Jordi Sánchez-Riera

Pascal Fua

Francesc Moreno-Noguer

Simultaneous Pose, Correspondence and Non-Rigid Shape *

Recent works have shown that 3D shape of non-rigid surfaces can be accurately retrieved from a single image given a set of 3D-to-2D correspondences between that image and another one for which the shape is known. However, existing approaches assume that such correspondences can be readily established, which is not necessarily true when large deformations produce significant appearance changes between the input and the reference images. Furthermore, it is either assumed that the pose of the camera is known, or the estimated solution is pose-ambiguous.

In this paper we relax all these assumptions and, given a set of 3D and 2D unmatched points, we present an approach to simultaneously solve their correspondences, compute the camera pose and retrieve the shape of the surface in the input image. This is achieved by introducing weak priors on the pose and shape that we model as Gaussian Mixtures. By combining them into a Kalman filter we can progressively reduce the number of 2D candidates that can be potentially matched to each 3D point, while pose and shape are refined. This lets us to perform a complete and efficient exploration of the solution space and retain the best solution.

Introduction

Reconstructing 3D deformable surfaces from single images is one of the central goals in computer vision with a large number of applications in related fields such as robotics, computer graphics or augmented reality. When 3D-to-2D correspondences between an input image and another one for which the shape is known can be established, monocular 3D non-rigid reconstruction is a well understood problem effectively addressed by many recent works [START_REF] Ecker | Semidefinite programming heuristics for surface reconstruction ambiguities[END_REF][START_REF] Moreno-Noguer | Capturing 3D stretchable surfaces from single images in closed form[END_REF][START_REF] Perriollat | Monocular template-based reconstruction of inextensible surfaces[END_REF][START_REF] Salzmann | Reconstructing sharply folding surfaces: A convex formulation[END_REF][START_REF] Salzmann | Closed-form solution to non-rigid 3D surface registration[END_REF]. However all these methods rely on the quality of the matches, which are usually established by means of local image descriptors that may become unre-Figure 1. Simultaneous Correspondence and Non-Rigid Shape Reconstruction. Left column: Reference configuration for which we know the shape and a set of 3D points lying on it. Right Column: Given a set of 2D points in an input image in which the shape is unknown and the camera pose is different from that in the reference configuration, we seek to simultaneously establish the 3D-to-2D matches (second row) and retrieve the pose of the camera and the shape of the surface in the input image (third row, right image). Our approach solves the matching problem without considering texture information, which in this case is unreliable because the scene contains many repetitive patterns. In addition, we can handle certain amount of outliers and clutter points, shown as red crosses and circles, respectively, in the second row images. liable when the shape deformation in the reference and input images is significantly different, or when the surface texture contains repetitive patterns, such as the example shown in Fig 1 . In addition, existing approaches assume the pose of the camera with respect to the reference shape to be known and that does not change when capturing the input image.

In order to obtain a solution that performs robustly when texture is not a reliable cue and when the camera can freely move, we propose an approach that simultaneously solves for the correspondences, pose and 3D shape without using image appearance information, that is, just considering the 1 3D position and the 2D location of two unmatched sets of points. Inspired on a recent paper on rigid object pose estimation [START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF], we reduce the complexity of this problem using weak priors on pose and shape, that we learn from training data, and model as Gaussian Mixture Models. These priors let us to define a region in the image where to seek for the potential 2D candidates that may be assigned to each 3D point. Using a Kalman filter strategy this search region is progressively shrunk while the estimations of the pose and shape are refined. This is repeated for different combinations of pose and shape priors, in order to guarantee a complete exploration of the solution space. As shown in Fig. 1, we can recover shape and correspondences even under the presence of outliers and clutter.

To the best of our knowledge, this is the first approach addressing the simultaneous pose and correspondence problem for non-rigid objects. Indeed, several works have already been proposed for retrieving matches and the six degrees of freedom of the pose in rigid objects [START_REF] David | Softposit: Simultaneous pose and correspondence determination[END_REF][START_REF] Grimson | Localizing overlapping parts by searching the interpretation tree[END_REF][START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF][START_REF] Olson | Efficient pose clustering using a randomized algorithm[END_REF]. However, when reconstructing non-rigid surfaces many more variables need to be considered to account for the deformation degrees of freedom, which require from a different solution to make the problem tractable.

Related Work

Monocular 3D reconstruction of non-rigid surfaces is known to be a highly under-constrained problem that requires from prior information to solve it. The most common approach to limit the set of possible solutions is to represent the shape as a weighted sum of modes, either physically-based ones [START_REF] Cohen | Finite-element methods for active contour models and balloons for 2D and 3D images[END_REF][START_REF] Mcinerney | A finite element model for 3D shape reconstruction and nonrigid motion tracking[END_REF][START_REF] Metaxas | Constrained deformable superquadrics and nonrigid motion tracking[END_REF] or learned from training data [2, 5]. Estimating shape, then amounts to retrieving the weights of this linear combination, by minimizing an image-based objective function. However, since such functions usually are non-convex and have many local minima, these methods require from good initializations.

Several approaches have shown that shape may be recovered from a set of 3D-to-2D correspondences, between the 3D points of a reference shape and the 2D points on the input image [START_REF] Ecker | Semidefinite programming heuristics for surface reconstruction ambiguities[END_REF][START_REF] Moreno-Noguer | Capturing 3D stretchable surfaces from single images in closed form[END_REF][START_REF] Perriollat | Monocular template-based reconstruction of inextensible surfaces[END_REF][START_REF] Salzmann | Reconstructing sharply folding surfaces: A convex formulation[END_REF][START_REF] Salzmann | Closed-form solution to non-rigid 3D surface registration[END_REF]. These point correspondences use to be computed by SIFT-like local image descriptors [START_REF] Belongie | Shape matching and object recognition using shape contexts[END_REF][START_REF] Lepetit | Keypoint recognition using randomized trees[END_REF][START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF][START_REF] Mikolajczyk | A performance evaluation of local descriptors[END_REF], which have been proved to be robust to certain transformations of the image, although they are prone to fail when dealing with the large nonlinearities produced in the deformation of non-rigid surfaces. Recent works have addressed the problem of general deformations [START_REF] Cheng | A deformable local image descriptor[END_REF][START_REF] Ling | Deformation invariant image matching[END_REF]. While these are promising approaches, they are still based on assumptions that can only rarely be satisfied, such as that intensity variations due to deformations are minimal.

This brings us to the situation where image intensity becomes an unreliable cue to establish correspondences, for example because the deformation itself creates artifacts on the image, such as self-shadowing or strong occlusions, or because the surface texture contains repetitive and undistinguishable patterns. In these cases it becomes necessary to simultaneously retrieve the shape and establish the correspondences. This problem has been deeply studied for rigid objects, in which case is then necessary to simultaneously estimate pose and correspondences. The most traditional approach in doing so is the RANSAC algorithm [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF], which, given a set of 3D and 2D points, iteratively hypothesizes and validates small subsets of correspondences until a good solution is found. Many variants of this strategy have been proposed over the years to reduce the computational cost [START_REF] David | Softposit: Simultaneous pose and correspondence determination[END_REF][START_REF] Grimson | Localizing overlapping parts by searching the interpretation tree[END_REF][START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF][START_REF] Olson | Efficient pose clustering using a randomized algorithm[END_REF].

In fact, our approach is inspired in [START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF], a recent work that uses priors on the camera pose to register rigid objects. However, in order to address the problem for non-rigid surfaces, besides pose and correspondences, we will have to estimate additional variables accounting for the degrees of freedom of the surface deformation. This, considerably increases the complexity of the problem and we believe that our approach is the first one to simultaneously solve for correspondences, pose and non-rigid shape from single images.

Shape and Pose Priors for Non-Rigid Surface Reconstruction

In this section we first show that the solution of our problem can be expressed as a minimization of an error function depending on the correspondences, pose, and shape parameters. We then show that weak priors on both pose and shape can be effectively used to retrieve the minimum of the error function, in spite of having to explore a high dimensional solution space.

Problem Statement and Notation

We assume we are given a set of 3D points P = {p ref 1 , . . . , p ref M } on a reference configuration with known shape, and a set of 2D points U = {u 1 , . . . , u N } on an input image of the same surface but with a different and unknown deformation. The correspondence between these two sets of points is also unknown.

We represent the surface as a triangulated 3D mesh with n v vertices v i concatenated in a vector x = [v ⊤ 1 , . . . , v ⊤ nv ] ⊤ , and denote by x ref the reference mesh, and x the mesh we seek to recover.

