Gaétan Bahl
email: gaetan.bahl@inria.fr

Lionel Daniel
email: lionel.daniel@irt-saintexupery.com

Florent Lafarge
email: florent.lafarge@inria.fr

SCR: Smooth Contour Regression with Geometric Priors

While object detection methods traditionally make use of pixel-level masks or bounding boxes, alternative representations such as polygons or active contours have recently emerged. Among them, methods based on the regression of Fourier or Chebyshev coefficients have shown high potential on freeform objects. By defining object shapes as polar functions, they are however limited to star-shaped domains. We address this issue with SCR: a method that captures resolution-free object contours as complex periodic functions. The method offers a good compromise between accuracy and compactness thanks to the design of efficient geometric shape priors. We benchmark SCR on the popular COCO 2017 instance segmentation dataset, and show its competitiveness against existing algorithms in the field. In addition, we design a compact version of our network, which we benchmark on embedded hardware with a wide range of power targets, achieving up to real-time performance.

Introduction

Over the recent years, the development of Convolutional Neural Networks [START_REF] Lecun | Optimal Brain Damage[END_REF], combined with the wide availability of powerful GPU hardware and large amounts of data for training [START_REF] Deng | ImageNet: A large-scale hierarchical image database[END_REF], has allowed great advances in various computer vision tasks, such as image classification [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], object detection [7], or instance segmentation [START_REF] He | Piotr Dollár, and Ross Girshick[END_REF].

Object detection is the task of finding bounding boxes of objects in images. However, an inherent flaw of bounding boxes is the fact that their IoU (Intersection over Union) with their actual underlying objects will always be limited by their rectangular shape. This is especially true for object that have holes or multiple sharp angles (e.g. animals, bicycles).

Instance segmentation tries to alleviate this problem by providing pixel-level masks for each detected object. Masks, however, also have some drawbacks as an object contour representation: they are resolution dependent, and have to be resized and interpolated to be used on images Figure 1. Our algorithm captures the silhouette of objects with a compact resolution-independent representation based on Fourier coefficients. Contours produced by our algorithm adequately approximate the general shape of free-form objects such as persons, animals, cars or pots while being defined in a simple parametric way with only a few complex Fourier coefficients, here 8. Images from COCO 2017 test-dev.

with different resolutions. Masks also take up a lot of storage space and require more computational power, which is especially problematic on embedded systems, such as drones or satellites. Furthermore, Geographic Information System (GIS) applications such as online mapping rely on vector formats, and the automatic conversion of raster masks to more compact representations such as polygons is a difficult scientific challenge [START_REF] Li | Approximating shapes in images with low-complexity polygons[END_REF].

A good middle ground between bounding-box regression and mask-based instance segmentation is probably the regression of object contours using shape encoding, the goal being to capture the boundary of objects using a simple parametric function. Methods such as PolarMask [START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF] regress the contour of objects directly at a fixed set of sampling points revolving around the object's center. Some other methods use active contours or snake algorithms [23]. Finally, methods such as ESE-Seg [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF] and FourierNet [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF], 1 make use of Chebyshev or Fourier coefficients to explicitly regress an encoding that represents the shape of the object.

Our work proposes a Fourier shape decoder that brings several important improvements over previous methods. First, we give insights and mathematical arguments about the fact that Fourier coefficients constitute a better representation than the Chebyshev coefficients used in [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF]. By careful design of the loss function, we also improve the visual quality of regressed contours with fewer Fourier coefficients, while also simplifying the training process. Moving to a complex representation for contours allows us halve the number of calls to the Inverse Fast Fourier Transform (IFFT) in the final shape decoding stage, while allowing more freedom in shape representation by alleviating the limits of polar coordinates-based methods. Finally, we propose a compact version of our architecture based on a lightweight backbone [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] for use on low-power systems, typically found on board satellites and UAVs, targeting applications such as real time on-board instance segmentation and efficient transmission within constellations of satellites.

Related Work

Detection neural networks R-CNN [7] and its followups [START_REF] Girshick | Fast r-cnn[END_REF]27], were the original neural networks for object detection. They are two-stage approaches, where object locations are first proposed by a Region Proposal Network (stage one) and then classified by a subsequent CNN (stage two). Later, fully convolutional single-stage approaches were proposed [START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF][START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF], which were significantly faster than their two-stage counterparts. These methods rely on a set of "anchor" bounding boxes, to which the detected boxes are assigned and regressed relatively. The simplicity and efficiency of these single stage detectors have made them very popular over the years. A large number of improvements to this design have been proposed, such as the Feature Pyramid Network (FPN) [START_REF] Lin | Feature pyramid networks for object detection[END_REF] and subsequent work [START_REF] Ghiasi | Nas-fpn: Learning scalable feature pyramid architecture for object detection[END_REF][START_REF] Tan | Efficientnet: Rethinking model scaling for convolutional neural networks[END_REF][START_REF] Wang | Carafe: Content-aware reassembly of features[END_REF], which aggregates features from the backbone at different levels in order to create semantically rich feature maps at each level. Recently, FCOS [START_REF] Zhi Tian | Fcos: Fully convolutional one-stage object detection[END_REF], an anchorfree detection neural network architecture, has been developed. By leveraging the pyramid structure given by the FPN, FCOS removes the problem of anchor box assignment, and thus streamlines the object detection task with a one-stage, proposal-free, anchor-free framework.

Instance segmentation neural networks Instance segmentation neural networks were pioneered by Mask-RCNN [START_REF] He | Piotr Dollár, and Ross Girshick[END_REF], a two-stage approach based on [7]. A significant number of architectures have been developed [START_REF] Cao | Gcnet: Non-local networks meet squeeze-excitation networks and beyond[END_REF]3,[START_REF] Liu | Path aggregation network for instance segmentation[END_REF][START_REF] Wang | Solo: Segmenting objects by locations[END_REF][START_REF] Wang | Solov2: Dynamic, faster and stronger[END_REF], each bringing incremental improvements in popular benchmarks. Some methods save storage space by regressing masks as low-dimensional embeddings, which are decoded using either a learned decoder [START_REF] Jetley | Straight to shapes: Real-time detection of encoded shapes[END_REF][START_REF] Zhang | Mask encoding for single shot instance segmentation[END_REF] which complexifies the model and its training, or fixed functions such as the Discrete Cosine Transform [START_REF] Shen | Dct-mask: Discrete cosine transform mask representation for instance segmentation[END_REF], in which case the encoding dimension is still quite large. Nonetheless, mask-based instance segmentation methods all encounter the drawbacks of the pixel-based representation, and are generally much slower than their bounding box-based counterparts.

Contour regression methods

Recently, the community has explored new ways of regressing the boundaries of objects. These methods aim to alleviate the aforementioned issues of the pixel-based mask representation, while still offering better IoU with detected objects than bounding box based architectures, which most of them take as a starting point.

Snake-based methods [START_REF] Liu | Dance: A deep attentive contour model for efficient instance segmentation[END_REF]23] represent shapes as polygons in pixel coordinates and deform them iteratively using circular convolutions. While these methods can reach good accuracy levels at reasonable speeds, they generally make use of a high number of contour points, which can be costly in terms of storage on embedded devices, especially compared to shape encoding methods described below.

ESE-Seg [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF] is based on the YOLOv3 detection neural network [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] and regresses an explicit shape encoding for each detected object in the form of either Chebyshev or Fourier coefficients and a center. The contour of the object is modeled as a function ρ(θ), θ ∈ [0, 2π], revolving around this center, as shown on the inset Figure . Corresponding ground truth coefficients have to be generated for each object of the training dataset. When used with Chebyshev coefficients, this representation is however not well suited to periodic functions, as we will demonstrate later. Moreover, this method is only able to fully represent star-shaped domains and thus cannot be used to regress less regular shapes.

In PolarMask [START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF], the authors modify the FCOS [START_REF] Zhi Tian | Fcos: Fully convolutional one-stage object detection[END_REF] architecture in order to regress the polar coordinates of the contour points directly, i.e. the values of the function ρ. Again, specific ground truth labels have to be created for each object, with a fixed number of rays cast from its center, and only star-shaped objects can be regressed, since only one coordinate is regressed for each ray.

FourierNet [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF], also based on FCOS, simplifies the contour regression problem by using the Inverse Fast Fourier Transform (IFFT) as a differentiable shape decoder. Thus, the neural network regresses Fourier coefficients, but the output is directly compared to ground truth polygons. The authors use either polar or Cartesian coordinates. In the first case, one set of complex coefficients is used to find ρ, and the regressed shapes have to be star-shaped. In the Cartesian case, the contour is seen as two functions, X(t) and Y (t) as shown on the inset Figure . Two sets of complex coefficients are used (one for each) to model these functions. While the authors manage to get good looking results in the polar case, their Cartesian version does not perform as well quantitatively, and fails to produce coherent shapes when using a high number of Fourier coefficients.

Method

Our network architecture is based on the FCOS detection neural network [START_REF] Zhi Tian | Fcos: Fully convolutional one-stage object detection[END_REF]. Like most recent detection architectures, FCOS is composed of a backbone, a neck (feature pyramid) and several detection heads with shared weights. As backbones, we use ResNet-50 [START_REF] He | Deep residual learning for image recognition[END_REF], ResNeXt-101 [START_REF] Xie | Aggregated residual transformations for deep neural networks[END_REF] and the lightweight DarkNet-53 [START_REF] Redmon | Yolov3: An incremental improvement[END_REF], in order to address different performance targets. We run experiments with both the classical FPN neck from [START_REF] Lin | Feature pyramid networks for object detection[END_REF], as well as the more recent FPN-CARAFE [START_REF] Wang | Carafe: Content-aware reassembly of features[END_REF], which provides a small accuracy improvement.

