Toby Collins

Pablo Mesejo

Adrien Bartoli

An Analysis of Errors in Graph-based Keypoint Matching and Proposed Solutions

An error occurs in graph-based keypoint matching when keypoints in two different images are matched by an algorithm but do not correspond to the same physical point. Most previous methods acquire keypoints in a black-box manner, and focus on developing better algorithms to match the provided points. However to study the complete performance of a matching system one has to study errors through the whole matching pipeline, from keypoint detection, candidate selection to graph optimisation. We show that in the full pipeline there are six different types of errors that cause mismatches. We then present a matching framework designed to reduce these errors. We achieve this by adapting keypoint detectors to better suit the needs of graph-based matching, and achieve better graph constraints by exploiting more information from their keypoints. Our framework is applicable in general images and can handle clutter and motion discontinuities. We also propose a method to identify many mismatches a posteriori based on Left-Right Consistency inspired by stereo matching due to the asymmetric way we detect keypoints and define the graph.

Introduction

Nonrigid keypoint-based image matching is the task of finding correspondences between keypoints detected in two images that are related by an unknown nonrigid transformation. This lies at the heart of several important computer vision problems and applications including nonrigid registration, object recognition and nonrigid 3D reconstruction. Graph-based methods solve this with a discrete optimisation on graphs whose edges encode geometric constraints between keypoints. This is NP hard in general and current research involves finding good approximate solutions [START_REF] Conte | Thirty years of Graph Matching in Pattern Recognition[END_REF][START_REF] Leordeanu | An integer projected fixed point method for graph matching and MAP inference[END_REF][START_REF] Cho | Reweighted Random Walks for Graph Matching[END_REF] or better ways to learn the graph's parameters [START_REF] Caetano | Learning graph matching[END_REF][START_REF] Leordeanu | Unsupervised learning for graph matching[END_REF][START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF]. Most methods tend to treat the underlying keypoint detector as a black box, however to improve the overall performance of a graph matching system one should design or adapt the keypoint detector to also reduce errors and ease the matching problem. We show that there are six types of errors that cause mismatches which occur throughout the whole matching pipeline from keypoint detection to graph optimisation. We then give a general framework for reducing these errors and show this greatly improves the end performance. This includes a method for detecting mismatches a posteriori that is based on the Left-Right Consistency (LRC) test [START_REF] Faugeras | Real time correlation-based stereo: Algorithm, implementations and applications[END_REF][START_REF] Kowdle | Combining monocular geometric cues with traditional stereo cues for consumer camera stereo[END_REF] originating in stereo matching. Although simple, this has not been used in graph-matching before because many prior formulations are symmetric (i.e. if the roles of the two images are reversed the solution remains the same). We argue that, to reduce errors, an asymmetric approach (where keypoints sets in either image are significantly different) has many advantages, and it also permits using the LRC 1 .

Previous work. Graph-based keypoint matching is typically formulated as a quadratic binary assignment problem [START_REF] Leordeanu | An integer projected fixed point method for graph matching and MAP inference[END_REF][START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF][START_REF] Umeyama | An eigendecomposition approach to weighted graph matching problems[END_REF][START_REF] Cho | Learning graphs to match[END_REF][START_REF] Zhou | Deformable graph matching[END_REF][START_REF] Zaslavskiy | Path following algorithm for the graph matching problem[END_REF][START_REF] Chertok | Efficient high order matching[END_REF][START_REF] Duchenne | A tensor-based algorithm for high-order graph matching[END_REF][START_REF] Leordeanu | A spectral technique for correspondence problems using pairwise constraints[END_REF][START_REF] Gold | A graduated assignment algorithm for graph matching[END_REF]. This can represent constraints on individual matches (i.e. unary terms) and pairs of matches (i.e. pairwise terms). Some works have incorporated higher-order constraints [START_REF] Chertok | Efficient high order matching[END_REF][START_REF] Duchenne | A tensor-based algorithm for high-order graph matching[END_REF] to handle the fact that pairwise constraints, such as preserving 2D Euclidean distances between points [START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF][START_REF] Leordeanu | A spectral technique for correspondence problems using pairwise constraints[END_REF] are not invariant to deformation and viewpoint change. However increasing the order to third and beyond increases exponentially the cost to compute and store the graph's terms. Thus second-order graphs remain the dominant approach. A limitation of many previous works is that they enforce all nodes in the graph to have a match. In some works this is done explicitly by using a permutation assignment matrix [START_REF] Umeyama | An eigendecomposition approach to weighted graph matching problems[END_REF][START_REF] Scott | An Algorithm for Associating the Features of Two Images[END_REF]. In the others this is done implicitly because their cost function is always reduced by making a match [10-14, 16, 4]. Works that deal with unmatchable keypoints include [START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF][START_REF] Hamid | Dense non-rigid point-matching using random projections[END_REF][START_REF] Albarelli | Imposing semi-local geometric constraints for accurate correspondences selection in structure from motion: A game-theoretic perspective[END_REF][START_REF] Leordeanu | A spectral technique for correspondence problems using pairwise constraints[END_REF]. [START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF] allows keypoints to be assigned to an unmatchable state in a Markov Random Field (MRF)-based energy function. However reported results required ground-truth training images of the scene to learn the right weighting of the energy terms, which limits applicability. Unmatchable keypoints are found in [START_REF] Hamid | Dense non-rigid point-matching using random projections[END_REF][START_REF] Leordeanu | A spectral technique for correspondence problems using pairwise constraints[END_REF] by detecting those which have poor geometric compatibility with high-confidence matches. [START_REF] Leordeanu | A spectral technique for correspondence problems using pairwise constraints[END_REF] requires a fully-connected graph and is very computationally demanding for large keypointsets. [START_REF] Hamid | Dense non-rigid point-matching using random projections[END_REF] iteratively computes correspondence subspace from highconfident matches and uses this as a spatial prior for other points. This was shown to work well for smooth deformable surfaces but may have difficulty with motion discontinuous or cluttered scenes. Other approaches are the Hard Matching with Outlier Detection (HMOD) methods [START_REF] Pilet | Fast non-rigid surface detection, registration and realistic augmentation[END_REF][START_REF] Pizarro | Feature-Based Deformable Surface Detection with Self-Occlusion Reasoning[END_REF][START_REF] Tran | In defence of RANSAC for outlier rejection in deformable registration[END_REF][START_REF] Chui | A new point matching algorithm for non-rigid registration[END_REF]. These match keypoints using only their texture descriptors, and because many of these matches will be false, use a secondary stage of detecting mismatches by computing their agreement with a deformation model fitted from the matches that are considered correct.

Types of Errors in Graph-based Keypoint Matching

We define R = {r 1 , ..., r m } and T = {t 1 , ..., t n } to be the keypoint-sets for the two images, which we refer to as the reference and target images respectively. Without loss of generality let m ≥ n. The goal of keypoint matching is to find, if it exists, a correct correspondence for each member of T in R (Fig. 1). Because keypoint detectors localise keypoints with some margin of error, a soft criteria is required to distinguish correct from incorrect correspondences.This means that for any member of T there may exist multiple correct matches in R. We define S i ⊂ R to be all correct matches for a keypoint t i . We define t i to be a matchable keypoint if S i is non-empty. We define t i to be an unmatchable keypoint if S i is empty. We define the visible keypoint-set V ⊆ T to be all keypoints in T that are visible in the reference image (but are not necessarily in R). In this paper we tackle problems where T and R are large unfiltered keypoint-sets generated by a keypoint detector such as SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] or SURF [START_REF] Bay | Speeded-up robust features (SURF)[END_REF]. For typical images m and n will be O(10 2 → 10 5 ). To keep the costs of graph-based matching manageable in terms of (i) storage, (ii) time to compute the graph's constraints and (iii) time to find the solution, the possible members of R that t i can be matched to should be pruned significantly. This has been done previously by taking the p members in R that have the most similar descriptors, with p being small (e.g. O(10 1 )). We define a keypoint's candidate match set C(t i ) ⊂ R to be the pruned set for t i . We define the output of a matching system by the assignment vector s ∈ {0, 1, ...m} n , where s(i) = 0 means that t i is matched to r s(i) ∈ C(t i ), and s(i) = 0 means that t i is unmatched and in the unmatchable state. In all prior works in graph-based keypoint matching performance is evaluated by measuring the number of correct matches made between T and R. This does not tell us the whole story for the complete performance of the system, nor does it provide a breakdown of where the error occurred. For a visible keypoint t i ∈ V a matching system will fail to establish a correct correspondence in the reference image according to four types of errors (Fig. 1). These are as follows:

Introduction: types of error

• Detection error : E d (t i ∈ V) def = (S i = ∅). t i is
visible in the reference image but there was no correct correspondence detected in the reference image.

• Candidate pruning error :

E cp (t i ∈ V) def = (S i = ∅, C i ∩ S i = ∅).
There is no detection error but the keypoint's candidates do not contain a correct correspondence.

• Candidate selection error :

E cs (t i ∈ V) def = (S i = ∅, C i ∩ S i = ∅, s i = 0, s i / ∈ S i ).
There is neither a detection nor candidate pruning error, but s i selects a wrong correspondence in the candidate-set.

• Unmatched error : E u (t i ∈ V) def = (S i = ∅, C i ∩ S i = ∅, s i = 0
). There is neither a detection nor candidate pruning error, but the keypoint is unmatched.