Let p i be a point on the mesh x corresponding to the point p ref i in the reference configuration. We can express p i in terms of the barycentric coordinates of the face it belongs:

p i = 3 j=1 a ij v [i] j , (1)
where the a ij are the barycentric coordinates and v

[i] j are the vertices of the face containing the point p i . Since we assume the mesh does not stretch while it deforms, these barycentric coordinates remain constant for each point and can be easily computed from points p ref i and the mesh x ref . Let us denote by A = {a 1 , . . . , a M } the set of barycentric coordinates associated to the 3D points, where

a i = [a i1 , a i2 , a i3 ] ⊤ .
Additionally, we assume we can model the mesh deformation as a linear combination of a mean shape x 0 and n m deformation modes Q = [q 1 , . . . ,

q nm ] x = x 0 + nm i=1 α i q i = x 0 + Qα , (2)
where α = [α 1 , . . . , α nm ] ⊤ are the unknown weights of the basis shapes. In our implementation, these modes were obtained by applying Principal Component Analysis (PCA) over a set of synthetic inextensible meshes generated with the modelling software Blender [11].

And finally, we assume we know the calibration matrix A of the camera, although we do not know its pose. As shown in Fig. 1, this may occur if the reference and input images were captured with different camera positions.

Given all these initial assumptions, our goal is to simultaneously retrieve the pose of the camera -parameterized by a rotation matrix R and a translation vector t-, the shape of the mesh x -parameterized by the vector α-, and as many 3D-to-2D correspondences as possible, considering that not all the 3D points must have a 2D match and viceversa.

Let us consider ũi to be the projection of a point p i given a pose and shape estimates {R, t, α}:

w i ũi 1 = A [R|t] p i 1 = AR 3 j=1 a ij x [i] 0j + Q [i] j α + At (3)
where w i is a scalar projective parameter and x

[i] 0j and Q

[i] j are the subvector of x 0 and submatrix of Q corresponding to the coordinates of the vertex v

[i] j . Note that in the second step we have used Eq. 1 and 2 to write the point p i in terms of the modal weights.

We can now formulate our problem as an optimization one, where we seek to retrieve the parameters R, t and α such that minimize the reprojection error between points p i projected on the image and their corresponding matches. This can be written as minimize

R,t,α M X i=1 Detected ( ũi -Match(ũi, U) ) (4)
where Match(ũ i , U) returns the 2D point of U that is closest to ũi , and Detected(d) =min(d, T ) is used to penalize points that are not correctly matched and avoid trivial solutions with just a few, but accurate, matches. In all our experiments we set T = 10 pixels.

Estimating Pose and Shape Priors

Trying to minimize Eq. 4 with no other information besides the deformation modes, becomes computationally prohibitive, even when using a small number of points. For example, the approach in [START_REF] David | Softposit: Simultaneous pose and correspondence determination[END_REF] is one of the most efficient algorithms in solving pose and correspondences in the rigid case and has a O(M N 2 ) complexity, which allows to handle problems with about a M ≈ 100 points in a reasonable amount of time. However, if on top of this complexity one has to consider the additional n m degrees of freedom introduced by the deformable model, the problem seems to be unsolvable in practice.

In this paper, we show that using weak priors on the camera pose and on the types of deformations the surface can have, we can turn our problem into a tractable one. To this end, we first compute pose and shape priors and parameterize them in terms of Gaussian Mixture Models. We then combine these priors, and using a Kalman filter strategy we guide the matching process for each 3D point. We show that the search space for potential 2D candidates is drastically reduced from the entire image, to a small region. We can then efficiently explore the space of possible solutions and keep the solution that minimizes Eq. 4.

Priors Computation

Pose priors are computed following a similar methodology as in [START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF]. Given the reference mesh, we initially generate sample positions on a large region where the camera center is expected to be. For example, for the 40 × 40 cm mesh shown in Fig. 2(Left), the region of possible camera positions we define is a 140 × 140 × 20 cm 3 volume above the mesh. Then, for each camera center sample, we consider a random direction of the optical axis by allowing the camera to point anywhere on the reference mesh. We also allow a random rotation of the camera around its optical axis. All these pose samples are then represented by 6-dimensional vectors accounting for the three degrees of freedom of the rotation and the three of translation. Using Expectation-Maximization (EM) [START_REF] Figueiredo | Unsupervised learning of finite mixture models[END_REF] we model them as a Gaussian Mixture Model, with {ρ p 1 , . . . , ρ p Gp } mean poses with associated 6 × 6 covariances {Σ p 1 , . . . , Σ p Gp }, and probabilities {p p 1 , . . . , p p Gp }. Fig. 2(Left) shows the part of the pose priors corresponding to the camera centers, with their associated 3 × 3 covariance matrices.

As mentioned above we represent the surface as a linear combination of n m basis shapes computed by applying PCA on a large training set of synthetically deformed meshes. Therefore, each shape is indeed represented by a n m -vector of weights α. Shape priors are then computed by applying EM over the weights of all training meshes. This results in G s mean shape vectors {ρ s 1 , . . . , ρ s Gs } with associated n m × n m covariances {Σ s 1 , . . . , Σ s Gs } and probabilities {p s 1 , . . . , p s Gs }. Fig. 2(Middle) shows the region The small green dots are the samples of camera centers we use to compute the pose priors. The ellipsoids represent the covariance of the priors on the translational space. We obtain similar covariances for the three-dimensional rotational space, although we do not plot them here. Middle-Right: Shape Priors. We represent the shape by a 30-dimensional vector of modal weights, and compute the shape priors in this space. The blue shaded area represents the region where shape samples are generated and the meshes on the right side are four of such samples. Expectation-Maximization is then applied over these samples to compute the shape priors. The color lines in the middle image represent the mean vector for three of these priors.

of the shape space where samples were generated and the mean vectors of three shape priors.

It is worth to mention that the kind of priors on the modal weights we use are more restrictive than the regularization terms introduced by current techniques to penalize the less meaningful modes. This lets us to retrieve the shape without imposing the additional constraints that these methods consider, based on local inextensibility [START_REF] Ecker | Semidefinite programming heuristics for surface reconstruction ambiguities[END_REF][START_REF] Perriollat | Monocular template-based reconstruction of inextensible surfaces[END_REF][START_REF] Salzmann | Reconstructing sharply folding surfaces: A convex formulation[END_REF][START_REF] Salzmann | Closed-form solution to non-rigid 3D surface registration[END_REF] or shading information [START_REF] Moreno-Noguer | Capturing 3D stretchable surfaces from single images in closed form[END_REF].

Weighting Joint Priors

In the next subsection we will use different combinations of pose and shape priors to simultaneously solve for the correspondences while pose and shape are refined. We could, in principle, explore all possible combinations of {(ρ p i , Σ p i ), (ρ s j , Σ s j )} for i = 1, . . . , G p and j = 1, . . . , G s . However, we prefer not doing so because this has a high computational cost, and to avoid having to explore all the G p • G s prior combinations, the algorithm will be stopped when Eq. 4 drops below a certain threshold. In order to further accelerate this convergence we will not explore the joint priors following a random ordering. Instead, we will explore them according to the ordering determined by a weight computed as follows.

Given the 2D image points U, a combination of the i-th pose and j-th shape priors will be assigned a weight proportional to the joint probability p(i, j|U) ∝ p(U|i, j)p p i p s j [START_REF] Cootes | Active appearance models[END_REF] where p(U|i, j) is computed by projecting the points {p 1 , . . . , p M } onto the image assuming a pose ρ p i and a shape ρ s j , and comparing these projected points with the actual distribution U. The comparison is done in terms of point cloud Hausdorff distance and, since 3D-to-2D correspondences are not explicitly solved, it can be computed very fast. Although this is just an approximate measurement, in the Results Section we will show that this ordering lets us to terminate the algorithm when only a small percentage of the joint priors are explored.

Using Priors to Simultaneously Solve for Pose, Shape and Correspondences

We will now use the joint priors to guide the 3D-to-2D matching process while pose and shape are progressively refined. Starting with the pair of priors {(ρ p , Σ p ), (ρ s , Σ s )} that was scored the most probable according to Eq. 5, our method is based on the following iterative algorithm.

Computing initial set of potential matches

We denote by ρ p,s = [ρ p⊤ , ρ s⊤ ] ⊤ the initial value of our state vector accounting for pose and shape. Given this initial estimate we use Eq. 3 to project the set of 3D points P, represented by their barycentric coordinates A, onto the image plane at positions Ũ = {ũ 1 , . . . , ũM }.

In order to define a search region for each point ũi in the image plane, we propagate the covariance matrices according to

Σ u i = J(a i )Σ p,s J(a i ) ⊤ (6)
where J(a i ) is the 2 × (6 + n m ) Jacobian of Eq. 3 with respect to the 2D coordinates evaluated on the barycentric coordinates of the point p i , and Σ p,s is the (6 + n m ) × (6 + n m ) block diagonal covariance matrix, in which the diagonal elements are Σ p and Σ s . We can then define an elliptical search region on the image plane by considering the potential matches for ũi , as the u j ∈ U such that

(u j -ũi ) ⊤ (Σ u i ) -1 (u j -ũi ) ≤ M 2 (7)
where M is a threshold chosen to guarantee a specified level of confidence, and its value can be computed using the cumulative chi-squared distribution. In our experiments we set M = 3 to achieve a 99% degree of confidence.

Ground Truth Iteration #0 Iteration #3 Iteration #6 Iteration #9 Error

Figure 3. Shape Convergence. When correct matches are iteratively assigned the shape estimated by the Kalman Filter converges to the ground truth solution. Observe that at iteration #9 only slight differences remain between the estimated shape and the ground truth one.

