Sarah Parisot

Hugues Duffau

Stéphane Chemouny

Nikos Paragios

Joint Tumor Segmentation and Dense Deformable Registration of Brain MR Images

In this paper we propose a novel graph-based concurrent registration and segmentation framework. Registration is modeled with a pairwise graphical model formulation that is modular with respect to the data and regularization term. Segmentation is addressed by adopting a similar graphical model, using image-based classification techniques while producing a smooth solution. The two problems are coupled via a relaxation of the registration criterion in the presence of tumors as well as a segmentation through a registration term aiming the separation between healthy and diseased tissues. Efficient linear programming is used to solve both problems simultaneously. State of the art results demonstrate the potential of our method on a large and challenging low-grade glioma data set.

Introduction

The automatic evaluation of the evolution of brain tumors, that are often monitored through MRI imaging, is of growing interest. The methods currently used to evaluate precisely the position, size and evolution of a tumor, involve a complete 3D manual segmentation by an expert. It is, however, extremely time consuming and highly dependent on the expert, particularly in the case of the diffusively infiltrative low grade gliomas, that have fuzzy boundaries and inhomogeneous appearances [START_REF] Kaus | Automated Segmentation of MR Images of Brain Tumors[END_REF]. As it has been shown that low-grade gliomas tend to appear in preferential locations [START_REF] Parisot | Graph Based Spatial Position Mapping of Low-Grade Gliomas[END_REF], the construction of statistical atlases enables the study of location dependent behaviour of tumors. As a result, both automatic segmentation and registration with a healthy atlas of tumor-bearing images are of great interest for the study of brain tumors.

State of the art segmentation methods combine efficient classification techniques [START_REF] Friedman | Additive Logistic Regression: a Statistical View of Boosting[END_REF] with low level segmentation methods [START_REF] Lee | Segmenting brain tumors using pseudo-conditional random fields[END_REF]. From such perspective, tumor detection is addressed as a classification problem where one aims at separating healthy from diseased tissues at the voxel level, while imposing smoothness constraints. Despite promising performance, these methods suffer from modularity and scalability. Visual and geometric appearance of tumors depend on organs, contrast agent, hardware acquisition device. The use of segmentationby-deformable registration of anatomical atlases [START_REF] Prastawa | Automatic Brain Tumor Segmentation by Subject Specific Modification of Atlas Priors[END_REF] can be an answer to the aforementioned limitations due to their ability to include global consistency and deal with isolated erroneous measurements. This kind of method however suffers from the inherent link between the two problems, making the segmentation dependent on the registration quality and vice versa. A common approach to deal with the presence of tumors for registration is to mask the pathology after segmentation [START_REF] Stefanescu | Non-rigid atlas to subject registration with pathologies for conformal brain radiotherapy[END_REF]. Another strategy is atlas seeding and simulating the tumor induced deformation in the atlas using growth models [START_REF] Bach Cuadra | Dense deformation field estimation for atlas-based segmentation of pathological MR brain images[END_REF].

The aim of this paper is to perform tumor segmentation and registration with missing correspondences in a one shot optimization approach. Concurrent segmentation and registration is a not a new objective [START_REF] Yezzi | A variational framework for integrating segmentation and registration through active contours[END_REF][START_REF] Wyatt | MAP MRF joint segmentation and registration of medical images[END_REF], however the problem becomes far more challenging in the presence of tumors. This is due to the fact that modeling tumor appearance and geometry is not as trivial as modeling anatomical regions, while at the same time registration is violated in the tumor area. [START_REF] Gooya | Joint Segmentation and Deformable Registration of Brain Scans Guided by a Tumor Growth Model[END_REF] proposes registration of an atlas, deformed using a complex tumor growth model, to the patient's space. The parameters are learned using the Expectation Maximization algorithm which can be very computationally expensive and sensitive to initialization. Our approach aims at combining the modeling of both problems where the unknown variables correspond to a two layer graphical model, one that represents the 3D deformation field and another that refers to the 3D binary segmentation map. This graphical model is superimposed to the volume domain. The deformation is obtained using the discrete formulation introduced in [START_REF] Glocker | Dense image registration through MRFs and efficient linear programming[END_REF]. Segmentation is obtained through a conventional graph-based formulation [START_REF] Boykov | Graph Cuts and Efficient ND Image Segmentation[END_REF] where tumors and healthy tissues are separated through the use of boosting. In order to reduce complexity we adopt a reduced label grid that estimates segmentation likelihoods in the whole image domain through interpolation of the associated variables. The two layers are interconnected with a combined cost that relaxes the registration in the presence of tumors, while at the same time performs a segmentation-by-registration using the segmentation costs as criterion. Linear programming and duality [START_REF] Komodakis | Performance vs computational efficiency for optimizing single and dynamic MRFs: Setting the state of the art with primaldual strategies[END_REF] are used to determine the optimal solution of the combined problem. The proposed formulation is modular with respect to the registration criterion and the segmentation likelihoods, scalable since the segmentation/registration grid can be adapted to the application domain and computationally efficient. The remainder of this paper is organized as follows, section 2 describes the combined registration and segmentation method, while experimental validation is presented in the following section. Discussion and future directions conclude the paper.

Graph-based Image Registration and Segmentation

Let us now first introduce the individual components of the method and then conclude the methodological part of the paper with their coupling. The resulting formulation refers to a two-layer interconnected graph where both the segmentation labels on the atlas space and the deformable registration field are determined through a one shot optimization. Such a formulation seeks both maximization of the posterior statistical likelihoods of the tumor versus healthy voxels, and optimal similarity matching between the atlas and the observed volume that is relaxed in the tumor area.

Graph-based image registration

In the task of image registration, we seek to align a source image A to a target image I defined on a domain Ω ⊂ R 3 . To this end, we seek to estimate a geometric deformation T (x) that will map A to I.

Let us consider a sparse grid G ⊂ Ω superimposed to the image. The deformation is evaluated on the grid's control points p as T (x) = x+ p∈G η ( xp ) d p , where d p is the displacement vector of the control point p. The idea is to deform the source image by moving the grid's control points. The displacement of a voxel x of the image will be determined by the control point's displacements and the influence of each control point on x, which is given by the projection function η.

A typical example of projection function would be cubic B-splines. The optimal deformation field should minimize the distance between the target image I(x) and deformed image A(T (x)), that is evaluated by a similarity function ρ:

E(T ) = 1 |G| p∈G Ω η ( x-p ) ρ (I(x) , A(T (x))) dx (1)
The similarity function being defined on the image domain Ω, a function η is introduced to back project the voxel wise information on the grid's control points. It determines how much voxel x influences the control point p.

We adopt the discrete approach described in [START_REF] Glocker | Dense image registration through MRFs and efficient linear programming[END_REF], in which problem (1) is reformulated as a labeling problem. To this end, we define a set of labels L = {l 1 , ..., l k } and a discrete displacement set ∆ = {d 1 , ..., d k }. Each label corresponds to a specific displacement vector so that assigning a label l p ∈ L to a control point p imposes the corresponding displacement d i of the node. The deformation field can be reformulated as:

T l (x) = x + p∈G η ( x -p ) d lp .
This enables to pose the problem as a discrete Markov Random Field (MRF) optimization with respect to the labeling l, where the goal is to minimize the following energy:

E def (l) = 1 |G| p∈G V p (l p ) + λ p∈G q∈N (p) V pq (l p , l q ) (2)
V p (l p ) can be approximated as

V p (l p ) ≈ Ω η ( x-p ) ρ I(x) , A x + d lp ) dx,
V pq is a pairwise constraint that imposes local smoothness of the deformation field and λ is a constant parameter balancing the single and pairwise costs. This formulation is not sufficient in the case of lacking correspondences between the images. In these areas, the source image will be heavily deformed and information will be lost.

Graph-based Image Segmentation

In order to automatically segment tumor voxels, we seek to construct a classifier that will describe the likelihood that each voxel belongs to a tumor. One simple and efficient way to do so is to learn the classifier based on a vector of image features and boosting. We use the Gentle Adaboost algorithm [START_REF] Friedman | Additive Logistic Regression: a Statistical View of Boosting[END_REF], that is known to be more robust to noise and outliers than other boosting algorithms. We adopt three feature spaces. First, we rely on the intensity values in the voxel of interest x i and its neighborhood. We include in the feature vector an intensity patch centered on x i , the median, entropy and standard deviation of patches of variable sizes. Second, we use Gabor features [START_REF] Michel | Boosted metric learning for 3D multi-modal deformable registration[END_REF] on 2 scales and 10 orientations. Last, the tumor's presence will introduce some lack of symmetry between the hemispheres. Assuming we know an approximate symmetry plane Π, we compute a symmetry feature as Sym(

x i ) = 1 k N (x i ) I(x i ) -1 k N (x i ) I(x iΠ ),
x iΠ being the symmetric of x i with respect to Π, and N a neighborhood around x i introduced to reduce the approximation error. The classifier's output can be converted into a tumor p tm (x) and a background probability p bg (x). Both probabilities are fixed after training.

The problem of tumor segmentation can be cast as an MRF optimization on the whole volume I, where each voxel gets a label l ∈ {0, 1} identifying it as tumor or background:

E seg (l) = x V x (l(x)) + λ seg x y∈N (x) V xy (l(x), l(y)) (3)
The single costs impose a labeling according to the classification likelihoods: V x (l(x)) = -log (p bg (x))(1-l((x)))log (p tm (x)) l((x)). V xy is a pairwise cost added to impose local consistency of the segmentation and λ seg balances the single and pairwise costs. The segmentation task could also benefit from the registration, since existing correspondences between the atlas and the image to be segmented could eliminate false detections of the classifier.

