gros | Color illumination models for image matching and indexing[END_REF], since the reflectance a a a has to vary smoothly in W , which is not true in general. Yet, the model ( 6) is available on small windows of interest, such as the windows that are used in the context of feature points tracking. The following section exposes a second way to cope with illumination changes, by using colour invariants.

Invariant attributes

Most colour invariants are based on the dichromatic model [START_REF] Shafer | Using color to separate reflection components[END_REF], which is equivalent to (1) when C C C a = 0. It supposes a single light source, no ambient lighting and no interreflection. Most invariants imply that the illuminant is white :

C C C s s s = C C C d d d = I(p).
Therefore, only the illuminant intensity I(p) (and not its colour) is supposed to vary. They also assume that a white balance has been achieved so that the integrals of each sensor sensibility over the wavelengths are the same on each sensor. We call Q this value. Consequently, the model (1) becomes :

f f f (p) = I(p)a a a(p)m d (p) + I(p)h f (p)Q (7)
Note that a a a(p) only depends on reflectance and sensor properties, which are supposed to be constant during the sequence. We distinguish two classes of colour invariants, which are related to lambertian or specular surfaces :

• Lambertian surfaces. In that case, the term I(p)Q(p)h f (p) vanishes in [START_REF] Heigl | Tracking points in sequences of color images[END_REF]. Thus, a ratio of two colour components cancels I(p) and m d (p) and depends only on the constant parameters a a a(p). In this article, we use L 1 and L 2 norms and c 1 c 2 c 3 invariants [START_REF] Gevers | Color based object recognition[END_REF].

• Specular surfaces. According to [START_REF] Heigl | Tracking points in sequences of color images[END_REF], the difference of two colour components cancels the specular term

I(p)Q(p)h f (p).
Then, the ratio of two differences cancels I(p) and m d (p). We use the following invariant a 1 a 2 a 3 :

a 1 = R -G D a , a 2 = R -B D a and a 3 = G -B D a with D a = |R -G| + |G -B| + |B -R|.
Unfortunately, colour invariants are undefined or are very noisy for low intensities (L 1 and L 2 norms, c 1 c 2 c 3 ) and/or low saturation (a 1 a 2 a 3 ).

L 1 , L 2 and c 1 c 2 c 3 do not make up for specular highlights but compensate well for lighting variations. By considering the L 1 values, we can express the part of illumination changes which is not compensated by the invariant attributes so that

f f f L 1 (p) = f f f L 1 + ψ ψ ψ L 1 ( 8
)
where ψ ψ ψ L 1 can be described with respect to the scene geometry and the material properties. Let be

K 1 (p) = I(p)m b (p) and K 2 = I(p)h f (p)Q. Then : ψ ψ ψ L 1 = K 2 (p)(a G (p) + a B (p) -2a R (p)) K 1 (p)Σ 2 + 3K 2 (p)Σ (9)
with Σ = (a R (p) + a G (p) + a B (p)). We assume that these residuals can be correctly approximated on a small window W by a continuous function. The latter can be estimated by a Taylor series around the point to be tracked p :

f f f L 1 (q) = f f f L 1 + α α α L 1 (x -x p ) + β β β L 1 (y -y p ) + γ γ γ L 1 . (10)
When the illuminant is white α α α L 1 , β β β L 1 and γ γ γ L 1 are scalar values. Nevertheless, the use of vectors can compensate for colour variations of the specular reflection component. Consequently, the white illuminant constraint is less restricting. The next section explains the use of the photometric model and the colour invariants in the context of feature points tracking.

Tracking methods

The first tracking method consists in computing the motion model δ (q, µ µ µ), parametrized by µ µ µ of a small window of interest W centered on p by minimizing the following relationship [START_REF] Lucas | An iterative image registration technique[END_REF][START_REF] Tomasi | Detection and tracking of point features[END_REF][START_REF] Heigl | Tracking points in sequences of color images[END_REF] :

ε 1 (µ µ µ) = ∑ q∈W || f f f (q) -f f f (δ (q, µ µ µ)) || 2 (11)
f f f and f f f can be any vectorial images. In this article, we consider the colour invariants described in previous section : L 1 and L 2 norms, c 1 c 2 c 3 and a 1 a 2 a 3 attributes. Indeed, they show a good adequacy between invariance with respect to illumination changes and discrimination between colours. These methods will be compared with the use of luminance and RGB components.

In addition, we propose a second technique which takes the models [START_REF] Harris | A combined corner and edge detector[END_REF] or [START_REF] Negahdaripour | Revised definition of optical flow : integration of radiometric and geometric cues for dynamic scene analysis[END_REF] into account :

ε 2 (µ µ µ) = ∑ q∈W || f f f (q) -f f f (δ (q, µ µ µ)) + B B BU U U || 2 (12
∑ m∈W V V V P V V V P T d d d = 1 3 ∑ m∈W ∑ i f i (q) -f i (δ (q, µ µ µ) -U U U T B B B i V V V i P ( 13
)
where B B B i = (α i , β i , γ i ) T where i is one of the components (RGB or colour invariants). The vector V V V i P P P and V V V P P P are expressed :

V V V P P Pi = f x i , f y i , x f x i , x f y i , y f x i , y f y i , x -x p , y -y p , 1 T (14)
V V V P = f x , f y , x f x , x f y , y f x , y f y , x -x p , y -y p , 1 T (15)
where f x and f y refer to the sum of the derivatives of the three colour components of f f f according to x and y. Let us notice that the matrix used to compute d d d is not well-conditionned and requires a pre-conditioning. These tracking methods will be compared in the following section.

Experiments

Several experiments are carried out on sequences where specular highlights and/or ligthing changes occur. In order to compare the tracking techniques, the points selected in the initial frame must be the same for each method, although the colour attributes are different.

Detecting points

The points detection in luminance images [START_REF] Harris | A combined corner and edge detector[END_REF] has been extended to colour [START_REF] Gouet | Evaluation de détecteurs de points d'intérêt pour la couleur[END_REF]. It consists in computing the matrix M(p) at each pixel p of the initial image :

M(p) = ∑ W (R 2 x +G 2 x +B 2 x ) ∑ W (R x R y +G x G y +B x B y ) ∑ W (R x R y +G x G y +B x B y ) ∑ W (R 2 y +G 2 y +B 2 y ) (16)
R x , B x , G x , R y , B y , G y are the spatial derivatives of the image RGB in directions x and y. After computation of the singular values λ 1 et λ 2 of this matrix, the point is accepted if min(λ 1 , λ 2 ) > S λ , where S λ is a threshold which is fixed by the user. In addition, in order to take the singularity of most colour invariants into account (most of them are not defined for unsaturated colours), we improve the detection by weighting each component of the matrix by the saturation value S = 1 -

3 min(R,G,B)
R+G+B of the pixel. This ponderation reduces the number of points that are selected (30 % on average in the sequences of this article).