We can aggregate these errors into what we call a visible-keypoint match error :

E v (t i ∈ V) = (E d (t i ) ∨ E cp (t i ) ∨ E cs (t i ) ∨ E u (t i )
). A matching system can make two other types of errors which involve matching keypoints that have no match in R. The first is an occluded match error : E oc (t i / ∈ V) = (s i = 0). The second is what we call a visible-but-unmatchable error :

E vu (t i ∈ V) = (E d ∨ E cp ), s i = 0.
We can also combine E oc and E vu into what we call an unmatchable-keypoint error : E u (t i ) = E oc (t i ) ∨ E vu (t i ). Developing a good graph-based matching system is very challenging because it must simultaneously reduce unmatchable keypoint errors yet try to match as many visible keypoints as possible. For visible keypoint matching errors, we will show that errors in detection and candidate pruning tend to have a compounding effect: not only does a visible keypoint become unmatchable, but also geometric constraints are lost between it and its neighbours in the graph. When this occurs too frequently the graph is weakened, which can then lead to more candidate selection errors.

How might one reduce visible-keypoint match errors? Current research in keypoint detection involves reducing the so-called repeatability error [START_REF] Mikolajczyk | Scale & affine invariant interest point detectors[END_REF]. A repeatability error occurs when a keypoint detector fails to find the same two keypoints in both images. Thus perfect repeatability implies V = V , but this is different from perfect detection error (which is what we want), which implies V ⊆ V . To improve matching performance, we should design or adapt keypoint detectors to reduce detection and candidate pruning errors. The challenge here is how to incorporate this into graph matching efficiently. For example, one could reduce detection errors by having keypoints positioned densely in scale-space in the reference image. This requires an expensive dense computation of keypoint descriptors, and heavy candidate pruning. Reducing candidate selection errors has been the main objective in prior graph matching papers, and this has mainly been used as the evaluation criteria. This ignores the other types of errors we have listed. We now propose the first matching framework designed to reduce all types of errors. We call this framework Graph-based Affine Invariant Matching (GAIM), and its main contributions are: 1. To reduce detection and candidate pruning errors by constructing R using a redundant set keypoints. We do this so that even if there is large deformation between the two images, there is a high likelihood of a correct correspondence in R with a similar descriptor ( §3.1). 2. To reduce candidate selection errors by constructing second-order graphconstraints that are fast to evaluate and deformation invariant ( §3.3). To achieve this we show that one can obtain automatically an estimate of the local affine transform for each candidate match from our solution to point 1. These can then be used to enforce piecewise smooth matches. We show that this reduces assignment errors compared with the most common secondorder constraint, which is to preserve Euclidean distance [START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF][START_REF] Leordeanu | An integer projected fixed point method for graph matching and MAP inference[END_REF]. 3. To handle unmatchable keypoints with a robust graph cost function, and show that many can be effectively detected a posteriori using an LRC procedure adapted to graph matching ( §3.4).

3 Proposed Framework: GAIM

Reference Keypoint-set Construction to Reduce Detection and Candidate Pruning Errors

We use the fact that keypoint detectors trigger at different image positions in scale-space depending to a large part on deformation and viewpoint change. Thus, rather than attempting to make a keypoint's descriptor affine-invariant (e.g. [START_REF] Mikolajczyk | Scale & affine invariant interest point detectors[END_REF]) we construct R by simulating many deformations of the reference image, and for each simulation we harvest keypoints and add them into R.

Although the deformation space is enormous, most deformations can be wellapproximated locally by an affine transformation. Thus we only need to simulate global affine deformations of the reference image (Fig. 2). We do not simulate all of its six DoFs because keypoint methods such as SIFT are reasonably invariant to 2D translation, isotropic scale and rotation. Thus we simulate deformations by varying the remaining two DoFs (shear and aspect ratio). A similar strategy was proposed in [START_REF] Morel | ASIFT: A New Framework for Fully Affine Invariant Image Comparison[END_REF] to make SIFT affine invariant. However, our goal is different because we want to have asymmetric keypoints (where all keypoints in V have a correspondence in R, but not necessarily all keypoints in R have a correspondence in V). We also want to obtain an estimate of the affine transform between t i and its candidate matches, as we will include these in the graph's cost. In [START_REF] Morel | ASIFT: A New Framework for Fully Affine Invariant Image Comparison[END_REF] the affine transforms were not inferred. We uniformly quantise shear in the range [-1.0 : 1.0] and anisotopic scale in the range [-0.25 : 4.0]. We use x intervals that we experimentally vary from 3 to 8. We denote the simulated image set by I s , s ∈ [1...S], and its simulated affine transform by A s . We represent each keypoint r k ∈ R by r k = (y k , q k , d k , σ k , θ k , p k ). We use y k to denote the index of the simulated reference image from which r k was harvested. We use q k ∈ R 2 to denote the 2D position of the keypoint in I y k and d k ∈ R d denotes its descriptor (using SIFT we have d = 128). We use p k ∈ R 2 to denote its 2D position in the reference image. This is given by p k = w(A -1 σ k , q k ) where w(A, •) : R 2 → R 2 transforms a 2D point by an affine transform A. We use σ k ∈ R + and θ k ∈ [0; 2π] to denote the scale and orientation of the keypoint in I y k respectively.

Target Keypoint-set, Candidate-sets, and Graph Construction

We construct T by running the keypoint detector on the target image with the same parameters as used to construct R. Unless otherwise stated we rescale the images to a default image diagonal resolution of 1,000 pixels before running the detector. For any keypoint that have multiple descriptors due to multiple orientation estimates, we use the descriptor that corresponds to the strongest orientation. However in R multiple descriptors are kept. We denote each keypoint t i ∈ T by t i = (p i , di , σi , θi ), which holds its 2D position, descriptor, scale and orientation respectively. We construct C(t i ) by running an Approximate Nearest Neighbour (ANN) search over R. Currently we use FLANN [START_REF] Muja | Scalable nearest neighbor algorithms for high dimensional data[END_REF] with default parameters with a forest of 8 KD trees. We modify the search to account for the fact that due to the redundancy in R, several keypoints are often returned as nearest neighbours but they are approximately at the same position in the reference image. If this occurs then slots in the candidate-set are wasted. We deal with this in a simple manner by sequentially selecting as a candidate the keypoint in R with the closest descriptor from the ANN method, then eliminating keypoints which are approximately at the same image location (we use a default threshold of 5 pixels). The graph is then constructed that connects spatial neighbours in T . In our experiments this is done with a simple Delaunay triangulation. We handle the fact that edges may cross motion discontinuities by including robustness into the graph's cost function. Because the geometric constraints are defined locally a natural way to express matching constraints is with a MRF. We use a second-order MRF of the form:

E(s) = i∈[1...|T |] Ui(s(i)) + λ (i,j)∈E Pij(s(i), s(j))) (1)
The first-order term U i ∈ R is used to penalise matches with dissimilar descriptors. We use the standard unary term

U i (s(i) > 0) def = d s(i) -di 1 .
The 2 nd -order term P ij ∈ R enforces geometric consistency. The term λ balances the influence of P ij , and can be set by hand or learned from training images.

Second-order Deformation-Invariant Graph Constraints

We now show how the full-affine transform A k i that maps the local region in the target image about a keypoint t i to the local region about a candidate match r k in the reference image can be computed very cheaply (Fig. 2). Unlike affine normalisation [START_REF] Mikolajczyk | Scale & affine invariant interest point detectors[END_REF] this requires no optimisation. We achieve this by making the following assumption. Suppose t i and r k is a correct correspondence. Because their descriptors are close (since r k was selected as a candidate match), the local transform between the simulated image I y k and the target image at these points is likely to not have undergone much anisotropic scaling or shearing, and can be approximated by a similarity transform S k i . We obtain S k i from the keypoint detector:

S k i ≈ σ k R(θ k ) q k 0 1 σiR( θi) pi 0 1 -1
where R(θ) is a 2D rotation by an angle θ. A k i is then given by composing this transform with A -1 yi as A k i ≈ A -1 yi S k i (Fig. 2). There are two points to note. The first is that we are exploiting the fact that the keypoint is not fully affine invariant to give us A k i . Two of its DoFs come from the simulated deformation transform A yi (shear and aspect ratio) and four come from the keypoint's built-in invariance to give S k i . The second is that computing it is virtually free given any candidate match. We then use these affine transforms to construct P ij in a way which makes the graph invariant to local affine deformation. By enforcing this robustly we can encourage matches that can be explained by a piecewise smooth deformation. This writes as follows:

Pij(s(i) > 0, s(j) > 0) def = min w(A s(i) i , pj) -p s(j) 2 , w(A s(j) j , pi) -p s(i) 2 , τ (2)
The first term in Eq. ( 2) penalises a pair of matches if A s(i) i (the affine transform between t i and r s(i) ) cannot predict well p s(j) (the 2D position of r s(j) ). Because we also have A s(j) j , the second term is similar but made by switching the roles of i and j. We take the minimum of these two terms to improve robustness, which allows tolerance if either A s(i) i or A s(j) j is estimated wrongly. The term τ is a constant which truncates the pairwise term. This is used to provide robustness. Currently we set τ manually using a small dataset of training images (see §4).

Reference image Target image

Images simulated from the reference by changing shear and aspect ratio Similarity: rotat Inferred from

Similarity: rotat Inferred from

Full affine transfo Full affine transfo shear and aspect ratio Fig. 2. Full affine transform A k i for a candidate match using simulation (for shear and aspect ratio) and keypoint-invariance (scale, rotation and translation).