Combined Registration and Segmentation

Let us now describe the main contribution of this paper: the joint segmentationregistration framework. Let us consider that the source image A, a healthy brain image, is to be registered to a target brain image I featuring a low-grade glioma.

In the method presented in section 2.1, the tumor's presence leads to heavy deformation of the brain and loss of structural information. The aim of our method is to amend this problem by detecting the tumor's position and not taking it into account during registration, while at the same time obtaining a precise segmentation of the tumor. We seek to evaluate the geometric deformation T (x) and the tumor's position S(x) through the same optimization. We adopt a discrete MRF formulation in which each grid node p is assigned a label l ∈ L = {l 1 , ..., l 2k }. Each label corresponds to a pair {s, d} ∈ {0, 1} × {d 1 , ..., d k }. The first term s lp will simply characterize the node as tumor or background, while the second one d lp determines the node's displacement. The segmentation and displacement information is propagated to the whole image by cubic B-spline interpolation:

T (x) = x + p∈G η( x -p )d lp and S(x) = H( p∈G η( x -p )s lp -0.5) (4)
Due to the interpolation, the segmentation map S(x) is not binary. We deal with this issue by thresholding the map using the heaviside function H.

Let us now proceed with the definition of the singleton cost V p (l p ). In the case of healthy voxels, the evaluation of the deformation is still based on a similarity measure between the source and target image. In the tumor area (s lp = 1), we cannot rely on the similarity measure and therefore introduce a constant cost C tm as suggested in [START_REF] Kang | Handling occlusions in dense multi-view stereo[END_REF] for stereo matching:

V def (l p ) = Ω η ( x-p ) 1-s lp ρ V (x), A(x + d lp ) + s lp C tm dx (5)
This potential enables registration without taking into account the area detected as tumor (highly dissimilar points in the image). While part of the tumor can be detected using this single potential, the lack of similarity alone is not sufficient to efficiently segment the tumor. Only tumors voxels with high contrast enhancement will be detected. Furthermore, dissimilarity doesn't necessarily correspond to a tumor. To enhance the segmentation and prevent false positives, we make use of the classifier built in section 2.1. We couple the deformation potential with a second one acting mainly on the segmentation space:

V seg (l p ) = Ω η ( x-p ) -log p bg (x + d lp ) 1-s lp -log p tm (x + d lp ) s lp dx (6)
This potential imposes that voxels with a high probability of being tumor (p tm ) are labeled accordingly. It takes the deformation into account since the classifier's score is defined on the target image: the better the source image is aligned to the target image, the more precise becomes the tumor's position. We can now define the singleton cost as V p (l p ) ≈ αV def (l p ) + (1α)V seg (l p ), where α is a constant coefficient of key importance balancing the 2 potentials' influence. If α is high, the registration will be prioritized, resulting in a good registration (except in the tumor area) and poor segmentation. On the contrary, imposing a low α yields a good segmentation (assuming the classifier is efficient), while the registration quality will be deteriorated. As for the pairwise cost, it is set as:

V pq (l p , l q ) = αλ d lp -d lq p -q + (1 -α) |s lp -s lq | p -q (7)
In order to recover the optimal labeling solution, we use Fast-PD [START_REF] Komodakis | Performance vs computational efficiency for optimizing single and dynamic MRFs: Setting the state of the art with primaldual strategies[END_REF], an efficient MRF optimization method based on linear programming.

We adopt a multi-scale incremental approach (both on the image and grid resolutions) were the tumor presence will have an increasing impact on the process. We evaluated the deformation and segmentation on 3 image levels and 4 grid levels (9 × 9 × 5 to 65 × 65 × 37). The parameter C tm was increased at each level, from 5 to 10 times the mean value of the similarity measure, while the parameter α is progressively reduced from 1 to 0.015. λ and λ seg were set at 20 and 3 respectively. At each level t, new deformation and segmentation fields are estimated w.r.t. the source image deformed in level t -1. The deformation field T t (x) is updated by composition with the one obtained at level t -1. Since false positives are less likely to appear in the coarse levels, we add a penalty term as V t pen (l p ) = -Ω η ( x-p ) (1s lp ) exp(-t)dx to voxels labeled as background (S t-1 (x) = 0). We use the Sum of Absolute Differences as similarity criterion.

The healthy brain template used for registration was a 3D MRI FLAIR image, of size 256 × 256 × 24, and resolution 0.9 × 0.9 × 5.45mm 3 . All images in our data-set (93 images) were 3D FLAIR images of different patients with low-grade gliomas that have been manually segmented by an expert. Their sizes ranged from 256×256×24 to 512×512×33, and resolution from 0.4×0.4 to.9×0.9mm 2 in the (x,y) plane and 5.3 to 6.4 mm in the z plane. The smallest tumor was 3.5 cm 3 , and the largest was 230cm 3 . Preprocessing involved skull stripping and intensity regularization. In order to avoid losing tumor vs healthy tissue contrast, we simply set all images to the same median and interquartile range as the reference pose, without taking into account background voxels. Eventually, all images were rigidly registered to the reference pose, which permitted to use the template's symmetry plane to evaluate the symmetry feature for the boosting classifier. 36 randomly selected volumes were used for boosting learning. The patches sizes for intensity statistics were k × k × 3, with k = {3, 5, 7}.

We tested our framework on the 57 remaining images. Overall computational time was approximately 6 minutes. The registration was compared qualitatively to individual registration where the pathology is masked using the manual segmentation masks (enabling comparison to a sequential approach). Some visual registration results, along with the corresponding deformation maps are shown in Fig. [START_REF] Parisot | Graph Based Spatial Position Mapping of Low-Grade Gliomas[END_REF]. We observe high visual correspondences between the target and registered images outside the tumor area for both methods. As shown by the individual registration's deformation maps, deformations inside the tumor area are very important and excessively unnatural. Using our method, the deformations are much smoother and the deformed image's anatomy is respected, showing that our method performs significantly better in the tumor area. The automatic segmentation (A) was evaluated w.r.t the manual segmentation (M) using the Dice coefficient, the rate of false positives (F P = A -M ∩A A ) and the Mean Absolute Distance (MAD) between contours. Results were compared to the voxel wise individual segmentations. Comparative boxplots are shown in Fig. [START_REF] Kaus | Automated Segmentation of MR Images of Brain Tumors[END_REF]. The median dice increases from 77% to 80%, while false positives significantly diminished (median 30% to 20%) and MAD values also diminished.

Discussion

In this paper we have proposed a novel, efficient and principled method for combined tumor segmentation and dense registration. This was achieved through a two-layer interconnected graphical model that was optimized using a single shot approach towards optimal recovery of both variable spaces. Extensive validation concerning the case of low-grade gliomas was considered to evaluate the performance of the method, leading to very promising results. Introducing prior knowledge in the reference space as suggested in [START_REF] Parisot | Graph Based Spatial Position Mapping of Low-Grade Gliomas[END_REF] is a straightforward extension of the proposed formulation which could further improve the performance of the method while allowing the characterization of tumors. Another possible extension of the method is the use of recent advances in MRF learning [START_REF] Komodakis | Efficient training for pairwise or higher order CRFs via dual decomposition[END_REF] towards encoding prior knowledge using spatially varying coefficients of the corresponding segmentation framework that could be learned from training examples.

Fig. 1 :

 1 Fig. 1: Boxplots of the (a) Dice values, (b) false positive rates and MAD (c) for the joint framework (1) and the individual segmentation method (2)

Fig. 2 :

 2 Fig. 2: Visual registration and segmentation results. (a) Segmented target image (manual (red) and automatic (blue) segmentations), (b) Individual registration, (d) our framework, (c,e) associated deformation fields.

⋆ This work was supported by ANRT (grant 147/2010), Intrasense and the European Research Council Starting Grant Diocles (ERC-STG-259112).


Pierre Yves Baudin
email: pierre-yves.baudin@ecp.fr

Noura Azzabou
email: n.azzabou@institut-myologie.org

Pierre Carlier
email: p.carlier@institut-myologie.org

Nikos Paragios
email: nikos.paragios@ecp.fr

Manifold-enhanced Segmentation through Random Walks on Linear Subspace Priors

In this paper we propose a novel method for knowledge-based segmentation. Our contribution lies on the introduction of linear sub-spaces constraints within the randomwalk segmentation framework. Prior knowledge is obtained through principal component analysis that is then combined with conventional boundary constraints for image segmentation. The approach is validated on a challenging clinical setting that is multicomponent segmentation of the human upper leg skeletal muscle in Magnetic Resonance Imaging, where there is limited visual differentiation support between muscle classes.

Introduction

Although there is an abundant computer vision literature on the subject of automatic image segmentation, many concrete problems still wait to be solved, in particular in segmentation of organs in medical applications. Our particular interest lies in segmentation of skeletal muscles in 3D Magnetic Resonance Imaging, which poses some very specific issues: simultaneous multi-object segmentation, non-discriminative appearance of the muscles, partial contours between them, large inter-subject variations, spurious contours due to fat infiltrations.