Iteratively refining pose and shape

Given the set of 3D points and their potential 2D candidates, we could now follow a RANSAC-based approach and hypothesize sets of 3D-to-2D correspondences and validate them with any of the current techniques to retrieve shape from such correspondences [START_REF] Ecker | Semidefinite programming heuristics for surface reconstruction ambiguities[END_REF][START_REF] Moreno-Noguer | Capturing 3D stretchable surfaces from single images in closed form[END_REF][START_REF] Perriollat | Monocular template-based reconstruction of inextensible surfaces[END_REF][START_REF] Salzmann | Reconstructing sharply folding surfaces: A convex formulation[END_REF][START_REF] Salzmann | Closed-form solution to non-rigid 3D surface registration[END_REF]. However, these methods are based on a relatively large number of correspondences -about n m -, which would require from an excessively large number of hypotheses to guarantee retrieving a correct solution.

More specifically, let us assume that all the 3D points have the same number n << N of potential 2D matches. If p d is the probability that a 3D point is detected and is not an outlier, the probability of n m correct matches is (p d /n) nm . Hence, the number of hypotheses-and-validations that are needed to ensure with a probability R that at least one of them is correct will be:

Number Trials Constant Search Region = n p d nm log 1 1 -R (8)
In our experiments we use n m = 30 deformation modes, and hence, even for relatively small search regions yielding only a few 2D potential candidates n for each 3D point, the number of trials would become prohibitive.

In order to reduce this theoretical high number of trials, we will use a Kalman filter formulation, where correspondences are progressively done for different 3D points, and after each match the number of potential 2D candidates for the rest of 3D points is reduced.

We start considering the 3D point with less potential candidates. Let ũi ∈ Ũ be the image projection of such a point, and u j ∈ U the 2D point closest to ũi in terms of the Mahalanobis distance of Eq. 7. By establishing an hypothetical match between ũi and u j , we can use the Kalman filter equations to update the state vector ρ p,s and reduce the covariance matrix Σ p,s in the pose-shape space:

ρ p,s+ = ρ p,s + K(u j -ũi ) (9) Σ p,s+ = (I -KJ(a i )) Σ p,s (10)
where K is the Kalman gain and I is the (6+n m )×(6+n m ) identity matrix. The new pose, shape and covariance matrices are used to project again the 3D points onto the image, with the re-newed search regions which are considerably smaller than those in the previous iteration. This allows to reduce the number n of potential 2D matches that can be associated to each 3D point. A second match is then established and the Kalman update equations applied again. After a few iterations -less than 10 in practice-of repeating this process the Kalman filter converges to a solution, although we do not know yet if it corresponds to the correct one.

To assess the accuracy of the solution we match the remaining 3D points. For that, we project them onto the image plane and match to their nearest neighbor from U. We then compute the error of Eq. 4, and if it is below a given threshold the algorithm is stopped. Otherwise we repeat the whole process of establishing correspondences and updating shape and pose with different combinations of the potential candidates, and by iterating over the various pose and shape joint priors according to the ordering determined in Section 3.2.2. Fig. 3 shows how the shape estimation converges to the true solution when correct matches are established.

Let us now analyze the computational complexity of the solution we propose. If we denote by n it << n m the number of iterations until convergence of the Kalman filter, and by Shrink( ) a monotonically decreasing function accounting for the reduction of the search region size, we can rewrite Eq. 8 by

Number Trials

Shrunk Search Region = " Q n it i=1 Shrink(i) • n p nm d « log " 1 1-R « (11)
Since the number of iterations n it ≈ 10 is considerably smaller than the number of modes n m = 30 and the product Shrink(i) • n rapidly drops to 1, the final amount of theoretical trials that our approach requires is much smaller than when considering search regions of constant size.

However, still one critical element remains in Eq. 11, which is the increase in complexity produced by the number of outliers, represented by the percentage p d of detected 3D points. For example, if we had 20% of outliers, that is p d = 0.8, the term (1/pd) nm would represent having to evaluate 870 times more hypotheses than when all the points are detected. And for p d = 0.6, this factor would grow up to 4.5 × 10 6 .

In order to handle this situation, we consider that each point ũi can be either matched to any of its potential matches within the search region of Eq. 7, or it can be an outlier. Since correspondences are iteratively established while pose and shape are refined, the probability that O consecutive 3D points are considered as outliers, decreases with O. Thus, we limit the maximum number n co of consecutive 3D points labelled as "outliers". In all our experiments we fixed this maximum value to n co = 0.25M p d . For M = 50 3D points, and p d = 0.6 this would mean that we only should have to evaluate (1/pd) nco = 46 times more hypotheses than for the case p d = 1.

Results

We now present the results on both synthetic and real data. In the synthetic results we compare our approach to [START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF], which we denote by BPnP. As discussed in Section 2 this work uses priors on the camera pose to simultaneously retrieve pose and correspondences for rigid objects.

Synthetic Experiments

Using the Blender software [11], we synthesized 25 frames of a deforming 40 × 40 cm flag approximated by a 32 × 32 mesh such as the one shown in Fig. 3(Left). In order to generate the input data we randomly distributed M 3D points on the surface of a reference planar mesh. Assuming a virtual camera placed approximately 70 cm upon the reference mesh, a percentage p d of the 3D pointsaccounting for the number of detected points-were projected onto the input image of a deformed mesh. A 2 pixel variance Gaussian noise was added to these projections. Furthermore, to account for clutter, a percentage p c of 2D points were added at random positions in the image plane.

Fig. 2 shows the kind of priors we used. For the pose, we generated G p =20 Gaussian priors distributed in a relatively large volume over the mesh. The shape priors were approximated by G s = 5 Gaussian distributions in the n m = 30dimensional space of the modal weights.

In order to evaluate our approach we proceeded similarly as in [START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF]. For each one of the 25 meshes, we performed several experiments by changing the number of 3D model points M = {30, 40, 50, 60}, the fraction p d = {0.8, 0.6} of detected points -corresponding to 20% and 40% occlusion rates, respectively-, and the percentage of clutter p c = {0.2, 0.4, 0.6, 0.8}. In addition, we did 10 different trials for each combination of these parameters, yielding a total of 25 × 4 × 2 × 4 × 10=8000 experiments.

We report the accuracy of our method in terms of a convergence ratio we define by comparing the pose and shape we retrieve with their ground truth values. More specifically, let ρ p true , ρ s true be the true pose and shape, and let ρ p , ρ s be our respective estimates. We computed the relative errors by E p = ρ p trueρ p / ρ p true and E s = ρ s trueρ s / ρ s true , and considered a solution did converge if both E p and E s were below 0.1. The convergence of the BPnP was computed by just considering the percentage of error in the pose.

The results of all experiments are summarized in Fig. 4. Each graph plots the mean convergence as a function of M , p c and p d . Observe that for most experimental conditions our approach guarantees a convergence ratio above 0.8. Only slightly lower values are obtained for large percentages of clutter and occlusions.

In Fig. 5 we compare the performance of our approach to that of the BPnP, in two meshes with different levels of deformation. As shown in the left column of Fig. 5, when the deformation of the input mesh is relatively similar to the planar reference configuration, BPnP still yields valid solutions with large percentages of convergence, almost comparable to those obtained with our approach. However, as shown in the right column of Fig. 5, when dealing with meshes whose deformation significantly differs from that of the reference configuration, the accuracy of the BPnP rapidly deteriorates, while we still obtain good results.

In terms of computation time our approach needs approximately 10, 15, 20 and 25 minutes per frame to compute a solution for M = 30, 40, 50 and 60 model points, respectively, when all the joint priors on pose and shape need to be explored. This is about 10 times more than the time required by BPnP, although this was indeed expected because we are seeking the solution in a space of much higher dimensionality and we are considering a much larger number of priors. Nevertheless, when exploring the priors combination following the ordering defined in Section 3.2.2, in many of the experiments we obtained a good solution without having to consider all of the joint priors. This let us terminate the algorithm much faster. Fig. 6

Percentage of Clutter

Our Approach-Occlusion 20% Our Approach-Occlusion 40% BPnP-Occlusion 20% BPnP-Occlusion 40%

Figure 5. Comparing the convergence rate of our approach and BPnP [START_REF] Moreno-Noguer | Pose priors for simultaneously solving alignment and correspondence[END_REF]. Left Column: Results over a mesh with small deformation compared to a planar reference shape. Right Column:

Results over a mesh with large deformation.

and G s = 5. Note that although this yields a total of 100 possible joint priors, for most of the experiments the algorithm converged to a solution after exploring less than 20 of them.

Real Experiments

We also evaluated our approach on a real 150 framessequence of the 16×16 cm bending paper already introduced in Fig. 1, which is textured with repetitive patterns of musical notes. This is a clear example where texture is not a reliable cue for establishing matches, and we can take advantage of a technique like ours that uses geometry alone.

