Enjie Ghorbel
email: enjie.ghorbel@mines-douai.fr

Rémi Boutteau

Jacques Boonaert

Xavier Savatier

Stéphane Lecoeuche

Kinematic Spline Curves: A temporal invariant descriptor for fast action recognition

Keywords: RBG-D cameras, action recognition, low computational latency, temporal normalization

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Introduction

Human action recognition has represented an active field of research over the last decades. Indeed, it has become important due to its wide range of applications. As example, it could be cited video surveillance, Human Machine Interaction (HMI), gaming, home automation and e-health.

The first generation of approaches dealing with human action recognition was generally based on Red Green Blue (RGB) videos. Many reviews of these methods can be found in the literature [START_REF] Poppe | A survey on vision-based human action recognition[END_REF][START_REF] Weinland | A survey of vision-based methods for action representation, segmentation and recognition[END_REF]. However, RGB videos present some major disadvantages such as the sensitivity to illumination changes, occlusions, viewpoint changes and background extraction. To overcome these limitations, new acquisition systems have recently been proposed to give additional information about the observed scene: Red Green Blue-Depth (RGB-D) cameras. Depth images have simplified and sped up the extraction of human skeletons via integrated libraries such as OpenNI and KinectSDK (around 45ms for skeleton extraction per frame according to [START_REF] Papadopoulos | Real-time skeleton-trackingbased human action recognition using kinect data[END_REF]), which was a very difficult task with the use of RGB images. Other acquisition systems such as motion capture systems provide more accurate skeleton sequences but remain unadapted due to their high cost and their non-portability.

In [START_REF] Ghorbel | 3D real-time human action recognition using a spline interpolation approach[END_REF], it has been demonstrated that depth-based methods are generally more accurate than skeleton-based methods since depth modality is more robust to noise and occlusions. However, skeleton-based descriptors are faster to compute because of their lower dimension and are therefore more suited for applications requiring a fast recognition.

Many recent papers have proposed RGB-D-based descriptors for human action recognition, showing their high accuracy of recognition in various datasets [START_REF] Li | Action recognition based on a bag of 3D points[END_REF][START_REF] Ohn-Bar | Joint angles similarities and HOG2 for action recognition[END_REF][START_REF] Oreifej | HON4D: Histogram of oriented 4D normals for activity recognition from depth sequences[END_REF][START_REF] Vemulapalli | Human action recognition by representing 3D skeletons as points in a Lie group[END_REF][START_REF] Yang | Super Normal Vector for activity recognition using depth sequences[END_REF][START_REF] Kim | View invariant action recognition using generalized 4D features[END_REF][START_REF] Luo | Spatio-temporal feature extraction and representation for RGB-D human action recognition[END_REF]. Nevertheless, a central challenge is often omitted in these earlier papers:

How long does it take to recognize an action? In fact, if an action recognition system does not work in real-time, its usability in real-life applications remains limited. Thus, the performance of motion descriptors can be viewed as a trade-off between accuracy of recognition and low latency as mentioned in [START_REF] Ellis | Exploring the trade-off between accuracy and observational latency in action recognition[END_REF], where latency is defined as the sum of computational latency (the time necessary for computation) and observational latency (the time of observation required to make a good decision). In this paper, we will focus mainly on computational latency because the actions are assumed to have already been segmented. Indeed, many off-line applications still require a quick recognition such as medical rehabilitation, gaming, coaching, etc.

Based on these observations, we propose a new skeleton-based human action descriptor called Kinematic Spline Curves (KSC) which operates with low computational latency and with acceptable accuracy compared with recent state-of-the-art methods.

To build this descriptor, a Skeleton Normalization (SN) is first carried out to overcome the anthropometric variation. Then, kinematic features including joint position, joint velocity and joint acceleration are calculated from the normalized skeleton data.

After this step, a Temporal Normalization (TN) is carried out in order to reduce the effect of execution rate variability. The TN introduced in this paper is called Time Variable Replacement (TVR). It is based on a change of variables of time by a variable called Normalized Action Time (NAT). Thus, this variable is invariant to execution rate and varies in a fixed range whatever the length of the action is. Functions that can be used to do this change of variables are called Time Variable Replacement Functions (TVRF). In this article, Two TVRF are introduced, namely, the Normalized Accumulated kinetic Energy (NAE) of the skeleton and the Normalized Pose Motion Signal Energy (NPMSE). This step is a novel, fast and simple way to make actions with different execution rate comparable, without the loss of the temporal information. Finally, features are interpolated via a cubic spline interpolation and are uniformly sampled in order to obtain vectors with the same size.