Vastly different approaches have been used in the relatively small literature addressing automatic muscle segmentation. In [START_REF] Gilles | Musculoskeletal MRI segmentation using multi-resolution simplex meshes with medial representations[END_REF][START_REF] Gilles | Fast musculoskeletal registration based on shape matching[END_REF], segmentation is achieved with multi-object deformable models. Deformable models ( [START_REF] Mcinerney | Deformable models in medical image analysis: a survey[END_REF][START_REF] Montagnat | A review of deformable surfaces: topology, geometry and deformation[END_REF]) are surface models superimposed to the c 2011. The copyright of this document resides with its authors. It may be distributed unchanged freely in print or electronic forms.

target image in an initial state and deformed by minimizing an objective function composed of two terms: a data-term -attracting the surface to detected image contours -and a regularization term, ensuring the surface remains realistically smooth or similar enough to a reference surface (as in [START_REF] Gilles | Fast musculoskeletal registration based on shape matching[END_REF]). In [START_REF] Essafi | Hierarchical 3D diffusion wavelet shape priors[END_REF], prior knowledge on the shape of one muscle is further enforced by imposing the model to reside in a hierarchical shape space built via diffusion wavelets decomposition on training examples. Relying on iterative local optimization procedures, deformable models are very sensitive to the proximity of the initialization state with the expected solution, and usually only yield non-global optima of the objective function. In [START_REF] Wang | 3D knowledge-based segmentation using pose-invariant higher-order graphs[END_REF], the muscle surface is represented with a set of landmarks in a graph framework, and a higher-order pose invariant shape prior is encoded by learning the distances within triplets of landmarks. Then, the surface is fit to the image though a discrete optimization procedure allowing to solve the objective function globally. Such method implies one can find consistent anatomical landmark points, an assumption which is hardly verified for muscles. Apart from surface models, a pixel-wise region-based approach was proposed in [START_REF] Andrews | Probabilistic multi-shape segmentation of knee extensor and flexor muscles[END_REF] for segmenting muscles with good results. In this framework, a segmentation is a labeling of the pixels of the target image. Shape prior is enforced by learning a low-dimensional space of allowed segmentations through Principal Component Analysis on a set of training segmentations projected in the Isometric Log-Ratio space. Optimization of the convex objective function is carried out through gradient descent. Like surface-based methods, this method explicitly relies on preliminary contour detection, a difficult and often unreliable process in muscle images.

Recent advances in numerical methods for computing globally large discrete regionsbased segmentation methods -Graph-Cuts ( [START_REF] Boykov | Interactive graph cuts for optimal boundary & region segmentation of objects in ND images[END_REF]) and Random Walks ( [START_REF] Grady | Random walks for image segmentation[END_REF]) -have made them practical for large images. Due to being graph-based, these methods are intrinsically bottomup -pixel-wise information has an influence on the whole segmentation. In particular, such property makes Random Walks notoriously robust to partial contours. On the other hand, few methods exist for enforcing prior knowledge on object shapes in such framework. Every method known to us proposes at best an affine deformation of a template shape [START_REF] Freedman | Interactive Graph Cut Based Segmentation with Shape Priors[END_REF][START_REF] Vu | Shape prior segmentation of multiple objects with graph cuts[END_REF].

In this paper, we propose a segmentation method based on Random Walks, in which shape deformation is constrained to remain close to a Principal Component Analysis shape space built from training examples. Using the PCA allows us to model complex non-rigid shape variations relying on a few eigen-modes. Our method also benefits from the high performances of the RW optimization process, as it only requires a simple addition to the RW objective function.

This paper is organized as follow: in section 2, we review the RW segmentation framework, and describe how to include a PCA shape prior. Then in section 3, we demonstrate the potential of our method on a challenging data set of 3D muscle MR volumes. Section 4 sums up and concludes this paper.

2 Random walk segmentation with prior knowledge

Random walk formulation

In this section we briefly review the random walks approach for segmentation, based on the fully detailed presentation of [START_REF] Grady | Random walks for image segmentation[END_REF]. A segmentation is formulated as a labeling problem of an undirected weighted graph G = (V, E), where V is a set of nodes and E is a set of edges. The i-th node v i corresponds to the i-th pixel of I, the image to be segmented. Given a set of K The RW method amounts to computing the probability x s i that the node v i ∈ V is assigned to the label s. Assume we possess a set V M of pre-labeled nodes for each label, also called seeds (typically, manually marked pixels). We denote the set of non pre-labeled nodes as V U . It was shown ( [START_REF] Grady | Random walks for image segmentation[END_REF]) that all unknown (non pre-labeled) entries of x s = [x s 1 , x s 2 , . . . , x s N ] Ti.e. the probability that each node v i ∈ V U is assigned to label s -minimize the functional:

E s RW (x s ) = x sT Lx s (1)
where the known entries of x s (the seeds) are set as follow:

∀v i ∈ V M , x s i = 1 if pixel i is marked with label s 0 if pixel i is marked with another label (2)
and where L is the combinatorial Laplacian matrix of the graph, defined as:

L i, j =      ∑ k w k j if i = j -w i j if (i, j) ∈ E 0 otherwise (3)
where w i j is the weight assigned to the edge spanning vertices i and j. It is common to define w i j as a Gaussian weighting function :

w i j = ω + exp -β (g i -g j ) 2 (4)
where g i is the gray-level of pixel i, β is a scaling parameter set accordingly to the image contrast, and ω is an regularization parameter which amounts to penalizing the gradient norm of x s (no regularization if ω = 0). Once computed x s for each label s, the segmentation is obtained by retaining the label of maximum probability: ŝi = arg max s x s i .

Since minimizing (1) is an independent process for each label, the whole RW process can be equivalently synthesized in one equation:

E RW (x) = x T Lx (5)
where

x T = x 1 T |x 2 T | . . . |x K T (x ∈ R KN×1
) is the concatenation of all the x s in one vector, and

L =    L • • • 0 . . . L . . . 0 • • • L    ∈ R KN×KN (6)
is a block diagonal matrix with L repeated K times on the diagonal -i.e. as many times as there are labels. We keep this formulation for the remaining of the paper.

Prior shape model

Prior knowledge to the RW formulation was introduced in [START_REF] Grady | Multilabel Random Walker Image Segmentation Using Prior Models[END_REF], through an estimate of the probability distribution of the gray-level intensity for each label. Such prior could be enforced by minimizing this cost function:

E s RWP (x s ) = x s T Lx s + γ x sT Λx s -2x sT λ s (7)
where λ s (i) is the probability that the intensity at pixel i belongs to the intensity distribution for label s, and Λ = diag (∑ s λ s ) (we refer the reader to [START_REF] Grady | Multilabel Random Walker Image Segmentation Using Prior Models[END_REF] for details).

In the context of muscle segmentation, since the intensity distributions for the labels (the muscles) are extremely similar, such prior would not be very efficient. As we could not find any discriminative features (textures, consistent anatomical points, etc.) to use within this framework, we designed a pixel-wise shape model, trained from a series of ground-truth segmentations.

Assume we possess such a shape space, in the form of an affine model (details in section 2.3) -i.e. any vector x inside this space takes the form:

x = Γγ + x
where x is the mean shape, Γ ∈ R KN×k is the shape space matrix (e.g. obtained through PCA), γ ∈ R k is the coordinate of x within the shape space. Thus, vector x can be expressed as:

x = dx + Γγ + x ( 8
)
where dx ∈ R KN×1 is the deviation of x from the shape space. In order to obtain a segmentation which remains close to the shape space, we want to minimize the objective function 5 with respect to both dx and γ, while keeping dx small. This leads us to the following functional:

E 1 (dx, γ) = (dx + Γγ + x) T L (dx + Γγ + x) + λ dx 2 (9)
where λ is a free parameter setting a constraint on the norm of dx. We reformulate 9 as:

E 2 (y) = (Ay + x) T L (Ay + x) + λ y T By (10)
with

y = dx γ , A = [I KN Γ] , B = I KN 0 0 0 . ( 11
)
where I KN is the identity matrix of size KN × KN.

The minimum of (10) verifies:

A T LA + λ B y = A T L x (12)
Although the system of equations ( 12) is invertible even when all entries of x are unknown (no seeds), as noted in [START_REF] Grady | Multilabel Random Walker Image Segmentation Using Prior Models[END_REF], it is still possible -and useful -to use seeds in combination with a prior to obtain more robust segmentations.

Shape space

Any segmentation corresponding to a vector of probability x belongs to R KN×1 . However, it is obvious that the entries of x are not independent from each other, but instead are highly tied to each other, at least for sets of close-by pixels. This implies that there exists an implicit lower dimensional space in which any true segmentations reside. The principle of a shape space is to design a low dimensional affine space approximating this implicit space. If we succeed, we expect any true segmentation to lie "not too far" from the shape space, and the projection of that segmentation into the shape space to be a good approximation of the segmentation itself. Principal Component Analysis notoriously provides a simple mean to compute such a shape space.

Assume we possess T co-registered segmented training volumes. We perform the PCA on vectors xi i=1...T , which are ground truth segmentations represented as probability vectors (actually binary vectors, since x s i = 1 if pixel i has label s, and 0 otherwise). Retaining the n eigen-modes of greatest variance, the projection of any segmentation in the shape space is represented as:

x = x + Γγ where Γ ∈ R KN×n is a rectangular matrix whose columns are the n retained eigen-vectors, and γ ∈ R n×1 is the vector of coordinates of x in the shape space.

Experimental validation

In this section, we present results obtained with our method on a set of 3D MR volumes of muscles. Our data set is comprised of 30 volumes of the right thigh of healthy subjects, covering a wide range of ages and morphologies. These volumes were acquired with a 3T Siemens scanner using a 3-point Dixon sequence (T R = 10ms, T E 1 = 2.75ms, T E 2 = 3.95ms, T E 3 = 5.15ms, rf flip angle = 3 • ) with resolution 1mm × 1mm × 5mm and initial size 224 × 224 × 128. Four clinically relevant muscles were manually segmented -rectus femoris, vastus lateralis, vastus intermedius, vastus medialis, providing the ground truth segmentation against which the output of our method will be compared.