Comparison of the methods

A point is rejected of the tracking process if the convergence residuals (ε 1 and ε 2 ) become greater than a threshold (15% according to the initial image, which represents a severe threshold according to the illumination changes provoked). The window size is N = 15. In order to evaluate the robustness and accuracy of the methods, we compute :

-the number of points that are correctly tracked by each tracker during the sequence, for each method (ε 1 or ε 2 ), and for each colour attribute (GL refers to the gray level image) ;

-the convergence residuals of some of the most relevant techniques. The lower the residuals are the more the lighting changes are compensated for. Moreover, when residuals remain low during the whole sequence, it is more probable that the estimated locations of the points are correct. First, the comparison is achieved on two sequences showing specular highlights. Secondly, lighting changes are caused.

Specular highlights variations

Sequence 1

The first sequence (see Fig. 2(a)) represents a painting under glass which is moved by an operator. The camera is motionless. The movement of the object with respect to the lighting (daylight and several fluorescent lights) induces the occurrence of specular highlights. No lighting change is caused.

The number of points which are correctly tracked (on 10 points selected initially) is counted in the table of Fig. 2.(b), with respect to the approach : minimization of ε 1 or ε 2 . By using the first technique (ε 1 ), L 2 norm and c 1 c 2 c 3 obtain the best robustness results in comparison to the other techniques including a 1 a 2 a 3 which are theorically dedicated to specular materials.

Then, the joint use of a photometric model with colour attributes provides some quite satisfying results (minimizing of ε 2 ). Two important points have to be underlined. First of all, the method using RGB components and the photometric model loses less points than the same approach in luminance images (GL with model), and it loses less points in comparison to the use of colour invariants (L 2 with model for example). Secondly, the joint use of the norm L 2 and the photometric model obtains the same results. It is also interesting to notice that these two methods are more relevant than the use of a 1 a 2 a 3 when specular highlights changes are concerned.

Fig. 2 (c) compares the convergence residuals obtained with the classical approach RGB, the use of the photometric model and the joint use of invariant with the model (L 2 +model). At 50 th frame, the classical method loses all of the points, whereas the methods envolving a photometric model (RGB+model et L 2 + model) get low residuals during the whole sequence. Finally, L 2 +model compensates for specular highlights more significantly.

Sequence 2

The second sequence (Fig. 3(a)) represents a hardback box whose surface is smooth and specular. Let us notice that the colours are less saturated compared to the previous sequence. Both camera and lighting are motionless. The intensity of light does not change but some specular highlights appear because of the motion undergone by the object with regard to the spotlights and to the camera.

Initially, 36 points are detected. As it is shown by the number of points that are correctly tracked (table of Fig. 3(b)), the use of the photometric model significantly improves the robustness. Moreover, these results are more convincing with RGB components in comparison with the use of colour invariants. That result can be explained by the fact that the colours are less saturated in comparison with the first sequence, and colour invariants are less efficient in those conditions. Thus, the use of method ε 2 is clearly justified in that case. Fig. 3(c) shows the residues obtained with RGB+model, c 1 c 2 c 3 +model and RGB. The latter loses all of the points at the 25 th frame. L 2 +model obtains the lowest residuals although the number of points that are correctly tracked is lower than RGB+model (see table of Fig. 3(c)). Indeed, despite the points detection method, the use of invariants comes up against the occurrence of noise for unsaturated colours during the image sequence.

Lighting changes

Sequence 3

The third sequence (Fig. 4(a)) shows several objects of various materials, which are lighted by one fluorescent lighting and one halogen lamp. Only the camera is moved. The scene is composed of specular surfaces, and illumination variations are provoked by the occurrence of shadows that are cast on the objects. Table of Fig. 4(b) registers the number of points that are correctly tracked, on 39 points initially selected. The joint use of invariants with a photometric model, and particularly the method L 2 +model, improves significantly the efficiency of the tracking, since it loses less points than any other approach. In a similar manner as in second sequence, method c 1 c 2 c 3 +model obtains the lowest convergence residuals, which proves its accuracy. Let us notice that the attributes a 1 a 2 a 3 ensure a correct robustness. In comparison with the previous experiments, their contribution is more relevant when lighting changes are caused than when only specular highlights are provoked.

Sequence 4

The image sequence of Fig. 5(a) represents a few objects of different materials under several lightings (daylight, fluorescent, halogen light). 15 points can be tracked during the sequence, the others go out of the field of view of the camera. A lamp is switched off at the 40 th image and the fluorescent lamp is switched off at its turn in the 60 th frame. Both are switched on at iteration 80. The camera undergoes a rotation motion. The number of points which are correctly tracked attests the robustness of methods which jointly envolve colour invariants and photometric model (see table of 5(d). First of all, the residuals of RGB+model reach lower values than L 2 . Thus, the model is more adapted to the photometric changes provoked, although it only compensates for a local approximation of these changes, unlike colour invariants. Finally, L 2 +model is more relevant, judging by the weak level of its residuals. Let us notice that the residuals obtained by the different approaches are especially high when the lamps are switched off (from frame 60 to frame 80). This remark highlights the weakness of these colour invariants which are not efficient when the saturation of the colours are drastically reduced.

Computation time

The computation time of the tracking is a crucial criterion for some computer vision applications. Unfortunately that is the main limitation of colour tracking. For a window size 11 the classical approach KLT [START_REF] Lucas | An iterative image registration technique[END_REF][START_REF] Tomasi | Detection and tracking of point features[END_REF] takes about 3 ms for a point, whereas the same approach takes 7ms in colour images. To finish, the minimization of (12) takes 15 ms in colour images, and only 5 ms in luminance images. These have been obtained with a processor AMD Athlon, 1,8 GHz, and a RAM of 512 Mb.

Discussions

A few remarks have to be underlined. First of all, colour invariants defined for specular surfaces have not proved to be really efficient in the context of points tracking. In the one hand, they are noisy when colours are not saturated enough (R G B). In the other hand, they are likely to reduce the discriminative power between colours.

In each sequence, the tracking methods using colour invariants, in particular the L 2 norm and c 1 c 2 c 3 attributes, is more robust than the use of the only luminance or the colour since less points are lost in general. Broadly speaking, the joint use of a photometric model and colour improves the tracking results in a more efficient way. It is even more efficient to jointly use L 2 ou c 1 c 2 c 3 , which are only valid for inhomogeneous and dull dielectric surfaces, with a photometric model that makes up for specular variations. However, when colours are not saturated enough (as in sequence 2) and when specular highlights are caused, we obtain better results by using RGB components with the photometric model.

To improve the tracking, the detection of points should be dedicated to the tracking method which is carried out. However, in order to compare the tracking techniques, we have selected the same points for each method.

Conclusion

In this article, we have proposed three methods to improve points tracking in colour images : by computation of a local photometric model, by a global compensation of illumination changes (use of colour invariants) and by the joint use of colour invariants with a photometric model. Even if the invariants improve the tracking results in comparison to the classical approach in RGB space, especially when lighting changes are produced, the joint method has proved to be the most efficient whatever the illumination changes are : specular highlights changes and lighting variations. Unlike the use of invariants defined for specular objects, the latter can compensate for illumination changes even if a white balance is not carried out.