Choice of shape representation

In [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF][START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF][START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF], the shape of an object is represented as a function ρ(θ) with θ ∈ [0, 2π], and thus revolve around a center, which is either regressed along with the coefficients in [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF], or is the center of output "cells" from FCOS in [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF][START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF]. ρ is a 2π-periodic function measuring the distance of each point of a contour to this center. By contrast, we choose to model the shape of objects with a more generic complex periodic function of the form C

(t) = X(t) + iY (t) with t ∈ [0, 1].
On Chebyshev coefficients The use of Chebyshev coefficients [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF] is not well suited to the regression of continuous periodic functions through a neural network -in this case, the ρ function. Indeed, we demonstrate in the appendix that ρ is continuous periodic, i.e. ρ(0) = ρ(2π), if and only if the regressed Chebyshev coefficients α i satisfy N/2 k=0 α 2k+1 = 0. It is unlikely that a neural network can learn this exact relation, even if optimized as a loss during training. Hence, discontinuities can be observed in nearly every shape regressed in this manner, as seen in Figure 11 of [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF] reproduced in appendix. A similar relationship can be found using the derivatives of Chebyshev polynomials if we also want shapes to be smooth, which should be the case for most objects.

Contours as a complex function, and Fourier series

This drawback of Chebyshev polynomials steers us toward the Fourier decomposition, which is better suited to our purposes simply due to its conception, since any set of Fourier coefficients will always represent a continuous smooth periodic function.

Our single complex function representation has several advantages over using two real functions as in [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF]: not only does it halve the number of IFFT calls in the shape decoding stage; it also ties these coefficients together as they represent a single function, which might help during the training of the neural network, as backpropagation occurs through a single IFFT call.

In order to evaluate how many complex coefficients are needed to represent shapes accurately, we interpolate all polygons of the COCO 2017 validation set using the scheme detailed in section 3, we then compute the FFT and its inverse after zeroing a certain number of coefficients, and finally compare the result to the original polygons using the Chamfer Distance as reconstruction error.

The inset figure shows the quickly diminishing returns of increasing the number of coefficients, highlighting the fact that only a small number of them suffice to represent a shape accurately. We decide to run our experiments using 8 to 32 complex coefficients.

SCR head

The SCR head, shown in Figure 2, is based on the FourierNet [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF] head. For both the class and contour regression branches, we make use of 3 deformable convolutions [START_REF] Zhu | Deformable convnets v2: More deformable, better results[END_REF] with shared weights among feature levels. Thanks to the use of the complex contour representation, we only require one call to the IFFT for shape decoding. We add two regularization losses which we will detail further below. The first one is applied directly to the regressed coefficients. The second one is applied to the decoded point coordinates before they are scaled according to the stride of the feature level and a learnable scaling coefficient.

Loss function

For training SCR, our loss function is defined as a sum of different terms:

L = L CD + L cent + L cls + L perim + L coeff (1)
where L CD is a polygon regression loss based on a symmetrized Chamfer Distance (similarly to [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF]), L cent is the centerness loss from FCOS [START_REF] Zhi Tian | Fcos: Fully convolutional one-stage object detection[END_REF] (binary cross-entropy), and

Contours

Weights shared between feature levels Figure 2. Our SCR head, which is applied to each output feature level of the FPN. Learned operators are in orange. Fixed operators are in blue. One branch of the head is used for the class prediction. The other branch is used for both centerness prediction and Fourier coefficient regression. All weights are shared between feature levels except the final scaling coefficient. An L 1 regularization is applied to the Fourier coefficients to direct training towards the regression of simpler shapes. An L 2 regularization is used on the perimeter of the regressed contours to encourage the regression of shapes that circle the object only once.

L cls is the Focal Loss for classification from RetinaNet [START_REF] Lin | Focal loss for dense object detection[END_REF]. L perim and L coeff are regularization terms on the shape perimeter and on Fourier coefficient respectively. Both are detailed below.

In contrast to [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF], note that we do not use any bounding box loss function as bounding boxes can be inferred from the regressed polygons and do not need to be learned separately.

L2 Perimeter regularization The use of the Chamfer distance loss as the main polygon regression loss has a major drawback that needs to be addressed: since ground truth points are only compared to the closest regressed point, the network does not necessarily learn to regress contours that go around the object only once. Indeed, circling the object multiple times is not penalized in L CD . One way to alleviate this problem, as done in [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF], is to first perform a warm-up of the shape regression with an L 1 loss, so that each output point is already tied to specific ground truth points. However, after being trained with Chamfer loss, their network still tries to regress shapes in an overly complex way, which creates lots of self-intersections in the output polygon, as seen in our comparison in section 4 (Figure 4).

We instead choose to apply a L 2 perimeter-based regularization to every output contour, even the ones that have not been assigned to any object. This, while also simplifying the training process, encourages simpler shapes that go around the object once and have a minimal number of self-intersections:

L perim = λ perim N i=0 (x i -x i+1 ) 2 (2)
The λ perim coefficient can be set to a relatively low number in order to avoid influencing training too much. Setting it too high will result in small round-ish shapes. It can be decayed over time or set to zero after L perim reaches a steady state, in order to retrieve more complex shapes while retaining the effect of regularization, as shown in our experimental results (Figure 3).

L1 Fourier coefficient regularization While representing contours using Fourier coefficients is already more storage-efficient than using masks, we may want to further reduce the storage space needs by ignoring very small coefficients (i.e. setting them to zero after regression). Thus, it would be interesting to enforce sparsity in the coefficients, in order to have as many of them be close to zero as possible.

In order to favor sparsity of the Fourier coefficients F i , we use an L 1 regularization term:

L coeff = λ coeff N c n i=-n,i / ∈{-1,0,1} |F i | (3)
Where λ coeff is an adjustable parameter (500.0 in our experiments). It has to be noted that we do not apply this penalty to the -1, 0 and 1 frequency coefficients, since these coefficients will have to be quite large in most cases. Having lots of Fourier coefficients close to zero, especially high frequency ones, also helps the network create simpler and smoother contours, as shown in section 4 (Figure 5).

Polygon ground truth processing Existing works [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF][START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF][START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF] typically convert the ground truth polygons to polar coordinates with respect to a center. This is done by casting a number of rays at regular angular intervals from this center to the farthest contour point. This process can hardly be done on-the-fly without slowing the data loading pipeline. We also note that the distance between two adjacent points created in this manner can vary greatly (e.g. in oblong objects). We believe that having a high number of evenly spaced out ground truth points is better for the stability of the Chamfer Distance loss. Thus, our interpolation is done in a "constant spacing" fashion for each ground truth polygon. Since the original ground truth polygons vary in number of points, we interpolate them such that the number of ground truth points N p matches the number of regressed points. This is easily done on the fly in the data loading pipeline and does not incur any performance loss.

Experiments

We implement our ResNet-50, ResNeXt-101 and DarkNet-53 based models using the MMDetection 2.7.0 object detection toolbox [START_REF] Chen | MMDetection: Open mmlab detection toolbox and benchmark[END_REF], based on MMCV 1.2.1 for easy and fair comparison with other methods. The underlying framework is PyTorch 1.7.0 with CUDA 10.1. We make use of the updated FFT package from PyTorch 1.7.0 which allows the use of complex numbers. We use the Chamfer Distance implementation from PyTorch3D 0.3.0 [START_REF] Ravi | Accelerating 3d deep learning with pytorch3d[END_REF]. The interpolation scheme described in 3 is implemented as a module in the MMDetection data loading pipeline using the "interp" module from Numpy. Notably, due to software incompatibilities during the conversion of our DarkNet-53 model for low-power hardware targets, we had to simplify the architecture by removing deformable convolutions and CARAFE. In order to evaluate our method and easily compare it to previous works, we choose to work on the widely used COCO 2017 Instance Segmentation dataset [START_REF] Lin | Microsoft coco: Common objects in context[END_REF].

Our networks are trained for 38 epochs using SGD with momentum (0.9), with an initial learning rate of 0.0003, batch-size of 16 (2 images per GPU), gradient clipping at a maximum L 2 norm of 45, and employ of a 500 step warmup with a ratio of 0.001 applied to the learning rate. We use the multiscale training feature found in MMDetection to train with image heights of 640 and 800 pixels. We balanced our loss function terms through trial-and-error, and settled on the following values: λ cls =1.0, λ cent =1.0, λ CD =1.0, λ perim =0.01, λ coeff =500.0. Qualitative results Figure 1 shows a selection of results from the COCO 2017 test-dev dataset, computed using our smallest DarkNet-53 based network. Our method properly regresses smooth artifact-free contours in these scenes varying in number of objects, classes and shapes. In particular, non-star-shaped objects can be accurately represented. An extended selection of results is provided in appendix.

Name-Backbone

Quantitative comparison We compare our method to existing shape encoding methods [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF][START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF][START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF], as well as state-of-the-art snake-based [START_REF] Liu | Dance: A deep attentive contour model for efficient instance segmentation[END_REF]23] and mask-based methods [START_REF] He | Piotr Dollár, and Ross Girshick[END_REF][START_REF] Liu | Path aggregation network for instance segmentation[END_REF][START_REF] Wang | Solov2: Dynamic, faster and stronger[END_REF]. The evaluation is done using three metrics: mAP for accuracy (as defined in [START_REF] Lin | Microsoft coco: Common objects in context[END_REF]), FPS (frames per second) reported on a single Nvidia GTX 1080Ti GPU for speed and memory usage per regressed object in bits, which we call SEC (for Shape Encoding Complexity). Run time and object size in memory or storage are important factors for embedded hardware applications, such as realtime on-board change detection. Thus, we propose OES (Overall Efficiency Score), which is simply the multiplication of these three scores and thus defined as OES = 100 × mAP × F P S × SEC -1 . This metric represents the trade-off between having slow but accurate models outputting detailed shapes, and having faster models outputting simpler shapes. The comparisons against other shape encoding methods are shown in Table 2 for the ResNeXt-101 backbone and 3 for the DarkNet-53 backbone. We observe that with the same backbone and at a similar number of coefficients, our method achieves good accuracy compared to other shape encoding methods.