To achieve a fair comparison of our descriptor with other approaches in terms of accuracy and computational latency, we report the accuracy of recognition and the execution time per descriptor on various challenging benchmarks. Available algorithms of recent state-of-the-art methods are tested with the same parameters and settings on four datasets: MSRAction3D dataset [START_REF] Li | Action recognition based on a bag of 3D points[END_REF] UTKinect dataset [START_REF] Xia | View invariant human action recognition using histograms of 3D joints[END_REF], MSRC12 dataset [START_REF] Fothergill | Instructing people for training gestural interactive systems[END_REF] and Multiview3D Action dataset [START_REF] Hammouche | Toward a real time viewinvariant 3d action recognition[END_REF]. Our descriptor is also tested on a large-scale dataset NTU RGB+D dataset [START_REF] Shahroudy | NTU RGB+D: A large scale dataset for 3D human activity analysis[END_REF].

Based on our previous work [START_REF] Ghorbel | A fast and accurate motion descriptor for human action recognition applications[END_REF], the paper presents a complete action recognition method including novelties such as: 1) The generalization and the extension of a fast and simple novel Temporal Normalization (TN) method called Time Variable Replacement (TVR) in order to avoid the execution rate variability. 2) The detailed analysis of the different component influence (kinetic values, temporal normalization, spatial normalization, etc). 3) A larger experimentation by including three other challenging benchmarks (MSRC-12, Multiview3D and NTU RGB+D datasets). This article follows this scheme: in Section 2, an overview of related methods is given. Section 3 presents the concept of cubic spline interpolation and Section 4 presents the methodology used to build our descriptor. In Section 5, the experiments and the results are discussed. Finally, Section 6 summarizes this paper and includes some ideas for future work.

Related work

In a recent survey [START_REF] Zhu | From handcrafted to learned representations for human action recognition: a survey[END_REF], action recognition methods have been categorized according to the nature of the used representation: learned representations and hand-crafted representations.

Hand-crafted methods are the most common approaches used in the literature [START_REF] Ohn-Bar | Joint angles similarities and HOG2 for action recognition[END_REF][START_REF] Vemulapalli | Human action recognition by representing 3D skeletons as points in a Lie group[END_REF][START_REF] Wang | Mining actionlet ensemble for action recognition with depth cameras[END_REF]. They are based on the classical schema of action recognition, where low-level features are first extracted, then the final descriptor is modeled using low level features and finally a classifier is used to train a classification model such as Support Vector Machine (SVM) or k Nearest Neighbors (kNN).

With the recent advances in deep learning, learning-based representations have been proposed. Instead of selecting specific features, this category of methods learn itself the appropriate ones as in [START_REF] Ke | A new representation of skeleton sequences for 3D action recognition[END_REF][START_REF] Liu | Spatio-temporal LSTM with trust gates for 3D human action recognition[END_REF][START_REF] Ke | Skeletonnet: Mining deep part features for 3-D action recognition[END_REF][START_REF] Liu | 3D-based deep convolutional neural network for action recognition with depth sequences[END_REF][START_REF] Qiao | Learning discriminative trajectorylet detector sets for accurate skeleton-based action recognition[END_REF]. These methods are very interesting and efficient. However, they require an important amount of data, as well as an important execution time for learning.

In this paper, we focus on hand-crafted methods, since we are interested in recognizing actions using small scale datasets.

The RGB-D-based human action descriptors are commonly divided into two categories, namely depth-based descriptors and skeleton-based descriptors.

Depth-based descriptors

The first tendency was to adapt motion descriptors, initially developed for RGB or gray-level images, to depth images. For example, we can cite the work of [START_REF] Xia | Spatio-temporal depth cuboid similarity feature for activity recognition using depth camera[END_REF] where the Spatio-Temporal Interest Points detector (STIP) [START_REF] Belongie | Monitoring animal behavior in the smart vivarium[END_REF] and the cuboid descriptor [START_REF] Dollár | Behavior recognition via sparse spatio-temporal features[END_REF] were adapted to depth images. They called them respectively the Depth-Spatio-Temporal Interest Points (DSTIP) and the Depth-Similarity Cuboid Features

(DSCF).