In order to compute the shape space (see section 2.3), we non-rigidly register all volumes and segmentation atlas to the same reference volume, using the method in [START_REF] Glocker | Dense image registration through MRFs and efficient linear programming[END_REF] and the related software (Drop, at www.mrf-registration.net). Adopting a leave-one-out validation protocol, all the volumes in the training set, bar one, are registered to the remaining volume, on which we test our segmentation method.

Given the large size of the matrices, we solved the system of equations 12 with iterative methods like Bi-Conjugate Gradient. Computation time is approximately 15 min per Figure 2: Effect of the PCA shape prior: (left) mean segmentation using x = x, (middle) shape space segmentation using x = Γγ + x, (right) segmentation with shape prior using x = dx + Γγ + x. Notice that the shape space segmentation fits the boundaries better than the mean segmentation, but has fuzzy contours due to the approximation of projecting complex shapes into a linear subspace. On figure 2, we show the effect of the PCA shape prior on one example of our dataset. We see that the PCA model (formulated as Γγ + x) is deformed from its initial shape (i.e. x) to fit the boundaries of the test image. Due to the rough approximation of modeling probabilistic atlases with a linear subspace, the muscles' boundaries are not well defined. However, when allowing a small deviation (i.e. dx) from the model, as does our method, the contours are more accurately defined. Notice there remains segmentation errors (the top red muscle in figure 2), as the PCA model is not capable of deforming enough to fit unusual shapes while retaining a realistic topology.

We assess the quality of the segmentation using Dice coefficients, whose expression is: Dice = 2 |T ∩ R| / (|T | + |R|), where T and R are the pixel sets for the same muscle in the computed segmentation and the ground truth segmentation respectively. We obtain an average Dice coefficient of 0.84 ± 0.08. On figure 3 we compare the results obtained with our method to those obtained when using x = x (We called it "registration only"). The latter method is equivalent to a segmentation through multi-atlas registration, with majority label voting. A previous method of ours [START_REF] Baudin | Automatic Skeletal Muscle Segmentation Through Random Walks and Graph-Based Seed Placement[END_REF] yielded an average Dice of 0.80 ± 0.19 for a different but similar dataset. In this method, we automatically generated appropriate seed positions with respect to the muscle classes. Unlabeled seeds were sampled across the test image, and were automatically labeled through a graph-based approach considering visual and topological properties. The resulting seeds were then fed to the standard RW algorithm [START_REF] Grady | Random walks for image segmentation[END_REF].

In the presented method, the main segmentation errors are due to the muscles with unusual shape -e.g. more elongated, smaller, etc. Modeling probabilistic segmentations with PCA model does not allow representing shapes which differ too much from standard shapes. However, the amount of deformation that is allowed by such modeling is enough to capture a large number of common shapes and thus is valid in many cases. Some segmentation results are shown in figure 4.

Conclusion

In this paper, we presented a novel multi-object segmentation method with prior knowledge, based on the Random Walks segmentation algorithm. We designed our approach to be capable of handling muscle segmentation in 3D MRI, where partial contours, large inter-subject variations, absence of texture differences, etc. render typical segmentation methods inefficient. Building from the RW formulation, we add a shape space prior term to the objective function. Our shape space is built from PCA on a set of training segmentations. To reveal the potential of our method, we presented segmentation results on MR volumes of the thigh on 30 subjects.

Our model is capable of fitting most shapes encountered in our dataset. However, due to the approximation induced by the projection into a linear subspace, some more extreme shapes are not segmented well. In the future, we intend to design a model not relying on PCA and capable of handling more varied shapes.

Figure 1 :

 1 Figure 1: (left) Cross-section of an MR volume of the thigh. (right) Manual segmentation of four muscles.

Figure 3 :

 3 Figure 3: Segmentation results as Dice coefficients. This paper's method is represented in red, and the mean segmentation (using x = x) is shown in white.

Figure 4 :

 4 Figure 4: Examples of segmentation results


P-Y Baudin

N Azzabou

P G Carlier

N Paragios

Prior Knowledge, Random Walks and Human Skeletal Muscle Segmentation

In this paper, we propose a novel approach for segmenting the skeletal muscles in MRI automatically. In order to deal with the absence of contrast between the different muscle classes, we proposed a principled mathematical formulation that integrates prior knowledge with a random walks graph-based formulation. Prior knowledge is represented using a statistical shape atlas that once coupled with the random walks segmentation leads to an efficient iterative linear optimization system. We reveal the potential of our approach on a challenging set of real clinical data.

Introduction

Segmentation of the skeletal muscles is of crucial interest when studding myopathies. Diseases understanding, patient monitoring, etc. rely on discriminating the muscles in anatomical images. However, delineating the muscle contours manually is an extremely long and tedious task, and thus often a bottleneck in clinical research. Simple automatic segmentation methods rely on finding discriminative visual properties between objects of interest, accurate contour detection or clinically interesting anatomical points. However, skeletal muscles show none of these features and as a result, automatic segmentation is a challenging problem. In spite of recent advances on segmentation methods, their application in clinical settings is difficult, and most of the times, manual segmentation/correction is still the only option.

Among the limited amount of work on this specific subject, in [START_REF] Gilles | Fast musculoskeletal registration based on shape matching[END_REF][START_REF] Gilles | Musculoskeletal MRI segmentation using multi-resolution simplex meshes with medial representations[END_REF] a method based on deformable models was proposed to perform the segmentation of all the muscles in one limb. Deformable models are surface models which are fitted to the target image by minimizing a functional balancing a data term -pushing the model towards the target contours -and a regularization term -which imposes a smooth solution along the curve. Such models only reach a local optimum of the functional, which can be far from the desired solution, and depend heavily on their initial position. In [START_REF] Essafi | Hierarchical 3D diffusion wavelet shape priors[END_REF], a more efficient shape representation was introduced where prior knowledge was encoded through diffusion wavelets to reduce the space of solutions and thus relax the smoothing constraints. The surface of one muscle was modeled through a hierarchical representation, and the set of allowed deformations at each scale was learned from an annotated training set. Modeling the surface of one muscle with landmark points is an efficient alternative, as proposed in [START_REF] Wang | 3D knowledge-based segmentation using pose-invariant higher-order graphs[END_REF] with a graph-based method. The shape variability was modeled through high-order pose-invariant priors and the data term relied on classification and detection of the landmark points. The graph-based framework allowed to perform an efficient non-local optimization, without initialization. However, such method requires to be able to learn consistent image features in order to detect the landmarks, which is difficult to insure in practice in the case of muscles. More generally, all surface models suffer from the absence of reliable contours in MRI images of skeletal muscles. Recently, in [START_REF] Andrews | Probabilistic multi-shape segmentation of knee extensor and flexor muscles[END_REF], a model-based method operating in the image domain with promising results was proposed. This approach consisted in modeling a segmentation though Principal Component Analysis in an Isometric Log-ratio space. Then, a gradient descent was performed with respect to the PCA coefficients to minimize an energy functional which allows label transition only along detected contours. In this method, contour detection is explicit, achieved in a pre-processing stage, and could be a weak link in the chain in cases of undetected or spurious contours.

Our approach builds upon the general Random Walker Segmentation algorithm proposed in [START_REF] Grady | Random walks for image segmentation[END_REF]. The strength of this method relies on its robustness in the case of incomplete contours and its efficient optimization. While originally this method required manual interaction -an user had to annotate a few pixels of each desired object -the possibility of using prior knowledge based on intensity distribution was introduced in [START_REF] Grady | Multilabel Random Walker Image Segmentation Using Prior Models[END_REF]. In this paper, we propose to build a prior model of the shape of the thigh muscles from a training data set, to be used in the RW framework. The prior term of our functional is derived from learning a Gaussian model of the RW unknown probability vector. We also propose to modulate the strength of the model constraints according to the strength of the contours found in the segmented image.

This paper is organized as follows: in section 2 we briefly recall the principle of the RW segmentation and detail the formulation of our model. Then, in section 3, we present segmentation results obtained on 3D MR volumes of the right thigh. Section 4 concludes the paper.

Random Walks Segmentation With Prior Knowledge

Notations Let us consider an image I with N pixels, and I i the gray-level of pixel i. The segmentation is formulated as a labeling problem of an undirected weighted graph G = (V, E), where V is the set of nodes and E is the set of edges. Given S, a set of labels, we want to assign a label s ∈ S to each node p ∈ V. In this framework, the node v i is the i-th pixel, and to each label corresponds a muscle.

Random Walks Formulation

Let us proceed with a review of the random walks algorithm for image segmentation. We refer the reader to [START_REF] Grady | Random walks for image segmentation[END_REF] for an extensive description of this method.

The RW approach provides the probability x s i that the node v i ∈ V is assigned to the label s. In its original formulation, one has to provide the algorithm with a few already labeled (marked) nodes, also called "seeds". Typically, the user will manually mark some pixels of each object to be segmented with a different label. Lets denote V M the set of marked nodes and V U the set of unmarked nodes, such that V U ∩V M = Ø and V U ∪V M = V. It was shown [START_REF] Grady | Random walks for image segmentation[END_REF] that all unknown entries of

x s = [x s 1 , x s 2 , . . . , x s N ]
T -i.e. the probabilities that each node v i ∈ V U is assigned to label s -can be obtained through the minimization of:

E s RW (x s ) = x sT Lx s (1)
where the known entries of x s (the seeds) are set as follow:

∀v i ∈ V M , x s i =
1 if pixel iis marked with label s 0 if pixel iis marked with another label [START_REF] Gilles | Musculoskeletal MRI segmentation using multi-resolution simplex meshes with medial representations[END_REF] and where L is the combinatorial Laplacian matrix of the graph, defined as:

L i,j =      k w kj if i = j -w ij if i = j 0 otherwise (3)
with

w ij = ω + exp -β (I i -I j ) 2 ( 4
)
where β is a scaling parameter to be set according to the contrast of the image, and ω is a regularization parameter which amounts to penalizing the gradient norm of x s (no regularization if ω = 0). After minimizing E s RW for each label s, the segmentation is obtained by retaining the label of maximum probability: l i = arg max s x s i .