We initially acquired a reference image under a planar configuration, as shown in the top-left image of Fig. 1, and using SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] we detected a set of interest points, and computed their 3D barycentric coordinates with respect to the mesh vertices. For each input image the set of 2D points was computed using the same detector. Since SIFT usually returns several of interest points per image but our algorithm only can handle in a reasonable amount of time about M = 60 model points, we just kept those with larger gradient values. Note however, that this is the typical amount of points that state-of-the-art algorithms solving the simultaneous pose and correspondence problem for rigid objects can handle.

Pose priors were defined in a region of about 1m 3 above the mesh, large enough to ensure it contained the real camera pose. Shape priors, were computed applying the methodology explained in Section 3.2.1 over synthetic sequences of meshes resembling paper deformations.

Figure 7 shows the reconstruction obtained in several frames of the sequence. Observe that with the proposed approach we are able to retrieve shape in situations where non-rigid deformations and repetitive patterns, might lead to failure the algorithms that just rely on texture for establishing correspondences.

Finally, we also applied our approach to simultaneously estimate the camera pose and shape of a sail. Fig. 8-Left shows the priors we used on the translational component of the pose. Fig. 8-Right shows the results obtained in two different frames, where the camera pose we retrieve is represented on a coordinate system fixed on the sail.

Conclusion

Many recent approaches to monocular 3D shape reconstruction rely on the fact that point correspondences can be readily established between the input image and a reference image in which the shape and pose are known. However, there are cases where it is difficult to compute such matches, for example, when the surface deformation in the input image highly differs from that in the reference one, or when the surface texture contains repetitive patterns. Furthermore, the camera may move, and hence, its pose can no longer be considered to be known.

In such cases it is necessary to simultaneously estimate the camera pose, shape and correspondences, which is a computationally prohibitive problem unless additional constraints are considered. In this paper we have presented an approach to turn this problem into a tractable one, by using weak priors on both pose and shape, and modelling them using Gaussian Mixture Models. We have shown that using a Kalman filter strategy such priors can be progressively refined while solving for the correspondences to an ever smaller number of possible matches.

In future work, we plan to integrate additional sources of information, such as texture and motion. For instance, although we have shown that texture may be in some occasions an unreliable cue, we could nevertheless use its information as additional prior with an associated uncertainty. We believe that this would drastically reduce the number of potential 2D matches for each 3D point, yielding faster and more accurate solutions.

Figure 2 .

 2 Figure2. Left: Pose Priors. The small green dots are the samples of camera centers we use to compute the pose priors. The ellipsoids represent the covariance of the priors on the translational space. We obtain similar covariances for the three-dimensional rotational space, although we do not plot them here. Middle-Right: Shape Priors. We represent the shape by a 30-dimensional vector of modal weights, and compute the shape priors in this space. The blue shaded area represents the region where shape samples are generated and the meshes on the right side are four of such samples. Expectation-Maximization is then applied over these samples to compute the shape priors. The color lines in the middle image represent the mean vector for three of these priors.

Figure 4 .

 4 Figure 4. Convergence ratios in the synthetic experiments.

Figure 6 .

 6 Figure 6. Number of Joint Priors Explored. If the combination of pose and shape priors are evaluated following the ordering defined in Section 3.2.2, our algorithm converges to a solution without having to explore all of the joint priors. The figure plots the distribution of number of explored priors over a maximum of 100, for 200 different experiments.

Figure 7 .Figure 8 .

 78 Figure 7. Reconstruction of a bending paper. Top: Mesh recovered using the proposed approach overlaid on the original image. Botton: Reconstructed mesh seen from a different viewpoint.

  plots a histogram of the number of joint priors explored in 200 experiments with G p = 20

			Small Deformation				Large Deformation
	Convergence Ratio	0.2 0.4 0.6 0.8 1	Our Approach-Occlusion 20% Our Approach-Occlusion 40% BPnP-Occlusion 20% BPnP-Occlusion 40%	Convergence Ratio	0.2 0.4 0.6 0.8 1
		0.2	0.4 Percentage of Clutter 0.6	0.8		0.2	0.4	0.6	0.8

This work has been partially funded by the Spanish Ministry of Science and Innovation under projects 200850I055, DPI2008-06022, and Consolider Ingenio 2010 CSD2007-00018, by EU GARNICS project FP7-247947, and by the Swiss National Science Foundation.


A Pichard

J Mrázek

M Assié

M Hass

M Honusek

G Lhersonneau

F De Oliveira

Santos

M.-G Saint-Laurent

E Šimečková

F De Oliveira De Santos

A new cross section measurement of reactions induced by 3 He-particles on a carbon target

Keywords: 25, 55, -e 3 H, 3 He

The production of intense beams of light radioactive nuclei can be achieved at the SPIRAL2 facility using intense stable beams accelerated by the driver accelerator and impinging on light targets.

The isotope 14 O is identified to be of high interest for future experiments. The excitation function of the production reaction 12 C( 3 He, n) 14 O was measured between 7 and 35 MeV. Results are compared with literature data. As an additional result, we report the first cross-section measurement for the 12 C( 3 He, α+n) 10 C reaction. Based on this new result, the potential in-target 14 O yield at SPIRAL2 was estimated: 2.4×10 11 pps, for 1 mA of 3 He at 35 MeV. This is a factor 140 higher than the in-target yield at SPIRAL1.

a very intense beam of 3 He. However, large discrepancies are observed in the various excitation functions available in the literature [START_REF] Cirilov | [END_REF][9][10][11]. Moreover, above 10 MeV a strong difference appears between the integrated yield deduced from Ref. [11] and a thick target saturation activity measurement [12]. A single measurement using a thick target in order to fully stop the primary beam of 3 He could seem sufficient in order to estimate the yield for SPIRAL2. However, the beam power withstood by a target depends on primary beam intensity and energy. Good knowledge of the excitation function helps to choose the best primary beam energy, considering also beam power. Consequently, we decided to perform a new measurement of this cross section by an activation method. The production rate is deduced from the residual activity of thin carbon foils irradiated by the 3 He primary beam. Special attention has been devoted to the measurement normalisation. Apart from a standard current measurement, we counted the activity of aluminium and cobalt foils irradiated simultaneously by the primary beam. As a matter of fact, excitation functions for 27 Al( 3 He, 2p) 28 Al [13] and 59 Co( 3 He, x) [14] - [20] reactions are reported in literature. A complementary results of this experiment is the first measurement of the excitation function of the 12 C( 3 He, α+n) 10 C reaction.

Experimental setup

The experiment took place at NPI (Nuclear Physics Institute), Řež, Czech Republic. The isochronous cyclotron U-120M can deliver 3 He 2+ beams at energies up to 38 MeV. Measurements have been performed with two differ- keV and 2319 keV with branching ratio and uncertainty of 92.7(0.5) % and 82 (3) % respectively [START_REF]Nuclear database of NNDC[END_REF]. This gave an efficiency point very close to the 2314 keV of interest.

Each target counting lasted 300 s. Figure 3 shows a γ-ray spectrum from 35.3 MeV 3 He irradiation. The Co foils were counted until several weeks after irradiation due to the long life-time of some of the elements produced. Their counting rate being consequently lower, the detector was in a lead castle in order to limit the natural background.

The distance between the target and the detector was 5.5

cm. There was no shielding between them. The Germanium counting rate was typically 50 pps.

Data Evaluation

Basic nuclear data and threshold of the contributing reactions are collected in table 1. The 1238 keV 56 Co γ-line

was not used because it was strongly contaminated by the room background. The number of incident particles was obtained by real time measurement of the current collected on a metallic plate placed at the back of the target and on the degrader rack. Simultaneous irradiation of aluminium and cobalt foils by the beam allowed a check of this measurement:

-The cross section for 27 Al( 3 He, 2p) 28 Al is known to good accuracy [13]. The 28 Al (2.2 m) yield measured in the 10-µm aluminium foils placed at the back of the carbon target allowed us to calculate the 3 He integral intensity received by the target at each irradiation.

-The excitation functions of reactions induced by 3 He on cobalt being known too [14][15][16][17][18][19][20], a stack of 3 µm-thick cobalt foils placed on the degrader rack also allowed for checking the normalisation.

The cross-section uncertainties have been estimated by summing quadratically the uncertainties of the different contributing processes:

-The dead time of counting is monitored by a pulse gen-erator. Its accuracy is always better than 1 %.

-Accuracies on branchings per decay of the monitored γ-rays, available from literature, are given in table 1.

-The accuracy on the efficiency of the γ detector, including transmission through the lead plates, is the main contribution to the total error. It is estimated to be about 6 % for the 2314 keV line of 14 O and the 1779 keV line of 28 Al. For the long counting of cobalt foils it is 10 % and about 25 % for the 718 keV line of 10 C.

-The number of target atoms in each disk is obtained by weighting. The accuracy is about 3 % for the aluminium and carbon disks and about 10 % for the thinner cobalt foils.

-The accuracy of the primary beam intensity is estimated to be 5 % from the current measurement device.

It is confirmed by the evaluation of activity generated by the beam in Aluminium foils, see section 4.4.1.