In addition, the experiments have highlighted the fact that the main drawback of colour invariants is their inefficiency when colours are not saturated. Thus, our future works will deal with the detection of points and the fusion of luminance information and colour invariants to achieve tracking whatever the saturation of the colours.

FIG. 1 .

 1 FIG. 1. Vectors and angles involved in the reflection description.

  ) with U U U = (xx p , yy p , 1) T . B B B is the matrix of the photometric parameters B B B = (α α α, β β β , γ γ γ). Let be D D D = [µ µ µ T , B B B T ] T . In this paper, we compute an affine motion model between the initial frame f f f and the current one f f f . Consequently, we assume a little variation d d d around an estimate D D D, D D D = D D D+d d d, so that (12) can be linearized in d d d. Consequently, the method consists in solving the following linear system

Fig. 5 (

 5 b), where c 1 c 2 c 3 +model and L 2 +model track a larger number of points in comparison with any other technique). Fig. 5(c) and Fig. 5(d) display the residuals as a function of the frame number. Only the most convincing approaches are shown : RGB+model and L 2 on Fig. 5(c), L 2 +model and c 1 c 2 c 3 +model on Fig.

Fig. 2 .

 2 Sequence 1. (a) : two images of the sequence. (b) : number of points which are correctly tracked (on 10 points intially selected). (c) and (d) : convergence residuals obtained for N = 15.

Fig. 3 .

 3 Sequence 2. (a) : two images of the sequence. (b) : number of points which are correctly tracked (on 36 points intially selected). (c) and (d) : convergence residuals obtained for N = 15.

Fig. 4 .

 4 Sequence 3. (a) : two images of the sequence. (b) : number of points which are correctly tracked (on 39 points intially selected). (c) and (d) : convergence residuals obtained for N = 15.

Fig. 5 .

 5 Sequence 4 (a) : two images of the sequence. (b) : number of points which are correctly tracked (on 15 points intially selected). (c) and (d) : convergence residuals obtained with N = 15.

Acknowledgments

The authors thank the Cemagref of Rennes, the OFIVAL ( Office national interprofessionnel des viandes, de l'élevage et de l'aviculture) and the county council of Poitou-Charentes for the funding of the global project including these researches.


Michèle Gouiffès
email: gouiffes@sic.sp2mi.univ-poitiers.fr

Christophe Collewet

Christine Fernandez-Maloigne

Alain Trémeau

Feature points tracking: robustness to specular highlights and lighting changes

Since the precise modeling of reflection is a difficult task, most feature points trackers assume that objects are lambertian and that no lighting change occurs. To some extent, a few approaches answer these issues by computing an affine photometric model or by achieving a photometric normalization. Through a study based on specular reflection models, we explain explicitly the assumptions on which these techniques are based. Then we propose a tracker that compensates for specular highlights and lighting variations more efficiently when small windows of interest are considered. Experimental results on image sequences prove the robustness and the accuracy of this technique in comparison with the existing trackers. Moreover, the computation time of the tracking is not significantly increased.

Introduction

Since many algorithms rely on the accurate computation of correspondences between two frames through an image sequence, feature tracking has proved to be an essential component of vision systems. Indeed, many high level tasks can depend highly on it, such as 3D reconstruction, active vision or visual servoing for example. Nevertheless, robust feature tracking is still a problem to be addressed. It becomes far more complicated when no mark (edges or lines for example), can be extracted from the observed object, such as in natural environment [START_REF] Espiau | Robust features tracking for robotic applications: towards 2d1/2 visual servoing with natural images[END_REF]. In such a context, only points, among other possible features, are likely to be easily detectable. However, tracking a point into an image sequence is not a trivial task since the only available information is the luminance of the point and of its neighboring pixels. The seminal works in this domain are due to Lucas and Kanade [START_REF] Lucas | An iterative image registration technique[END_REF][START_REF] Tomasi | Detection and tracking of point features[END_REF] who assume the conservation of the point luminance during the image sequence [START_REF] Horn | Determinig optical flow[END_REF]. The measure of a correlation function between two successive frames determines the translation motion undergone by the point. Thereafter, some more robust tracking approaches have been proposed [START_REF] Shi | Good features to track[END_REF][START_REF] Tommasini | Improving feature tracking with robust statistics[END_REF]. However, such methods still assume that the luminance remains constant between two successive frames, which is often wrong. Indeed, most surfaces are not Lambertian and lighting conditions are mostly variable during an image sequence. To solve this problem, Hager and Belhumeur [START_REF] Hager | Efficient region tracking with parametric models of geometry and illumination[END_REF] acquire an image data base of the scene under several illuminations and use these data to improve the tracking. This method is efficient but requires a prior learning step, which can be seen as too restrictive. An easier way to cope with illumination changes is to achieve a photometric normalization as in [START_REF] Tommasini | Improving feature tracking with robust statistics[END_REF] for example. In [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF], the tracking task compensates for affine illumination changes by computing the contrast and illumination variations during the whole image sequence. These two methods will be detailed in Section 3.

In this paper, we propose a new feature point tracking algorithm, based on the study of reflection models, which is robust to specular highlights occurrence and lighting changes. In addition, we show clearly on which assumptions the approaches mentioned above are based. Besides, we will see that the proposed algorithm provides a more appropriate model of the illumination changes, particularly for specular surfaces.

This article is structured as follows. Section 2 focuses on the modeling of luminance changes, especially in cases of specular reflections and lighting variations. Section 3 details some of the existing tracking approaches: the Shi-Tomasi-Kanade tracker [START_REF] Lucas | An iterative image registration technique[END_REF][START_REF] Shi | Good features to track[END_REF][START_REF] Tomasi | Detection and tracking of point features[END_REF], the normalized one [START_REF] Tommasini | Improving feature tracking with robust statistics[END_REF] and the tracker with an affine illumination compensation [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF]. Thereafter, section 4 describes the proposed tracking method. To finish, section 5 shows experimental results, in order to compare the different tracking techniques according to their robustness and accuracy. In addition, this section will prove the efficiency of our approach.