In Table 2, our method has a competitive efficiency score (OES). While [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF] is a little bit faster, our method reaches a better accuracy. By contrast, PolarMask [START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF] can achieve higher mAP at the cost of a higher number of coefficients, and thus has a lower overall efficiency.

Table 3 shows that our method is faster and more efficient than ESE-Seg [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF]. In addition, even though its mAP score is a little bit lower, our method will not visually suffer from the drawbacks of Chebyshev coefficients detailed in section 3.

Method

N coeff mAP FPS SEC OES PolarMask [START_REF] Xie | Polarmask: Single shot instance segmentation with polar representation[END_REF] 36 32.9 4.1 1152 11.7 FourierNet-Cartesian [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF] The comparison against other types of approaches is shown in Table 4. Snake-based methods typically offer a better accuracy than shape encoding methods, however their overall efficiency suffers from a high shape encoding complexity and are slower than our method when using the same processing size and backbone. Mask-based methods reach very high mAP scores, but are also hindered by their speed and SEC and thus generally have a lower overall efficiency.

Method

N coeff mAP FPS SEC OES Snake DeepSnake [23] 4. COCO 2017 test-dev results against mask and snakebased methods with a ResNet-50 backbone and an image height of 800 pixels. Ncoeff in real numbers for shape encodings: one complex coefficient counts as two. SEC: size of a single detected object in memory (bits). For snake-based methods, we assume pixel coordinates are stored as 16-bit integers. For mask-based methods, we assume masks are binary. OES: Overall Efficiency Score. *: Number extrapolated from [START_REF] Liu | Dance: A deep attentive contour model for efficient instance segmentation[END_REF] While our method does not necessarily give the best score for each of the evaluation metrics, it offers a good compromise between them. In particular, the OES scores obtained by our method is competitive compared to existing methods under the same backbone. Effect of perimeter penalty decay As discussed in section 3, it is possible to set λ perim to zero after L perim stabilizes. Removing the perimeter regularization after a few iterations (2000 here) leads to more complex and accurate contours, while retaining the effect of regularization. On figure 3, the network learns to properly follow both front and back legs of the giraffe (left), which is not the case without perimeter penalty decay (right). This is also a typical example of non-star-shaped contour that cannot be regressed by previous works that opt for a single polar function. Setting λ perim = 0 right away, i.e. removing L perim entirely, will let the network circle objects multiple times, leading to poor results, as shown in our ablation study and on Figure 5(a). Fourier coefficient regularization The benefits of our coefficient regularization can be observed on figure 4, where our network (left) learned to regress smooth contours compared to the erratic shape regressed by the Cartesian version of FourierNet [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF] (right), even when using a high Resnet-50 CARAFE DCN 20 128 800 5.8 14.5 3.5 3.1 6.6 7.8 #13 5.4 13.9 3.1 2.9 6. number of coefficients. Figure 5(b) shows that disabling our regularization leads to the same kind of behavior shown in [START_REF] Hamd | Fouriernet: Compact mask representation for instance segmentation using differentiable shape decoders[END_REF].

Ablation study We now evaluate the impact of our design choices through an ablation study. The results are summarized in Table 5. Numbers in parentheses in the following paragraph refer to corresponding lines of that Table .  (#2,#3) We trained the ResNet-50 based model with 60 and 128 contour points, and found that for this set of hyperparameters, there was a performance degradation when using 128 points. We thus kept this parameter at 60, a finding consistent with the literature. (#1,#3) We show that the new FPN-CARAFE from [START_REF] Wang | Carafe: Content-aware reassembly of features[END_REF] improves the scores of SCR compared to the vanilla FPN, for which it is a dropin replacement. (#3,#4,#5; #6,#7; #14,#15) Increasing the number of complex coefficients from 10 (20 real values) to 20 made no significant difference, while increasing it further to 32 yielded a small performance increase of 1.4 mAP.

Depending on the requirements of the application, this increase in detail might be worth the extra storage space. Note that it is still possible to regress a high number of coefficients, then decide to set some of them to zero after the fact. (#6,#7,#8,#9) We find that the models used on smaller image resolutions do not perform very well on small objects (AP S ). (#14,#15) Using a larger ResNeXt-101 backbone yields a small accuracy improvement, but not to the point where we would think it is worth the decrease in FPS and the up to two times longer training times. (#10) DarkNet-53, on the other hand, seems to perform as well as ResNet-50 at high resolutions, especially on larger objects. (#2,#11,#12,#13) Disabling the L 1 coefficient penalty leads to a very significant drop in mAP. This is probably due to the erratic shapes regressed without it, as shown on Figure 5. Disabling the L 2 perimeter penalty also significantly affects the scores, as the contours circle the objects multiple times and thus miss a lot of detail.

Limitations Our regularization terms tend to direct the training of our network towards simple shapes without actively preventing self-intersections. Thus, such errors might appear occasionally. On figure 6, for instance, we see that while the two leftmost horse contours seem properly regressed, they each present one self-intersection. Nevertheless, in that particular case, the masks given by these contours are still usable. One other limitation is that by design, our method is more tailored towards free-form objects than the regression of piece-wise linear shapes, for which polygon-based methods are better suited. Performance benchmarks on low-power hardware We evaluate the speed of our fastest DarkNet-53 based network on a variety of devices, ranging from powerful discrete GPUs to low-power edge devices (Table 6).

Our first test platform is the Nvidia Jetson Nano, which is a popular, affordable and widely available yet reasonably powerful family of development kits. Our method runs at 9.5 FPS on these boards, which is close enough to realtime to be usable in many applications. We also run the same benchmarks on the more powerful Jetson Xavier NX, featuring INT8 compute capabilities, thus yielding a significant performance boost by bringing the throughput up to 115 FPS. It achieves 5.75 FPS/Watt, making it by far the most efficient combination. The Intel Movidius Myriad X VPU is an ultra low-power AI accelerator with a power consumption of only 1W. The Myriad VPUs have been used in ESA's φ-sat-1 CubeSat, as well as in smart security cameras, UAVs and industrial machine vision equipment. We benchmark SCR on the Myriad X using the imposed Open-Vino toolkit and achieve 2.8 FPS, which is very satisfactory for such a low-power device. As a point of comparison with other works, we also run this network on an Nvidia GeForce GTX 1080Ti GPU, which is still one of the most popular discrete GPUs. For the sake of completeness, we also provide numbers on an Intel Xeon Silver 4114 CPU.

Conclusion

In this paper, we proposed SCR, a method that captures object contours with a compact resolution-free representation based on a low number of Fourier coefficients. Using a complex representation and geometric priors, we were able to observe qualitative and quantitative improvements in the regressed shapes over previous methods.

We also implemented a downsized version of our model which, while retaining competitive accuracy, is able to run on a wide range of low-power hardware at very reasonable speeds (up to 115 FPS on a 30W device), and is therefore suitable for edge computing applications such as on-board processing of UAV or satellite images, autonomous vehicles and robotics.

Future work will include testing and evaluating SCR on other applicative scenarios, such as autonomous driving and remote sensing. Matching the "shape signature" of detected objects to an on-board database is an example of how our work might be used for embedded change detection. This work could be also extended by adding a polygon simplification scheme that takes advantage of properties of the Fourier decomposition, such as ease of differentiation. momentum (0.9), with an initial learning rate of 0.0003, batch-size of 16 (2 images per GPU), gradient clipping at a maximum L 2 norm of 45, and employ of a 500 step warmup with a ratio of 0.001 applied to the learning rate. We use the multiscale training feature found in MMDetection to train with image heights of 640 and 800 pixels. We balanced our loss function terms through trial-and-error, and settled on the following values: λ cls =1.0, λ cent =1.0, λ CD =1.0, λ perim =0.01, λ coeff =500.0.

Due to software incompatibilities during the conversion of our DarkNet-53 model for low-power hardware targets, we had to simplify the architecture by removing deformable convolutions and CARAFE, as shown in Table 7.

C. Hardware configuration

This section contains details regarding the embedded hardware configuration used for the throughput benchmarks presented in section 4 of the main text.

Our Nvidia Jetson Nano 2GB, Nano 4GB and Xavier NX development kits are running JetPack 4.5, ONNXRuntime 1.4.0 and TensorRT 7.1.3. The Nano SoC features a quad-core ARM A53 CPU and a 128 CUDA core Maxwell GPU. In order to obtain consistent results, we lock the power consumption at 10W for benchmarking using the jetson clocks tool and cool the heatsink with a Noctua A4x20 5V PWM fan using the default fan curve. We also ran our benchmarks on a Jetson Nano 4GB model and found no performance difference with the 2 GB model. Thus, Table 6 of the main text refers to both of them as "Nano".

The Jetson Xavier NX features a 6-core ARM processor, a 384 CUDA core Volta GPU and 8GB RAM, with a power consumption of 30W and is running the same setup.

The Intel Movidius Myriad X Vision Processing Unit is an ultra low-power AI accelerator with a power consumption of only 1W. It has been used in ESA's φ-sat-1 Cube-Sat, as well as in smart security cameras, UAVs and industrial machine vision equipment. We benchmark SCR on the Myriad X using OpenVino and FP16 precision.