Other examples can be mentioned such as the extension of Histogram of Oriented Gradient (HOG) [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] to the HOG3D [START_REF] Klaser | A spatio-temporal descriptor based on 3Dgradients[END_REF] and also its extension to HOG2 [START_REF] Ohn-Bar | Joint angles similarities and HOG2 for action recognition[END_REF]. However, some researchers rapidly pointed out the limitations of these adapted methods justifying their claims by the fact that depth modality has different properties and a different structure from color modality [START_REF] Oreifej | HON4D: Histogram of oriented 4D normals for activity recognition from depth sequences[END_REF][START_REF] Yang | Super Normal Vector for activity recognition using depth sequences[END_REF].

Because of the popularity of the Bag-Of-Words approach and its variants in the field of RGB-based action recognition [START_REF] Sabri | Human action classification using surf based spatio-temporal correlated descriptors[END_REF][START_REF] Bettadapura | Augmenting bag-of-words: Datadriven discovery of temporal and structural information for activity recognition[END_REF], many attempts have been made to extend this kind of methods to depth modality [START_REF] Foggia | Recognizing human actions by a bag of visual words[END_REF][START_REF] Shukla | Action recognition using temporal bag-ofwords from depth maps[END_REF][START_REF] Brun | Action recognition by using kernels on aclets sequences[END_REF].

Less conventional depth-based descriptors were also proposed [START_REF] Oreifej | HON4D: Histogram of oriented 4D normals for activity recognition from depth sequences[END_REF][START_REF] Rahmani | Histogram of oriented principal components for cross-view action recognition[END_REF][START_REF] Slama | Grassmannian representation of motion depth for 3D human gesture and action recognition[END_REF][START_REF] Yang | Super Normal Vector for activity recognition using depth sequences[END_REF]. One of the most efficient representations that we can cite is the 4D normals. This original idea was introduced in the work of [START_REF] Oreifej | HON4D: Histogram of oriented 4D normals for activity recognition from depth sequences[END_REF] where the motion of the human body was considered as a hyper-surface of R 4 (three dimensions for the spatial components and one dimension for the temporal component). In differential geometry, one of the wellknown ways to characterize a hyper-surface is to find its normals. Thus, Oreifej and Liu chose these features and built the descriptors by quantifying a 4D histogram of normals using polychrons (a 4D extension of simple polygons [START_REF] Coxeter | Regular polytopes[END_REF]).

Many authors were inspired by this relevant representation, but changed the quantization step (because of the neighboring information loss with the use of HON4D). Slama et al. [START_REF] Slama | Grassmannian representation of motion depth for 3D human gesture and action recognition[END_REF] modeled 4D normals as subspaces lying on Grassmann manifold. At the same time, Yang et al. [START_REF] Yang | Super Normal Vector for activity recognition using depth sequences[END_REF] used the same representation by learning a sparse dictionary and then by coding each action as a word of the learned dictionary. The obtained descriptor was called the Super Normal Vector (SNV). This new generation of features based on depth modality has shown its high accuracy compared to other representations. Notwithstanding this important advantage, they generally require considerable computation time as shown in Section 5.

Skeleton-based descriptors

One of the pioneer work proposed to build a 3D Histogram Oriented of Joints (HOJ) where the 3D position of joints is used [START_REF] Xia | View invariant human action recognition using histograms of 3D joints[END_REF]. Thus, each posture is represented by a specific histogram. The evolution of these postures is included in a Hidden Markov Model (HMM) which is based on a bag of postures. These features are interesting but are very sensitive to anthropometric variability.

To reduce the effect of this body variation, new methods emerged, where the relative position of joints were used instead of the absolute position as in [START_REF] Yang | Eigenjoints-based action recognition using Naive-bayes-Nearest-Neighbor[END_REF]. These authors [START_REF] Yang | Eigenjoints-based action recognition using Naive-bayes-Nearest-Neighbor[END_REF] proposed the use of the spatial and temporal distances between joints as features. In [START_REF] Devanne | 3-D human action recognition by shape analysis of motion trajectories on Riemannian manifold[END_REF][START_REF] Devanne | Motion segment decomposition of RGB-D sequences for human behavior understanding[END_REF], the main idea is to calculate a similarity shape measure between joint trajectories. To realize that, these curves are reparametrized using the Square-Root Velocity Function (SRVF), leading to a representation in a Riemannian manifold.

Recently, new skeleton-based descriptors have been proposed, inspired by earlier bio-mechanical studies [START_REF] Johansson | Visual perception of biological motion and a model for its analysis[END_REF] where the skeleton was a very commonly-used representation. Our work is principally inspired by these descriptors more particularly by the two following papers.