Detecting Unmatchable Keypoints a posteriori

The unmatchable state is difficult to handle as a graph constraint. To define the unary term U i (s(i) = 0) we would require to associate some cost (otherwise the graph would prefer a solution with all nodes in the unmatchable state [START_REF] Torresani | Nonrigid structure-from-motion: Estimating shape and motion with hierarchical priors[END_REF]). Balancing this cost ideally is non-trivial because V can vary significantly between image pairs depending on clutter, different backgrounds and surface self-occlusions. We also do not usually know a priori the expected number of detection and candidate pruning errors, as these vary between scenes and imaging conditions. We face the same difficulty with defining the pairwise terms involving the unmatchable state. To address this problem we note that our matching framework is not a symmetric process. When searching for matches, only the detected keypoints in the target image are matched. If the role of the reference and target images is reversed, a second set of matches would be found, and this should be consistent with the first set. This is the so-called LRC test [START_REF] Faugeras | Real time correlation-based stereo: Algorithm, implementations and applications[END_REF][START_REF] Kowdle | Combining monocular geometric cues with traditional stereo cues for consumer camera stereo[END_REF] applied to graph-based keypoint matching. We note that the LRC is also equivalent to the uniqueness constraint which ensures each point in one image can match at most one point in the other image [START_REF] Faugeras | Real time correlation-based stereo: Algorithm, implementations and applications[END_REF].

Our solution to handle the unmatchable state and the uniqueness constraint is to apply the LRC constraint (Fig. 3). First the MRF is defined in the target image without utilising the unmatchable state (we call this the target MRF ). The robustness of the graph's terms are used to prevent unmatchable keypoints adversely affecting the solution. The target MRF is then solved and we denote its solution by ŝ and the matched pairs by M T = {(t i , r ŝ(i) )}, i ∈ {1...n} (Fig. 3, top right). We then form the set R def = {r ŝ(i) } ⊂ R, i ∈ {1...n}, and define a second MRF called the reference MRF using R as its nodes. Duplicates or near-duplicates may exist in R because two keypoints in T may have been matched to approximately the same location in the reference image. We detect these using agglomerative clustering (we use a cluster threshold of 5 pixels), and collapse reference MRF nodes to have one node per cluster. A new MRF is then constructed using the cluster centres and their neighbours. Then the target image is treated as the reference image and the reverse match is computed (Fig. 3, bottom left). We denote its solution by M T = {(r ŝ(i) , t i )}. The LRC is then applied by ensuring that t i and t i are close, and we assign t i to the unmatchable state if p i -pi 2 ≥ τ c . We use a default threshold of τ c = 15 pixels.

Matching from reference to target image

Experimental Evaluation

We divide the experimental evaluation into three parts. The first part evaluates GAIM for reducing detection and candidate pruning errors. The second part evaluates GAIM for reducing assignment errors. The third part evaluates our method for detecting incorrect matches after graph optimisation. We compare against two state-of-the-art HMOD methods: PB12 [START_REF] Pizarro | Feature-Based Deformable Surface Detection with Self-Occlusion Reasoning[END_REF] and TCCBS12 [START_REF] Tran | In defence of RANSAC for outlier rejection in deformable registration[END_REF]. Because we deal with such large keypoint-sets (O(n) ≈ 10 3 and O(m) ≈ 10 4 ), most factorisation-based methods are unsuitable to tackle the problem. As a baseline graph-based method we use [START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF] which can handle large m and n via candidate match pruning. We use the naming scheme F/G/O/M to describe a graph matching method configuration. This denotes a method that computes keypoints with a method F, computes graph constraints with a method G, optimises the graph with an MRF optimiser O, and performs mismatch detection a posterior with a method M. In our experiments F is either SIFT, AC-SIFT (Affine-Covariant SIFT), or Gx-SIFT (our proposed asymmetric method by adapting SIFT in §3.1, where x denotes the number of synthesised views plus the original image). SIFT and AC-SIFT are computed with VLFeat's implementation [START_REF] Vedaldi | VLFeat: An open and portable library of computer vision algorithms[END_REF]. We use SIFT because it is still very popular, has matching accuracy that is competitive with state-of-the-art, and currently supports switching between standard and affine-Covariant versions of the descriptor. G is either GAIM (our proposed graph constraint) or RE (which stands for Robust Euclidean). RE is the Euclidean distance-preserving constraint from [START_REF] Torresani | Feature Correspondence Via Graph Matching: Models and Global Optimization[END_REF] but made robust by introducing the truncation term τ . This allows a fair comparison with our constraint, and we manually tune τ using the same training images. For O we use fast methods known to work well on correspondence problems. These are AE (α-expansion [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF] with QPBO [START_REF] Rother | Optimizing binary MRFs via extended roof duality[END_REF][START_REF] Hammer | Roof duality, complementation and persistency in quadratic 01 optimization[END_REF]) or BP (loopy Belief Propagation). M is either PB12 [START_REF] Pizarro | Feature-Based Deformable Surface Detection with Self-Occlusion Reasoning[END_REF], TCCBS12 [START_REF] Tran | In defence of RANSAC for outlier rejection in deformable registration[END_REF] or LRC+ (our proposed Left-Right Consistency test). We use GAIM to be shorthand for G26-SIFT/GAIM/AE/LRC+.

Test Datasets

We evaluate using public datasets from the deformable surface registration literature, and some new challenging datasets that we have created (Fig. 4). We divide the datasets into 6 groups (Fig. 4). Group 1 involves 8 representative images from CVLAB's paper-bend and paper-crease sequences [START_REF] Salzmann | Convex optimization for deformable surface 3-d tracking[END_REF]. The first frame is the reference image and five other frames were used as target images.

We also swapped the roles of the reference and target images, giving a total of 20 reference/target pairs. Group 2 involves 8 reference/target pairs of an open deforming book with strong viewpoint change and optic blur that we shot with a 720p smartphone. Group 3 involves 8 reference/target pairs taken from CVLAB's multiview 3D reconstruction dataset with GT computed by [START_REF] Strecha | LDAHash: Improved matching with smaller descriptors[END_REF]. This is a rigid scene, but it was used since GT optic flow is available and the scene has motion and surface discontinuities. Although the epipolar constraint is available due to rigidity, we did not allow methods to use this constraint. Group 4 involves 16 reference/target pairs from CVLAB's cardboard dataset [START_REF] Salzmann | Local deformation models for monocular 3d shape recovery[END_REF] that we constructed in a similar manner to Group 1. This is challenging due to texture sparseness and local texture ambiguity. Groups 5 and 6 involve 20 reference/target pairs taken from Oxford's wall and graffiti datasets respectively. We used the first frame as reference image, and the target images were generated from the first and third frames by applying randomised synthetic deformations to them using perspective and TPS transformations. In total there are 92 reference/target pairs. We trained λ and τ by hand on a training set comprising CVLAB's bedsheet and cushion datasets [START_REF] Salzmann | Convex optimization for deformable surface 3-d tracking[END_REF]. GT optic-flow is not provided for Groups 1,2 and 4. We computed this carefully using an interactive warping tool. All images were scaled to a resolution with diagonal 1,000 pixels and a match was considered correct if it agreed to the optic flow by less than 12 pixels.

Experiment 1: Reduction of Candidate-set Errors

The first source of errors in graph-based keypoint matching is a candidate-set error, which is when either a detection error or a candidate pruning error occurs (see §2). In Fig. 5 we plot the mean candidate-set error rate as a function of the candidate-set size p for SIFT, AC-SIFT and Gx-SIFT, with p varying from 1 to Fig. 4. Top: The six groups of test images used in evaluation. In total we test 92 reference/target image pairs. Bottom: average size of R and T and time (in ms) to construct, indexing and ANN querying R on an i7-3820 PC with FLANN [START_REF] Bartoli | Maximizing the predictivity of smooth deformable image warps through cross-validation[END_REF] with a default maximum search depth of 15. Although the time to index G26-SIFT is considerably lager (taking a second or more) the time to query is only approximately a factor of two/three slower than SIFT and AC-SIFT. In tracking tasks where indexing only needs to be done once the benefits of using Gx-SIFT are very strong.

200 and x varying from 10 to 65. In solid lines we plot the error rates when using the default detection parameters for building R and T . Across all datasets we find a clear advantage in using Gx-SIFT. The benefits in a larger x is stronger for smaller p. For p > 100 we see no real benefit in G8-SIFT over G26-SIFT in any testset. For SIFT and AC-SIFT one might expect lower error rates by keeping in R all detections without filtering those with low edge and scale-space responses (and so to potentially reduce detection errors). In dashed we show the error rates when R was constructed without the post-filtering. However there is no clear error reduction in general, and in some cases this actually increased the error. The large improvement in using Gx-SIFT tells us that a major cause for candidate-set errors are the lack of viewpoint and local deformation invariance. Despite AC-SIFT being designed to handle this, the results show that it lags quite far behind Gx-SIFT.

Experiment 2: Reduction of Candidate Selection Errors

After detection and candidate pruning the next source of errors are candidate selection errors (we refer the reader to §2 for the formal definition). We compared 10 matching configurations. These are listed in the first row of Fig. 6.

For our proposed keypoint detection method we use G26-SIFT. Our proposed matching configurations in full is G26-SIFT/GAIM/BP & AE. Because AC-SIFT also provides local affine transforms between candidate matches we also test the configuration AC-SIFT/GAIM by using the affine transforms from AC-SIFT in Eq. ( 2). In all experiments we use a default value of p = 90. The results on the test sets are shown in rows 1 and 2 of Fig. 6. The best results are obtained by the four configurations on the left (i.e. the ones where the reference keypoint-set was built using G26-SIFT keypoints). This result is important because it tells us that when we use keypoints with higher candidate-set errors (i.e. SIFT and AC-SIFT), this weakens the graph and causes keypoints that do not have candidate-set errors to be matched incorrectly. With the exception of Cardboard, there is a clear performance improvement with using GAIM constraints over RE. We believe the reason is because in Cardboard there is (i) little scale change between the views and (ii) the texture is much sparser than other datasets, which means that the local affine model used by GAIM has to be valid between distant surface points. There is little difference in the performance of G26-SIFT/GAIM/BP and G26-SIFT/GAIM/AE, which indicates both graph optimisers tend to find the same solution (although more tests are required). With respect to AC-SIFT, although this performs significantly worse than G26-SIFT we do see a performance improvement with using GAIM graph constraints over RE. We also measure the visible keypoint match error rates for each configuration. Recall from §2 that a visible keypoint match error occurs when a keypoint is visible in the target image, but the graph's solution does provide a correct correspondence. This is a combination of both candidate-set and assignment errors. The results are shown in rows 3 and 4 of Fig. 6.