Nvidia GeForce GTX 1080Ti (11GB VRAM, 250W) benchmarks were run using either the same MMDetection setup used for training, or TensorRT 7.1.3.

Intel Xeon Silver 4114 CPU benchmarks were run using the Intel OpenVino toolkit.

D. Carbon Impact Statement

While the focus of our work is to create efficient neural networks that are able to run on very low power devices, we cannot help but notice that these networks are still created and trained using power-hungry multi-GPU machines and wonder about the environmental impact. For this paper, we missed the opportunity to track the real power consumption of our experiments, and to estimate its global environmental footprint, but we are trying as much as we can to publish these estimations, as recommended in [START_REF] Anthony | Carbontracker: Tracking and predicting the carbon footprint of training deep learning models[END_REF][START_REF] Henderson | Towards the systematic reporting of the energy and carbon footprints of machine learning[END_REF][START_REF] Lannelongue | Green algorithms: Quantifying the carbon emissions of computation[END_REF].

In order to run the experiments required for our main results and ablation study, we have used 5864.55 GPU hours on Nvidia Tesla V100 GPUs, which are rated for a power consumption of 300W. This, not counting CPUs, cooling, PSU efficiency, storage of datasets and results, as well as different trials or hyper-parameter searches on workstations, amounts to 1759kWh. Since the carbon intensity of our electricity grid is 10 gCO2/kWh, we estimate an emission of 17590 gCO2, which is equivalent to 146 km traveled by car according to [START_REF] Anthony | Carbontracker: Tracking and predicting the carbon footprint of training deep learning models[END_REF].

E. Qualitative results

In this section, we provide an extended selection of qualitative results that could not be part of the main paper for space reasons. Figures 9 and8 show results from our smallest and fastest DarkNet-53-based model.

Figure 3 .

 3 Figure3. SCR-D53 result with (left) and without perimeter penalty decay (right). The perimeter regularization loss coefficient λperim is set to zero once Lperim reaches steady state (2000 iterations here), thus leading to a more detailed shape while retaining the effect of regularization.

Figure 4 .

 4 Figure 4. Thanks to our coefficient regularization, our method (left) learns to regress smooth shapes compared to the erratic ones regressed the cartesian version of [31] (right) when using a high number of coefficients (here 32).

Figure 5 .

 5 Figure 5. Disabling our perimeter regularization (a) lets the network circle the object multiple times, while disabling the coefficient regularization (b) leads to erratic shapes with lots of self-intersections. When both regularizers are enabled (c), the network outputs smooth and accurate shapes with a low number of self-intersections, here zero.

Figure 6 .

 6 Figure 6. Examples of self-intersections which can appear occasionally as they are not actively prevented. The contours still follow the shape of the animals and are thus still somewhat usable.

Table 1 .

 1 DCNv2 CARAFE N in N f eat Time Characteristics of our main networks. Variations are further explored in our ablation study. DCN refers to the use of Deformable Convolution v2 [39] layers in the head. CARAFE refers to the use of the FPN-CARAFE [32] neck. Training time on 8 Tesla V100 GPUs.

	SCR-D53	No	No	128 128 20h
	SCR-R50	Yes	Yes	256 256 40h
	SCR-X101	Yes	Yes	256 256 80h

Table 2 .

 2 16 22.9 4.9 512 21.9 COCO 2017 test-dev results against other shape encoding methods based on ResNeXt-101 with an image height of 800 pixels. Ncoeff in real numbers for shape encodings: one complex coefficient counts as two. SEC: size of a single detected object in memory (bits). OES: Overall Efficiency Score.

	FourierNet [31]		16 23.3 4.9 512 22.3
	Ours (SCR-RX101)	16 27.3 4.2 512 22.4
	Method	N coeff mAP FPS SEC OES
	ESE-Seg [37]	20	21.6 38.5 640	130
	Ours (SCR-D53)	16	21.2 39.1 512	162

Table 3

 3

. COCO 2017 val results for models based on DarkNet-53 with an image height of 416 pixels. Ncoeff in real numbers for shape encodings: one complex coefficient counts as two. SEC (Shape Encoding Complexity): size of a single detected object in memory (bits). OES: Overall Efficiency Score.

Table

			256 31.0 6.68* 4096	5
		DANCE [22]	196 34.6 7.6 3136	8
	Mask	Mask R-CNN [8] 784 33.6 10.8 784 PANet [20] 784 38.2 4.5 784 SOLOv2 [34] 64000 38.8 12.4 64000 0.75 46 22
		Ours (SCR-R50) 16	24.2 10.7 512	51

  Exp. Backbone Neck Head N coeff N pts H Perim. reg Coeff. reg. mAP AP 50 AP 75 AP S AP M AP L

	#1	Resnet-50	FPN	DCN 20	60 800	22.4 43.0 21.0 10.8 24.2 29.7
	#2	Resnet-50 CARAFE DCN 20 128 800	21.4 44.2 19.0 10.9 22.9 28.4
	#3				20		23.8 45.0 22.4 11.6 25.6 31.9
	#4	Resnet-50 CARAFE DCN 40	60 800	23.8 46.4 21.8 11.7 25.4 32.3
	#5				64		25.2 48.1 23.6 13.1 26.9 33.6
	#6	Resnet-50 CARAFE DCN 20	60 360	19.1 36.2 18.1 4.0 19.9 31.3
	#7	Resnet-50 CARAFE DCN 64	60 360	18.9 37.3 17.0 4.3 19.5 30.9
	#8					320	17.6 33.8 16.4 1.5 17.0 32.1
	#9 DarkNet-53	FPN	-	16	60 416	21.2 39.3 20.4 3.7 22.6 35.5
	#10					608	23.

9 42.2 24.0 6.2 27.5 36.3 #11 10.8 22.6 9.3 6.3 12.5 14.2 #12

Table 5 .

 5 Impact of different design choices on our models on COCO test-dev. H = image height. Ncoeff in real numbers (one complex coefficient counts as two). Npts = number of contour points in ground truth and after IFFT. DCN = Deformable Convolution v2.

Table 6 .

 6 Throughput of SCR with Darknet-53 backbone on a wide range of devices with varying levels of power. H = image height.

		Device Power Net. H	Tool	Prec. FPS fps/W
		Nano	10W Ours 320 TensorRT FP32 5.4 0.54
				TensorRT FP16 9.5 0.95
	Low-power	Nano Xavier NX 30W Ours 320 TensorRT FP16 66.7 2.22 10W Ours 608 TensorRT FP32 1.72 0.17 TensorRT FP16 3.04 0.30 TensorRT FP32 17.2 0.57
				TensorRT INT8 115 5.75
		Myriad X 1.5W Ours 320 OpenVino FP16 2.8 1.87
		Xeon 4114 85W Ours 608 OpenVino FP16 5.3 0.06
	High-power	1080Ti 250W Ours 320 MMDet FP32 40.6 0.16 1080Ti 250W Ours 320 TensorRT FP32 186.6 0.74 1080Ti 250W Ours 608 MMDet FP32 32.8 0.13 1080Ti 250W Ours 608 TensorRT FP32 68.2 0.27 1080Ti 250W Ours 416 MMDet FP32 39.1 0.15
		1080Ti 250W [37] 416 PyTorch FP32 38.5 0.15
		1080Ti 250W [31] 800 MMDet FP32 4.9 0.02
		1080Ti 250W [31] 360 MMDet FP32 16.5 0.07

Figure 8. More results from our DarkNet-53 based model selected from COCO 2017 test-dev.

Figure 9. More results from our DarkNet-53 based model selected from COCO 2017 test-dev.

Acknowledgements

This project is funded by the CIAR project at IRT Saint Exupéry. The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support. This work was granted access to the HPC resources of IDRIS under the allocation 2020-AD011011311R2 made by GENCI. We thank Maxime Nabon for running our network on Jetson Xavier NX.

A. Chebyshev coefficients

We demonstrate why Chebyshev coefficients, as used in [START_REF] Xu | Explicit shape encoding for real-time instance segmentation[END_REF], are not well suited to the regression of continuous periodic functions through a neural network.

Chebyshev polynomials are obtained through the following recurrence relation:

Let N be the number of degrees used for interpolation of ρ using the truncated Chebyshev series. Then for x ∈ [-1, 1] we have:

Since the contour ρ is a continuous periodic function on [-1, 1], in order to perfectly interpolate it, the Chebyshev coefficients α i must satisfy:

We also know that Chebyshev polynomials satisfy the following properties, for all n ∈ N:

Then, we have the following:

Thus, we have shown that in order to perfectly interpolate a continuous periodic function, the Chebyshev coefficients must satisfy the following relationship:

Figure 7 shows the discontinuities created when this relationship is not strictly enforced, e.g. when the Chebyshev coefficients are regressed by a neural network. They appear at θ = 2π on every regressed shape.