On the one hand, Vemulapalli et al. [START_REF] Vemulapalli | Human action recognition by representing 3D skeletons as points in a Lie group[END_REF][START_REF] Vemulapalli | R3DG features: Relative 3D geometrybased skeletal representations for human action recognition[END_REF] proposed to define the rotations and translations between adjacent skeleton segments using transformation matrices of the Special Euclidean group SE(3). Therefore, the obtained features represent a concatenation of transformation matrices expressed in SE n (3), where n is equal to the number of segment connections. To switch from a discrete space to a continuous one and to represent descriptors as curves of the Lie group SE n (3), an interpolation is done on the points of SE n (3) after the transition to se n (3), the Lie algebra associated with SE n (3). This method considers the evolution of the static pose but does not take into account some important physical measures such as velocity and acceleration.

On the other hand, Zanfir et al. [START_REF] Zanfir | The moving pose: An efficient 3D kinematics descriptor for low-latency action recognition and detection[END_REF] chose to use the kinematics of skeleton joints as features: joint positions, joint velocities and joint accelerations. To overcome anthropometric variability, a skeleton normalization is done. After that, joint velocities and joint accelerations are computed using the information from joint positions. The three kinematic values are then concatenated in a single vector, where the importance of each kinematic value is weighted by optimized parameters. Using this kinematic-based descriptor, the action recognition is done using a kNN classifier (k Nearest-Neighbors).

Even if this method has given good results according to the experimentation of [START_REF] Zanfir | The moving pose: An efficient 3D kinematics descriptor for low-latency action recognition and detection[END_REF], the discontinuity of features can generate some limitations. Indeed, the non-uniformity of velocity during acquisition or the presence of noisy skeletons can negatively impact the results.

Cubic spline Interpolation: A review

As presented in Introduction, a cubic spline interpolation will be used to interpolate kinematic features. We reject the idea of using approximation instead of interpolation because of the low number of frames (generally between 20 and 50 per action in wellknown datasets) favoring an inaccurate approximation. This method is a well-known numerical method which connects coherently a set of N points (α k , y k )) k∈ 1,N with α 1 < α 2 < ... < α k < ... < α N . The goal is to estimate the function f which is assumed to be a continuous C 1 piecewise function of N -1 cubic third degree polynomials f k as described by equation [START_REF] Poppe | A survey on vision-based human action recognition[END_REF].

f (t) = (f k (t)) f or α k < t < α k+1 (1)
Each function f k is defined on the slice [α k , α k + 1] (2). In order to respect the continuity of f , each f k is joined at α k by f k-1 (the first derivative y k = y k+1 -y k x k+1 -x k and the second derivative y k = y k+1 -y k x k+1 -x k of y k are also assumed to be continuous). The coefficients a k , b k , c k , d k of each polynomial f k are computed following the equations (3).

f k (α) = a k (α -α k ) 3 + b k (α -α k ) 2 + c k (α -α k ) + d k (2)
where

                     a k = 1 (α k+1 -α k ) 2 (-2 y k+1 -y k α k+1 -α k + y k + y k+1 ) b k = 1 α k+1 -α k (3 y k+1 -y k α k+1 -α k -2y k -y k+1 ) c k = y k d k = y k (3)
We choose this kind of interpolation because of the particular behavior of third degree polynomial which presents a maximum of one inflection point. Therefore, this kind of curve is at the same time realistic (in our case) and does not present undesired oscillations. Furthermore, this algorithm presents low complexity, compared to other interesting interpolation techniques such as B-spline.

To carry out the cubic spline interpolation, a matlab toolbox implementing the algorithm proposed by [START_REF] Boor | Spline toolbox for use with MATLAB: user's guide, version 3[END_REF] is directly used.

Kinematic Spline Curves (KSC)

In this section, we describe the different procedures allowing the calculation of the proposed human action descriptors KSC. Figure 1 illustrates these different processes.

Each subsection details one of these steps.

Spatial Normalization (S.N.) via skeleton normalization

In this study, an action is represented by a skeleton sequence varying over time.

At each instant t, the skeleton is represented by the pose P(t), which is composed of n joints with the knowledge of their 3D position p j (t) = [x j (t), y j (t), z j (t)] with j ∈ 1, n .

P(t) = [p 1 (t), ..., p j (t), ..., p n (t)] (4)
Thus, a skeleton sequence, which represents a specific action (segmented actions), can be seen as a multidimensional time series. As in the majority of bio-mechanical studies, the initial position of human hip joint is assumed to be the origin. This is why we subtract the hip joint coordinates p hip from each joint coordinates. Figure 2 gives an example of skeleton and describes hip joint location.