Modeling of luminance changes

Suppose f and f to be respectively the images of an object acquired at two different times. A point P of this object projects in image f to p of coordinates (x p , y p ) and to p of coordinates (x p , y p ) in the image f after a relative motion between the camera and the scene. The luminance at p depends on the scene geometry. Fig. 1 describes the vectors and the angles used in this paper. V and L are respectively the viewing and the lighting directions, which form the angles θ r and θ i with the normal n in P . B is the bisecting line between V and L, it forms an angle ρ with the normal n. According to the most widely used reflection models, such as the Torrance-Sparrow [START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF] and the Phong [START_REF] Phong | Illumination for computer generated images[END_REF] ones, the luminance at p can be described as follows

f (p) = K d (p)a(p) cos θ i (P ) + h f (p) + K a (1)
where K a is the intensity of ambient lighting and K d a diffuse coefficient corresponding to the direct lighting intensity. These values depend also on the gain of the camera. The term a(p) is related to the albedo4 in P . The function h f expresses the contribution of the specular reflection, which vanishes in case of a pure diffuse reflection, that is for Lambertian objects. Consequently, with such objects and for a given lighting direction L, the luminance at p remains constant whatever the viewing direction V is. Phong describes in [START_REF] Phong | Illumination for computer generated images[END_REF] the specular reflection as a cosine function of ρ

h f (p) = K s (p) cos n (ρ(P )) (2)
where n is inversely proportional to the roughness of the surface and K s is the specular coefficient of the direct lighting. Torrance-Sparrow [START_REF] Torrance | Theory for off-specular reflection from roughened surfaces[END_REF] describes h f by an exponential function depending on ρ and on the surface roughness ς. For each model, h f reaches a maximum value for ρ(P ) = 0, that is when B coincides with n. Let us also notice that the specular reflection depends on the roughness, the lighting and the viewing directions. After a relative motion between the camera and the scene, when no lighting change occurs, θ i , K d and K a are constant at P during the time. In the same way, the albedo is constant at P leading to a(p ) = a(p). However, the specular component h f , which depends on the viewing direction, may vary strongly during the motion of the camera. In those conditions, the luminance f is given by

f (p ) = K d (p)a(p) cos θ i (P ) + h f (p ) + K a (3)
where h f is the specular function. Now, let us consider that some lighting shifts ∆K a , ∆K d and ∆θ i are respectively provoked on K a , K d and θ i . Thus, the luminance can be expressed as

f (p ) = K d (p)a(p) cos θ i (P ) + K a + h f (p ) (4) with K d (p) = K d (p) + ∆K d (p), θ i (P ) = θ i (P ) + ∆θ i (P ) and K a = K a + ∆K a .
The specular term h f (p ) includes the intensity change of the specular coefficient K s if necessary. From (4), the next section will clearly show the assumptions on which the most widely used tracking methods are based.

Analysis of existing tracking methods according to their robustness to illumination changes

Let be m a point located in a window of interest W of size N × N centered around p. Point m is the projection of a physical point M in f and m is its projection in f . The tracking process consists in computing a motion model δ parameterized by a vector A between f and f . According to the tracking method, the assumptions about the photometric model are different. By eliminating a(m) between ( 1) and ( 4), it yields to the following relationship between two images of the same sequence f (m ) = λ(m)f (m) + η(m). This relationship is found in [START_REF] Negahdaripour | Revised definition of optical flow:integration of radiometric and geometric cues for dynamic scene analysis[END_REF] in an optical flow context, but λ and η are supposed to be constant locally. According to our analysis based on the reflection models, λ(m) and η(m) are expressed by:

λ(m) = (K d (m) + ∆K d (m)) cos(θi(M ) + ∆θi(M )) K d (m) cos θi(M ) (5)
η(m) = -(h f (m) + Ka)λ(m) + h f (m ) + Ka + ∆Ka (6)
From these relations we deduce the assumptions on which the classical trackers are based.

The classical approach. The classical point feature tracker [START_REF] Lucas | An iterative image registration technique[END_REF][START_REF] Shi | Good features to track[END_REF][START_REF] Tomasi | Detection and tracking of point features[END_REF] assumes a perfect conservation of luminance at point M during the sequence: f (m ) = f (m), ∀m ∈ W and ∀W. Owing to owing to ( 5) and ( 6), that implies λ(m) = 1 and η(m) = 0 ∀m ∈ W, which is correct when no lighting change occurs (∆θ i (M ) = 0, ∆K d (m) = 0 ∀m ∈ W) and when objects are strictly lambertian (h f (m) = h f (m ) = ∆K a = 0, ∀m ∈ W and ∀W). Because of noise and because of the strong assumptions considered on the motion and photometric models, it is more suitable to minimize the following criterion

1 (A) = m∈W (f (m) -f (δ(m, A)) 2 (7)
This approach leads to good results in most cases but can suffer from lighting changes and specular highlights occurrence. In order to cope with this problem, a photometric normalization can be performed.

Use of an affine photometric model. In [START_REF] Jin | Real-time feature tracking and outlier rejection with changes in illumination[END_REF] the authors propose a tracking method in order to compensate for contrast and intensity changes by computing the parameters of the following criterion

2 (A, λ, η) = m∈W (λf (m) -f (δ(m, A)) -η) 2 . ( 8
)
According to [START_REF] Lucas | An iterative image registration technique[END_REF], λ(m) is supposed to be constant at each point of W. That is correct for any surface curvature (and then ∀∆θ i and ∀θ i ) and for any lighting

(∀K d and ∀∆K d ) only if each function ∆θ i , θ i , ∆K d , K d is constant in W.
Then, it assumes that η(m) is constant at each point of W, which is correct for any surface curvature and for any roughness when both h f (m) and h f (m) are constant at each point of W.

As a conclusion, this technique assumes that the illumination changes are the same at each point of W. Practically, that can be a coarse assumption when the surface projected onto W is non planar.

The photometric normalization approach. This approach is based on the minimization of the criterion 2 (A) except that λ and η are measured at each step of the minimization process instead of being computed simultaneously with the motion parameter A. Their values are λ =

σ f σ f and η = µ f - σ f µ f σ f
, where µ f and µ f are the average values respectively of f and f in W and σ f , σ f are the standard deviations. λ and η are supposed to be constant at each point m of W, so that the same assumptions as the previous technique hold. Indeed, in those conditions, the values µ f and σ f are given by

µ f = K d cos(θ i )µ a + K a + h f σ f = K d cos(θ i )σ a (9)
where µ a is the average value of a and σ a its standard deviation. Consequently, f (m) -µ f , a(m) -µ a and f (m ) -µ f are invariant to highlights occurrence. Finally we show easily that the following ratios are also invariant to ambient and direct lighting changes, and to gain variation

(f (m) -µ f )/σ f = (a(m) -µ a )/σ a = (f (m ) -µ f )/σ f , ∀m ∈ W. (10)
Nevertheless, these properties are true only if the specular reflection and the lighting changes are the same at each point of W, as it as been mentioned above. In some cases, these assumptions are not realistic, particularly when W is the projection of a non planar surface of the scene. In addition, these values can be ill-defined when σ a ≈ 0, that is when the intensities almost saturate or more generally when they are almost homogeneous in W.

Our method is described in the next section. It states the assumption that the illumination changes can be approximated by a continuous function on W.