Experiment 3: Complete Performance Evaluation

We now evaluate the complete performance of our approach, and show that unmatchable keypoints can be successfully detected a posteriori using the LRC.

Space limitations prevent a full performance breakdown, so we present here the complete system recall/precision performance. This is given by the proportion of matches that a method computes correctly versus the proportion of visible keypoints that have been matched. We compare against PB12 and TCCBS12 using hard matches from G26-SIFT, AC-SIFT and SIFT. We also investigate if PB12 and TCCBS12 can detect incorrect matches given the solution of a graph-based method. This is an interesting test and has not been done in the literature. We plot these results in rows 5 and 6 in Fig. 6. The trend we see is that the HMOD methods perform significantly better with G26-SIFT, and the reason is that many more hard matches are correct with G26-SIFT than with SIFT or AC-SIFT. For both G26-SIFT/RE/AE and G26-SIFT/GAIM/AE the performance when using PB12 to detect incorrect matches in their outputs is not significantly greater than hard matching using G26-SIFT, and in some instances is worse. The reason for this is because incorrect matches outputted by a graph method tend spatially correlated, and this makes them hard to distinguish from correct matches. The best performing method across all test sets is G26-SIFT/GAIM/AE/LRC. In Fig. 7 we give some representative visual results of the methods.

Conclusions

We have given a comprehensive breakdown of errors in graph-based keypoint matching into six different types. These errors occur at various stages of matching; from keypoint detection, candidate selection to final graph optimisation. In previous works keypoint detectors have been used in a rather black-box style, however there is a deep interplay between the keypoint detector and graph-based matching that should not be ignored. We hope the results of this paper will stimulate the design of new keypoint methods that specifically reduce candidate-set errors in graph-matching rather than the commonly used repeatability metric. We have presented the first matching system that has been designed to reduce all six error types. Candidate-set errors have been considerably reduced by Gx-SIFT features. These also provide automatic information about a keypoint's local affine transform that can be used as a second-order deformation invariant matching constraint. This produces lower candidate-selection errors than the commonly-used euclidean distance-preserving constraint. We have provided a method to detect mismatches a posteriori using a Left-Right Consistency procedure adapted to asymmetric deformable graph matching, and shown that the full framework outperforms state-of-the-art HMOD methods.

Fig. 1 .

 1 Fig. 1. The six types of errors in graph-based matching. Reference and target keypointsets are illustrated by circles and candidate matches are given at the bottom. Orange arrows illustrate matches made by a matching algorithm. Best viewed in colour.

Fig. 3 .

 3 Fig. 3. Proposed Left-Right Consistency test for keypoint-based graph-matching.

P a p e

  r -b e n d & p a p e r -c r e a s e ( CV L A B ) Co o k b o o k ( A L Co V ) F o u n t a i n & He r z j e s u ( CV L A B ) Ca r d b o a r d ( CV L A B ) De f o r me d wa l l ( Ox f o r d & A L Co V ) De f o r me d g r a f f i t i ( Ox f o r d & A L Co V )

Fig. 5 .

 5 Fig. 5. Candidate-set error rate versus candidate-set size across the six test-sets. For a fair comparison all methods use the same detected keypoints in the target image.

mismatch occurs

  in keypoint matching when keypoints in two different images are matched by an algorithm but they do not correspond to the same physical 3D point up to noise.

		Correct matches		Detection error Detection error
		Unmatched error Unmatched error	Candidate pruning error Candidate pruning error
				Candidate selection error Candidate selection error Candidate selection error Candidate selection error
	Reference image	Target image	Occluded match error Occluded match error	Visible-but-unmatchable error Visible-but-unmatchable error
		Correct matches:

3 A Note: if keypoint visible in reference but not in target No problem! We are never try to match it, because we do not see it!

  19 2.82 16.89 997 21.53 2.83 16.06 54279 1248.24 9.30 34.57

	Cookbook	635	1352 28.67 3.83 22.57 1482 32.84 3.86 22.29 45137 1051.12 10.91 41.54
	Fountain &	602	1047 25.81 3.71 22.10 1123 25.81 3.76 23.02 50461 1199.47 9.87 37.37
	Herzjesu
	Cardboard	65	88	1.93	0.94	8.17	96	2.66	0.87	9.30 4891 116.15 1.19	6.15
	Wall with	748	1688 37.84 6.04 28.34 1778 43.82 6.29 28.73 81587 1882.41 14.25 50.77
	deformation
	Graffiti with	848.03 1235 26.54 5.43 31.22 1325 27.94 6.07 31.40 41490 974.01 22.61 61.90
	deformation

Acknowledgments. This research has received funding from the EU's FP7 through the ERC research grant 307483 FLEXABLE.

CMs stands for the number of Correct matches (higher is better, in blue), UVKs stands for the number of Unmatched Visible Keypoints (lower is better, in green) and UOKs stands for the number of unmatched occluded keypoints (higher is better, in yellow). In red are false matches. In each target image we show the Region-Of-Interest (ROI) within which we had GT optic flow. To ease evaluation we restricted R to be keypoints within the ROI. Therefore any keypoint in the target image that does not have a correct match in the ROI is an occluded keypoint. No ROI was used to filter keypoints in the target images.


Pablo Mesejo
email: <pablomesejo@gmail.com>

Stefano Cagnoni

Automatic Segmentation of Anatomical Structures using Deformable Models and Bio-Inspired/Soft Computing

This PhD dissertation, carried out within the Marie Curie project MIBISOC ("Medical Imaging using Bio-Inspired and SOft Computing", EU FP7 G.A. n. 238819), describes the accomplishments of our research on the development of algorithms for the automatic and accurate segmentation of anatomical structures in biomedical images. Firstly, hybridizations between soft computing (SC) and computer vision (CV) techniques have been investigated to help in biomedical imaging-related tasks. Secondly, in collaboration with the Molecular Biotecnology Center of Torino, we have developed a specific tool aimed at extending the current understanding of the complex signaling events occurring in dendritic spines (DS), by integrating state of the art computational and experimental approaches. DS are neuronal protrusions, which receive input typically from one excitatory synapse, containing neurotransmitter receptors, organelles, and signaling systems essential for synaptic function and plasticity.

We initially aimed at developing new recognition algorithms for brain anatomical structures, capable of automatically extracting two types of information: 1) the relative abundance of the mRNA of every gene in the main brain areas; and 2) the identity of mRNAs that are particularly enriched in dendrites, which are very likely to be involved in DS biogenesis/functions. The main motivation can be found in the great interest that neurobiologist have for RNA molecules specifically enriched in the neuropil of neuronal cells (and, in particular, in DS), in virtue of their involvement in synaptic structure, plasticity and neuropsychiatric disorders (like fragile-X syndrome, autism spectrum disorders, or schizophrenia). The systematic identification of these molecules is, therefore, a very important task and can be based on visual features, such as image texture, which recur in the regions where they are more densely present. In that regard, the high-resolution images of RNA In Situ Hybridization (ISH) * experiments contained in the Allen Brain Atlas (ABA) represent a very rich resource to identify them and, for this task, they have been so far exploited based on human-expert analysis. However, software tools that automatically address the same objective are not easily available or effective. Since the regions of interest, where the molecules under study are more densely present, occupy only a small part of the whole images, a tool for automatically focusing on such regions is essential if a huge number of images is available for processing. To accomplish such a goal, three different segmentation methods that allow for the accurate delineation of a specific anatomical region have been developed.

The first method [1, 2, 3] preliminarily processes a training set of images, from which the deformation range and the main modes of variation of a parametric deformable model (DM) are empirically assessed. Based on these data one can define very simple templates, managed by a fast method which allows genome-wide experiments on thousands of images. This approach is divided into four different stages (initialization, localization

using DMs and metaheuristics (MHs), segmentation, and expansion of the segmentation using ensemble classifiers), and can be seen as a smart way of selecting promising regions to be subsequently processed by a fast and well-established image segmentation/analysis technique. The main advantages of this approach are its execution time and accuracy, and the simplicity in the definition of the models, while its main disadvantages are its ad-hoc nature (it needs a training set of shapes/textures, and suitable parametric models of the object to locate), and the impossibility of managing topological changes.

The second method [START_REF] Mesejo | Segmentation of Histological Images using a Metaheuristic-based Level Set Approach[END_REF] uses the Level Set (LS) method to allow topological changes. The templates are created automatically using principal components analysis to derive texture, mean shape and shape variability from the training set. After that, a MH is used to evolve the segmenting contour by optimizing the weights of a linear combination of the different eigenshapes and the mean shape, based on an intensity-and texture-based fitness function. This method has obtained reasonably good results, even if it is less precise than the previous approach and requires a longer execution time. Thus, it can be looked upon as an intermediate stage that led us to develop the third segmentation method.