Prior Knowledge

In [START_REF] Grady | Multilabel Random Walker Image Segmentation Using Prior Models[END_REF], prior appearance knowledge to the RW formulation was introduced through an estimate of the probability distribution of the gray-level intensity for each label. A prior appearance term is simply added to the RW cost function, balanced by a parameter γ:

E s RWP (x s ) = x s T Lx s + γ x sT Dx s -2x sT d s (5)
where d s (i) is the probability that the intensity at pixel i belongs to the intensity distribution for label s, and D = diag ( s d s ) (we refer the reader to [START_REF] Grady | Multilabel Random Walker Image Segmentation Using Prior Models[END_REF] for details). In the context of muscle segmentation, the intensity distributions of the labels (the muscles) are extremely similar resulting in an inefficient prior. Moreover, we could think of no other discernible and discriminative features (textures, remarkable points, etc.) to use within this framework. Thus, we decided to learn a pixel-based model of the shape based on previous segmentations of images in a training set D.

Assume we know xs i and σ s 2 i , respectively the mean and the variance of x s i . Our model simply penalizes the deviation of vector x s i from xs i , weighted by the inverse of σ s 2 i . In vector form, we obtain the following functional:

E s model (x s ) = (x s -xs ) T Λ s σ (x s -xs ) (6)
where Λ s σ is a diagonal matrix such that Λ s σ (i, i) = 1/σ s 2 i . This is equivalent to modeling x s i as a random variable with Normal distribution N xs i , σ s 2 i , and maximizing the log probability of x s i . Since x s i is a probability, such Gaussian modeling can only be a rough approximation. The mean and variance are estimated by computing, respectively, the empirical mean and the empirical variance over a training base of non-rigidly registered segmented images. When one owns only a small number of training examples, the empirical variance is known to be a particularly inefficient estimator. In [START_REF] Rousson | Prior knowledge, level set representations & visual grouping[END_REF], an improved locally-smooth estimator was proposed, for using as a similar shape prior in the level-set framework. The new estimate is computed, through a gradient descent, as the minimum of a functional which combines the log-likelihood of the training data and a spatial regularization term :

σs = arg min σ N i=1    d∈D log σ 2 i + x s d,i -xs i 2 σ 2 i    + α N i,j=1 δ i,j σ 2 i -σ 2 j 2 (7)
where x s d,i = 1 if pixel i of training data d has label s (x s d,i = 0 otherwise), δ i,j = 1 if pixels i and j are neighbors (δ ij = 0 otherwise), and α is a weighting parameter setting the degree of smoothing.

We combine energy functionals (1) and ( 6) by introducing a balancing parameter λ s :

E s total1 (x s ) = E s RW (x) + λ s E s model (x) (8)
It is possible to set a different value of λ s for each label s, as some muscles may require a stronger influence from the prior model than others. The solution which minimizes (8) verifies:

(L + λ s Λ s σ ) x = λ s Λ s σ xs (9)
As noted in [START_REF] Grady | Multilabel Random Walker Image Segmentation Using Prior Models[END_REF], when one adds such a prior term to the RW functional, it is no longer necessary to own pre-labeled nodes (seeds) in order to compute the segmentation. Indeed, the system of equations ( 9) is directly invertible, even when all entries of x are unknown. However, it is still possible -and useful -to use seeds to obtain more robust segmentations.

Confidence Map

As we saw previously , the functional E s model (x s ) penalizes the deviation of x s from the mean xs . Such prior is all the more useful as the local uncertainty of contour presence is large. One can impose such a condition by adjusting the influence of the model according to the strength of the contours in the test image: the stronger the contours, the least we should rely on the model. Assume we possess such a "confidence map" c, with values close to 0 on strong contours, and values close to 1 in homogeneous regions, we replace the term ( 6) by the following:

E s model (x s ) = (x s -xs ) T Λ c Λ s σ (x s -xs ) T ( 10
)
where Λ c is a diagonal matrix with c on the diagonal. The local confidence of the image can be easily determined using a decreasing function inversely proportional to the image variance (see figure 1):

c i = exp -k v σ 2 r (i) ( 11
)
where σ 2 r (i) is the variance at pixel i computed on a patch with radius r, and k v is a free parameter. The system to solve is now:

(L + λ s Λ c Λ s σ ) x s = λ s Λ c Λ s σ xs (12)

Experimental Validation

Our data set comprises 14 3D volumes of the right thigh of healthy subjects, covering a wide range of morphologies (7 females, 7 males, ages range: 14 to 60), acquired with a 3T Siemens scanner and using 3pt Dixon sequence (TR=10ms, TE1=2.75 ms TE2=3.95 ms TE3=5.15 ms, rf flip angle =3°) of resolution: 1mm×1mm×5mm. We manually segmented each volume in order to evaluate the quality of the segmentation results. We focused our evaluation on clinically relevant muscles of the thigh (13 muscles).

In order to compute the empirical mean xs and the empirical variance σ s 2 i , we non-rigidly register all the volumes and their segmentation map in the training set to the same target volume. The registration process is achieved using the method presented in [START_REF] Glocker | Dense image registration through MRFs and efficient linear programming[END_REF] and the related registration software (Drop © , www.mrf-registration.net). This way, we ensure that all the registered volumes belong to the same space. We adopted a leave-one-out cross-validation protocol: each test volume is used as the target volume for the registration of the 13 other volumes. Then we compute the estimates of xs and σ s 2 i on the 13 registered volumes, and perform the segmentation of the test volume. After the registration process, all volumes had the size: 191 × 178 × 63.

Due to the size of the matrices involved in the RW process, the system of equation cannot be solved directly. Instead, we used iterative algorithms, such as Bi-conjugate Gradient, to minimize (12). Computing the segmentation takes around 5 min on a 2.8 GHz Intel® processor with 4 GB of RAM. From a series of tests, we computed the best value for parameter λ s = 10 -3 , except for the Gracilis muscle which we had to constrain more: λ s = 10 -1 .

The quality of the segmentation is measured by computing Dice coefficients with the box-plot presentation 1 (See figure 2). The expression of the Dice coefficient is: D = 2 |T ∩ R| / (|T | + |R|), where T and R are the pixel sets for the same muscle in the computed segmentation and the ground truth segmentation respectively. We also compare the different methods with p-values obtained using the non-parametric statistical test Wilcoxon rank-sum (cf. scipy.stats for more information). In figure 2, for comparision with a simple segmentation by atlas registration method, we compute the Dice coefficients of the segmentation which we obtain when retaining the label of maximum probability of the mean probability: l i = arg max s xs i . This method yielded inferior results as compared to our method (without confidence map) with a p-value of 2 × 10 -10 . In figure 3, we show cross-sections of segmentation results obtained with the RW method using the prior model and the confidence map. Segmentation errors tend to affect primarily small muscles (e.g. Gracilis) and muscles located on the extreme upper part of the volumes (e.g Tensor Fasciae Latae) which reveals the limitations of the mean model. These errors are due to the large registration errors on the same muscles. This shows that our model is too constraining, as it does not allow the segmentation to deviate enough from the mean. In figure 2, we compared the segmentations obtained when adding the confidence map, which slightly improves the segmentation results (p-value: 6.6 × 10 -2 ), yielding an average Dice coefficient value of 0.86 ± 0.07. Due to the few number of training examples, we noted that the variance estimate had little influence on the results: replacing Λ s σ with the identity matrix gave us no significantly different results. This suggests that we should add more data to the training set in order to improve the statistical validity of our estimates.

In figure 2, we compare our results with a previous method of ours [9] which has been accepted for publication. This method consisted in automatically determining appropriate seed positions with respect to the different muscle classes. A number of unlabeled seeds were generated in the image domain in a sampling process, and the labeling of these seeds was achieved through a graph-based approach considering visual and geometric properties. The output of this optimization process was then fed to the standard RW algorithm (1). We obtained inferior results to the method presented here (average Dice: 0.80 ± 0.19; p-value when compared to confidence map method: 7.0 × 10 -4 ).

The inherent difficulties of segmenting the skeletal muscles in MR imagesnamely: partial contours, no discernible texture differences, large variation interindividuals and unremarkable shapes -render standard segmentation methods inoperative on this issue. One has to search for methods which perform robustly on inconsistent images cues (partial contours), and flexible models, allowing enough freedom to account for large inter-subject shape variations. We propose a prior model method, resting on the strengths of the Random Walks segmentation algorithm. Due to its robustness when faced with missing contours, the RW algorithm appears to be a good candidate for combination with a trained shape model. We believe to have achieved promising results which demonstrate the potential of our fully automatic approach.

Future work will consist in designing a model which allows more shape variability. We could obtain such model by building a low dimensional space through computing a PCA on the training base. Furthermore, the similarities between the RW algorithm and Markov Random Fields formulations let us envision applying the recent advances in MRF learning to the estimation of the Laplacian matrix.

Fig. 1 .

 1 Fig. 1. (left) Cross-section of an MR volume of the thigh. (center) Manual segmentation of the muscles. (right) Confidence map.