B. Implementation details

We use the MMDetection 2.7.0 object detection toolbox [START_REF] Chen | MMDetection: Open mmlab detection toolbox and benchmark[END_REF], based on MMCV 1.2.1 for easy and fair comparison with other methods. The underlying framework is PyTorch 1.7.0 with CUDA 10.1. We make use of the updated FFT package from PyTorch 1.7.0 which allows the use of complex numbers. We use the Chamfer Distance implementation from PyTorch3D 0.3.0 [START_REF] Ravi | Accelerating 3d deep learning with pytorch3d[END_REF]. The interpolation scheme described in section 3 of the main text is implemented as a module in the MMDetection data loading pipeline using the interp module from Numpy. Our networks are trained for 38 epochs using SGD with


Nicolas Thibault

Fabrice Minoletti

Silvia Gardin

Offsets in the early Danian recovery phase in carbon isotopes: evidence from the biometrics and phylogeny of the Cruciplacolithus lineage

Keywords: calcareous nannofossils, biostratigraphy, Paleogene, ecosystem recovery

Changes in the size and in the shape of the cross of the early Danian Cruciplacolithus lineage have been studied along with bulk carbon isotope data at two reference Cretaceous-Paleogene boundary sections: Bidart, SW France (Basque Basin) and Elles, central Tunisia (SW Tethys). Our study documents the progressive increase in the size of this lineage in the early Danian and allows for the definition of a new sub-lineage with a narrow central area and a broad axial cross composed of two species: a small, primitive and rare form , Cruciplacolithus praebornemannii n.sp. and a large common form Cruciplacolithus bornemannii n.sp. Successive first occurrences in the lineage of "axial cross" Cruciplacolithus can be correlated between the two sections. This correlation shows that the recovery phase in carbon isotopes following the Cretaceous-Paleogene boundary negative excursion is delayed in the southwestern Tethys compared to the Basque Basin. The delayed recovery in carbon isotopes at the Elles section is likely related to the lengthened environmental stress in shelfal settings as compared to the pelagic settings of the Basque-Cantabrian basin. The emergence of Cruciplacolithus, the occurrence of the different lineages in this genus and the change in dominance from small to large forms are all related to the progressive improvement in early Danian environmental conditions.

Introduction

The calcareous nannofossil genus Cruciplacolithus constitutes an important component of early Paleogene assemblages, first occurring very early within the first Danian nannofossil zone NP1 and evolving rapidly within the Danian, with a wide geographic distribution, from continental margins o open ocean and low to high latitudes (Bralower and Parrow, 1996). Significant taxonomic revisions by van Heck andPrins (1987), Varol (1992) and Bralower and Parrow (1996) have led to a number of recombinations of species within other sister genera Chiasmolithus and Sullivania. For instance, Bralower and Parrow (1996) have recombined Cruciplacolithus asymmetricus van Heck & Prins (1987) into Sullivania asymmetrica, while Cruciplacolithus edwardsii Romein (1979) has been recombined as Chiasmolithus edwardsii by van Heck and Prins (1987). These two recombinations have not been followed in recent literature and both forms are generally considered as species of Cruciplacolithus [START_REF] Bown | Paleocene calcareous nannofossils from Tanzania (TDP Sites 19, 27 and 38)[END_REF] and/or are considered to belong to the same species and included in C. edwardsii (Fornaciari et al., 2007). These examples highlight some of the taxonomic difficulties and confusion inherent to this lineage. Much controversy and confusion are also involved in the definition of axial cross forms C. primus, C. intermedius and C. tenuis as commented further and such controversies affect the biostratigraphy of the early Paleogene (Agnini et al., 2014). Because the Cruciplacolithus lineage shows a size increase over time and appears as an important group for the biostratigraphy of the earliest Paleogene, a critical interval with recovery from the Cretaceous-Paleogene (K-Pg) mass extinction, there is a need for biometric studies of this genus [START_REF] Kim | Recovery of plankton cell and coccolith size after the Cretaceous-Paleogene mass extinction (IODP Expedition 342 Sites 1403 and 1407, North Atlantic)[END_REF]. The purpose of this study is thus to compare the evolution in size of the Cruciplacolithus lineage in the early Danian of two famous outcropping sections of the Cretaceous-Paleogene interval: Bidart (SW France) and Elles (central Tunisia). The results are compared to changes in bulk carbonate carbon isotopes and a correlation of early Danian horizons is proposed between the two sections.

Figure 1 about here: paleogeography

Material and Methods

Both studied sections are representative of low latitude deposits but belong to two distinct basins. Bidart is situated in the Basque-Cantabric Basin which was connected to the North Atlantic while Elles was situated on the southwestern margin of the Tethys (Fig. 1).

Bidart, Basque country, SW France

Bidart is a famous Cretaceous-Paleogene boundary (K-PgB) section, situated on the coast of the Basque country, France, in the town of Bidart, next to Biarritz. The section is well-exposed along the shore and has been the subject of various studies (Perch-Nielsen, 1979;Delacotte, 1982;Seyve, 1984;Rocchia et al., 1987;Clauser, 1994;Gorostidi and Lamolda, 1995;Galbrun, 1997;Robin and Rocchia, 1998;Galbrun and Gardin, 2004;Thibault et al., 2004;[START_REF] Minoletti | Minoletti et al[END_REF][START_REF] Minoletti | Minoletti et al[END_REF][START_REF] Font | A new sedimentary benchmark for the Deccan Traps volcanism?[END_REF][START_REF] Font | Mercury anomaly, Deccan volcanism, and the end-Cretaceous mass extinction[END_REF]. The succession at Bidart was deposited in a lower to middle bathyal setting, in the hemipelagic part of the Basque-Cantabrian basin, under the influence of deep-sea fans as attested by many turbiditic and slump deposits observed in all the sections of this basin (Delacotte, 1982;[START_REF] Rat | The Basque-Cantabrian basin between the Iberian and European plates, some facts but still many problems[END_REF][START_REF] Payros | The Relevance of Iberian Sedimentary Successions for Paleogene Stratigraphy and Timescales[END_REF]. Despite the presence of slumps and turbidites, the outcrops of the Basque-Cantabrian Basin have delivered exquisite continuous successions which constitute today references in stratigraphy [START_REF] Batenburg | [END_REF][START_REF] Batenburg | [END_REF]Dinarès-Turell et al., 2014;Husson et al., 2014;[START_REF] Payros | The Relevance of Iberian Sedimentary Successions for Paleogene Stratigraphy and Timescales[END_REF]. The K-PgB interval at Bidart also constitutes one of the classic sections for this boundary but the early Danian at Bidart is likely discontinuous due to the presence of two slumped intervals at 1.7 and 5.6 m (Fig. 2). The magnetostratigraphy of the succession was revised by Galbrun and Gardin (2004) who showed that the C29r to C30n chron reversal lies at 3.5 m above the boundary, in-between the two slumps (Fig. 2), thus providing a useful age anchor (Table 1).

Elles, central Tunisia

Elles is one of many K-PgB sections in central Tunisia, situated ~75 km southeast of El Kef, the Global Stratotype Section and Point (GSSP) of the boundary. The section is a marly succession within the El Haria Formation (Fig. 3), accessible on a flank of the Karma valley near the hamlet of Elles (Said, 1978;Karoui-Yakoub et al., 2002). The studied section corresponds to Elles II as described in [START_REF] Adatte | Agnini et al[END_REF] and studied samples are those of the Gardin (2002) dataset. Deposition took place in a middle neritic environment at Elles and it has been considered as slightly shallower conditions than at the GSSP of El Kef (Burollet, 1956;Burollet et Oudin, 1980). Thanks to its well-exposed, expanded and continuous succession, and helped by the constitution of a workshop in 2001 [START_REF] Remane | Foreword[END_REF], the K-PgB interval at Elles has already been the subject of numerous studies (Li et al., 1998(Li et al., , 1999;;[START_REF] Adatte | Agnini et al[END_REF]Karoui-Yakoub et al., 2002;Gardin, 2002;Stüben et al., 2002;Thibault and Husson, 2016;Thibault et al., 2016).

Methods

A total of 21 samples were studied for biometrics at Bidart in the 2 m to 5.7 m interval (Appendix 1). 50 specimens of Cruciplacolithus were measured for their length and width and along with the length of the coccolith, morphological criteria on the axial cross (thin versus thick) and central area (open/close) allowed species assignment (see chapters 4 and 8). A total of 14 samples were studied at Elles between 6.5 and 13 m (Appendix 2). Cruciplacolithus specimens were rare at 6.5 m and therefore, we could only measure 6 specimens in this sample. For all other samples, 50 specimens were measured and identified in the same fashion as for the Bidart section. These morphometric measurements allowed us to report the evolution of the mean length of the group on both sections (Figs 2-3). Density plots are also reported and were drawn in Matlab® by calculating percentages in bins of 1 µm. Frequency distribution is also reported for the main two distinct lineages of axial-cross Cruciplacolithus that we identified (see chapter 8).

Bulk carbonate stable isotopes were measured for 16 samples at the Bidart section from 3.5 to 5.7 m and complemented previous analyses reported by Delacotte (1982) and [START_REF] Minoletti | Mise au point d'un protocole de séparation des assemblages de nannofossiles calcaires. Apport à la micropaléontologie et à la géochimie des producteurs carbonatés pélagiques[END_REF]. In total, the compiled bulk carbonate stable isotope record at Bidart is based on 44 samples in the Danian and 10 samples in the Maastrichtian (Fig. 2). Bulk carbonate stable isotope measurements were measured every m from 1.5 to 6.5 m, then every 0.5 m from 7 to 13 m at Elles and complemented previous highresolution analyses across the K-PgB interval of [START_REF] Minoletti | Mise au point d'un protocole de séparation des assemblages de nannofossiles calcaires. Apport à la micropaléontologie et à la géochimie des producteurs carbonatés pélagiques[END_REF]. In total, the compiled bulk carbonate stable isotope record at Elles is based on 32 samples in the Danian and 8 samples in the Maastrichtian (Fig. 3). All samples were measured in year 2000 on a Finnigan Delta E mass spectrometer at the LODYC (UPMC Paris 6, Paris) with analytical precision of 0.05‰ for carbon and 0.1‰ for oxygen.