Results

Table 2 shows the measured cross-sections obtained for the Hahn and Ricci [9] seem to be consistent. For [START_REF] Cirilov | [END_REF], we used data from NNDC [START_REF]Nuclear database of NNDC[END_REF]. They show a 15 mb maximum at 5

MeV. We note that these values were modified compared to those of the original paper. The excitation function presented by Osgood et al. [10] has quite a similar structure but, while the absolute value is comparable for energies lower than 4 MeV, it then rises to reach a 35 mb maximum at 7 MeV. The only measurement at higher energies was done by Singh [11]

Thick target yield of 14 O

The thick-target yields are obtained by folding the energy loss of the primary beam in the target with the excitation functions of the 12 C( 3 He, n) 14 O reaction. Figure 6 shows a comparison of these yields with the target saturation activities measured by Nozaki and Iwamoto [12] (full dots).

The dashed line is calculated from the excitation function given by Singh [11]. As already mentioned in the in-

Cross-section for the production of 10 C

A by-product of this experiment is the measurement of the 10 C production cross-section. To our knowledge, this is the first report of experimental data for this reaction.

In the energy range used, the reaction mainly contributing to the production is 12 C( 3 He, α+ n) 10 C (see the threshold value in table 1). In figure 7, they are compared with values given by TENDL-09. The measured cross section is zero for the lowest energies and starts to increase at about 18 MeV. This is due to the fact that the threshold of the 12 C( 3 He,a+n) 10 C reaction is equal to 14 MeV. The cross section reaches about 12 mb at the highest energy; this corresponds to the maximum of the compound nucleus

Monitor reactions

Special care was taken for the normalisation. For this goal, aluminium and cobalt foils were irradiated simultaneously with the targets to monitor the primary beam intensity.

Aluminium foils

The aluminium foil was the last disk of the target stack. A part of 28 Al produced via 27 Al( 3 He, 2p) 28 Al had enough kinetic energy to escape from the 10 µm-Al disk. This fraction was calculated with the ISOL Catcher utility [START_REF]LISE++ : a simulation of fragment separators -Last modifications[END_REF] of the code LISE++. This fraction depends on the primary beam energy. At 34.6 MeV, it represents 23.5 % of Homma and Murakami [18] which are already known to be wrong [14], [16]. Indeed all these published results are not in agreement, as demonstrated by the values given in the right column of table 3 which exhibit large discrepancies. Thus, the ratio between all these published values (except [18]) and our measured values fluctuates between 0.25 and 1. 4).

Outlook

The present measurement of the 12 C( 3 He, n) 14 O crosssection confirm a strong reduction of the cross section at energies higher than 6 MeV. For comparison, the TALYS code was used to predict the cross sections. Despite the fact it is not made for these light nuclei, predictions are in rather good agreement with the measurements. The shapes of the excitation functions are well reproduced. It seems that the preequilibrium contribution is often underestimated, but no correction was made to correct this fact.

An 14 O beam is currently available at GANIL within the SPIRAL1 facility. It is produced via the fragmentation of an 16 O 95 MeV/A primary beam impinging onto a graphite target. An intensity of 3.2×10 5 particles per sec-ond after extraction and ionisation is available for experiments in an energy range from 3.2 up to 12.5 MeV/A [START_REF]Spiral beams -GANIL WEB PRO USERS[END_REF].

With the new facility SPIRAL2, the intense 3 He (1 mA / 43.5 MeV) beam delivered by the LINAC driver offers opportunities to raise the available 14 O beam intensity.

Based on our new result, we can estimate the potential in-target 14 O yield: 2.4×10 11 pps, for 1 mA of 3 He at 35

MeV. This is 140 times higher than the in-target yield at SPIRAL1. Preliminary simulations of the thermal conditions did show that the maximal intensity the production target can withstand is indeed close to 1 mA. A production system is currently being designed at GANIL and simultaneously more precise thermal simulations will be carried out with the designed target.

Acknowledgement

We thank the Řež cyclotron and technical crew for delivering the 3 He beam and for the nice working conditions.

We are also grateful to Dr. Lebeda for providing us the

Fig. 1 .

 1 Fig. 1. Schematic view of the irradiation station.

Fig. 3 .

 3 Fig. 3. γ-ray spectrum from 35.3 MeV 3 He irradiation on C/Al target.

12 C( 3

 3 He, n)14 O and 12 C( 3 He, α+n)10 C reactions and for the27 Al( 3 He, 2p)28 Al monitor reaction. Each target was irradiated and counted twice (three times at 35.3 MeV)to check for consistency. The values listed as σ give the measured cross-section for each irradiation, which are subsequently combined to obtain the resultant mean value σ r . The energy degradation ∆E inside the target and the mean primary beam energy < E > were calculated with the code LISE++[START_REF] Tarasov | [END_REF] using the Ziegler[START_REF] Littmark | Handbook of range, distributions for energetic ions in all elements[END_REF] low-energy model for both C and Al foils. A measurement with a car-

  who, however has normalized his values to those of Osgood et al. His excitation function decreases smoothly from 27 mb at 6 MeV to 12 mb at 30 MeV. Our only measurement at low energy (7 MeV) places sigma between the values given by Osgood et al., on the one hand, and those given by Cirilov et al. and by Hahn and Ricci, on the other hand. At higher energies, our values are a factor 4 lower than those of Singh.

Figure 5

 5 Figure5compares experimental and various TALYS[START_REF] Talys | [END_REF] calculations. The dotted line shows the cross section from

Fig. 4 .

 4 Fig. 4. Excitation functions for the 12 C( 3 He,n) 14 O reaction. : our data. •: Hahn and Ricci. +: Osgood et al.. : Singh.

Fig. 5 .

 5 Fig. 5. Excitation functions for the 12 C( 3 He,n) 14 O reaction. : our data. Dotted line: TENDL-09 data base calculated with TALYS using default parameters. Solid line: population of the 14 O ground state only, calculated with TALYS and default parameters. Long dashed line: the same but calculated with larger number of discrete states considered in all nuclei.

Fig. 6 .

 6 Fig. 6. Comparison of the thick target production yields for the 12 C( 3 He,n) 14 O reaction as a function of the primary beam energy. Dashed line: Singh. Solid lines: lower and upper estimates using our and combined data (see text for details). Full dots are Nozaki and Iwamoto target saturation activity measurements.

Fig. 7 .

 7 Fig. 7. Excitation functions for the 12 C( 3 He, α+n) 10 C reaction. : our data. Dotted line: TENDL-09 data base calculated with TALYS using default parameters. Solid line: ground state and first excited state calculated with TALYS and default parameters. Long dashed line: ground state with increasing the number of discrete levels considered with Hauser-Feshbach decay.

Fig. 8 .

 8 Fig. 8. Excitation function for the 27 Al( 3 He, 2p) 28 Al beam monitor reaction; : our data. : Frantsvog et al., : our data averaged on the same energy spread for comparison with the ones of Frantsvog et al.. Dashed line: TENDL-09 data base. Solid line: a eye guide of our data.

90

  Nb calibration source and to Dr. J. Kiener for helping us with the TALYS Code. This work has been supported by the IN2P3 -ASCR, by the European Community FP7 -Capacities -SPIRAL2 Preparatory Phase N o 212692, and by the Région Basse Normandie for a PhD thesis (A.P.).

Table 1 .

 1 Investigated nuclear reactions induced in C, Al and Co by 3 He particles irradiation

	Foil	Isotope	Half-life Eγ (keV) [21] Iγ (%) [21]	Reactions	Threshold (MeV)
	12 C	14 O	70.61 s	2312.6	99.388(11)	12 C( 3 He, n) 14 O	1.4
		10 C	19.26 s	718.4	100.	12 C( 3 He, α+n) 10 C	14.1
						12 C( 3 He, d+t) 10 C	36.1
						12 C( 3 He, n+p+t) 10 C	38.9
						12 C( 3 He, 3 He+2n) 10 C	39.8
	27 Al	28 Al	2.24 m	1778.85	100	27 Al( 3 He, 2p) 28 Al	0.
	59 Co	60 Cu	23.7 m	826.4	21.7(11)	59 Co( 3 He, 2n) 60 Cu	5.3
				1791.6	45.4(23)
		61 Cu	3.33 h	283.0	12.2(22)	59 Co( 3 He, n) 61 Cu	0.0
				656.0	10.8(20)
		56 Co	77.23 d	846.8	99.94(3)	59 Co( 3 He, 2n+α) 56 Co	10.3
				1037.8	14.17(13)	59 Co( 3 He, 2t) 56 Co	22.2
						59 Co( 3 He, n+d+t) 56 Co	28.8
						59 Co( 3 He, 2n+p+t) 56 Co	31.1
						59 Co( 3 He, 3n+ 3 He) 56 Co	31.9
		57 Co	271.7 d	122.1	85.60(17)	59 Co( 3 He, α+n) 57 Co	0.
				136.5	10.68(8)	59 Co( 3 He, d+t) 57 Co	16.8
						59 Co( 3 He, n+p+t) 57 Co	19.2
						59 Co( 3 He, 3 He+2n) 57 Co	20.0
						59 Co( 3 He, n+2d) 57 Co	23.4
						59 Co( 3 He, 2n+p+d) 57 Co	25.8
						59 Co( 3 He, 3n+2p) 57 Co	28.1
		58m+g Co 70.86 d	810.8	99.45	59 Co( 3 He, α) 58 Co	0.
						59 Co( 3 He, p+t) 58 Co	10.2
						59 Co( 3 He, 3 He+n) 58 Co	11.0
						59 Co( 3 He, 2d) 58 Co	14.4

59 Co( 3 He, n+p+d) 58 Co 16.7 59 Co( 3 He, 2n+2p) 58 Co 19.1 6 A. Pichard et al.: A new cross section measurement of reactions induced by 3 He-particles on a carbon target

Table 2 .