P(t) = [p 1 (t) -p hip , ..., p j (t) -p hip , ..., p n (t) -p hip ] (5)
Although we consider that the hip joint is the absolute origin, an important spatial variability continues to be present, mainly due to anthropometric variability. To overcome this variability, we propose a skeleton normalization that is very similar to the normalization used in [START_REF] Zanfir | The moving pose: An efficient 3D kinematics descriptor for low-latency action recognition and detection[END_REF]. However, these latter chose to learn an average skeleton for each dataset and then to constrain each skeleton of the dataset to have the same limb sizes. The problem is that in real-world applications, the average skeleton of a specific dataset cannot be representative. Thus, we propose the Euclidean normalization of each segment length without imposing a specific length. Hence, we obtain skeletons with unitary segments. Each skeleton is normalized successively starting with the root (hip joint) and moving gradually to the connected segments. This approach preserves the shape of the skeleton and its performance will be discussed in the Section 5. Algorithm 1 describes the skeleton normalization used. All joint positions are normalized except the hip joint position which is assumed to be the root and is therefore unchanged.

p norm hip = p hip .
We use the normalized skeleton sequence P norm obtained after applying Algorithm 1 to p hip to design our descriptor as depicted by equation [START_REF] Ohn-Bar | Joint angles similarities and HOG2 for action recognition[END_REF]. Since all joints are interconnected, we obtain a normalized position for each joint i, (p

norm a i , p norm b i with a i , b i ∈ 1, n ). P norm = [p norm 1 , p norm 2 , ..., p norm n ] (6)

Kinematic Features (KF)

As in [START_REF] Zanfir | The moving pose: An efficient 3D kinematics descriptor for low-latency action recognition and detection[END_REF], we assume that human motion can be described by three important mechanical values: joint positions (normalized) P norm , joint velocities V and finally Features (KF), noted KF in Equation [START_REF] Oreifej | HON4D: Histogram of oriented 4D normals for activity recognition from depth sequences[END_REF].

KF(t) = [P norm (t), V(t), A(t))] (7)
In this subsection, we describe how velocity (8) and acceleration (9) are numerically computed from the discrete data of joint positions as in [START_REF] Zanfir | The moving pose: An efficient 3D kinematics descriptor for low-latency action recognition and detection[END_REF]. In reality, the skeleton is composed of a skeleton sequence of N frames. k ∈ 1, N denotes the frame index and t k represents its associated instant.

V(t k ) = P norm (t k+1 ) -P norm (t k-1 ) (8)
A(t k ) = P norm (t k+2 ) + P norm (t k-2 ) -2 × P norm (t k ) (9)
The dimension of the KF extracted from each frame is equal to M = 9 × n. We recall that n is the number of joints.

Time Variable Replacement (TVR) : a new approach for Temporal Normalization (TN)

Temporal variability is principally caused by execution rate variability (the temporal differences that exist when a subject repeats a same action or when the different subject performs a same action), which implies changeable action duration and different distribution of motion. This means that actions are not performed in the same time slice as illustrated on the top panel of the Figure 3. Consequently, recognizing actions with features varying in different time slices proves to be difficult. For this reason, Temporal Normalization (T.N.) is needed to improve the efficiency of our method. We propose to interpolate the components of the features, according to a novel variable called Normalized Action Time (NAT), instead of time. This change of variable is done thanks to a function TVRF which places the actions in a space that is invariant to execution rate variability as shown in Figure 3. Therefore, the KF can be expressed as a variable depending on NAT which is rate-invariant [START_REF] Kim | View invariant action recognition using generalized 4D features[END_REF].

KF(N AT ) = [P norm (N AT ), V(N AT ), A(N AT )] (10)
The choice of the function TVFR is crucial for the good functioning of the normalization.

1) First, this function should have a physical meaning which makes features invariant to the execution rate variability.

2) Second, the used function should be increasing and have to realize a one-to-one correspondence with time.

3) Finally, to compare actions with different temporal length, the latter function should vary in a fixed range independently from the length of the skeleton sequence as described by Figure 4.

In this way, let consider the T V RF instance (instance represents the index of the action instance) as an increasing one-to-one correspondence between the variable interval of time and a fixed interval where NAT varies: The Normalized Accumulated kinetic Energy (NAE) of the skeleton is proposed as a Time Replacement Variable Function (TVRF). At an instant t and for each instant the human body on the whole skeleton sequence composed of N frames. Equation [START_REF] Luo | Spatio-temporal feature extraction and representation for RGB-D human action recognition[END_REF] describes this new term where E(t) represents the kinetic energy consumed by the human body at an instant t.