Biostratigraphy and age model

The planktic foraminifer biostratigraphy of the Bidart section has been the subject of several lowresolution studies through the years, leading to disagreement on the position of early Danian zone boundaries (Haslett, 1994;Fondecave-Wallez et al., 1995[START_REF] Apellaniz | [END_REF]. Gallala et al. (2009) have recently revised the foraminifer biostratigraphy through a high-resolution of the K-PgB interval (Table 1). The calcareous nannofossil biostratigraphy of Bidart reported here is that of Galbrun and Gardin (2004) with some additions from this study. A detailed planktic foraminifer biostratigraphy of the early Danian succession at Elles was published by Karoui-Yakoub et al. (2002) and recently revised by Gallala (2013). The calcareous nannofossil biostratigraphy is from Gardin (2002).

The subsequent age model applied here to biostratigraphic results (Table 1) arises from various sources. For most calcareous nannofossil horizons, the reference is ODP site 1262, South Atlantic, due to its Paleogene astronomical time scale [START_REF] Bernaola | [END_REF]Wade et al., 2011;Westerhold et al., 2008;[START_REF] Barnet | [END_REF]. For planktic foraminifer horizons, the age model of the early Danian in Wade et al. (2011) is not concordant with the work of [START_REF] Arenillas | [END_REF] who reviewed several classic sections of the K-PgB interval. Therefore, results from [START_REF] Arenillas | [END_REF] were preferred and used to recalculate ages of foraminifer horizons with an age of 66.022 for the K-PgB. None of these biostratigrpaphic age models is without complications and they are difficult to apply to any of the two studied sections due to differences in the stratigraphic order of biohorizons. In addition, more confusion may arise from the definition of nannofossil Zone NP2. The base of [START_REF] Martini | [END_REF] nannofossil zone NP2 (or Cruciplacolithus tenuis Zone) has been originally defined by the first occurrence (FO) of C. tenuis sensu Hay and Mohler (1967) which actually corresponds today to specimens included in C. intermedius. Therefore the strict application of the zonation implies that the base of NP2 is defined by the FO of C. intermedius and not by that of C. tenuis which first occurs later in the early Danian. By far, the most robust horizons of the age model are the K-PgB and the base of chron C29n which has been identified with precision at Bidart (Galbrun and Gardin, 2004; Table 1).

Table 1 about here

A graphic correlation of planktic foraminifer and calcareous nannofossil biohorizons of the two sections highlights the misalignments of horizons in the two respective microfossil groups (Fig. 4). Out of the four planktic foraminiferal horizons that are misaligned with the nannofossil ones (PF3, PF4, PF5, PF6), three of them (PF4, PF5 and PF6) have been recently identified at Elles between 2 and 3.5 m (G. Keller, pers. comm.), a result that would actually fit much better our line of correlation between the K/Pg boundary and nannofossil horizon N1 (Fig. 4). Therefore, it seems like so far, the foraminifer biostratigraphy of Elles is not fully reliable. For this reason, we decided to base our correlations between the two sections on nannofossil biohorizons only.

Figure 4 about here 4. Preliminary remarks on the taxonomy of Cruciplacolithus and Prinsius species Some of the taxonomic confusion inherent to the Cruciplacolithus lineage has been sorted out by Fornaciari et al. (2007) andAgnini et al. (2014) whose concepts are reminded here and expanded by considerations from Gardin (2002) and Galbrun and Gardin (2004):

-Cruciplacolithus primus (< 7 µm) includes small specimens with an axial cross that correspond to the "small" C. primus concept of Perch-Nielsen (1977). Two distinct forms can however be distinguished following the work of Gardin (2002) and Galbrun and Gardin (2004): C. primus small comprises very small specimens generally comprised between 3 and 5 µm, but as shown in this study, a few rare specimens may even be smaller down to 2.5 µm in length. C. primus large comprises specimens between 5 and 7 µm (Fig. 5).

-Cruciplacolithus intermedius includes specimens with an axial cross and length > 7 µm (Fig. 5), which actually corresponds to the concept of the "large" forms of C. primus of Romein (1979) and Perch-Nielsen (1981) and to the concept of C. tenuis s.l. of Hay and Mohler (1967).

-Cruciplacolithus tenuis (Heliorthus tenuis Stradner 1961, emend. Hay and Mohler 1967emend. Romein 1979) includes forms with a robust axial cross bearing triangular expansions or "feet", often pointing towards a counter clockwise direction in distal view. In other words, the cross has a right hand swastika shape. Rare specimens with a left hand swastika shape may also be observed. C. notus of Perch-Nielsen (1977) is a synonym of C. tenuis. Size comprised between 7 and 14 µm (Fig. 5).

-Cruciplacolithus asymmetricus (van Heck and Prins 1987) includes specimens whose cross is deviated from the axis of the coccolith in a clockwise direction in distal view. The angle between the the longest arm of the cross and the long axis is below 20°. We could actually observe two distinct morphotypes in this study: one where only the short arms are rotated with respect to the short axis but long arms essentially remain parallel to the long axis, while a second morphotype show both short and long arms deviated. Size comprised between 7 and 12 µm (Fig. 6). C. asymmetricus was considered here as a stand-alone species and not regrouped along within Cruciplacolithus edwardsii s.l. such as in Agnini et al. (2014) because no specimens corresponding to the strict definition of C. edwardsii could be observed in this study (see below).

-Cruciplacolithus edwardsii (Romein, 1979) includes specimens with both arms of the cross deviated in a clockwise direction from the axis of the coccolith by more than 20°, thus forming a X rather than a cross. Size comprised between 6 and 10 µm (Fig. 6). No specimens of this species have been observed in this study.

Bulk carbonate carbon isotopes

Bidart section

Late Maastrichtian carbon isotope values at Bidart fluctuate steadily between 1.8 and 2.2 ‰. The drop in carbon isotopes at the K-PgB is of a total extant of 4‰ reaching a minimal value of -1.8 ‰. This is rapidly followed by a progressive increase through the first 80 cm of the Danian. At 0.8 m, carbon isotopes reach a value of 2.2 ‰, thus marking a first recovery back to Maastrichtian values. However, values decrease again down to 1.3 ‰ in the following 40 cm and full recovery back up to 2.3 ‰ at 1.25 m is followed by relatively constant values higher than 2 ‰ in the remaining of chron C29r. Values then decrease a bit and tend to fluctuate around 1.7 ‰ in chron C30n (Fig. 2).

Elles section

At Elles, late Maastrichtian values fluctuate around 0.85 ‰. The drop in carbon isotope at the K-PgB is less abrupt than at Bidart, stabilizing around -0.6 ‰ between 0.4 and 0.7 m in the Danian (except for one value reaching down to -0.9 ‰ at 0.5 m). Values continue to decrease steadily down to a minimum of -1.1‰ at 2.5 m. Then a positive trend is observed, and values tend to stabilize around 0‰ between 3.5 and 6.5 m, followed by a second increase up to a maximum of 0.9‰ at 7 m, marking a first return and recovery to Maastrichtian values. Between 7 and 10.5 m, two small negative excursions are observed with values down to 0‰ for the first one and 0.4‰ for the second. A return to values around 0.9 ‰ is once again observed at 10.5 m, after which values stabilize around this average (Fig. 3).

Size evolution of the Cruciplacolithus lineage in the early Danian

Bidart section

At Bidart, from 2.1 m to 4.5 m, the mean length of Cruciplacolithus fluctuates slightly around an average of 6 µm with samples reaching sometimes down to minima of 5 µm or maxima of 7 µm. A marked increase is then observed with a maximum mean of 9 µm reached at 5.25 m, after which values decrease but remain higher than before, reaching down to 6.8 µm in our last sample at 5.7 m (Fig. 2). The density plot shows that a large percentage of specimens are actually concentrated around 5 µm, except for two anomalous samples at the base which are situated within the first slump. At 3.5 m, the density plot shows the emergence of another size group of Cruciplacolithus at 9 µm. Higher up in the section, the two groups of size at around 5 and 9 µm are markedly present but the small group generally predominates. Around 5.2 m, the group with the largest size appears to become predominant, explaining the maximum of 9 µm observed in the mean length, then this group shows a slight decrease in length dropping down to a mean of 8 µm and then 7 µm while also remaining predominant. The decrease observed in the mean size of Cruciplacolithus between 5.2 and 5.7 thus does not appear to originate from an increase in the relative abundance of the smaller group but rather marks the dominance of specimens with an intermediate size (Fig. 2).

Elles section

At 6.5 m, only 5 specimens of Cruciplacolithus could be found and measured in 100 fields of view. All these specimens have size varying between 2.85 and 3.2 µm and are thus assigned to C. primus small. A mean length of 3 µm was calculated at this level and this sample was also considered in the density plot but statistics are poorly reliable at that level due to too few measured specimens. However, results observed in following samples confirm an important increasing trend in the mean size of Cruciplacolithus from specimens around 3 µm when the group first occurs to a mean size of 8 µm at 9.5 m (Fig. 3). The density plot shows the presence of two distinct size groups, with a small group whose size tends to increase from 3 µm at 6.5 m to 4 µm at 9 m then stabilizing around 5.5 µm from 9.5 to 12 m while the two samples at the top show this small group returning to a mean density of 4.5 µm. A second group with much bigger size emerges around 7.5 m and is probably responsible for the shift in the density plot to specimens being concentrated around 6 µm. This second size group then appears very well separated as a distinct branch from 8.5 m up to the top with specimens regrouped around a stable mean length of 9 µm (Fig. 3).