The proposed approach

It has been shown in section 2 how each kind of illumination changes can be expressed. By considering (1) and ( 4), we immediately obtain the relationship between f and f f

(δ(m, A)) = f (m) + ψ(m) (11)
In the general case the photometric change is written

ψ(m) = a(m)(K d (m) cos(θi(M ) + ∆θi(M )) -K d (m) cos θi(M )) + h f (δ(m, A)) -h f (m) + ∆Ka (12)
When the lighting (or the gain of the camera) does not vary, the total temporal change ψ in the specular component at a fixed scene point m is equal to

ψ(m) = h f (δ(m, A)) -h f (m) (13)
According to the most widely used reflection models (see [START_REF] Hager | Efficient region tracking with parametric models of geometry and illumination[END_REF] for example), the function ψ is variable on W since it depends on the viewing and lighting angles and therefore on the normal n at each point of W. When lighting changes are caused, it depends also on the albedo. We suppose that this function can be approximated by a continuous and derivable function φ on W. In that case, a Taylor series expansion can be performed at m in a neighborhood of p leading to the following expression for φ(m) by neglecting the higher order terms

φ(m) = φ(p) + ∂φ ∂x p (x -x p ) + ∂φ ∂y p (y -y p ). ( 14
)
Finally, by using ( 14) in ( 11) with m = (x, y) T and noting α = ∂φ ∂x p

, β = ∂φ ∂y p et γ = φ(p), the proposed tracker consists in computing the motion parameter A and the reflection parameters B = (α, β, γ) T by minimizing the following criterion:

3 (A, B) = m∈W f (m) -f (δ(m, A)) -U T B 2 (15)
where U = (x-x p , y -y p , 1) T . Contrary to the previous approaches, this method does not make assumptions about the scene. In particular, the incident angle θ i , the viewing angle θ r , the parameter K s and the roughness n (or ς) can vary. Therefore, specular highlights and lighting changes can be different at each point of W. However, when lighting changes occur, the method assumes that the values of the albedo can be approximated by a polynomial of first degree in W. Only in that case, the proposed approach is more adapted for small windows of interest W. In the next section, the different tracking methods are compared through experiments.

Experimental results

In the following experiments, the tracking methods are based on the computation of an affine motion model between the first frame and the current one. The tracking algorithm integrates an outlier rejection module, based on the analysis of the residues convergence i , i = 1 . . . 3. A point is rejected of the tracking process as soon as its residues become greater than a threshold S conv = N 2 E 2 ave , where E ave is the tolerated intensity variation for each point in W between f and f . In these experiments, E ave = 15. We consider some sizes from N = 9 to N = 13. In each case, the sequence is played from the first image to the last one and then from the last image to the first one, in order to evaluate the symmetry of the residuals and photometric curves. To compare the trackers we compute several criteria: 1) the robustness of the tracking, that is to say the number of points that have been tracked during the whole sequence; 2) the accuracy of the tracking: we compute the average of the residuals for all the points that have been tracked during the whole sequence by the considered method. This criterion provides information about the relevance of the photometric model. The lower the residues are, the better the illumination variations are compensated for; 3) the reflection parameters α, β and γ, computed by the parametric method. We will also compare the computation time and the conditioning of the matrices used in the minimization algorithm. Moreover, simulations results compare the accuracy of the tracking in terms of position estimation. In order to simplify the explanations of results, we introduce the following notations: C (classical approach), J (affine model compensation), T (photometric normalization) and P (proposed approach).

First, the tracking algorithms are tested on scenes showing specular highlights. For each image sequence, the scene (objects and lighting) is motionless, only the camera moves. The scenes are lighted by a direct and an ambient lighting.

• Experiment Book . The first sequence (see Fig. 2a) shows a book. The motion of the camera leads to specular highlights which appear and disappear during the sequence. A number of 97 points has been selected in the first frame but 11 points are lost because they are occluded by the book or because they get out of the camera field of view. The number of points correctly tracked is counted in table 1a. These results prove that the proposed approach (P ) is far more robust than the existing ones (C, J and T ) since the number of tracked points is always much larger. Fig. 2b compares the average residuals obtained by the trackers during the sequence for N = 9. Until the 100 th frame, P obtains the lowest convergence residuals: the proposed photometric model fits best to the specular occurrence. After the 100 th frame, the residuals of T become lower than the P ones. However, these values are computed by averaging the residuals of 33 points for T and 68 points for P . In order to compare correctly these two approaches, let us consider Fig. 2c, which shows the average residuals computed only on the few points that are correctly tracked by T and P simultaneously. Here, the residuals are lower for the proposed approach. That shows the good adequacy of the local model proposed to compensate for specular highlights. Let us notice that J is less convincing than T even though these two methods are based on the same photometric assumptions. As it will be shown later on, this is due to the ill-conditioning of J. Fig. 2d depicts the behavior of the photometric parameters that have been computed by P . Let us notice that these curves are perfectly symmetric, in agreement with the symmetry of the sequence and therefore with the symmetry of the illumination changes.

• Experiment Marylin. The image sequence Marylin (see Fig. 3a) shows different specular objects, planar or not, lighted by the daylight and the spotlights of the room. Different types of material are considered (ceramic, glass, glossy paper, metal). This sequence is particularly noisy since the camera used has an interlaced scan mode. Besides, we can see some specular reflections es- pecially on the glass of the photograph. A total number of 56 points has been selected in the first frame, but a large number of points (20 points) is lost because of occlusions and noise. Table 1b collects the number of points that have been tracked until the end of the sequence by each technique. Contrary to the previous sequence, T is less powerful than J, which could mean that T is less robust to noise than J. P tracks a larger number of points in each case. As it can be seen from the residuals obtained on the points that have been tracked by the whole of the approaches (see Fig. 3b), this technique is also more accurate.

Now, let us compare the techniques on a sequence showing lighting changes.

• Experiment Lighting changes. The scene consists of a specular planar object lighted by the daylight and by one direct spotlight (see Fig. 4a). Strong variations are produced on the direct lighting intensity since it varies periodically, from a minimum value to a maximum one each 10 frames. Table 1c collects the number of points that have been tracked, on 58 points selected initially. Obviously, the proposed method provides a better robustness of the tracking since the whole of the points are tracked. Fig. 4b shows the average residuals obtained. For this kind of illumination changes, P and T are the most accurate techniques and obtain quite similar residuals. However, Fig. 4c, compares the average residuals computed for the points that have been tracked by the two techniques simultaneously. P gets the lowest residuals. The temporal evolution of the illumination parameters is depicted on Fig. 4d. Their periodical variations correspond to the lighting changes that have been provoked.

Accuracy of the tracking. In order to evaluate the accuracy of the points positioning, we use a software that simulates the appearence of an object lighted by an ambient light and a direct spotlight, by using a Phong model (see [START_REF] Hager | Efficient region tracking with parametric models of geometry and illumination[END_REF]). In the simulated sequence from Fig. 5a and5b, a motionless cylinder is viewed by a moving camera. The direct light is moved, inducing some specular highlights changes. A point is rejected as soon as its error (the euclidian distance between the real position of the point, computed by the software, and its estimated position by the tracker) is greater than 0.5 pixel. 13 points are initially selected, the method C loses 13 points, T 7 points, J 6 points and P loses no point. The figure 5c shows the evolution of the average of positioning errors obtained by each technique. P obtains the lowest errors all along the sequence and the number of points that are correctly tracked is higher.