∀instance, T V RF instance : [0, N instance ] -→ [a, b] t -→ T V RF instance (t) = N AT
250

N AT = T V RF inst (t) = N AE(t) = E kinetic acc (t) E kinetic total (11)
Since the data structure is discrete. For each punctual instant t k , NAE is calculated as follows in Equation [START_REF] Ellis | Exploring the trade-off between accuracy and observational latency in action recognition[END_REF]. We recall that N represents the number of frames contained in the skeleton sequence.

N AT = T V RF inst (t k ) = N AE(t k ) = E kinetic acc (t k ) E kinetic total = t k c1=1 E kinetic (c 1 ) N c2=1 E kinetic (c 2 ) (12)
Indeed, the NAE variable increases when the velocity of joints increases and consequently when the displacement quantity increases as well. If there is no motion, NAE does not increase. We use NAE for two essential reasons. First, all actions must be expressed in the same space which is guaranteed by the normalization of the energy (varying between 0 and 1). Secondly, to perform a coherent interpolation, a growing variable is necessary. This is why accumulation is used.

Kinetic Energy Calculation: Instead of considering the skeleton as a body, it can be assimilated to a set of n points where each one represents a skeleton joint. In many earlier papers [START_REF] Onuma | FMDistance: A fast and effective distance function for motion capture data[END_REF][START_REF] Shan | 3D human action segmentation and recognition using pose kinetic energy[END_REF][START_REF] Shi | A local feature descriptor based on energy information for human activity recognition[END_REF], the kinetic energy of the human skeleton E kinetic (t k ), at an instant t k , is expressed by the equation [START_REF] Xia | View invariant human action recognition using histograms of 3D joints[END_REF] where n represents the number of joints, m i and V i , respectively, the mass and the velocity of the joint i. Since the skeleton joints are fictitious, we assume that they have a unitary mass (m i = 1, ∀i).

E kinetic (t k ) = n i=1 1 2 m i V 2 i (t k ) = n i=1 1 2 V 2 i (t k ) (13)
The NAE term N AE(t k ) is calculated, thanks to the kinetic energy term E kinetic (t k ) (13) described in the previous paragraph following equation [START_REF] Luo | Spatio-temporal feature extraction and representation for RGB-D human action recognition[END_REF]. After a change of variables, the KF can be expressed as a variable depending on NAE [START_REF] Kim | View invariant action recognition using generalized 4D features[END_REF]. Hence, Kinematic Features are expressed in the same range [0, 1] and become therefore less sensitive to execution rate variability.

Normalized Pose Motion Signal Energy (NPMSE) as a Time Variable Replacement Function (TVRF)

A second TVRF is proposed to replace the time variable called Normalized Pose Motion Signal Energy (NPMSE).

Before defining the notion of Pose Motion, we need to define the rest pose. The rest pose represents the position of the spatially normalized skeleton when any gesture can be visually detected as in Figure 2.

To obtain this rest pose, a statistical model could be learnt using the training data.

For the moment, since we are still working with temporally segmented data and since we know that all actions in the used datasets start with the rest pose, we can use the first skeleton of the sequence to define the rest pose.

We define the Pose Motion Signal s(t) as the distance d between the current pose P norm (t) and the rest pose P norm (rest) (first pose in our experimental conditions) [START_REF] Fothergill | Instructing people for training gestural interactive systems[END_REF].

s(t) = d(P norm (t), P norm (rest)) (14)
This used distance can be l 2 , l 1 , etc. More details about the nature of the distance will be given in the experimental section. Since the available data has a discrete nature, the energy of the discrete signal s(t k ) at an instant t k is calculated as follows [START_REF] Hammouche | Toward a real time viewinvariant 3d action recognition[END_REF].

This term is called Pose Motion Signal Energy.

E signal (t k ) = k i=1 |s(t i )| 2 (15)
It is important to specify that in this paragraph the energy calculated is the signal energy and not the kinetic one, which are two different notions.

The Pose Motion Signal Energy is a growing term, thanks to the use of the sum in the energy calculation. However, to ensure that all the actions will be expressed in the same range, a normalization of this term has to be done. For an instance instance, the TVRF is therefore the Normalized Pose Motion Signal Energy (NPMSE) which is calculated by dividing the Pose Motion Signal Energy at an instant t by the Pose Motion Signal Energy at the final instant of the sequence t f as described by equation [START_REF] Shahroudy | NTU RGB+D: A large scale dataset for 3D human activity analysis[END_REF].