Succession of early Danian evolutionary events and observed morphological trends in Cruciplacolithus

In both sections, the observed increase in the mean size of Cruciplacolithus through time is also accompanied by the emergence of specimens that bear a thick cross that fills and closes the central area (Fig. 7, specimens 16 to 20). No significant difference in the preservation of the assemblage has been observed between before and after the occurrence of these specimens and these forms do not bear any signs of enhanced recrystallization as compared to other Cruciplacolithus forms. Therefore, the presence of this thick cross filling the central area likely represents an evolutionary feature. Moreover, distinct species of Cruciplacolithus with a thick cross and a close central area are know from distinct Paleogene stratigraphic intervals such as late Danian to early Ypresian for C. latipons and Lutetian for C. klausii (Fig. 6). However, the thick-cross specimens observed here lack any clear median extinction lines on the cross, a distinct feature in C. latipons, and differ from the two latter Paleogene species by their broader inner cycle (Fig. 6 and Fig. 7, specimens 16 to 20). Interestingly, some of the specimens that bear such a thick cross filling the central area are comprised between 3.5 and 7 µm and are thus relatively small (Figs. 89). At Bidart, a distribution plot of all specimens of this thick-cross lineage bears two distinct modes, suggesting the presence of two distinct size morphotypes with a separation around 7 µm (Fig. 8C and9A). This feature is not observed in specimens from the Elles section but too few specimens of the small morphotype have been observed there to provide reliable statistics (Fig. 8D). Moreover, the distribution histogram of this lineage at Elles shows a distinct skewness towards the left (Fig. 8D) and the largest specimens with a thick cross appear to be regrouped as one cluster (blue diamonds in Figure 9B). Such features suggest a deviation from a normal distribution and tend to highlight the presence of two distinct morphogroups (Fig. 8D). These two morphogroups with a thick cross have been defined in the following as two distinct species, the small C. praebornemannii and the large C. bornemannii (see chapter 8). Both species first occur at the same stratigraphic level in both sections (Fig. 9), either in coincidence with the FO of C. asymmetricus (at Bidart) or immediately below the FO of C. asymmetricus (at Elles). In both sections, the sequence of evolutionary events in Cruciplacolithus appears quite similar (Fig. 10) allowing to construct a phylogeny of the group in the early Danian (Fig. 6). Two major evolutionary trends can be highlighted through the early Danian in the genus: an increase in the size of the coccolith and a thickening of the cross together with the closing of the central area (Fig. 7). These two evolutionary trends are responsible for giving rise relatively fast to a great diversity of distinct morphogroups (Fig. 6). As already mentioned in chapter 3, it appears difficult to apply strictly a relevant age-model to any of our two sections. Both at Bidart and Elles, the FO of F. petalosa is much above the K-PgB and is unlikely to have occurred only 1 kyr above the deposition of the boundary clay (Table 1). Moreover the FO of F. petalosa is coincident with that of C. primus at Bidart but recorded 3.5 m below that of C. primus at Elles (Fig. 10). Interestingly, the FO of C. primus at Elles coincides with a marked increase and consistent occurrence of F. petalosa which would therefore likely correlate well with the FO of the species at Bidart (Fig. 10). In both sections, the FO of P. dimorphosus is recorded after that of C. intermedius in contrast with the age-model. In both sections, the FO of C. pelagicus is recorded above that of C. intermedius in contrast to published age-models. Moreover, the FO of C. pelagicus does not seem to correlate between the two sections, this biohorizon being much delayed at Elles, recorded just one meter prior to the FO of C. tenuis whereas it is about 20 cm above the FO of C. intermedius at Bidart (Fig. 10). Previously proposed calculated ages of early Danian nannofossil biohorizons essentially derive from astronomical age-models erected either the deep-sea record of Shatsky Rise (ODP Site 1209, central Pacific, Bralower et al., 2005) or from Walvis ridge (ODP Site 1262, South Atlantic, [START_REF] Bernaola | [END_REF]. The early Danian calcareous nannofossil biostratigraphy of site 1209 remains too coarse so far to allow for any useful comparison with the astronomically calibrated record of Shatsky Rise (Bralower et al., 2005). [START_REF] Bown | [END_REF] studied in detail the first 5 m of the Paleocene at ODP Site 1210, a nearby borehole of Shatsky Rise. This author records very early rare occurrences of Praeprinsius sp. within the interval of acme of N. parvulum that immediately follows the boundary event in the lowermost part of NP1. The FO of C. primus is also recorded very early within the lower half of NP1 at the top acme of N. parvulum. The FO of small C. pelagicus is recorded in the second half of NP1, followed by the FO of C. intermedius and the FO of abundant F. petalosa. Large C. pelagicus and P. tenuiculus s.s. first occur successively within NP2. While examining the results of [START_REF] Bown | [END_REF], the early Danian nannofossil record of Shatsky Rise actually appears distinct from that of Walvis Ridge (Bernaola and Monechi, 2007, reported on Table 1) with respect to the stratigraphic order of biohorizons although these two boreholes are both tropical deep-sea sites. Ages of early Danian biohorizons occurring in the South Atlantic are inapplicable to Shatsky Rise and are neither applicable to the Bidart and Elles sections within Zone NP1. This discrepancy can be explained by the fact that within NP1, harsh unstable conditions essentially prevailed and the small emergent rselected species of the Prinsiaceae and Coccolithaceae competed for resources and favorably thrived depending on local conditions of the distinct oceanic basins and shelf seas. The base of NP2 as defined by the FO of C. intermedius correlates with or is above the top of the first recovery phase in carbon isotopes (Fig. 10). The conformable succession of biohorizons in the Cruciplacolithus lineage observed within NP2 between Bidart and Elles (Fig. 10) suggests that more stable environmental conditions were globally widespread in NP2, allowing for a synchronous colonization of k-selected taxa between the two distinct basins (Fig. 6). Age models based on calcareous nannofossil biohorizons appear poorly reliable within NP1 but appear more promising within NP2.

Correlation of biohorizons between Bidart and Elles: implications for the recovery in carbon isotopes

Despite the presence of a large slumped interval at Bidart which likely signs a hiatus as can be interpreted from our graphic correlation between the two sections (Fig. 4), a similar sequence of evolutionary events in Cruciplacolithus has been recorded in the two sections and allows for a correlation of isotopic trends (Fig. 10). Both sections tend to show a first recovery phase in carbon isotopes with values reaching back levels of the late Maastrichtian. This trend is followed by one or two additional episodes of lighter carbon isotope values with excursions amplitude of 0.5 to 1 ‰ before a full recovery phase, after which values tend to be stabilized (Fig. 10). However, while both the first recovery and full recovery phases take place before the FO of C. primus at Bidart, the 1 st recovery phase at Elles appears slightly delayed, coincident with the FO of C. intermedius and the full recovery phase occurs much later at Elles, some 50 cm below the FO of C. tenuis. Our correlation thus supports an offset between the two sections in the recovery phase in carbon isotopes following the K-Pg crash in productivity. It is proposed here that an earlier recovery in productivity levels was reached in the Basque-Cantabrian basin, while delayed recovery is observed in the Tethyan, shallow Saharan platform. Trends in nannoplankton assemblages of the two sections after the K-Pg mass extinction are very similar but a clear separation was noted in nannoplankton recovery between shelf and deep-ocean settings with the existence of a pronounced environmental stress gradient from shelf to deep ocean in the crisis interval [START_REF] Jiang | Environmental controls on calcareous nannoplankton response to the Cretaceous/Paleogene mass extinction in the Tethys realm[END_REF]. The latter authors actually included in their study nannoplankton assemblages of Bidart and El Kef, a section nearby Elles deposited in a slightly deeper setting on the Saharan shelf platform. Based on their observations of statistically meaningful differences observed in a collection of shelf to deep-sea sites, [START_REF] Jiang | Environmental controls on calcareous nannoplankton response to the Cretaceous/Paleogene mass extinction in the Tethys realm[END_REF] have proposed that shelf environments had limited capacity to buffer the lethal effects of the K-Pg crisis. Environmental stress was likely more intense and extended in time on the shelf than in pelagic settings. Therefore, the delayed recovery in carbon isotopes observed in the Elles section is likely related to the lengthened environmental stress on shelves.

Replacing the evolution of the Cruciplacolithus lineage in the early Danian context of ecosystem recovery