 2 Measured cross-sections for the 12 C( 3 He, n) 14 O, 12 C( 3 He, α+n) 10 C and 27 Al( 3 He, 2p) 28 Al reactions 12 C 27 Al 12 C( 3 He, n) 14 O 12 C( 3 He, α+n) 10 C 27 Al( 3 He, 2p) 28 Al The 12 C( 3 He,n) 14 O cross-section Figure 4 shows a comparison of our measurement with literature. Three authors report measurements at low en-

	mb)	σ (mb) σr (mb)

<E> / ∆E (MeV) <E> / ∆E (MeV) σ (mb) σr (mb) σ (mb) σr (ergy (2 to 12 MeV). The shape and normalization of the excitation functions presented by Cirilov et al.

[START_REF] Cirilov | [END_REF]

and by

Table 3 .

 3 Measured cross-sections σ (mb) for 3 He-induced reactions on 59 Co. Different values correspond to different Co foils 59 Co( 3 He, 2n) 60 Cu, 59 Co( 3 He, n) 61 Cu,

	<E> (MeV) ∆E (MeV)	Our work	Mean	Literature (see text)
	60 Cu	21.3	0.4	77(11)	77(11)	61.8(7.2) [14], 63.[15], 73.2(8.0)[16]
	61 Cu	21.3	0.4	6.7(2.0), 7.6(2)	7.1(2)	4.4(1.1)[14], 5[15], 6.4(0.8)[16],
						4.3(0.5)[17], 5.6(0.5)[20]
	56 Co	21.3	0.4	9.4(2.0), 9.6(2.0)	9.5(2.0)	3.6(0.5)[14], 3.8[15], 6.9(0.5)[17],
						6.2(0.5)[19], 2.4(0.3)[20]
		35.8	0.3	73(10), 64(9), 69(10)	69(6)	45.[15], 52.0(3.4) [17],
						61(4)[19], 47.5(4.0) [20]
	57 Co	21.3	0.4	116(14), 117(14)	116.5(10.0)	84.(10)[14], 67[15], 85(6) [17],
						100(7) [19], 78(4) [20]
		35.8	0.3	50(8), 40(8), 44(8)	45(5)	20.4 [15], 30.4(2.1) [17],
						37.8(2.9)[19], 31.6(1.9 )[20]
	58 Co	21.3	0.4	24(4), 24(4)	24(3)	16(2)[14], 16.4[15], 17.8(1.3)[17],
						20(2)[19], 26.6(2.5)[20]
		35.8	0.3	289(39), 271(36), 284(39)	281(22)	192[15], 179.(13) [17], 268(20) [19],
						128(8 )[20]
	4.4.2 Cobalt foils
	Cobalt foils provided another set of cross-sections of the
	following reactions: 59 Co( 3 He, x) 56,57,58 Co, with x representing the various
	open channels (see table 1). The measured cross sections
	are shown in table 3. The proportion of reaction fragments
	leaving the Co foil is taken into account using the products
	recoil range measured by Nagame et al. [15] and Jastrzeb-
	ski et al. [20]. It was determined that this proportion never

exceeds 10 % at 21.3 MeV and 16 % at 38 MeV of the total production, in good agreement with factors calculated using the LISE++ code. Our measured cross sections were compared with those of Fenyvesi et al. [14], Nagame et al. [15], Szelecsényi et al. [16], Michel and Galas [17], Homma and Murakami [18], Kondratyev et al. [19] and Jastrzebski et al. [20]. The measured cross sections from the literature were interpolated and values corresponding to our energies are shown in table 3, with the exception of data from

  1. On the average, the published values are 30 % smaller than our values. We note that our results are scaled according to the primary beam intensity, these are absolute values. A part of the discrepancy could be ascribed to the thickness of the Co foils or to the estimate of products recoiling from the extremely thin Co targets.Moreover, beam intensity (100 nA) and irradiation time (200 s), optimized for 14 O and 28 Al, were not optimum for the decay measurements of the long lived 60,61 Cu and 56-58 Co isotopes. It resulted in very low counting rates going from 0.001 pps up to 2.4 pps. In spite of the above mentioned limitations, if we scale our 12 C( 3 He, n)14 O data according to our Co cross-sections, in no case we confirm the cross section of Singh (triangles in figure

A collaboration between GANIL (


Laurent S R Simon
email: laurent.s.simon@inria.fr

Russell Mason

Spaciousness Rating of 8-channel Stereophony-Based Microphone Arrays

In previous studies, the localisation accuracy and the spatial impression of 3-2 stereo microphone arrays were discussed. These showed that 3-2 stereo cannot produce stable images to the side and to the rear of the listener. An octagon loudspeaker array was therefore proposed. Microphone array design for this loudspeaker configuration was studied in terms of localisation accuracy, locatedness and sound image width. This paper describes an experiment conducted to evaluate the spaciousness of 10 different microphone arrays used in different acoustical environments. Spaciousness was analyzed as a function of sound signal, acoustical environment and microphone array's characteristics. It showed that the height of the microphone array and the original acoustical environment are the two variables that have the most influence on the perceived spaciousness, but that microphone directivity and the position of sound sources is also important.

BACKGROUND

3-2 stereophony (also known as 5.1) has been shown to produce unstable images, as well as a poor localisation accuracy, to the side and to the rear of the listener [START_REF] Rumsey | Spatial audio[END_REF].

Research has been undertaken to create an 8loudspeaker array, resulting in a horizontal octago-nal arrangement, as shown in fig. 1, and this demonstrates more even localisation around the 360 • of the horizontal plane compared to a 5.1 ITU-R BS.775-1 loudspeaker array [START_REF] Simon | Developement of multichannel recording and reproduction techniques based on auditory perceptual cues[END_REF]. In order to successfully develop microphone techniques for this reproduction system based on optimum localisation, it was necessary to elicit localisation curves for each of the eight segments of the octagon. The aim of this experiment was to create a map of angle perception and The microphone array were designed for this octagon loudspeaker array on the bases of localisation accuracy. It was shown that the localisation profile of the microphone arrays could be predicted relatively accurately from the localisation curves and an estimation of crosstalk levels and delays. However, other spatial attributes resulting from these 8-channel microphone arrays were not yet investigated as a function of the microphone array's design.

Whilst a microphone array's localisation profile may appear to be an important factor in choosing one microphone array over another, sound engineers sometimes prefer microphone arrays that offer greater spaciousness despite a poor localisation of sound images [START_REF] Rumsey | Spatial audio[END_REF]. In addition, it was shown by Ceoen [START_REF] Ceoen | Comparative Stereophonic Listening Tests[END_REF] that near coincident or spaced microphone arrays offer greater spaciousness than coincident microphone techniques. This shows the need for a microphone array design tool that can predict the spaciousness as well as the localisation profile of a microphone array.

Therefore, after studying 8-channel microphone array design in terms of localisation profiles, as reported in [START_REF] Simon | Developement of multichannel recording and reproduction techniques based on auditory perceptual cues[END_REF], the influence of the microphone array design on spaciousness was evaluated. It was expected that the perception of spaciousness would be influenced by the time and level differences between the channels.

The first section of this paper describes the experiment set-up. The the results are then discussed in comparison to previous studies on microphone arrays and spaciousness.

EXPERIMENTAL DESIGN

Selection of experimental conditions

In the same manner as variations in physical parameters of a microphone array, such as the diameter of the array or the directivity of the microphones, caused changes to the resulting localisation profiles, it was expected that these would also alter the perceived spaciousness.

Ten 8-channel microphone arrays were therefore designed and are described in table 1. The different geometries caused changes in terms of maximum In-terChannel Time Differences (ICTDs), InterChannel Level Differences (ICLDs) 1 , crosstalk levels and crosstalk delays.

Two families of microphone arrays were used: regular and irregular microphone arrays. Regular microphone arrays make the assumption that the listener is free to move his head and might therefore face any direction, whereas the irregular microphone arrays are optimised in terms of localisation profile for a given direction of orientation.