Discussions. As expected, the classical tracker described in section 3 is not robust neither to specular highlights occurrence nor to lighting variations. A large number of points are lost, sometimes the whole of the points. Obviously, the tracking with photometric normalization and the approach with an affine compensation roughly improve the results. Besides, their efficiency increases for large windows of interest. Indeed, for small windows, the computation of σ f , σ g and λ are sensitive to noise. Because these values are multiplied or divided by the luminance values f , an error caused on these parameters have a huge influence, and can yield to the computation of an incorrect motion parameter A. On the other hand, it could seem surprising that J and T behave differently although they are based on the same photometric assumptions. Actually, J suffers from a bad convergence for small windows. As an example, table 2 contains the ratios CN given by the ratio of the Condition Number of the tracker to the Condition Number of the classical tracker (or the T approach since it has the same condition number), computed on 10 points of the sequence Book and for different sizes of window. It collects the maximum (Max), the minimum (Min) and the average values (Ave) obtained. These values show clearly that J is less well-conditioned than the other approaches included ours. To finish, the method P tracks a larger number of points than the other methods. It really compensates for the variability of specular reflection and lighting variations on W, which proves the relevance of the modeling of ψ by ( 14). However, let us notice that the difference of performance between the three methods (P , J and T ) is less significant when lighting changes are caused than when only specular highlights occur, in particular for large windows of interest. J and T are well adapted to contrast changes, and not to specular highlights, which are generally not constant in W. However, let us note once again that J is less efficient because of its ill-conditioning. The proposed method is perfectly adapted to specular highlights since their variations in W is well modeled. In case of lighting variations, the albedo must be approximated by a polynomial of first degree. However this approximation is satisfying for small windows of interest.

Up to now, we did not compare the computation time of the methods. For example, in sequence Book, the average tracking time of one point is 1.3ms for classical method, 1.7ms for the J technique, 4.6ms for the T one (let us note that the computation of the means and standard deviations are costly) and 1.4ms for our approach5 . As a conclusion, the computation time is not significantly increased in comparison to the classical approach.

Conclusion

The existing tracking methods are based on several assumptions that had never been, or partially, specified explicitly before. In this paper, the analysis of these methods is led according to specular reflection models. Contrary to the classical approach, the tracking methods based on an affine photometric compensation are, to some extent, robust to illumination changes. However, the illumination parameters are assumed to be constant around the point to be tracked. This can be incorrect when the surfaces are non planar or when specular highlights occur. Our approach overcomes these issues. We assume that an illumination change can be approximated by a continuous and derivable function around the points to be tracked. This model is well adapted for small windows of interest, since it improves the robustness of the tracking against highlights occurrence and lighting changes. The computation duration of this method is not significantly increased in comparison with the classical technique and the accuracy of tracking is improved. In addition, its convergence properties are more satisfying than the technique involving an affine model since better conditioning numbers have been obtained. Our future work will focus on a more appropriate modeling of illumination changes for larger windows of interest.

Fig. 1 .

 1 Fig. 1. Vectors and angles involved in the reflection description.

Fig. 2 .

 2 Fig. 2. Experiment Book (a) Three images of the sequence. (b) and (c): Residuals for N =9, (b) is the average of the residual computed on all the points tracked and (c) the average of the residuals for the points that are correctly tracked by T et P simultaneously. (d) Illumination parameters computed with the proposed approach.

(a) 1 Fig. 3 .

 13 Fig. 3. Experiment Marilyn (a) and (b) Two images of the sequence. (c) Average of the residuals for N =9 obtained by J and P .

Fig. 4 .Fig. 5 .

 45 Fig. 4. Experiment Lighting changes. (a) Three images of the sequence. (b) and (c) Residuals for N =9, (b) is the average of the residuals computed on all the points tracked and (c) the average of the residuals for the points that are correctly tracked by T et P at the same time. (d) Illumination parameters of point A computed with the proposed method.

Table 1 .

 1 Number of points tracked during the whole sequence versus N . (a) Book (86 points to track) (b) Marylin (36 points) (c) Lighting changes (58 points)

	Tracker N =9 N =11 N =13	Tracker N =9 N =11 N =13	Tracker N =9 N =11 N =13
	C	27	20	16	C	0	0	0	C	37	29	23
	T	33	49	53	T	1	3	7	T	45	51	53
	J	15	25	48	J	4	8	15	J	39	48	51
	P	68	68	69	P	23	21	21	P	58	58	58

Table 2 .

 2 Conditioning number on 10 points selected on sequence Book. Maximum (Max), minimum (Min) and average value (Ave).

	Tracker	Max Min	Ave
	J	9	1	3.5
	P	2	0.5	1.4

The albedo is the ratio of the amount of light reflected by a small surface in P to the amount of incident light. It depends only on the material and its texture.

With a processor Pentium III, 1.8GHz, 512Mo RAM.


Laurent Tapie
email: laurent.tapie@lurpa.ens-cachan.fr

Bernardin Kwamivi Mawussi

Decomposition of forging die for high speed machining

Keywords: Forging die, HSM, Machining preparation, CAM 1-Introduction

Today's forging die manufacturing process must be adapted to several evolutions in machining process generation: CAD/CAM models, CAM software solutions and High Speed Machining (HSM). In this context, the adequacy between die shape and HSM process is in the core of machining preparation and process planning approaches. This paper deals with an original approach of machining preparation integrating this adequacy in the main tasks carried out. In this approach, the design of the machining process is based on two levels of decomposition of the geometrical model of a given die with respect to HSM cutting conditions (cutting speed and feed rate) and technological constrains (tool selection, features accessibility). This decomposition assists machining assistant to generate an HSM process. The result of this decomposition is the identification of machining features.

Today, design of forged part and forging die is done in the engineering and design department. Die analysis is performed on the CAD model of die cavities with metal flow simulation [T1] [START_REF] Kulon | A knowledge-based engineering design tool for metal forging[END_REF]. Thus, die designers define forged part and dies according to CAD models and forging process simulation. The die model obtained is transferred to the machining process department. Recent die models have more complex shape. The introduction of close die forging process in order to limit re-machining operation (Fig. 1) increases geometrical complexity of die shapes. Because of the evolution of die shapes, the tasks carried out by the machining assistant are different insofar as he must work out the machining process starting from die model given by the engineering and design department. This model is considered in most of the cases as a void geometry, since it involves poor technological and topological information required by machining process assistant. So, machining process preparation must be based on geometrical and topological die shape analysis to associate machining information. This information must be useful to choose machining strategy and resources according to die shape with respect to the minimal machining time, the part quality and surface roughness. Integration of High Speed Machining (HSM) technology is also crucial for the processing of technical and economical constrains.

Current forging die machining process is composed of three main steps: roughing, semi-finishing and finishing. Roughing operations objective is to reach as close as possible the desired finished shape with high level of cutting material flow and tool path series as smooth as possible. Semi-finishing operations objective is to warrant a constant depth of cut or constant tool engagement during finishing operation. Finishing operations objectives are to obtain the final die shape up to the geometrical quality specified. The die geometry and topology must be taken into account during these machining operations with a high leveled refinement to reach these objectives.