Our data allowed for illustrating the evolution in the dominance (%) of the different Cruciplacolithus morpho-species through time (Fig. 10). The first 3 samples with abundance data at Bidart appear to provide an unexpected picture with an important contribution of C. intermedius immediately following its first occurrence whereas samples above 2.7 m tend to show the dominance of the two forms of C. primus followed by an increasing contribution of large forms through the rest of the Danian. These three samples being either within or immediately above the first slump, it is possible that the signal observed below 2.7 m is biased by the influence of gravity flows. Low-energy turbid currents during and/or following the slump event may have acted as a sieve favoring the enrichment in larger nannofossil particles and hence in C. intermedius over C. primus. Alternatively, enhanced bioturbation after slump deposition can also be responsible for acting as a particle sieve that induces a strong bias in biometric studies. In any case, apart from this strange interval at the base of our studied interval, our results appear to show the ultra-dominance of small forms within chron C29r, followed by an increasing contribution of large forms C. intermedius and C. bornemarnnii culminating at 5.25 m by the dominance of C. bornemannii, after which the contribution of C. primus remains below 50%. At Elles, the first studied sample at 6.5 m is dominated by C. primus 3µm, rapidly followed by a dominance of C. primus 5 µm and an increasing contribution of C. intermedius, followed by two episodes where the contribution of large forms (C. bornemanii, C. intermedius and C. asymmetricus) reach more than 60% of the assemblage (Fig. 10). These data illustrate the shift in the dominance of small towards large forms amongst emergent Danian species in a context of recovery from the mass extinction. One of the most interesting features of the K-Pg crisis is indeed the "Lilliput" effect observed in early Danian planktic foraminifer and calcareous nannofossil species [START_REF] Macleod | Phenotypic response of foraminifera to episodes of global environmental change[END_REF][START_REF] Keller | The end-Cretaceous mass extinction in the marine realm: year 2000 assessment[END_REF][START_REF] Keller | Biotic effects of volcanism and impacts[END_REF]Gardin, 2002). This effect corresponding to a drastic size reduction observed across the K-Pg environmental crisis can be essentially explained by a selective extinction of large Kstrategy species during the mass extinction and survival of small r-strategists [START_REF] Keller | The end-Cretaceous mass extinction in the marine realm: year 2000 assessment[END_REF]. This Lilliput effect also concerns newly emerged nannofossil species of the early Danian which are all minute species that successively dominate the Danian assemblage within the first ~1 myr after the KPgB (Fig. 11). The high-resolution data of Gardin (2002) on the Elles section allow for a reexamination of the most important features of this Danian nannofossil assemblage. From the original data of Gardin, we recalculated here the cumulative percentage of the main families and genera within the Danian assemblage as well as the contribution of all different Cruciplacolithus morphospecies (Fig. 11). This exercise illustrates the succession of all the dwarf, r-selected incoming species in the early Danian, first essentially dominated by Neobiscutum, then with a major contribution of Futyana petalosa and, rapidly, with a first important contribution of the small forms of Cruciplacolithus (C. primus 3 and 5 µm). Interestingly, within the peculiar environmental context of the Saharan shelf at Elles, there appears to be a strong coupling between evolution of carbon isotopes and these evolutionary events. For instance, the onset of the rise in carbon isotopes after the negative shift, coincides with a small but significant increase in the total nannofossil abundance that also corresponds to a significant increase in the relative abundance of Neobiscutum, thus suggesting enhanced productivity accompanying as slightly enhanced nannoplankton carbonate production. Also, the 1 st recovery in carbon isotopes occurs right after the peak in total nannofossil abundance which is essentially due to a Neobiscutum bloom (Fig. 11). This bloom of r-selected species thus favored a return to higher productivity levels after the mass extinction, and Cruciplacolithus first occurred within this content of environmental recovery. However, it is likely that harsh conditions prevailed in early Danian sensitive environments such as the Saharan shelf, as attested by the two successive negative peaks in carbon isotopes between 7 and 10.5 m and, within this context, small r-selected species thrived favorably, as illustrated by the ultra-dominance of C. primus in this interval, although large C. intermedius actually first occurred and could survive within this interval (Fig. 11). The contribution of large Cruciplacolithus species became only significant around 9.5 m after a second recovery phase in carbon isotopes and corresponding to the onset of a full recovery interval reached at 10.5 m after which carbon isotope values remain nearly unchanged (Fig. 11). C. primus remained the dominant form among Cruciplacolithus species through the whole studied interval but the contribution of large forms is non-negligible from 9.5 m up to the top, which suggests a progressive return to more stable environmental conditions that allowed k-selected species to thrive. The diversification and shifts in the dominance and contribution of species within Cruciplacolithusthus appears to reflects the emergence of this new lineage at a time of a first ecosystem recovery, followed by the dominance of small, r-selected forms in a sensitive environment that was still characterized by episodes of harsh environmental conditions, and a progressive increase in the proportion of large k-selected forms when the ecosystem fully recovered. Diagnosis: a medium to large (7 to 13 µm) species of Cruciplacolithus with a broad, birefringent axial cross that sometimes show median extinction lines in XPL. Both the inner and outer cycles are broad. The arms of the cross aligned along the long axis of the ellipse are broader and more developed than those aligned along the short axis. The short axis arms of the cross typically show a triangular or losange shape. Differentiation: This species exhibits similar features to C. latipons and C. klausii, two species of Cruciplacolithus that bear a broad axial cross and a narrow central area filled by the cross. C. klausii (Bown andDunkley-Jones 2012, syn. C. opacus, Shamrock andWatkins 2012) differs from C. bornemannii by its medium size (6-8 µm), wide and low birefringence outer rim, low birefringence axial cross, and the presence of a brighter tube cycle that may be indistinct. C. latipons differs from C. bornemannii by its much smaller size (3 to 7 µm) and narrower inner cycle. The median extinction lines of the cross appear as a distinct feature in illustrated specimens of C. klausii and C. latipons (Bown and Dunkley-Jones, 2012, Plate 5-1 to 5-10; Shamrock and Watkins, 2012, Plate 5-2 to 5-4).

Conclusions

We documented changes in the mean size and shape of the cross of specimens of Cruciplacolithus in the early Danian of the Bidart and Elles sections. These results illustrated the increase in the mean size of the group through the early Danian and allowed for the definition of two new species with a thick cross filling the central area, the small, <7 µm in length C. praebornemannii and the large, >7 µm in length C. bornemannii. Correlation of successive nannofossil evolutionary events and bulk carbonate carbon isotopes between the two sections highlights a delayed recovery in carbon isotopes at Elles after the K-Pg negative shift which is best explained by a lengthened environmental stress, and hence delayed productivity rise in shelf environments as compared to pelagic environments. The early Danian evolution of the Cruciplacolithus lineage and the shift in dominance from small, r-selected species to an increasing contribution of large, k-selected species parallels successive recovery episodes in carbon isotopes at the Elles section and thus provide a typical example of how nannofossil assemblages illustrate a stepwise ecosystem recovery from the mass extinction in sensitive shelf environments.

Thibault and Husson, 2016Thibault et al., 2004Thibault et al., 2016van Heck and Prins 1987Varol 1992Westerhold et al., 2008Zachos et al., 2004 Figure captions (N) and planktic foraminifer biohorizons (PF) of Table 1. Note that horizons PF4, PF5 and PF6 have all recently been revised and repositioned between 2 and 3.5 m at Elles (G. Keller, pers. comm.), which agrees much better with our nannofossil line of correlation. Consequently, PF horizons are not considered in this study for correlation of the two sections. Figure 10: Correlation of bulk carbonate stable isotopes, relative abundance of the distinct Cruciplacolithus species and major evolutionary events within this lineage between the sections of Bidart and Elles.

Figure 11: Emergence of new species per lineage, cumulative relative abundances of major groups within the Danian nannofossil assemblage, total nannofossil absolute abundance and cumulative contribution of all Cruciplacolithus species to the total assemblage compared to the evolution in bulk carbonate carbon isotopes in the early Danian of the Elles section.

Table 1: Biohorizons at Bidart and Elles sections with their respective age. Calcareous nannofossil horizons (1) come from Galbrun and Gardin (2004) for the Bidart section and Gardin (2002) for the Elles section with update from the present study. Planktic foraminifer horizons (2) come from Gallala et al. (2009) for the Bidart section and Gallala (2013) for the Elles section. (3) NP zones of [START_REF] Martini | [END_REF]. CNP zones of Agnini et al. (2014). ( 4) Zones of [START_REF] Berggren | [END_REF]. Note that zones P0 to P1c of Keller et al. (1995) are not reported here. The Keller zones do not coincide with that of Berggren and are defined differently, which adds to the general confusion in the high-resolution foraminifer stratigraphy of the early Danian.

Figure 2

 2 Figure 2 about here: Bidart results

-

  Prinsius dimorphosus s.l. as used in the recently revised Cenozoic Biostratigraphy of Agnini et al. (2014) comprises both Prinsius dimorphosus s.s. and Prinsius tenuiculus.

Figure

  Figure 5 about here: Cruci lineage

Figure 7

 7 Figure 7 about here, pics of species with trends

Figure

  Figure 10 about here, correlation Bidart-Elles

Figure 11

 11 Figure 11 about here: ecosystem recovery

Figs. 7 -

 7 Figs. 7-18 to 7-20

  Figs. 7-16 and 7-17

Figure 1 :

 1 Figure 1: Simplified palaeogeography for the late Maastrichtian-early Danian of the western Tethys with locations of the two studied sections. Modified from Philip and Floquet (2000) and Sues (2010).

Figure 2 :

 2 Figure 2: Bulk carbonate carbon isotopes alongside results on the mean length of Cruciplacolithus and density plot for the distribution of the length of Cruciplacolithus specimens on the Bidart section.

Figure 3 :

 3 Figure 3: Bulk carbonate carbon isotopes alongside results on the mean length of Cruciplacolithus and density plot for the distribution of the length of Cruciplacolithus specimens on the Elles section.

Figure 4 :

 4 Figure4: Graphic correlation between the Bidart and Elles section with reports of all calcareous nannofossil(N) and planktic foraminifer biohorizons (PF) of Table1. Note that horizons PF4, PF5 and PF6 have all recently been revised and repositioned between 2 and 3.5 m at Elles (G.Keller, pers. comm.), which agrees much better with our nannofossil line of correlation. Consequently, PF horizons are not considered in this study for correlation of the two sections.

Figure 5 :

 5 Figure 5: Schematic drawings and main criteria of all the different morpho-groups within the "axialcross" Cruciplacolithus lineage encountered in the early Danian of Bidart and Elles.

Figure 6 :

 6 Figure 6: Phylogeny of the Cruciplacolithus lineage with focus on its early Danian evolutionary history.

Figure 7 :

 7 Figure7: Illustrations of the most relevant morpho-groups studied here within the Cruciplacolithus "axial-cross" lineage and general evolutionary trends.

Figure 8 :

 8 Figure 8: Distribution histograms of various Cruciplacolithus species. Lineages with a thin (A-B) and thick axial-cross (C-D) have been treated separately for the Bidart (A-C) and Elles (B-D) sections.

Figure 9 :

 9 Figure 9: Cross-plot of Width versus Length of all specimens measured at Bidart (A) and Elles (B) with their respective systematics.

Appendix 1 :

 1 Specimens of Cruciplacolithus measured in samples of the Bidart section.Appendix 2: Specimens of Cruciplacolithus measured in samples of the Elles section.

  7 µm C. intermedius 8 µm C. bornemannii 9 µm