As discussed previously, to the side of the listener, ICTDs lead to less stable phantom images and using solely ICTDs means that the phantom image cannot be positioned inside one of the loudspeakers [START_REF] Simon | Time and Level Localisation Curves for a Regularly-Spaced Octagon Loudspeaker Array[END_REF]. Therefore, when designing a microphone array for precise localisation, it is better to use a technique mainly based on ICLDs to the side of the listener and mainly ICTD to the front and rear of the listener, leading to the design shown in fig. 2. An alternative microphone array producing mainly ICLDs to the front and rear of the listener was designed for 3. For these two microphone arrays, the two omnidirectional microphones can be replaced by cardioid microphones in order to reduce the front / back (for the array designed with with mainly side ICLDs) or side (for the array designed with with mainly front / rear ICLDs) crosstalk.

Original acoustical environments

According to [START_REF] Griesinger | Objective Measures of Spaciousness and Envelopment[END_REF] and [START_REF] Hartmann | Localization of a Source of Sound in a Room[END_REF], room reflections can play an important role in perception of space and source localisation. Spaciousness is often thought to be strongly correlated to the room's reverberation time, as a long reverberation time tends to indicate that the original acoustic environment (room) was large.

Each microphone array was therefore used in two different rooms: a small classical studio (room 1), having an RT60 of 1.03s, and a size of approximately 17m width x 14m depth x 7m height and a concert room (room 2), having an RT60 of 2.47s, and a size of approximately 20m width x 30m depth x 20m height.

Source positions

10 loudspeakers were positioned around the microphone array, every 36 • . The first loudspeaker was located 5 • off-centre, in order to avoid having any sound source at the same angle as a loudspeaker. The loudspeakers were located 3 metres away from the centre of the microphone array, as a 6m diameter circle is a likely size for a music ensemble.

The loudspeakers were 10 Genelec 8020As,positioned at a height of 1.35m, which is the expected height for a small musical instrument played in an orchestra by a seated musician.

Microphone array's height

In order to vary the direct-to-reverberant ratio for a given microphone array, it was necessary to vary the distance of the microphone array to the sound sources. In order to retain the microphone array configuration and to retain the relative angular position of each sound source, the direct-to-reverberant ratio was altered by adjusting the height of the array.

The microphone arrays were therefore used at a height of 1.70m, above the orchestra, as it was the minimum height possible for the microphone structure used to build the microphone arrays, and at a height of 3.75m.

Recording the audio content

In order to offer the freedom of using any type of source signal, the impulse responses from each loudsepaker to each microphone were captured, and were then convolved with the desired source signals in MATLAB.

The impulse responses were captured on a Macbook Pro laptop using an RME Fireface sound card. Adobe Audition and the Aurora plug-in were used to generate a sine sweep and its inverse response [START_REF] Farina | Simultaneous Measurement of Impulse Response and Distortion with a Swept-Sine Technique[END_REF]. Then for each microphone array at each microphone height in each room, sine sweeps were played sequentially from each loudspeaker and recorded using all the microphones.

In MATLAB, the impulse responses of each loudspeaker to each microphone of each microphone array were derived, enabling the experimenter to simulate any source signal at any loudspeaker position using any of the microphone arrays under test.

Sound stimuli

It was thought that the perception of spaciousness would depend on the type of music, on its temporal and spectral characteristics and on the arrangement of the sound images around the listener. Four types of source signal were therefore used:

• A frontal, single female voice.

• A frontal, single drum track.

• A 10-instrument pop music track, each instrument being played by one of the 10 virtual sound sources.

• A 9-instrument classical music track, each instrument being played by one of the 10 virtual sound sources.

Summary of the recording conditions

The independent variables tested in this experiment were therefore:

• 2 original acoustical environments.

• 2 microphone array heights.

• 4 source signals.

• 10 8-channel microphone arrays, including:

-4 irregular microphone arrays.

-6 regular microphone arrays, combining 2 different diameters and 3 different directivities.

Listening test interface

10 trained listeners were asked to rate the spaciousness of the stimuli using a MUSHRA-like interface.

The instructions specified that a long reverberation time does not imply that spaciousness is high, in order to limit the bias of misinterpreting the meaning of spaciousness. Spaciousness was rated between not spacious (0% of the scale) and very spacious (100% of the scale). For each source signal, two references were given for the 20% spaciousness and the 80% spaciousness. The references were chosen through an informal test which determined the stimuli thought to have the lowest and highest spaciousness respectively, though the listeners were informed that they were free to rate the spaciousness higher than the high anchor or lower than the low anchor if they felt it was necessary. In order to evaluate the consistency of the listeners, the two references were also hidden in each set of stimuli.

ANALYSIS OF THE RESULTS

In order to check that the data met the assumptions of parametric statistical analysis methods, a Kolmogorov-Smirnov test was carried out for each experimental condition. It showed that all of the cases were normally distributed. This means that the results are suitable for parametric statistical analysis (such as ANOVA) [START_REF] Field | Discovering Statistics Using SPSS[END_REF].

Several univariate ANOVAs were conducted on the results of the experiment. A first univariate ANOVA was conducted using the type of microphone array, the room, the source signal and the height of the microphone array as dependent variables. It was decided not to include the diameter of the microphone array as it is only relevant to regular microphone arrays. The microphone's directivity was also excluded as it does not apply to the same number of microphones for each type of microphone array (for regular microphone arrays, the microphone's directivity applies to all of the microphones, whereas in the irregular microphone arrays, it only applies to two microphones). This ANOVA showed that the interactions between the source signal and the microphone array height (sig. = 0.000, F = 17.754), the source signal and the room (sig. = 0.000, F = 32.644), the source signal and the type of microphone array (sig. = 0.000, F = 7.646), the type of microphone array and the room (sig. = 0.001, F = 6.847) and between the type of microphone array and the microphone height (sig. = 0.001, F = 6.783) were significant, as can be seen in table 2.

While the source signal and the type of microphone array (respectively sig. = 0.003, F = 4.647 and sig. = 0.000, F = 50.947) were found to be significant, the room and the microphone height were found to have a greater F-value by a factor of 5 to 15(respectively sig. = 0.000, F = 256.766 and sig. = 0.000, F = 747.591). According to [START_REF] Field | Discovering Statistics Using SPSS[END_REF], this means that only the room and the height of the microphone array should be examined further. However, as explained in section 1, the spaciousness was expected to be strongly correlated to the reverberation time (hence strongly correlated to the room) and to the direct-toreverberant ratio (hence strongly correlated to the microphone array height), as can be seen in fig. 4 and5. While this test confirms this assumption, this paper aims to study the effect of the interaction between these two variables and the microphone array geometry on the perception of spaciousness.

An univariate ANOVA was then conducted for each type of microphone array in order to analyse the influence of the remaining microphone geometry variables on the perception of spaciousness.

For regular microphone arrays, the interactions between the directivity of the microphone, the room, the height of the microphone array and the source signal (sig. = 0.005, F = 3.108), between the diameter of the microphone array, the height of the microphone array and the source signal (sig. = 0.005, F = 3.108) and between the diameter of the microphone array, the height of the microphone array and the directivity of the microphone (sig. = 0.012, F = 4.460) were found significant.

The univariate ANOVA conducted on type 2 microphone arrays showed that for microphone arrays producing mainly ICLDs to the side of the listener, the interaction between the directivity, the height of the microphone array and the source signal (sig. = 0.010, F = 3.861), between the room and the source signal (sig. = 0.000, F = 10.807) and between the room and the height of the microphone array (sig. = 0.000, F = 8.594) were found significant.

Finally, the univariate ANOVA conducted on type 3 microphone arrays showed that for microphone arrays producing mainly ICLDs to the front of the listener, the interaction between the directivity, the room and the source signal (sig. = 0.003, F = 4.644), between the height of the microphone array and the source signal (sig. = 0.000, F = 6.520) and between the directivity and the height of the microphone array (sig. = 0.000, F = 13.486) were found to have a significant influence on the perceived spaciousness.

It can be seen in fig. 6 that the difference of spaciousness for all microphone arrays between the studio room and the concert room is more significant for single source stimuli than for the multiple sound sources stimuli (the figure for the voice stimulus is similar to that of the drum signal and the figure for the classic music stimulus is similar to that of the pop music signal). This could be caused either by the masking of the diffuse field by the sound sources arranged around the listener or because spaciousness could be influenced by the source envelopment. It is also likely that sources with a high source envelopment mask part of the diffuse sound. Fig. 7 shows that whilst for a low height and a small array diameter, the voice is perceived to be significantly less spacious than the other signals, and that for a low height and a large diameter, the pop signal is perceived significantly more spacious than the other signals, for a high height, all signals are perceived as spacious as the others.

As can be seen in fig. 8, cardioid regular arrays are perceived to be as spacious as supercardioid regular arrays. However, microphone arrays using omnidirectional microphones produce more spacious recordings. This could be explained by the fact that omnidirectional microphones capture more of the diffuse field than cardioid and supercardioid microphones.

Fig. 9 shows that type 2 microphone arrays, (i.e. irregular microphone arrays producing mainly ICLDs to the side of the listener) are perceived less spacious when the two single microphones are omnidirectional microphones and not cardioid microphones, and that as for the rest of this study, the higher microphone arrays are rated as being more spacious than the lower microphone arrays. It is however unclear why one type of source signal can sometimes be perceived more or less spacious than the three others.