The third method [START_REF] Mesejo | Biomedical Image Segmentation using Geometric Deformable Models and Metaheuristics[END_REF] tries to solve all the previous problems found. Firstly, the LS method is again used to manage possible topological changes. Secondly, three different sources of information are taken into account (intensity, boundaries, and shape). Thirdly, the automatic MH-based tuning procedure allows one to apply the same method to different image modalities and anatomical structures, and makes this a general approach (tested on CT, MRI and microscopy histological images). Finally, only a single previously segmented reference image is necessary to perform the segmentation instead of a complete training set, as in the previous methods. This is possible thanks to a deformable registration process in which the target image is aligned with the corresponding image of an Atlas. In terms of generalization and accuracy, this method obtained the best results compared with seven state of the art segmentation approaches.


Mélise Jaud
email: mjaud@worldbank.org

Youssouf Kiendrebeogo
email: ykiendrebeogo@worldbank.org

Marie-Ange Veganzones-Varoudakis

Catherine Araujo Bonjean

Keywords: Financial crises, Financial vulnerability, Export dynamics JEL codes F12, F14, G01, O16

 to remove any endogeneity bias, the results point to a negative and economically large effect of financial vulnerability on export diversification. Financial crises reduce export dynamics disproportionately more in financially dependent industries. This effect is less pronounced in countries with initially more open capital account, suggesting that portfolio inflows are good substitutes for underdeveloped domestic financial markets.

Introduction

Diversification maters for economic development. To reduce vulnerability to external shocks, a country needs to build its capacity to reorient and redirect some of its exports towards new products and markets. The gains from trade diversification go beyond the benefits of specialization due to comparative advantage. By reducing price instability and output volatility, export diversification plays a central role in driving long-run growth and macroeconomic stability. Yet, developing countries have long been dependent on a narrow set of export products, despite some recent progress toward diversifying products and partners. Figure 1 shows that over the period 1997-2011, export concentration has been on average more than two times higher in developing countries than in advanced countries, although the gap has decreased in recent years. 1 Herfindahl-Hirschman Index

Interestingly, however, this relatively low export diversification in developing countries suggests the existence of more room for upgrading the existing export basket. Also, export concentration is likely to be heterogeneous across both developing countries and geographical regions. Country-specific characteristics, circumstances, institutions and policies might have played a role in the process of structural transformation. In light of the implications of "sunk costs" for export participation discussed in the literature, one of the main drivers of this process is the vulnerability of firms to adverse financial shocks. Due to 1. Export diversification is measured here by the Herfindahl-Hirschman index. Notice that higher values of this index indicate lower diversification.

the existence of substantial sunk costs of entry into exporting (see, for instance, [START_REF] Roberts | The Decision to Export in Colombia : An Empirical Model of Entry with Sunk Costs[END_REF][START_REF] Bernard | Export, Entry, and Exit by German Firms[END_REF][START_REF] Melitz | The impact of trade on intra-industry reallocations and aggregate industry productivity[END_REF], only more productive firms or firms with a certain level of financial health are able to export. Furthermore, financial vulnerability has been identified as one of the major trade barriers [START_REF] Manova | Credit Constraints, Equity Market Liberalizations and International Trade[END_REF][START_REF] Berman | Financial Factors and the Margins of Trade : Evidence from Cross-country Firm-level Data[END_REF][START_REF] Minetti | Credit constraints and firm export : Microeconomic evidence from Italy[END_REF][START_REF] Chor | Off the Cliff and Back ? Credit Conditions and International Trade during the Global Financial Crisis[END_REF] and the evidence suggests that the effect of credit rationing on exports is likely to be more pronounced in developing countries. The World Bank Enterprise Surveys country reports show that the percentage of firms identifying access to finance as a major constraint is typically higher in developing countries. 2At the macro level, compelling evidence supports the connection between financial development and export performance [START_REF] Beck | Financial Development and International Trade : Is there a Link[END_REF][START_REF] Beck | Financial Dependence and International Trade[END_REF][START_REF] Svaleryd | Financial Markets, the Pattern of Industrial Specialization and Comparative Advantage : Evidence from OECD Countries[END_REF][START_REF] Manova | Credit Constraints, Heterogeneous Firms, and International Trade[END_REF][START_REF] Manova | Credit Constraints, Equity Market Liberalizations and International Trade[END_REF][START_REF] Becker | Financial Development, Fixed Costs and International Trade[END_REF]. The idea is that countries with less developed financial sectors are likely to export goods not requiring external funding. Also, the empirical literature described how banking crises reduce export volumes but the underlying issue of the effect of crises on export diversification remains unresolved. In particular, it is still unclear whether the crisis effect of trade is the result of a reduced number of products, a reduced number of destination, or both. In addition, while there is a large literature on the implications of financial vulnerability for export-market participation, country samples have varied substantially across studies and very few papers have focused exclusively on developing countries.

This paper is, to the best of our knowledge, the first contribution to understand the extent to which financial vulnerability affects export dynamics. We focus on the developing world for three main reasons : (i) it experiences relatively highly-concentrated exports, (ii) it displays substantial heterogeneity across countries in export diversification, (iii) it is more likely to experience financial vulnerability than the advanced world. In part due to the lack of cross-country data on export diversification for developing countries, few studies have quantified the impact of financial vulnerability on export dynamics in these countries. This study also makes an important contribution to the literature by using a comprehensive cross-country database on exporter dynamics compiled at the industry level. The identification strategy follows the procedure introduced by Rajan & Zingales (1998) (henceforth RZ), which is well-known to be convenient in correcting for a potential endogeneity bias. The paper is concerned with the following specific questions : How do financial crises affect firm, product and destination dynamics ? Do firms in industries with higher financial dependence suffer more ? Does this effect vary across countries with different levels of capital account openness ?

The main result suggests that financial crises disproportionately increase export concentration in financially-vulnerable industries. Financial crises tend to reduce firm, product and destination entry rates, while increasing the corresponding exit rates. These effects are relatively more important in industries with higher financial dependence. Countries are also affected differently, with regard to the capital account openness. In the remainder of the paper, we present the identification strategy and the data used in Section 2. The results are discussed in Section 3, while Section 4 provides conclusions and implications for policymaking.

Empirical Strategy and Data

Baseline Estimating Equation

The central idea behind this study is that financial crises disproportionately hurt export concentration in industries that are more dependent on external financing. This is tested by estimating the following econometric specification :

Export cpkt = F inV ul k J j=0 (β 1 Crisis c(t-j) + β 2 Crisis p(t-j) + β 3 Crisis cp(t-j) )+ αX cpt + d ct + d pt + d ck + cpkt (1)
where Export cpkt is the indicator of export diversification in country c for trading partner p, in the industry k during the year t. F inV ul k is the index of financial vulnerability that captures the degree of external financial dependence in industry k. This is the RZ index of external financial dependence.3 Crisis c(t-j) is a dummy indicating whether a financial crisis happened in country i year t -j. We include both the contemporaneous and lag crisis dummies to capture the average effect of a crisis on export diversification during its onset and in the following years. Three dimensions of financial crises are considered, namely banking, currency, and sovereign debt crises. Figure 2 provides the number of countries experiencing a financial crisis over time for our period of interest.

X cpt is a set of conditioning information to control for other country-level factors influencing export diversification. This includes the Log of real GDP per capita and the Log of real exchange rate. The other potentially omitted macro factors will be captured through country-year and partner-year fixed effects (d ct and d pt ). We also include countrypartner fixed effects to account for other gravity controls such as distance and common language. d ck are country-industry fixed effects to control for industry-level unobserved heterogeneity across countries. 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1

Banking Crisis Currency Crisis Sovereign Debt Crisis

Source : [START_REF] Laeven | Systemic Banking Crises Database[END_REF]

Identification

In estimating the baseline specification (Equation 1), the immediate concern is the endogeneity bias arising from reverse causality, omitted variables or measurement errors on the regressor variables. Although the omitted variable bias is importantly limited by the introduction of our set of fixed effects, reverse causality from exports to financial vulnerability remains a central concern.

4
This issue is addressed by building on the generalized procedure proposed by [START_REF] Rajan | Financial dependence and growth[END_REF]. This approach has been largely used to estimate sectoral differential effects (see, for instance, Dell [START_REF] Dell' Ariccia | The Real Effect of Banking Crises[END_REF][START_REF] Manova | Credit Constraints, Equity Market Liberalizations and International Trade[END_REF][START_REF] Chor | Off the Cliff and Back ? Credit Conditions and International Trade during the Global Financial Crisis[END_REF].

In practice, this method consists in using the financial dependence of U.S. firms on external financing as a proxy for the demand for external finance in developing countries.

The main rationale is that any need for external finance in a steady-state equilibrium is the result of worldwide technical shocks. This amounts to saying that the demand for external financing of U.S. firms is therefore a good proxy for firms' financial dependence in developing countries.

Data, Country sample, and sample period

The set of countries covered in this study is motivated by the consideration of focusing Export diversification data, compiled at the HS 6-digit level of disaggregation, come from the Exporter Dynamics Database [START_REF] Cebeci | Exporter Dynamics Database[END_REF]. 6 Export diversification is measured by three dimensions of export dynamics : (i) firm dynamics, (ii) product dynamics, and (iii) destination dynamics. For each dimension, we use both the entry/exit rates into/from export markets. Notice that when using destination entry and exit rates, one only have a country-industry-year dimension, since the destination dimension disappears.

To match these data with the financial vulnerability indicator, we aggregate them to the 3-digit ISIC industry level, using the Haveman's concordance tables.