N AT = T V RF instance (t) = N P SM E(t) = E signal (t) E signal (t f ) (16)
Since the time data are discrete, the NPMSE at an instant t k is calculated as follows [START_REF] Ghorbel | A fast and accurate motion descriptor for human action recognition applications[END_REF].

N AT = T V RF instance (t k ) = N P M SE(t k ) = E signal (t k ) E signal (t N ) = k i=1 |s(t i )| 2 N i=1 |s(t i )| 2 (17)
Relation with key poses: The idea of using the distance between the current pose and the rest pose provides from the fact that the more the poses are different from the rest state, the more they can discriminate actions. It is in a certain way related to the notion of key poses presented in many papers [START_REF] Miranda | Real-time gesture recognition from depth data through key poses learning and decision forests[END_REF][START_REF] Cuntoor | Key frame-based activity representation using antieigenvalues[END_REF][START_REF] Raptis | Real-time classification of dance gestures from skeleton animation[END_REF], where the key pose is defined as the extreme pose of an action. In [START_REF] Miranda | Real-time gesture recognition from depth data through key poses learning and decision forests[END_REF], authors show that this kind of pose is sufficient to obtain interesting results. However, this kind of method does not include the spatio-temporal aspect which is also important. For this reason, we propose this TVRF function which exploits the notion of keypose without removing the spatiotemporal aspect.

Cubic spline interpolation of Kinematic Features (KF)

Based on the continuous nature of human actions, we assume that the kinematic values (position, velocity, acceleration) also represent continuous functions of time.

To switch from a numerical space to a continuous one, we propose to interpolate KF components depending on NAT as described in Section 4.3. For the interpolation, we use the cubic spline interpolation described in Section 3. Using the discrete information of each KF component, we obtain continuous functions KF c depending on a chosen TVRF, as described in equation [START_REF] Zhu | From handcrafted to learned representations for human action recognition: a survey[END_REF], where Spline refers to the cubic spline operator.

We recall that M represents the dimension of KF.

KF c i (N AT ) = KF c i (T V RF instance (t)) = Spline(KF i (T V RF (t k ))) k∈ 0,N , ∀i ∈ 1, M (18)

Uniform sampling

To build the final descriptor KSC of an instance instance, the obtained functions are uniformly sampled, choosing a fixed number of samples s. The choice of s will be depicted by equation [START_REF] Wang | Mining actionlet ensemble for action recognition with depth cameras[END_REF].

KSC = ∪ i=1..M ∪ e=0..s-1 KF c i (N AT (a + e(b -a) s -1 )) (19)
Figure 5 illustrates the interest of uniform sampling after interpolation of curves expressed as functions of a TVRF. The idea is to maximize the Euclidean distance be-tween two feature vectors when the represented actions are different and to maximize it when they represent the same action. We summarize the different processes that allow us to compute a KSC descriptor from a skeleton sequence in Algorithm 2.

Algorithm 2: Computation of KSC from an instance instance Input : skeleton sequence (P j (t k )) 1≤j≤n,1≤k≤N

Output: KSC

1 Normalize Skeleton (P norm j (t k )) 1≤j≤n,1≤k≤N (6)
2 Compute Kinematic Features KF i (t k )) 1≤i≤M [START_REF] Kim | View invariant action recognition using generalized 4D features[END_REF] 3 Compute (T V RF instance (t k )) 1≤k≤N (11)

4 for i ← 1 to M do 5 Interpolation: KF c i (N AT ) = KF c i (T V RF instance (t)) := Spline(KF i (T V RF instance (t k ))) 1≤k≤N 6 end
7 Uniform sampling with a sampling rate s:

KSC := ∪ i=1..M ∪ e=0..s-1 KF c i (N AT (a + (b-a)e s-1
))

Action recognition via linear Support Vector Machine

To perform action recognition, the KSC is used with a linear Support Vector Machine (SVM) classifier provided by the SVM library LibSVM [START_REF] Chang | LIBSVM: A library for Support Vector Machines, Transactions on Intelligent Systems and Technology[END_REF]. The multi-class SVM proposed by [START_REF] Crammer | On the learnability and design of output codes for multiclass problems[END_REF] is carried out with a cost which is equal to 10. The main advantage of the linear kernel classifier is its low computational latency compared to non-linear kernel ones [START_REF] Fanello | Keep it simple and sparse: Realtime action recognition[END_REF].