As can be seen in fig. 10, for type 3 arrays, while there is a large difference of spaciousness between the arrays using cardioid microphones and the arrays using omnidirectional microphones at a low microphone array height, the difference is smaller at a high microphone array height.

DISCUSSION

It was seen in this experiment that there is a strong correlation between the room in which the recordings were made and the perceived spaciousness of the recording. It was expected that the bigger the recording room, the more spacious the recording.

Similarly, the microphone height appeared to have a large effect on the results, at least as much as the room. For the low microphone arrays, the direct to reverberant ratio is high. The spaciousness of the direct sound might therefore dominate. On the contrary, for the high microphone arrays, the direct to reverberant ratio is low. The spaciousness of the reverberation might then dominate. Hence, for the low microphone arrays the direct sound is dominant, and with a small array diameter there is insufficient decorrelation between the channels. This leaves the spectral characteristics as the main cue to spaciousness, and hence based on research that indicates that low frequencies are important to spaciousness [START_REF] Griesinger | Objective Measures of Spaciousness and Envelopment[END_REF], those lacking in low frequencies are perceived to be least spacious. This results in the voice being rated as least spacious. For the low microphone arrays with a large array diameter, there is a greater decorrelation between the channels. In this case, the spatial arrangement of the sources is more important. This results in the pop extract being rated as most spacious. For the high microphone arrays, the reverberant sound is dominant, so the characteristics of the direct sound are less important.

It can also be seen that whilst for type 2 microphone arrays, using omnidirectional microphones causes the recording to be perceived less spacious, using omnidirectional microphones for regular arrays and type 3 arrays causes the recording to be perceived more spacious, as can be seen in fig. 8, fig. 9 and fig. 10. This could be caused by the fact that part of the signals recorded by the two cardioid / om-nidirectional microphones, in the case of type 3 or regular microphone arrays, are played by the side loudspeakers. Using cardioid microphones therefore means that the sound played to the side of the listener is less diffuse than when using omnidirectional microphones, and thus perceived as being less spacious, according to Griesinger. However, using omnidirectional microphones for the type 2 microphone arrays is likely to increase front-back confusions [START_REF] Simon | Developement of multichannel recording and reproduction techniques based on auditory perceptual cues[END_REF], which might appear un-natural and cause the listeners to give a lower spaciousness rating to the recordings, despite the absence of a previously known relationship between spaciousness and front-back confusions.

It was also seen that microphone arrays producing only ICLDs to the side of the listener appear less spacious than the two other types of microphone arrays when sound sources are located all around the listener. According to [START_REF] Griesinger | Objective Measures of Spaciousness and Envelopment[END_REF], envelopment is caused by the early reflections coming from the side of the listener. Griesinger assumes that spaciousness and envelopment both define the same perceptual phenomenon. Though this is arguable, as the link between the perception of how spacious a recording space sounds (spaciousness) and the feeling of being enveloped by the sound (envelopment) is not intrinsic. However, while the difference between both definitions is clear, the perceptual difference might not be. Some confusion might therefore happen between rating the spaciousness and the envelopment. With a type 2 microphone array, the six side microphones are closer to the side sound sources than with the other two types of microphones. The direct-toreverberant ratio in the side channels is therefore larger than for the two other types of microphones arrays. This limits the effect of the reflections coming from the side of the listener, thus diminishing the perceived envelopment, and possibly the perceived spaciousness.

Similarly, it was shown that stimuli using multiple sound sources located around the microphone array led to smaller differences of spaciousness between the studio room and the concert room, but that the number of sound sources has little influence on the difference of spaciousness between the high and low height microphone arrays. The number of sound sources was not the only variable: the program material with multiple sources were classical music and pop music whereas the program material with single sources were a drum track and a voice track; the sound level was more constant over time with the multiple sources stimuli than with the single sources stimuli, thus masking more of the reverberation of the room, particularly the end of the reverberation tail. The direct-to-diffuse ratio might have been less affected by this. It could be hypothesised that the presence of virtual sound sources to the side of the listener masked more particularly the side reflections that are important for the perception of envelopment, as discussed above, and that this could cause a smaller difference of spaciousness between the two rooms for multiple source stimuli than for single source stimuli. However, there should then have also been a smaller difference of spaciousness between the two heights for multiple sources stimuli than for single sources stimuli, which was not the case.

CONCLUSION

In this paper, it was shown that spaciousness is highly correlated to both the recording room and to the height of the microphone array. It was hypothesised that spaciousness is therefore correlated to the direct-to-reverberant ratio.

Moreover, it was shown that microphone arrays that produce mainly ICTDs to the front and rear of the listener and ICLDs to the side of the listener cause the recordings to be perceived less spacious than the other microphone techniques tested, although this might be caused by the proximity of the side microphones to the side sound sources.

It was also shown than when recording a single frontal sound source, the recording room has a stronger influence on the perceived spaciousness than when recording multiple sound sources around the microphone array.

1 :

 1 The octagon loudspeaker setup used in the experiment locatedness for each segment of the system depending on the interchannel level difference (ICLD) and interchannel time difference (ICTD).

Fig. 2 :Fig. 3 :

 23 Fig. 2: Microphone array configuration of the side-ICLD microphone array.

Fig. 4 :

 4 Fig. 4: Spaciousness as a function of the recording room.

Fig. 5 :

 5 Fig. 5: Spaciousness as a function of the microphone array height.

Fig. 6 :

 6 Fig.6: Spaciousness as a function of the room for both the drums stimuli and the pop music stimuli.

  Height: Low height, Diameter: 0,5 Height: Low height, Diameter: 2,5 Height: High height, Diameter: 0,5 Height: High height, Diameter: 2,5 Spaciousness as a function of source signal, microphone array height and microphone array diameter

Fig. 7 :

 7 Fig. 7: Spaciousness as a function of the stimuli, microphone array height and microphone array diameter.

Fig. 8 :

 8 Fig. 8: Spaciousness as a function of the height of the microphone array and directivity of the microphones for regular microphone arrays.

Fig. 9 :

 9 Fig.9: Spaciousness as a function of the stimuli, microphone array height and directivity of the two single microphones, for irregular microphone arrays producing mainly ICLDs to the side of the listener.

  Fig. 10: Spaciousness as a function of the microphone array height and of the two single microphones' directivity, for irregular microphone arrays producing mainly ICLDs to the front and rear of the listener.

Table 1 :

 1 List

	Type of array Diameter (m)	Directivity
	1	0.5	Cardioid
	1	2.5	Cardioid
	1	0.5	Omnidirectional
	1	2.5	Omnidirectional
	1	0.5	Supercardioid
	1	2.5	Supercardioid
	2	N.A.	Cardioid
	2	N.A.	Omnidirectional
	3	N.A.	Cardioid
	3	N.A.	Omnidirectional
		of the microphone array conditions
	used in the experiment. Type 1 arrays are regular
	microphone arrays. Type 2 arrays are irregular mi-
	crophone arrays producing mainly ICLDs to the side
	of the listener and type 3 microphone arrays are ir-
	regular microphone arrays producing mainly ICLDs
	to the front of the listener. For irregular arrays, the
	directivity specified is that of the two single micro-
	phones; the other microphones in the array are as
	shown in fig. 2 and 3
	comparison purposes, as shown in fig.

Table 2 :

 2 Results of the univariate ANOVA conducted on all types of microphone arrays

		Tests of Between-Subjects Effects
		Dependent Variable:Spaciousness
	Source Type III Sum of Squares	df Mean Square	F Sig.
	Corrected Model	599727,739	47	12760,165	36,204 ,000
	Intercept	4609833,334	1	4609833,334 13079,156 ,000
	Type	35913,507	2	17956,754	50,947 ,000
	Room	90498,941	1	90498,941	256,766 ,000
	Height	263493,253	1	263493,253	747,591 ,000
	Signal	4913,312	3	1637,771	4,647 ,003
	Type * Room	4826,657	2	2413,329	6,847 ,001
	Type * Height	4781,482	2	2390,741	6,783 ,001
	Type * Signal	16169,125	6	2694,854	7,646 ,000
	Room * Height	1283,549	1	1283,549	3,642 ,056
	Room * Signal	34517,086	3	11505,695	32,644 ,000
	Height * Signal	18772,545	3	6257,515	17,754 ,000
	Type * Room * Height	1214,893	2	607,447	1,723 ,179
	Type * Room * Signal	1962,820	6	327,137	,928 ,473
	Type * Height * Signal	3427,774	6	571,296	1,621 ,137
	Room * Height * Signal	726,943	3	242,314	,688 ,560
	Type * Room * Height * Signal	573,541	6	95,590	,271 ,951
	Error	772584,607 2192	352,456
	Total	7639587,000 2240
	Corrected Total	1372312,346 2239
	a. R Squared = ,437 (Adjusted R Squared = ,425)

Interchannel differences being used in this paper for differences between adjacent microphone channels only.

ACKNOWLEDGEMENTS

The research documented in this paper was supported by EPSRC, grant number EP/E064507/1. Its presentation was supported by the EUREKA Eurostars i3Dmusic project funded by Oseo.