Financial vulnerability and export diversification : facts

The data suggest that export dynamics remain relatively weak in developing countries, though there is significant heterogeneity across countries. In these countries, the firm entry rate is 53%, as compared with 62% in developed countries (Table 1). At the same time, the firm exit rate is in developing countries (59%) is more than two times higher than that in advanced economies (21%). Similarly, the product entry rate is relatively higher in developed countries (31% vs. 27%) but they experience a relatively lower product exit rate (18% vs. 23%), as compared to the corresponding rates in the developing world. The same picture is observed with regard to destination dynamics, where the entry/exit rates are again relatively higher/lower in developed countries (44% vs. 25% and 23% vs. 38%) than in developing countries. More interestingly, the standard deviation of firm, product, and destination entry and exit rates is always higher in the developing sample, suggesting a relatively higher heterogeneity. This could be due to the fact that developing countries are more represented in the sample (87% of the country sample is developing) but does not eliminate the reasons why focusing on the developing sample is interesting.

Furthermore, regarding the financial vulnerability issue, financial dependence tend to be a more important determinant of export dynamics in developing country than in advanced countries mainly due to market failures. These main reasons led us to put our focus on the developing sample in the remaining empirical excises.

The correlation between external financial dependence and export dynamics can be described using the RZ index. On the one hand, industries at the top 50% of the distribution of financial dependence (industries more dependent on external finance) experience lower firm, product, and destination entry rates than industries at the bottom 50% of this distribution (Table 2). On the other hand, exit rates in financially dependent industries appear to be relatively more important than in other industries.

Alternatively, when we use the [START_REF] Braun | Financial Contractibility and Asset Hardness[END_REF] index of asset tangibility, the data consistently show that the firm, product and destination entry rates are always relatively higher in industries with higher collateralizable assets. At the same time, industries with higher collateralizable assets observe relatively lower exit rates than industries with fewer collateralizable assets. 8

9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1
Entry rate at the bottom 50% of financial dependence Entry rate at the top 50% of financial dependence 0 .2 .4 .6

.8

1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 Exit rate at the bottom 50% of financial dependence Exit rate at the top 50% of financial dependence 8. To save space, these statistics are not presented here but are available upon request.

The statistical link between the external financial dependence and export dynamics in developing countries is also evidenced for each year of our sample period (Figure 3).

The average firm, product, and destination entry rates in industries at the bottom 50% of the distribution of financial dependence is always higher than that of industries at the top 50% of this distribution. As for the average firm, product, and destination exit rate, it appears however to be relatively higher in industries with higher financial dependence, with the exceptions of years 2000 and 2006.

These facts suggest a possible association between financial vulnerability and export dynamics in developing countries. In the remaining part of the paper, we further investigate the impact of financial crises on exporter dynamics.

Main findings

Previous studies on the impact of financial crises on international trade have paid little attention to the extent to which crises affect exporter dynamics. In this section we investigate the effects of financial crises on the three following dimensions of export dynamics : (i) firm dynamics, (ii) product dynamics, and (iii) destination dynamics.

Financial crises and firm dynamics

Table 3 presents the estimates of the baseline specification, Equation 1, using firm entry and exit rates as dependent variables. 9 The main variables of interest are the dummies of crisis in exporter and importer, their lags, and their corresponding interaction terms.

Without controlling for the crisis interaction, the coefficient estimates on the contemporaneous and lagged crisis dummies in both exporter and importer have the anticipated signs and are statistically significant. The contemporaneous effect suggests that financial crises are associated with relatively higher declines in the firm entry rate in financially-dependent industries (columns 1 and 3). The magnitude of these coefficients is economically meaningful, indicating that the firm entry effects of crises in exporter and importer are -7.1 and -2.5 percentage points more pronounced for industries with higher financial dependence (column 1).

These results point to a relative importance of the supply-side shock, compared to 9. The variation of firm dynamics explained by the model is more than 80 percent, suggesting a good fit.

the demand-side shock. The results also confirm the presence of substantial sunk costs in export entry in times of financial crisis. 10

Regarding the firm exit rate, the results on the contemporaneous crisis dummies show that experiencing a financial crisis in the partner country is associated with 10.6 percentage points relatively higher increase, for financially-dependent industries, in the rate at which firms exit the export market (column 3). The coefficient on the exporter crisis dummy is small and statistically indistinguishable from zero, reflecting a relative importance of demand shocks for the decision of firms to exit foreign markets. On the other hand, the average impact of supply-side shocks on the exit rate is relatively more important than that on the entry rate.

When controlling for the interacting effect of crises, these effects remain significant with very similar magnitudes. The negative and significant coefficient on the crisis interaction in column 2 indicates that the supply-side shocks and demand-side shocks complement and reinforce each other in decreasing the firm entry rate when both countries are in crisis. But the exacerbating effect is statistically insignificant on the firm exit rate (column 4). These effects of crises on firm entry/exit into/from exporting are sizable given the cross-country variation of firm dynamics in the data (see Table 1).

Turning to the lagged crisis dummies, their estimated effects have the expected signs as well. The results in column 1 show that the medium-run impact of crises on the firm entry is economically significant and disproportionately severe for financially-vulnerable industries. On average, the firm entry rate falls relatively higher in these industries by 7.8 and 11.3 percent in the 3 subsequent years after a financial crisis in the exporter and in the importer, respectively. Conversely, the firm exit rate raises relatively strongly in vulnerable industries by 9.3 and 6.8 percent in the medium-term, respectively for crisis in exporter and importer (column 3). The lagged crisis interaction also enters with the expected sign but its coefficient is only significant for the one-period lag. This is the sign that the exacerbating effect of crises on firm dynamics is not persistent over time.

The results on the control variables are quite intuitive as well. Real GDP per capita for both exporting and importing countries enter positively and significantly in explaining the 10. The sunk costs of entry into foreign markets include learning about foreign markets, administrative standards, and establishing distribution networks (e.g., [START_REF] Roberts | The Decision to Export in Colombia : An Empirical Model of Entry with Sunk Costs[END_REF][START_REF] Bernard | Export, Entry, and Exit by German Firms[END_REF]. These sunk costs are likely to be sensitive to firm financial conditions and are amplified during crises.

firm entry rate. Their impact on the firm exit rate displays the right sign but is insignificant at any conventional level. An increase in the real exchange rate, which represents a real appreciation of the exporter currency vis-à-vis the importer, reduces the entry rate of exporting firms but fails to show any significant effect on their exit rate. and * denote significance at the 1-percent, 5-percent, and 10-percent levels, respectively. All the crisis-related dummies are interacted with the sector-level financial dependence index.

Financial crises and product dynamics

This section complements the previous one by focusing on the second dimension of export dynamics : the product entry/exit rates into/from export markets. Product dynamics is one of the aspects of the intensive margin of exports. Indeed, firms entry and exit do not tell us enough about the number new products exported. However, financial crises may affect product dynamics through the disruption effect and the income effect.

The results from estimating Equation 1 using product entry and exit rates as dependent variables are presented in Table 4. As before, we first focus on the impact of crises on export dynamics while ignoring the crisis interaction (columns 1 and 3). As anticipated, financially-vulnerable industries in crisis-hit countries tend to experience lower levels of bilaterally exported and imported products, reflecting the balance-sheet problems-increased fixed costs of exporting a new product-in the wake of financial crises. The neous estimated effects of financial crises on the product entry rate are statistically and economically significant (column 1). and * denote significance at the 1-percent, 5-percent, and 10-percent levels, respectively. All the crisis-related dummies are interacted with the sector-level financial dependence index.

In times of crisis, firms tend to delay exporting new products or abandon exporting some existing products owing to the limited access to working capital and reduced foreign demand. Other things being equal, financial crises are associated with 2 and 9.5 percentage points more pronounced collapses of the product entry rate in financially-vulnerable industries, respectively for crises in exporter and importer. In addition, the coefficients in column 3 indicate that the product exit rate raises disproportionately higher by 13.3 percent in financially-dependent industries during financial crises in the exporting country. The coefficient on the contemporaneous effect of crises on the product exit rate is positive but statistically insignificant, suggesting that demand shocks matter more than supply shocks for product exits. This is consistent with the results presented in Table 3 and signals that the disruption of trade credit at the height of a crisis in the exporting country introduces important sunk costs of entry of new products without necessarily causing the exit of existing products.

Once again, the estimated coefficients on the lagged crisis dummies indicate the gradual nature of the product dynamics adjustment. Nearly 50 percent (0.036/0.072) and 80 percent (0.097/0.119) of the effect of a financial crisis would remain three years after the event, respectively when the crisis occurs in the exporting and importing country.

Financial crises in both exporter and importer countries are associated with relatively higher declines of the product entry rate and increases of the product exit rate in vulnerable industries. On average, the number of new exported products as a percentage of all products exported falls relatively higher in vulnerable industries by 7.2 and 11.9 percent over three years following the onset of the crisis, respectively for a crisis in the exporter and importer countries.

By contrast, a financial crisis in the importing country leads to 9.3 percentage points relatively higher increase of the product exit rate over the three following years in financiallydependent industries. The crisis in the exporting country positively affects the product exit rate but its effect is statistically insignificant. As regards the estimated coefficients on the crisis interaction terms, it is apparent that the impact of financial crises on product dynamics is amplified when both exporter and importer countries are in crisis. However, this exacerbating effect becomes statistically insignificant two years after the crisis. As before, controlling for this interaction does not alter our main results on the detrimental impact of crises on product dynamics in financially-vulnerable sectors.

Regarding the conditioning information, higher levels of real GDP per capita of both exporter and importer raise the product entry rate and reduce its exit rate. An increase in the bilateral real exchange rate is associated with lower levels of product entry rate but its effect on the product exit rate is statistically insignificant.