Fig. 2 .

 2 Fig. 2. Box-plot presentation of the Dice coefficients of our segmentation algorithm. (Right) Individual muscles; (Left) All muscles. Average dice values for: registration only: 0.81 ± 0.08; RW + shape prior: 0.84 ± 0.08; RW + shape prior + c. map: 0.86 ± 0.07; automatic seeds ([9]): 0.80 ± 0.19.

Fig. 3 .

 3 Fig. 3. Segmentation results obtained with the RW algorithm with shape prior and confidence map. Segmentation errors are shown in white.

Box-plot presentation: the boxes contain the middle 50% of the data and the median value, and the extremities of the lines indicate the min and max values, excluding the outliers (for more details, see the documentation of Matplotlib).


Anne-Laure Le Nadant
email: anne-laure.lenadant@unicaen.fr

Frédéric Perdreau
email: frederic.perdreau@univ-st-etienne.fr

Do Private Equity Firms Foster Innovation? Evidence from French LBOs

Keywords: Buyouts, Innovation, Private Equity Firms, Strategic Entrepreneurship

Agency theory has historically presented buyouts as a superior governance framework that generates economic efficiencies in the short term but these transactions might have a negative impact on long-term growth and innovation. In this study, we use a strategic entrepreneurship perspective to argue that private equity firm's extensive network and relationships, and expertise and competencies help managers to innovate. Using a propensity score methodology, we provide an empirical analysis of the innovative efforts of a sample of 89 French manufacturing firms that underwent a buyout between 2001 and 2005. The matching estimates (average treatment on the treated, ATT) of the effect of LBOs on firm level of innovation expenditures in 2006 show no significant differences between LBO targets and comparable companies that did not go through an LBO. In contrast, we find significant effects of LBOs on both service innovation and marketing innovations in design and packaging and product promotion. Results suggest that private equity firms provide marketing capabilities or encourage managers of LBO targets to build new innovation strategies.

Introduction

Innovation is an essential preoccupation for a firm as it affects its competitiveness. Agency theory has traditionally presented buyouts as a superior governance framework that generates economic efficiencies in the short term but LBO funds, driven by short-term profit motives, might sacrifice long-term growth and innovation to boost short-term performance.

LBOs involve investments in which investors and a management team pool their own money (together with debt finance) to buy shares in a target company from its current owners [START_REF] Meuleman | Agency, Strategic Entrepreneurship, and the Performance of Private Equity-Backed Buyouts[END_REF]. Although the change in governance resulting from LBOs is generally found to exert a positive impact on firm"s economic and financial performance [START_REF] Cumming | Private Equity, Leveraged Buyouts and Governance[END_REF], these transactions have mainly been associated with cost-cutting activities and shorttermism, to the detriment of innovation and Research and Development (R&D) investments. However, in a recent study, [START_REF] Boucly | Growth LBOs[END_REF] show that, instead of reinforcing credit constraints, as was the case in the 1980s transactions, today"s LBOs can alleviate them.

Whether LBOs have an impact on innovation is not clear, a priori. Both positive and negative effects are likely to occur. From the agency theory perspective, after an LBO, technological matters may be delayed or set aside because managers may be more oriented to day to day operations resulting from the transaction [START_REF] Hitt | The Market for Corporate Control and Firm Innovation[END_REF] or because private equity firms exert pressure on management to focus on investment opportunities that are less uncertain and more rewarding in the short term [START_REF] Ughetto | Assessing the Contribution to Innovation of Private Equity Investors: a Study on European Buyouts[END_REF].

Alternatively, from the strategic entrepreneurship perspective, LBOs may foster entrepreneurial initiatives, enabling managers to better and more thoroughly exploit firm resources for new innovation projects (Wright, Hoskisson and Busenitz, 2001). Innovation requires the entrepreneurial capabilities of opportunity recognition and opportunity exploitation [START_REF] Withers | Doing More with Less: The Disordinal Implications of Firm Age for Leveraging Capabilities for Innovation Activity[END_REF]. The private equity firm's expertise and competencies with regard to strategy, operational and financial management, human resources, marketing policy, and mergers and acquisitions help identify an opportunity for innovation and create value for the target firm [START_REF] Lee | Internal Capabilities, External Networks, and Performance: A Study on Technology-based Ventures[END_REF][START_REF] Wright | Finance and Management Buyouts: Agency versus Entrepreneurship Perspectives[END_REF]. Private equity-backed buyouts can also make use of the private equity firm's extensive network and relationships (customers, suppliers, other investors, access to more sophisticated resources in banking, and legal and other areas) to leverage their capabilities for innovation. In particular, private equity firms" networks may put them in a position to provide resources and capabilities the management of the buyout firm is currently missing [START_REF] Meuleman | Agency, Strategic Entrepreneurship, and the Performance of Private Equity-Backed Buyouts[END_REF]. Moreover, inside management does not always own the tacit knowledge and idiosyncratic skills required to seize new opportunities [START_REF] Hendry | The Principal's Other Problems: Honest Incompetence and the Specification of Objectives[END_REF]. If major innovation is required, it may be necessary to introduce outside managers who do own these skills (Wright, Hoskisson, and Busenitz, 2001). In this situation, the private equity firm plays an important role in assessing the skills of the incumbent managers and their potential replacements [START_REF] Meuleman | Agency, Strategic Entrepreneurship, and the Performance of Private Equity-Backed Buyouts[END_REF].

Evidence regarding the impact of LBOs on investment in innovation and R&D is so far limited and rather mixed. Some studies show a decline in R&D spending [START_REF] Long | LBOs, Debt and R&D Intensity[END_REF] whereas others find no decline [START_REF] Lichtenberg | The Effects of Leveraged Buyouts on Productivity and Related Aspects of Firm Behavior[END_REF] or stability [START_REF] Smith | Corporate Ownership Structure and Performance: the Case of Management Buyouts[END_REF] of research spending after the LBO. However, as most firms involved in LBOs do not belong to technology-intensive industries, the impact of LBOs on cumulative innovation is likely slight [START_REF] Hall | The Impact of Corporate Restructuring on Industrial Research and Development[END_REF].

In industries where R&D requirements are more crucial, these expenditures are used more effectively. [START_REF] Zahra | Corporate Entrepreneurship and Financial Performance: The Case of Management Leveraged Buy-Outs[END_REF] finds, on a sample of 47 LBOs, that firms involved develop more new products and intensify their efforts in terms of innovation and productivity (even if at the same time the level of R&D expenditure doesn"t change). Wright et al. (2001) provide several examples of buyouts in technology-based industries followed by significant increases in product and technology development, R&D and patenting. [START_REF] Malone | Characteristics of Smaller Company Leveraged Buyouts[END_REF] and [START_REF] Wright | Venture Capital and Management-led, Leveraged Buy-outs: A European perspective[END_REF] also cite evidence of new product innovation following buyouts.

More recently, [START_REF] Lerner | Private Equity and Long-Run Investment: The Case of Innovation[END_REF] investigate 472 LBOs with a focus on investments in innovation as measured by patenting activity. They find no evidence that LBOs are associated with a decrease in these activities. Contrary to the frequent argument that private equity firms have short-term horizon and little incentive to favor long-term investment opportunities of target companies, this study shows that LBOs lead in fact to significant increases in long-term innovation. The authors find that patents granted to firms involved in LBOs are more cited (a proxy for economic importance) and show no significant shifts in the fundamental nature of the research. Hence, if some US and UK based studies show a decline in investment expenditure after LBOs, critical investments in R&D seem to be maintained. [START_REF] Ughetto | Assessing the Contribution to Innovation of Private Equity Investors: a Study on European Buyouts[END_REF] has focused on innovation of Western European manufacturing firms undergoing an LBO. She finds that innovation activity of portfolio firms (measured by the number of patents granted) is affected by different types of investors, pursuing different objectives.

Through this study, we seek to contribute to the entrepreneurship and strategic management literatures on innovation and networks in several ways. First, we revisit the questions in the previous studies because the private equity industry is more substantial today than it was in the 1980s. Changes in the industry-such as the increased competition between and greater operational orientation of private equity firms-suggest that the earlier relationships may no longer hold. In addition, transactions involving technology-intensive industries have become more common recently and it is also desirable to look beyond public-to-private transactions, since these transactions represent a fairly small fraction of the private equity universe. Second, we contribute by using an innovation survey that has never been used to analyze buyouts. This survey substantially enhances our ability to measure and study the impact on innovation. Third, nearly all studies on LBOs and innovation have concentrated on the US and the UK (the only exception we are aware of is the study of Western Europe by [START_REF] Ughetto | Assessing the Contribution to Innovation of Private Equity Investors: a Study on European Buyouts[END_REF]. By shifting the focus to France and following [START_REF] Boucly | Growth LBOs[END_REF], this study investigates the possibility that some LBOs aim to seize innovation opportunities and expand the scale and scope of the target"s activities. France is an interesting context to study LBOs and innovation because it is a country with many family-managed firms that tend to be, on average, smaller than non family firms and for which access to external finance may be more difficult than in the US or the UK.

The remainder of the paper is as follows. In Section 2, we describe the construction of the sample. Section 3 reviews the methodology employed in the study. We present the empirical analyses in Section 4. The final section concludes the paper and discusses future work.