The approach proposed in this paper is limited to finishing operation of die cavity. In the industrial context, machining resources often used are 3-axes high speed machine tools associated with ball nose or corner rounded end mill. CAM strategies used during this operation are very common: parallel planes guidance, Z level guidance, parallel curve guidance.

Technical and economical studies highlighted the importance of CAM preparation in the die machining process [AL 1]. During this preparation step, the machining process assistant know how have a great influence on the part quality. Thus, the tasks carried out by the machining process assistant must be adapted to the technical and economical context which involves evolutions on CAD models, CAM strategies and machining process (HSM). This adaptation depends on the good adequacy between die shape and HSM process. Indeed, a bad adequacy involves increases in machining time and decreases in part quality. A well known cause of part quality and surface integrity decreases is the difference between real cutting conditions and HSM cutting conditions set point [START_REF] Lamikiz | Cutting force integration at the CAM stage in the high-speed milling of complex surfaces[END_REF] [WK1] [MH1]: keep up a constant feed rate (deceleration and high breakages), avoid high tool engagement variation, keep up a constant cutting speed (avoid small effective cutting diameter).

To solve this problem the work presented in this paper proposes to decompose the die geometrical model in order to adapt locally a HSM process to die shape. This decomposition is based on the difficulties involved by HSM technology.

2-Proposed approach

Tasks carried out by the machining assistant are generally based on CAM/CAPP software modules and machining feature concept. Nevertheless, few works are focused on complex shape machining features. The approach presented in this paper is in keeping with the general pattern of these features creation [START_REF] Van Den Berg | Freeform feature modeling: concepts and prospects[END_REF].

The main objective of the work presented in this paper is to define machining features suitable for HSM process starting from the die model (3D CAD model). The CAD model decomposition based on difficulties induced by HSM process is an interesting concept to obtain a good adequacy with the part geometry. This concept is already used in the project "Usiquick" [U1]. The proposed approach is carried out starting from the analysis of the difficulties related to the machining of die shapes (Fig. 2). Then, results obtained at this level are superposed to obtain machining features and their topological relationship. A machining process is generated for each machining feature and then all the local machining processes are adapted for the entire forging die. This paper is focused on the topological decomposition of die model from machining difficulties.

3-Decomposition method

The CAD model decomposition for machining features creation is very efficient. Besides, machining difficulties criteria are very relevant as it is highlighted in [START_REF] Derigent | 3D digitising strategy planning approach based on a CAD Model[END_REF].

The die shape is decomposed in machining difficulties features. These features helped the machining process assistant to identify machining features. Two types of HSM process difficulties encountered are studied: to keep the set point of the cutting speed and to find better sequences of machined surface suitable for the application of standard machining strategies.

-Cutting speed respect

As it is highlighted in several academic papers, to maintain the cutting speed set point is a key factor in the cutting mechanism. Indeed, thermal variations and cutting force variations in the cutting feature are observed when the effective cutting speed is significantly reduced. These variations can induce premature tool wear [T2], dimensional error, bad surface roughness [T2] and part metallographic variation [START_REF] El Wardany | Surface integrity of die material in high speed hard machining Part 1: Micrographical analysis[END_REF]. For the toolpath generated according to the Z-level strategy, Fig. 5, the tool and part contact area shape and position are constant in the whole part. Yet, in some features the tool engagement is higher, particularly when the part shape curvature is equal or lower than the tool radius. These two analysis highlight the more significant causes of machining process or part shape alterations: no respect of cutting speed and some phenomena linked with the toolpath choice.

The two iso-plane tool path generated by the slicing approach for the whole part highlight the difficulty to limit the effective cutting speed variations.

-Contact area type

The analysis of 3-axes machining of forging dies makes it possible to define two types of contact areas: end contact and flank contact (Fig. 6). The first type of contact area "end contact" is located to the tool tip. A great reduction of effective cutting speed occurs in this feature because of the proximity of the spindle axis. As the second type of contact area "flank contact" is located in the periphery of the cutting tool. A cutting speed close to theoretical one can be reached. The contact area type is linked to the inclination angle β between tool axis and part material direction (Fig. 7). Thus, this contact area type is analysed using this angular parameter. CAD forging die model are very often designed with a B-Rep model. Nevertheless, the data used in this kind of models (vertex, edge, surface) are complex to be manipulated by machining assistant. This complexity is highlighted in the Usiquick Project [D1]. Geometrical and topological refinement of a B-Rep model involves designing complex algorithm in order to recognize automatically pre-defined machining features.

« End contact » «Flank contact» « End contact » «Flank contact»
To simplify the B-Rep data in our approach, the forging die B-Rep model is converted in a polygonal model. The STL model is chosen because it is a very common model associated with computer aided conversion algorithm in several CAD, CAM and CAPP software.

Starting from the STL model we generate a map which represents the cutting speed variation. The main objective of this map is to search shapes having the same type of contact areas between the cutting tool and the part. Then the map is obtained by colouration of each facet according to the criterion δ (Fig. 8, δ=cos(β), β angle between tool axis direction and material direction). The tool axis is definied according to 3 axis machining. Thus the cutting tool axis is colinear to the Z axis of the process. The material direction defines the free material side of the shape. Each material direction is given by the corresponding STL facet normal. According to the analysis of several industrial dies different machining features can be extracted. The example shown in Fig. 9 illustrates these features :

In dark red unique shade, δ =1, the tool tip is used ("end contact" configuration), the contact area between the tool and the part is constant. This type of contact induces a low effective cutting speed associated to a ball end mill tool. This feature type is qualified as "horizontal" feature.

In red, δ≈1, the tool tip is used ("end contact" configuration Fig. 6). This feature type is qualified as "quasi-horizontal" feature. The limiting threshold with a "flank configuration" is fixed by the machining assistant. Yet, the colour map allows to visualize the shapes which have the same contact area. In Fig. 9, two "horizontal" features are identified at the die top (δ=1, dark red) in relation with a red feature, δ≈1. Thus, this feature is necessarily "quasi-horizontal".

In deep blue unique shade, δ =cos(f), f is the die draft angle, the tool flank is used ("flank contact"configuration), the contact area between the tool and the part is constant. This type of contact induces an effective cutting speed close to the required cutting speed. This feature type is qualified "draft" feature.

In blue, δ≈cos(f), the tool flank is ued ("flank contact" configuration). The limiting threshold with a "flank configuration" is decided by the machining assistant. This feature type is qualified "quasi vertical" feature.

In colour gradation between red and blue, the tool is sollicited with the "end contact" and "flank contact" configuration. These features appear between "horizontal" or "quasi-horizontal" feature and "draft " or "quasivertical" feature. In this case, colour graduation features are qualified "transition" features. . They underline the fact that surfaces must be grouped according to the same direction series in order to limit cutting tool material engagements and exits. Indeed, each cutting tool engagement or exit in the part induces mechanical and thermal problems.