Financial crises and destination dynamics

In this section our interest is on the impact of financial crises on the extensive margin of exports to a given destination. The destination entry and exit rates are alternatively used as the indicators of destination dynamics. One advantage of considering destination dynamics is that independently from firm and product dynamics, the disruption effect could matter for the decision to exit a destination or to export to a new destination, as suggested by [START_REF] Muuls | Exporters and credit constraints. A firm-level approach[END_REF]. Since the interest is in the number destinations, we estimate a modified version of Equation 1 in the full panel of the measures of destination dynamics for the 34 countries used in subsections 3.1 and 3.2 over the 1997-2011 period.

Export cpkt = F inV ul k J j=0 (β 1 Crisis c(t-j) + β 2 Crisis p(t-j) + β 3 Crisis cp(t-j) )+ αX cpt + d ct + d ck + cpkt (2)
Crisis p(t-j) is now defined as a dummy variable taking 1 when at least one of the top 5 trading partners are in crisis.11 Similarly, the regressors included in X are measured relative to the top 5 trading partners.

The regression results in Table 5 show that financially-dependent industries in crisishit countries experience lower levels of participation to new export destinations. The estimated coefficients on the crisis dummies are all negative in the destination entry rate equation, reflecting the detrimental nature of financial crises for the entry of exporting firms in new markets (columns 1 and 2). A contemporaneous crisis leads to a relatively lower destination entry rate, by 4.9 percent in vulnerable industries, confirming that financial crises are associated with increased sunk costs of exporting. This negative impact is persistently significant four years after the crisis. Additionally, during financial crises, the destination exit is disproportionately raised by 0.5 percent in financially-vulnerable industries but this effect becomes statistically insignificant in the years following the event.

Given that we are no longer on a bilateral specification we control for relative GDP per capita. This is the deviation of the country's real GDP per capita from the average level of its trading-partner. The coefficients of 0.149 on this variable implies that, other things being equal, a country whose GDP per capita is twice the average GDP of its trading-partners will have a destination entry rate that is 29.8 percentage points higher than that of the trading-partners, on average (Column 1). Consistently with this result, the coefficient on the relative GDP PC in column 3 indicates that an increase in relative GDP reduces the destination exit rate of incumbents. Furthermore, a real appreciation is associated with lower destination entry rate and higher destination exit rate but the latter effect is not statistically significant. ***, **, and * denote significance at the 1-percent, 5-percent, and 10-percent levels, respectively. Here the crisis interaction is defined to take 1 if the country is simultaneously in crisis with at least one of the its five top trading-partners. The relative GDP PC is the deviation of the country's real GDP per capita from its trading-partner average. All the crisis-related dummies are interacted with the sector-level financial dependence index.

We further investigate the variation of the magnitude of the crisis effect across coun-tries by splitting our sample between countries with higher and lower financial account openness. The first group includes countries at the top 50% of the distribution of financial openness, averaged over the 1997-2008 period, 12 whereas the second group includes the bottom 50% of the distribution. Financial openness is measured by the [START_REF] Chinn | A New Measure of Financial Openness[END_REF] index of capital account openness. 13 The results show that exporters are relatively more resilient to both supply-side and demand-side shocks in countries with higher financial openness (Tables 6 and7 in the appendix). 14 Also, the supply-side effect is always insignificant for the sample of countries with higher financial account openness (Tables 6).

By contrast, this effect is large and statistically significant for the sample of lower financial openness, though insignificant on the destination exit rate (Tables 6). As regard the demand-side effect, it is significant in the case of more financially-open countries only for the entry rates. It has no significant effect on the exit rates in these countries. On average, this negative impact of crises in partner countries on the firm entry rate, for which both coefficients are significant and therefor comparable, is more than five times higher in countries with lower financial openness. Similarly, the interacting effect is always lower and statistically insignificant in countries with higher financial openness.

These regression results contribute to reconcile the two existing views in the literature.

The debate on the relative importance of supply-side shocks versus demand-side shocks is now better understood through this decomposition of the impact of crises on the three dimensions of export dynamics. The results on the product and destination dynamics reveal the importance of both supply-side and demand-side shocks for export diversification.

Conclusions

Export diversification, as a core element of countries structural transformation, is now a priority on the policymakers' agenda. This study mainly revealed that financial vulnerability is negatively related to export diversification. Financial crises reduce export firm, product, and destination entry rates and increase the corresponding exit rates disproportionately more in financially-vulnerable industries. This detrimental effect is less pronounced in countries with initially more open capital account.

These empirical results have important policy implications in developing countries in 12. 2008 is the last year for which the data on financial openness are available. 13. See the list of countries in the appendix. 14. We report only the coefficients on our variables of interest to save space.

promoting export diversification. Reducing the dependence of firms to external financing may be a key for strengthening export dynamics and diversification, since this would result in increase entry rates into exporting and a reduced exit rates. Alternatively, measures to facilitate portfolio flows may be a second best option, given that increased capital inflows is found to be a good substitutes for domestic funding.

A. Appendix

Top 50% of the distribution of financial openness : Albania, Botswana, Burkina Faso, Cameroon, Costa Rica, Ecuador, Guatemala, Iran, Jordan, Lebanon, Macedonia, Mali, Mexico, Nicaragua, Niger, Peru, Senegal, Yemen Bottom 50% of the distribution of financial openness : Bangladesh, Bulgaria, Cambodia, Chile, Colombia, Dominican Republic, El Salvador, Kenya, Lao PDR, Malawi, Mauritius, Morocco, Pakistan, South Africa, Tanzania, Uganda. Note : The dependent variable is the rate of export entry or exit of products by country-destination. The regressions include year and exporter-importer fixed effects. Heteroskedasticity-robust standard errors are reported in parentheses. ***, **, and * denote significance at the 1-percent, 5-percent, and 10-percent levels, respectively. All the crisis-related dummies are interacted with the sector-level financial dependence index. The other regressors used previously are also included but not reported.

Figure 1 -

 1 Figure 1 -Evolution of export concentration : 1997-2011

Figure 2 -

 2 Figure 2 -Number of countries experiencing a financial crisis (starting date)

Figure 3 -

 3 Figure 3 -Financial dependence and export diversification over time

Note:

  The dependent variable is the rate of export entry or exit of products by country-destination. The regressions include year and exporter-importer fixed effects. Heteroskedasticity-robust standard errors are reported in parentheses. ***, **, and * denote significance at the 1-percent, 5-percent, and 10-percent levels, respectively. All the crisis-related dummies are interacted with the sector-level financial dependence index. The other regressors used previously are also included but not reported.

Table 1 -

 1 Summary statistics : Export diversification

	Variable	Obs. Mean p50 Std. Dev. Minimum Maximum
			Developed countries
	Firm entry rate	12587	0.62	0.50	0.27	0	1
	Firm exit rate	12606	0.21	0.24	15.13	0	1
	Product entry rate	11161	0.31	0.31	0.16	0	1
	Product exit rate	6816	0.18	0.18	0.15	0	1
	Destination entry rate 6345	0.44	0.46	0.10	0	1
	Destination exit rate	4257	0.23	0.23	0.10	0	1
			Developing countries
	Firm entry rate	28235	0.53	0.58	0.19	0	1
	Firm exit rate	27768	0.59	0.54	0.28	0	1
	Product entry rate	21538	0.27	0.25	0.23	0	1
	Product exit rate	21451	0.23	0.22	0.31	0	1
	Destination entry rate 17329	0.25	0.23	0.18	0	1
	Destination exit rate	17171	0.38	0.37	0.29	0	1

Note : These summary statistics are based on sector-level data in the 3-digit ISIC industry classification.

Table 2 -

 2 Export diversification and financial dependence

	Variable	Obs. Mean Std. Dev. p50	Minimum	Maximum
		Top 50% of the distribution of financial dependence
	Firm entry rate	12616	0.72	0.72	0.68	0	1
	Firm exit rate	12402	0.48	0.48	0.84	0	1
	Product entry rate	9410	0.29	0.28	0.23	0	1
	Product exit rate	9410	0.24	0.22	0.23	0	1
	Destination entry rate 8639	0.27	0.25	0.17	0	1
	Destination exit rate	8532	0.23	0.20	0.18	0	1
		Bottom 50% of the distribution of financial dependence
	Firm entry rate	15619	0.32	0.31	0.54	0	1
	Firm exit rate	15366	0.70	0.72	0.91	0	1
	Product entry rate	12128	0.24	0.23	0.23	0	1
	Product exit rate	12041	0.29	0.28	0.22	0	1
	Destination entry rate 8690	0.23	0.21	0.18	0	1
	Destination exit rate	8639	0.26	0.25	0.17	0	1

Note : These summary statistics are based on sector-level data in the 3-digit ISIC industry classification.