Dataset

To analyze the impact of LBOs on innovation at the company level, we use a new database built from three different databases: Capital IQ (to isolate transactions), CIS 2006 (for innovation data) and DIANE (for financial statements). To our knowledge, this is the first study to use a CIS survey (Community Innovation Survey) in relation with LBOs. Community Innovation Surveys are conducted at regular intervals in Europe. Questions are based on the Oslo Manual guidelines, which distinguishes four types of innovations: product innovations, process innovations, organizational innovations and marketing innovations. The Oslo Manual opted for collecting data at the firm level, including all its innovation outputs and activities, which is also the level of available accounting and financial data that can be merged with the innovation data for richer analyses. CIS 2006 was launched in 2007, based on the reference period 2006, with the observation period 2004 to 2006. The population of the CIS is determined by the size of the enterprise and its principal activity. All enterprises with 20 or more employees in any of the specified sectors were included in the statistical population. The following industries were included in the population of the CIS 2006: mining and quarrying (NACE 10-14), manufacturing (NACE 15-37), electricity, gas and water supply . Three reference periods were used in the questionnaire:

 The first relates to a set of questions for the whole of the period 2004-2006, for example whether the enterprise introduced an innovation at any time during this three-year period.

 The second set of questions refers uniquely to the reference year 2006, for example, indicators such as innovation expenditure.

 Finally, a limited number of basic economic indicators were requested for both 2004 and 2006, for example the turnover and employment figures.

We first identify 944 French deals over 2001-2005 reported as being "LBOs" from Capital IQ. More precisely, we retrieve all the deals from Capital IQ with the following characteristics: (i) they are announced between 2001 and 2005 (ii) either "closed" or "effective" (iii) reported by Capital IQ as being "LBOs". Most of the targets are medium sized, privately held firms. We then obtain innovation data from CIS 2006. Our transaction and innovation data do not have the same identifier so we match them by company name. Names are not always identical in both databases, so in case of ambiguity we resort to company websites and annual reports. The matching process reduces sample size to 109 transactions, of which 89 have available financial statements in Diane (Bureau Van Dijk) for the year preceding the transaction.

Table 1 summarizes the characteristics of the final sample. 47% of deals take place since 2004. Slightly less than 40% of the sample is composed of companies in intermediate goods (52% in terms of value of the deals). In terms of size, the sample is mostly constituted of relatively small companies: 34% of targets have less than 20 million (M) Euros in sales at the time of the deal, and 69% have less than 75M. Companies with sales above 75M constitute 31% of number but 74% of the value of deals.

Methodology

Gauging effects of LBOs on innovation is not trivial because LBOs do not occur randomly across the population of firms. LBO targets are selected by investors presumably because of their value creation potential. If an outside observer concludes that the average level of innovation expenditure of firms targeted by an LBO is higher than in other firms, one cannot rule out the possibility that this finding is due to the fact that LBO investors tend to select better firms on average relative to the population.

We address the problem of sample selection bias using a propensity score methodology (PSM) [START_REF] Rubin | Estimating Causal Effects of Treatments in Randomized and Non Randomized Studies[END_REF][START_REF] Heckman | The Economics and Econometrics of Active Labour Market Programs[END_REF]. We benchmark the level of innovation of LBO firms by selecting appropriate matching control firms to each LBO. The set of matching control firms is composed of firms that share the same financial characteristics as the LBO firm prior to the transaction. To select matching firms that have ex-ante the same probability of being selected by LBO investors, we implement a probit model to estimate the likelihood of being an LBO target in a given year and we use the use the probability estimate from that model to find a matching control for a firm that indeed was the target of an LBO deal1 .

The main steps of the PSM procedure are as follows. First, we introduce filters to obtain a dataset composed of about 1,200 companies. We need to do this because fitting a discrete choice regression model where the number of "zeros" (that is, observations where the firm is not an LBO target in a given year) is very high relative to the number of "ones" (that is, observations where the firm is an LBO target in a given year) results in poor estimates. This is the case since CIS 2006 contains data for about 5,200 companies, and the sample contains 89 LBOs (2 % of the dataset). We therefore introduce filters to obtain a manageable number of non-LBO observations. A matching company belongs to the same 4-digit sector as the target. If there are more than ten twins, we just keep the ten nearest neighbors to the target with the nearest turnover the year preceding the buyout. The matching methodology allows us to retain 1,144 "twin" companies to the sample, i.e. 12.85 twins by target. We choose 1,200 as a number that seems reasonable because it means that LBOs constitute about 8% of the regression sample.

Second, we run a probit regression that models the likelihood of a firm being the target of an LBO in a particular year. Denote by h k * the latent unobservable variable that represents the net present value of the acquisition of firm k by a bidder and h k,t a dummy that takes the value of 1 if an LBO bid is made in year t: h k,t = 1 if h k,t * >0 or h k,t = 0 if h k,t * < 0. The probit regression to be estimated for the probability of Pr(h k,t = 1) is:

h k,t * = α +W k,t δ + υ k,t (1)
The matrix W k,t contains firm-specific variables that the literature has identified as determinants of the likelihood that a firm is an LBO target, namely firm size (measured by turnover), the debtequity ratio, the level of income taxes, the firm"s profitability (measured by ROIC), liquidity (proxied by cash divided by assets) and level of working capital (Le [START_REF] Nadant | Financial Profile of Leveraged Buyout Targets: Some French Evidence[END_REF].

The predicted value from regression model ( 1) is called the propensity score. Its interpretation is that it measures the probability, as predicted by the model, that a firm becomes an LBO target in a given year. In other words, firms with similar propensity scores share similar characteristics that lead to being an LBO target. They constitute therefore adequate benchmarks for LBO innovation capacity.

Third, we use propensity score to match comparison units with treated units. Smith and Todd (2005) note that measuring the proximity of cases as the absolute difference in the propensity score is not an approach that is robust to "choice-based sampling," where the treated are oversampled relative to their frequency in the population of eligible individuals [START_REF] Caliendo | Some Practical Guidance For The Implementation Of Propensity Score Matching[END_REF]. As a consequence, we match on the log odds of the propensity score, defined as p/(1-p), to assure that results are invariant to choice-based sampling.

Different matching algorithms can be used: kernel matching, nearest-neighbor or radius matching. As we have many firms not involved in LBOs in our dataset, the radius matching algorithm (with a caliper of 0.06) is more appropriate: it enables us to compare firms with very close predictions of probit models. To avoid the risk of poor matches, radius matching specifies a "caliper" or maximum propensity score distance by which a match can be made. The basic idea of radius matching is that it uses not only the nearest neighbor within each caliper, but all of the comparison group members within the caliper. In other words, it uses as many comparison cases as are available within the caliper, but not those that are poor matches (based on the specified distance). The robustness of our results is tested using other matching methods. We then measure, for each LBO, its level of innovation relative to the level of innovation of its matching control pair.

Post LBO innovation: evidence and robustness a) Main results

Table 2 presents summary statistics for accounting measures of LBOs and the non-LBOs firms. Relative to their potential controls, LBO firms are larger (65 M in average turnover vs. 40M) and slightly more profitable (23% vs. 20% in terms of average ROIC), and they have higher income taxes expenses the year before the deal (2.39% of sales for LBO vs. 1.63% for non-LBO). Only differences in size and income taxes expenses are significant (see Appendix 2).

Table 2 -Summary statistics

This table shows the summary statistics for sample deals for the year before the deal. «LBO companies» refers to statistics of the sample of LBO firms. «Non-LBO companies» refers to statistics of the sample of all non-LBO companies from which matched controls are chosen using a propensity score model. All these accounting variables are obtained from DIANE. Turnover is in thousands of Euros. Income taxes expense and working capital are divided by turnover. Working Capital is divided by net fixed assets. Debt-to-equity is measured by financial debt divided by shareholders" equity (in %).

Variable

Obs The results of the probit model show that the largest firms and those that have the highest level of income taxes have a higher probability of being an LBO target (Table 3). In contrast, firms" financial structure, profitability, liquidity and level of working capital do not seem to explain LBO likelihood. The matching estimates (average treatment on the treated, ATT) of the effect of LBOs on firm level of innovation in 2006 show no significant differences between LBO targets and comparable companies that did not go through an LBO. Moreover, we find no significant effect of LBOs on innovation expenditure at different periods of time (one-year after transaction for LBOs in 2005, two-year after transaction for LBOs in 2004, etc.). Evidence thus suggests that private equity intervention is not detrimental to long-term investments in innovation.

Table 4 -Effect of LBOs on the level of innovation expenditure in 2006

Innovation expenditure is measured by all innovation expenditure in 2006 declared by firms as a percentage of turnover the same year.

Variable

Number of observations 89 1144

Notes: Significance levels: ***(1%), **(5%) and *(10%).

Even if LBO targets have the same level of innovation expenditure as their matched counterparts, LBOs might have an effect on other innovation outcomes. Indeed, LBOs might foster entrepreneurial initiatives and innovation projects that are not costly and do not require important expenditure. Table 5 presents PSM estimations on innovation outcomes for LBOs that occurred from 2001 to 2004 (69 LBOs) whereas Appendix 1 presents the innovation outcomes and innovation variables definitions. As questions in CIS 2006 relate to the 2004-2006 period, we exclude 20 LBOs that occurred in 2005 from the sample to run these estimates. The comparison of unmatched and ATT estimates (not reported here) shows the efficiency of the matching method, which significantly reduces the selection bias. Indeed, differences between LBOs and non-LBOs on several outcome variables appear to be significant when they are estimated with unmatched units. For instance, for product innovation (inpdgd) and innovation in external relations (oorgexr), t-statistic values are respectively 1.68 and 1.78 (significant at 10% level) but, after matching, the differences are no longer statistically significant, suggesting that matching helps reduce the bias associated with observable characteristics.