Three types of sequences of machined surfaces can be defined in the case of 3 axis forging die machining (Fig. 10). They are defined according to the tool axis direction and the material direction.

For the "simple sequence" type, a privileged planar direction P (Fig. 10.a) perpendicular to the tool axis direction allows connecting surfaces together which have the same material direction and are topological adjacent.

For the "oriented sequence" a privileged planar direction P (Fig. 10.b.) containing the tool axis direction allows connecting surfaces together which have the material direction contained in this specified plane and are topological adjacent.

For the "indifferent series" type any privileged planar direction can be found (Fig. 10.c.). The main objective of these maps is to identify part features corresponding to the same surface sequence type.

According to these maps, privileged directions for the parallel planes strategy are given to machining process assistant. These maps are obtained by colouration of each facet of the STL model according to the three sequence types previously definied.

The first map is designed for the identification of surface sequence in planes perpendicular to the tool axis which are associated to the Z-level strategy (Fig. 11). This map is already obtained by the same way from the cutting speed map. Nevertheless, the map interpretation is different. "Simple" sequences are obtained only if the criterion δ (δ=cos(β) is constant for the whole connected surfaces.

To identify the "oriented sequences", several directions are evaluated (Fig. 12). For a given direction, each facet is colourised according to the criterion of the angle χ in Fig. 12 between the fictive direction (line in green Fig. 12) and the material direction projection (doted line in blue Fig. 12) in the plane normal to the tool axis (plane P, Fig. 12). The material direction is given by the normal of the STL facet (Fig. 12). In the examples (Fig. 13), ten tested directions are mapped. The direction range is 90° with a step of 10°. This choice is done by machining assistant. The step can be refined around pertinent direction. Machining process assistant has to choose machining equipments (machine tools and cutting tools) and CAM machining strategies to compute tool trajectories during machining process preparation. Thus, he needs some information to make these choices. These infromation are given by the decomposistion maps. This paragraph highlights the decomposition maps exploitation and complementarity's in order to help machining process assistant to make choices.

-Cutting tool choice

The cutting speed distribution map is used to associate identified machining features with cutting tool (Fig. 14).

For instance, ball end mills tools are associated with "drafted" and "quasi-vertical" features. During the machining of these features, the effective cutting speed is closed to theorical one.

Ball end mills are also associated with "transition" features to limitate tool trajectories compared with corner end mill.

For "horizontal" features end mill or corner end mill are associated. These tool types allow to keep an effective cutting speed close to the theorical one. Yet, in the case of corner end mill use the torus radius must be small to limitate cutting speed decrease.

For "quasi-horizontal" features, corner end mill are associated for the same reason for "horizontal" features. Besides, if a "quasi-horizontal" feature is composed of a portion of "horizontal feature", ball end mill must be excluded because of a too small effective cutting radius. The machining strategy choice associated with decomposition maps (cutting speed distribution, oriented sequences) is focus on planar machining strategies as it is presented previously. Indeed, these machining strategies are the core of the majority of CAM software. Besides, these strategies are often the first choices done by machining assistant because of they are easy to generate.

For "drafted" and "quasi-vertical" features Z-level strategy is associated (Fig. 15). These features are machined by Zlevel strategy limiting strong variations of tool and part contact area. For the same reasons "transition" features are also associated with Z-level strategy (Fig. 15).

For "horizontal" features a surfacing machining strategy is associated (Fig. 15).

For "quasi-horizontal" features information given by cutting speed distribution associated map may be crossed with information given by oriented sequences associated maps. If a "quasi-horizontal" feature is "indifferent" any privileged machining direction is needed thus a surfacing strategy is associated (Fig. 15). If a "quasi-horizontal" feature is "oriented" a privileged vertical planar machining direction is suitable thus a parallel plan strategy is associated in the privileged oriented vertical plan (Fig. 15).

5-Conclusions

Today's forging die manufacturing approach must be adapted to economical choices which involve evolution and substantial modification in die machining preparation, die design and CAD models, CAM software, machining process HSM. Thus, the adequacy between part geometry and HSM process must be the core of machining process assistant approach.

In this paper, we proposed a new approach intented to help machining process assistant during HSM process generation. This approach is based on the decomposition of forging die geometrical model and topological analysis of machining sequences. The topological decomposition is based on machining process difficulties decomposition concept. This approach is particularly focused on HSM cutting conditions respect (cutting speed, feed rate). Otherwise, the geometrical model refinement is another important concept used in our approach. Thus, according to these concepts the machining process assistant has machining information linked with the die geometrical model. He can use these information to generate the die cavity HSM process during the topological analysis of machining sequences.

The die topology being the core of HSM process difficulties, the CAD model is decomposed according to specific topological information extracted from machining difficulties presented by maps: cutting speed map and machining direction sequences maps. Then these maps are superposed to decompose geometrical die model and extract machining features.

According to several industrial forging die studied with this topological decomposition five machining features are defined: flank feature, simple floor feature, oriented floor feature, indifferent floor feature, and transition feature. Yet, the boundaries between machining features are not automatically identified. Machining assistant has to make the correspondence between the STL and B-Rep CAD model.

Topological decomposition by machining difficulties of freeform parts is appearing to be an efficient systematic approach for machining process assistant. This new concept will be presented in the future.

Figure 1 :

 1 Figure 1: Conventional versus close die forging process [DB1]

Figure 2 :

 2 Figure 2: Structure of the decomposition processThe effective cutting speed is proportional to the effective diameter of the cutting tool. During finishing operations of complex shapes or sculptured surfaces with ball-end mill and corner rounded end mill with a large corner radius, the effective diameter varies permanently. To avoid calculating the effective cutting speed at any point, its variation can be associated to different contact areas defined by Lamikiz & al. (Fig.3)[LL1 & 2]. The critical feature for a ball end mill is the centre. In this feature the cutting speed is quasi null and the chip evacuation is critical.

Figure 3 :

 3 Figure 3: Variations of the contact area between tool and part [LL1 & 2] 3.1.1 -HSM of forging die analysis HSM of forging die is very often performed in a 3 axis

Figure 4 :

 4 Figure 4: Tool and part contact area analysis for parallel planes strategy

Figure 5 :

 5 Figure 5: Tool and part contact area analysis for Z-level planes strategy

Figure 6 :

 6 Figure 6: Contact area type

Figure 7 :

 7 Figure 7: Contact area type analysis

Figure 8 :

 8 Figure 8: Cutting speed map construction

Figure 9 :

 9 Figure 9: Cutting speed distribution map

Figure 10 :

 10 Figure 10: Surfaces sequence types

FigureFigure 12 :

 12 Figure 11: Simple sequence associated map

Figure 13 :

 13 Figure 13: Oriented sequences associated maps examples

Figure 14 :

 14 Figure 14: Cutting tool association

Figure 15 :

 15 Figure 15: Machining strategy association