Table 3 -

 3 Crises and firm dynamics

		(1)	(2)	(3)	(4)
	Dependent variable	Firm entry rate of incumbents Firm exit rate of incumbents
	Crisis in exporter	-0.071***	-0.075***	0.010	0.017
		(0.025)	(0.024)	(0.012)	(0.023)
	Crisis in importer	-0.025***	-0.041***	0.106***	0.053***
		(0.007)	(0.013)	(0.046)	(0.016)
	Crisis interaction		-0.283**		0.452
			(0.128)		(0.483)
	Crisis in exporter (t-1)	-0.118***	-0.073**	0.039*	0.022*
		(0.040)	(0.032)	(0.022)	(0.012)
	Crisis in importer (t-1)	-0.024***	0.013***	0.093***	0.057***
		(0.008)	(0.004)	(0.027)	(0.019)
	Crisis interaction (t-1)		-0.092**		0.151***
			(0.004)		(0.048)
	Crisis in exporter (t-2)	-0.183***	-0.101***	0.085**	0.057**
		(0.060)	(0.029)	(0.036)	(0.022)
	Crisis in importer (t-2)	-0.009**	-0.010*	0.040***	0.066***
		(0.004)	(0.005)	(0.013)	(0.022)
	Crisis interaction (t-2)		-0.016		0.191
			(0.050)		(0.216)
	Crisis in exporter (t-3)	-0.129***	-0.107***	0.082**	0.073**
		(0.044)	(0.033)	(0.035)	(0.031)
	Crisis in importer (t-3)	-0.002*	-0.001	0.051***	0.013*
		(0.001)	(0.003)	(0.015)	(0.007)
	Crisis interaction (t-3)		-0.000		0.019
			(0.004)		(0.030)
	Log exporter's real GDP PC	0.260***	0.318***	-0.154	-0.102
		(0.089)	(0.108)	(0.167)	(0.150)
	Log importer's real GDP PC 0.319***	0.310***	-0.189	-0.172
		(0.106)	(0.100)	(0.195)	(0.199)
	Log real exchange rate	-0.015*	-0.015*	0.003	0.007
		(0.008)	(0.009)	(0.010)	(0.013)
	Average supply shock	-0.078**	-0.102*	0.093***	0.016
		(0.033)	(0.059)	(0.027)	(0.033)
	Average demand shock	-0.113**	-0.044	0.068***	0.028
		(0.049)	(0.107)	(0.021)	(0.035)
	Observations	27981	27981	27225	27225
	R-squared	0.808	0.871	0.819	0.869

Note : The dependent variable is the rate of export entry or exit of firms by country-destination. The regressions include year and exporter-importer fixed effects. Heteroskedasticity-robust standard errors are reported in parentheses. ***, **,

Table 4 -

 4 Crises and product dynamics

		(1)	(2)	(3)	(4)
	Dependent variable	Product entry rate of incumbents Product exit rate of incumbents
	Crisis in exporter	-0.020**	-0.020**	0.184	0.182
		(0.010)	(0.009)	(0.205)	(0.206)
	Crisis in importer	-0.095***	-0.086*	0.133**	0.129**
		(0.032)	(0.028)	(0.060)	(0.057)
	Crisis interaction		-0.101*		0.002**
			(0.059)		(0.000)
	Crisis in exporter (t-1)	-0.145**	-0.146**	0.204	0.204
		(0.045)	(0.048)	(0.213)	(0.194)
	Crisis in importer (t-1)	-0.171***	-0.170***	0.237***	0.228***
		(0.050)	(0.051)	(0.065)	(0.073)
	Crisis interaction (t-1)		-0.128**		0.032*
			(0.058)		(0.018)
	Crisis in exporter (t-2)	-0.088***	-0.083***	0.107	0.105
		(0.029)	(0.026)	(0.148)	(0.146)
	Crisis in importer (t-2)	-0.115*	0.098*	0.125***	0.125***
		(0.066)	(0.056)	(0.043)	(0.039)
	Crisis interaction (t-2)		-0.041		0.219
			(0.052)		(0.287)
	Crisis in exporter (t-3)	-0.036**	-0.037**	0.060	0.068
		(0.017)	(0.017)	(0.075)	(0.078)
	Crisis in importer (t-3)	-0.097*	-0.094*	0.103***	0.101***
		(0.056)	(0.056)	(0.035)	(0.033)
	Crisis interaction (t-3)		-0.003		0.144
			(0.003)		(0.205)
	Log exporter's GDP PC	0.448***	0.439***	-0.099***	-0.092***
		(0.131)	(0.133)	(0.033)	(0.030)
	Log importer's GDP PC 0.281***	0.280***	-0.108**	-0.108**
		(0.080)	(0.087)	(0.049)	(0.048)
	Log real exchange rate	-0.017	-0.023*	0.062	0.063
		(0.038)	(0.013)	(0.116)	(0.129)
	Average supply shock	-0.072***	-0.102*	0.068	0.016
		(0.033)	(0.059)	(0.071)	(0.033)
	Average demand shock	-0.119***	-0.044	0.093***	0.028
		(0.049)	(0.107)	(0.027)	(0.035)
	Observations	21006	21006	20989	20989
	R-squared	0.740	0.752	0.717	0.719

Note : The dependent variable is the rate of export entry or exit of products by country-destination. The regressions include year and exporter-importer fixed effects. Heteroskedasticity-robust standard errors are reported in parentheses. ***, **,

Table 5 -

 5 Crises and destination dynamics

		(1)	(2)	(3)	(4)
	Dependent variable	Destination entry rate Destination exit rate
	Crisis	-0.044*** -0.049*** 0.005**	0.004**
		(0.012)	(0.014)	(0.002)	(0.001)
	Crises (t-1)	-0.102***	-0.094*	0.014	0.014
		(0.036)	(0.026)	(0.094)	(0.083)
			(0.271)		(0.008)
	Crises (t-2)	-0.015	-0.014	0.028	0.034
		(0.063)	(0.061)	(0.077)	(0.061)
	Crisis (t-3)	-0.080*** -0.083***	0.009	0.011
		(0.023)	(0.025)	(0.037)	(0.022)
	Crises (t-4)	-0.041**	-0.044** 0.085*** 0.079***
		(0.018)	(0.019)	(0.028)	(0.020)
	Relative GDP PC	0.149*	0.136*	-0.018**	-0.012*
		(0.087)	(0.079)	(0.007)	(0.006)
	Log REER	-0.133**	-0.133**	0.099*	0.087
		(0.060)	(0.061)	(0.056)	(0.089)
	Average supply shock	-0.078**	-0.102*	0.093***	0.016
		(0.033)	(0.059)	(0.027)	(0.033)
	Average demand shock -0.113**	-0.044	0.068***	0.028
		(0.049)	(0.107)	(0.021)	(0.035)
	Observations	17126	17126	17004	17004
	R-squared	0.783	0.792	0.749	0.765
	Note : The dependent variable is the entry or exit rate of products in exports by country and product
	category (HS-3 digit). REER stands for Real Effective Exchange Rate. The regressions include year and
	exporter-importer fixed effects. Heteroskedasticity-robust standard errors are reported in parentheses.

Table 6 -

 6 Crises and product dynamics : countries at the top 50% of capital account

	openness
	Dependent variable	Firm entry rate of incumbents	Firm exit rate of incumbents
	Crisis in exporter	-0.005	-0.011	0.066	0.110
		(0.006)	(0.083)	(0.105)	(0.174)
	Crisis in importer	-0.024*	-0.011**	0.029	0.106
		(0.013)	(0.004)	(0.083)	(0.209)
	Crisis interaction		-0.112		0.038
			(0.145)		(0.073)
	Dependent variable	Product entry rate of incumbents	Product exit rate of incumbents
	Crisis in exporter	-0.003	-0.011	0.098	0.205
		(0.014)	(0.083)	(0.193)	(0.219)
	Crisis in importer	-0.026	-0.012*	0.037	0.049
		(0.105)	(0.006)	(0.082)	(0.128)
	Crisis interaction		-0.034		0.011
			(0.119)		(0.018)
	Dependent variable Destination entry rate of incumbents Destination exit rate of incumbents
	Crisis in exporter	-0.012	-0.011	0.184	0.182
		(0.101)	(0.109)	(0.205)	(0.206)
	Crisis in importer	-0.073	-0.102*	0.028	0.029
		(0.088)	(0.058)	(0.054)	(0.061)
	Crisis interaction		-0.038		0.077
			(0.116)		(0.131)

Table 7 -

 7 Crises and product dynamics : countries at the top 50% of capital account

	openness
	Dependent variable	Firm entry rate of incumbents	Firm exit rate of incumbents
	Crisis in exporter	-0.104**	-0.139***	0.018*	0.026
		(0.045)	(0.042)	(0.010)	(0.102)
	Crisis in importer	-0.127**	-0.073	0.172	0.178
		(0.011)	(0.80)	(0.199)	(0.204)
	Crisis interaction		-0.295*		0.306
			(0.170)		(0.327)
	Dependent variable	Product entry rate of incumbents	Product exit rate of incumbents
	Crisis in exporter	-0.029**	-0.034**	0.217*	0.177
		(0.012)	(0.014)	(0.127)	(0.200)
	Crisis in importer	-0.100***	-0.109**	0.096	0.134*
		(0.030)	(0.047)	(0.107)	(0.078)
	Crisis interaction		-0.168*		0.206
			(0.097)		(0.204)
	Dependent variable Destination entry rate of incumbents Destination exit rate of incumbents
	Crisis in exporter	-0.080*	-0.080	0.118	0.110
		(0.046)	(0.040)	(0.135)	(0.201)
	Crisis in importer	-0.116**	-0.098*	0.093**	0.088
		(0.050)	(0.056)	(0.040)	(0.107)
	Crisis interaction		-1.127**		0.864*
			(0.341)		(0.499)

These country reports are available at http://www.enterprisesurveys.org/Reports.

We also used the[START_REF] Braun | Financial Contractibility and Asset Hardness[END_REF] index of asset tangibility and the results are consistent with those that are presented.

This is consistent with the argument of[START_REF] Do | The Direction of Causality between Financial Developent and Economic Growth[END_REF], that is changes in trade patterns result in changes in changes in demand for external financing, thus influencing developments in the financial sector.

Études et Documents n°26, CERDI, 2015

Information on the top trading partners is taken from the IMF DOT Statistics available at : http://elibrary-data.imf.org/DataExplorer.aspx.

Youssouf Kiendrebeogo

This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). http://www.cerdi.org/