Table in Appendix 2 shows the results of the balancing test for the radius model. It confirms the validity of the matching method, which significantly reduces the bias for all covariates. The balancing test measures the "similarity" of observations remaining after the matching process. Regarding product and process innovations, we find that LBO targets are more likely to implement service innovations. Hence, results suggest that LBOs lead to significant improvements in how services are provided (for example, in terms of their efficiency or speed), the addition of new functions or characteristics to existing services, or the introduction of entirely new services. However, as we also find an effect on innovations that are new to the firm but no effect on innovations that are new to the market, we can conclude that LBOs lead to minor product innovations that have already been implemented by other firms. Hence, it is likely that the nature of the innovation in LBOs is based upon incrementally improving current products for existing markets rather than developing new products and processes [START_REF] Zahra | Will Leveraged Buyouts kill U.S. Corporate Research and Development?[END_REF].

Regarding organisational and marketing innovations, we find significant effects for two variables of marketing innovation only: innovations in design and packaging and in products promotion. Marketing innovations are aimed at better addressing customer needs, opening up new markets, or newly positioning a firm"s product on the market, with the objective of increasing the firm"s turnover. LBOs have a significant effect on changes in product design and packaging that are intended to change or enhance the product"s appeal or to target a new market or market segment. They also have a positive impact on promotional efforts made by firms to improve their products" image or to increase awareness of their products. This evidence confirms that private equity firms provide superior managerial and technical expertise that enables target firms to seize innovative opportunities.

In contrast, we find no effect of LBOs on the methods of protecting innovations. In particular, we show that LBOs have no impact on patenting activity. This result is in line with [START_REF] Lerner | Private Equity and Long-Run Investment: The Case of Innovation[END_REF].

Last, innovation activity may be hampered by a number of factors. There may be reasons for not starting innovation activities at all, or factors that slow innovation activity or have a negative effect on expected results. These include economic factors, such as high costs or lack of demand, and enterprise factors, such as a lack of skilled personnel or knowledge. Regarding obstacles to innovation, results show an effect of LBOs on two factors hampering innovation activities: lack of skilled personnel and lack of external financial resource. These results suggest that LBO targets do not have the skilled personnel needed to engage in innovation activities, or their innovation activities may be slowed because they are unable to find the necessary personnel on the labour market. Moreover, contrary to [START_REF] Boucly | Growth LBOs[END_REF] who show that LBOs can alleviate financial constraints and foster growth, our results suggest that LBO targets lack external financial resource for their innovation activities.

b) Robustness check

We evaluate the robustness of the estimations by changing the matching algorithms. The matching algorithms used are nearest neighbor (with one and three neighbors), radius (with a caliper of 0.06) and kernel. The impact on innovation variables does not appear to depend critically on the algorithm used, since both the value of the coefficients and its significance are very similar using different alternatives. In particular, whatever the algorithm used, we find no impact of LBOs on innovation expenditure and a significant impact on marketing innovation and product innovation that is new only to the firm.

Conclusion

Hence, although most commentators in the public debate and many financial economists consider LBOs as a way to implement drastic, "cost cutting" measures, this paper provides evidence that ex-post innovation expenditure is similar in LBO targets and comparable firms in France. In addition, we find no evidence that, as for contribution to innovation, LBOs alleviate financial constraints or facilitate business relations.

In contrast, we find an effect of LBOs on service innovation but this effect is not related to radical or disruptive innovation as LBOs have an impact on product innovation that is new only to the firm. Moreover, results show that LBO targets are more likely to implement marketing innovations (design and packaging and product promotion) in order to increase turnover and market share. This result is in line with the idea that, following an LBO, management is not only focused on "cost cutting" activities that aim to increase value creation. Private equity firms also encourage managers of LBO targets to build new strategies to find and exploit value creation potentials. Further, these results emphasize the resources and capabilities that buyout specialists bring in terms of contribution to innovation to their portfolio companies as they suggest that LBOs do not lead to the acquisition of skills or resources that enable the introduction of disruptive innovation. Adversely, results show that post-LBO innovations, which are mainly based upon marketing and service innovations, rely on existing resources with only the addition of a marketing competency. They suggest that private equity firms provide marketing capabilities or encourage managers of LBO targets to build new innovation strategies based on these capabilities.

Contrary to [START_REF] Boucly | Growth LBOs[END_REF], results also suggest that LBOs do not alleviate financial constraints as we find a positive effect of LBOs on the lack of financial resources as a factor hampering innovation. Moreover, as we find no effect of LBOs on radical innovations or new products or services, it seems that LBOs do not help to achieve such radical innovation. This result could be a clue that LBOs lead to the strengthening of financial and human resources constraints that hamper radical innovation. But another interpretation is also possible. It could be that, as better innovators, LBO targets feel more acutely the financial and human resources gap that has to be filled to achieve radical innovations. Further research is needed to disentangle these possible explanations.

Table 1 -Descriptive statistics of final sample

 1 This table shows the number and value of deals in the sample. Value is measured using the sum of sales revenue of companies in each category, in thousands of Euros, for the year prior to the deal. Breakdown by sector follows the French classification named NES16 (Nomenclature Economique de Synthèse).

	Panel A : Breakdown by year
		Number	%	Value	%
	2001	18	20.22%	1 099 628	19.07%
	2002	12	13.48%	1 000 782	17.35%
	2003	17	19.10%	1 139 305	19.76%
	2004	22	24.72%	1 162 113	20.15%
	2005	20	22.47%	1 364 967	23.67%
	Total	89		5 766 795
	Panel B : Breakdown by Sector
		Number	%	Value	%
	Agricultural and food industries	7	7.87%	511 133	8.86%
	Consumer Goods Industry	13	14.61%	596 591	10.35%
	Automotive Industry	6	6.74%	241 696	4.19%
	Capital goods industries	24	26.97%	1 232 990	21.38%
	Intermediate goods	35	39.33%	3 019 754	52.36%
	Energy	4	4.49%	164 630	2.85%
	Total	89		5 766 795
	Panel C : Sample Breakdown by Sales revenues
	(Sales in Thousand of Euros)	Number	%	Value	%
	(0;20000]	31	34.83%	303046	5.26%
	(20000;75000]	30	33.71%	1201816	20.84%
	(75000;150000]	18	20.22%	1762976	30.57%
	(150000; max]	10	11.24%	2498958	43.33%
	Total	89		5 766 795

Table 3 -Estimations of the Probit model

 3 Turnover is in thousands of Euros. Income taxes and working capital variables are divided by turnover. Liquidity variable is measured by cash and cash equivalents divided by total assets. Debt-to-equity is financial debt divided by shareholders" equity.

		Probit regression on 2001-2004	Probit regression on 2001-2005
	Indep.variables	Coef	Std. Error	Coef	Std. Error
	Turnover	0.1278***	0.0427	0.1521***	0.0389
	ROIC	0.0629	0.1063	0.0234	0.1040
	Debt-to-equity	-0.0000	0.0001	-0.0000	0.0001
	Income taxes	6.2601***	2.3278	4.3494**	1.9705
	Liquidity	-0.0103	0.0296	-0.0267	0.0554
	Working capital	0.0698	0.1245	0.0744	0.1171
	Constant	-3.7741***	0.7318	-4.1082***	0.6705
		Number of Obs. = 1004 (69 LBOs)	Number of Obs. = 1233 (89 LBOs)
		LR Chi2(6) = 18.31		LR Chi2(6) = 23.01
		Prob > Chi2 = 0.005		Prob > Chi2 = 0.001
		Pseudo R2 = 0.036		Pseudo R2 = 0.036
	Notes: Significance levels: ***(1%) and **(5%).

Table 4

 4 reports both unmatched and matching estimates (average treatment on the treated, ATT) of the effect of LBOs on firm level of innovation, measured as innovation expenditure in 2006 divided by turnover in 2006.

Table 5 -ATT estimates of innovation outcomes during 2004-2006

 5

	Variable type	Variable code	Treated	Controls	Difference	S.E.
		inpdgd	.5147	.4385	.0761	.0632
		inpdsv	.2500	.1594	.0905*	.0542
		newmkt	.4264	.3249	.1014	.0623
	Product innovations	newfrm	.4411	.3237	.1174*	.0626
		turnmar	.0613	.0645	-.0031	.0197
		turnin	.0544	.0483	.0060	.0172
		turnung	.8842	.8871	-.0028	.0277
		inpspd	.3676	.3704	-.0027	.0610
	Process innovations	inpslg	.1911	.1619	.02918	.0495
		inpssu	.1764	.1718	.0045	.0482
		oorgbup	.3823	.3905	-.0081	.0615
	Organisational	oorgkms	.2205	.2517	-.0312	.0527
	innovations	oorgwkp	.3088	.3183	-.0095	.0585
		oorgexr	.2500	.1762	.0737	.0544
		mktdgp	.3235	.1869	.1365**	.0585
	Marketing	mktpdp	.2647	.1686	.0960*	.0553
	innovations	mktpdl	.1029	.0988	.0040	.0383
		mktpri	.1617	.1044	.0573	.0461
		ProPat	.2941	.3245	-.0304	.0577
		ProDsg	.2352	.1954	.0398	.0534
		ProTm	.3823	.2847	.0976	.0611
	Patents and other	ProCp	.1029	.0676	.0353	.0380
	protection
	methods	prosol	.1029	.0666	.0363	.0380
		ProSct	.1764	.1479	.0285	.0479
		ProCon	.2647	.2407	.0239	.0557
		ProTech	.2352	.1776	.0576	.0533

For a binary treatment variable, there is no strong advantage to use the logit vs. probit model. We also ran a logit model and found similar results.