Experimental evaluation

In this section, we propose to evaluate the proposed method for action recognition on five benchmarks: MSRAction3D [START_REF] Li | Action recognition based on a bag of 3D points[END_REF], UTKinect [START_REF] Xia | View invariant human action recognition using histograms of 3D joints[END_REF], MSRC12 [START_REF] Fothergill | Instructing people for training gestural interactive systems[END_REF], Multiview3D

Action dataset [START_REF] Hammouche | Toward a real time viewinvariant 3d action recognition[END_REF] and NTU RGB+D dataset [START_REF] Shahroudy | NTU RGB+D: A large scale dataset for 3D human activity analysis[END_REF].

Experimental Settings

In this subsection, we specify the conditions of experimentation settings chosen for the four first benchmarks. All calculations were run on the same machine, a Dell Inspiron N5010 laptop computer with intel Core i7 processor, Windows 7 operating 1 Multiview3D dataset can be download from the following link:

http://english.stackexchange.com/questions/55898/related-work-or-related-works system and 4GB RAM. To evaluate a given method in terms of computational latency, we propose to report the Mean Execution Time (MET) per descriptor. Indeed, this value which represents the average time necessary to compute a descriptor is an interesting indicator. We underline that each descriptor corresponds to the feature vector extracted from a whole action.

It could be noted that in the literature, papers generally do not report the execution time per descriptor, providing only the recognition accuracy. On the other hand, the parameters of evaluation vary greatly from one paper to another as discussed in the work of [START_REF] Padilla-López | A discussion on the validation tests employed to compare human action recognition methods using the MSR Action3D dataset[END_REF]. Since the goal of this work is to realize a trade-off between computational latency and accuracy, we propose to recover and to evaluate some available descriptors in terms of MET and accuracy. These latter can be divided into two main groups. For MSRAction3D, the parameters used [START_REF] Li | Action recognition based on a bag of 3D points[END_REF] are followed. The dataset is divided into three groups (AS1, AS2, AS3) where each one is composed of 8 actions. Hence, the classification is done on each group separately. A cross-splitting is carried out to separate the data in training and testing samples as in [START_REF] Yang | Super Normal Vector for activity recognition using depth sequences[END_REF]: the data performed by the subjects 1,3,5,7,9 have been used for training and the rest has been used for testing.

For UTKinect, we used the settings given in the experimentation of [START_REF] Vemulapalli | Human action recognition by representing 3D skeletons as points in a Lie group[END_REF], where the actions performed by half of the subjects were considered as training data and the rest as testing data.

For MSRC12, we followed the same protocol used in the paper of [START_REF] Hussein | Human action recognition using a temporal hierarchy of covariance descriptors on 3D joint locations[END_REF] where the end points of actions have been added to the dataset. Thanks to these points, it became possible to benefit from the large amount of data in order to test an action recognition method without detecting actions. Thus, a cross-splitting is done where the actions performed by half of the subjects are used for training, while the rest of data is used for testing. AS3 represents the three groups proposed in the protocol experimentation of [START_REF] Li | Action recognition based on a bag of 3D points[END_REF]. *The values have been recovered from the state-of-the-art For Multiview3D, we followed the experimental settings proposed in [START_REF] Hammouche | Toward a real time viewinvariant 3d action recognition[END_REF]. A crosssplitting is also done dividing the training and the testing samples. Then, the classification is done several times taking each time a specific orientation for the training and another specific orientation for the testing. This procedure allows us to analyze the effect of view-point changes on the accuracy of recognition.

In the experiments, the two TVRF presented in Section 4.3 are tested: the NAE and the NPMSE. When the KSC is calculated based on the temporal normalization using the NAE, our descriptor is noted KSC-NAE. To calculate NPMSE, two kind of distances are used l 1 and l 2 . When the function NMPSE (using the distances l 1 and l 2 )

during the temporal normalization, our descriptor is respectively noted KSC-NMPSE-

l 1 and KSC-NMPSE-l 2 .
The first benchmark allows us to test our algorithm against others in terms of accuracy and rapidity. Testing our descriptor in the other three datasets shows its robustness. 3, Table 4 and Table 5 compare our approach with state-of-the-art methods in terms of computational latency by reporting the MET per descriptor in respectively MSRAction3D, UTKinect, MSRC12 and Multiview3D datasets. As mentioned in [START_REF] Papadopoulos | Real-time skeleton-trackingbased human action recognition using kinect data[END_REF], it has been shown that the skeleton extraction process lasts approximately 45ms/per frame with the knowledge that actions are generally contained in a window composed of 12 to 50 frames. Thus, a fair comparison of execution time would have consisted in adding a penalty of 2250 ms for skeleton methods. But, this would have no influence on the fact that our approach remains more suited to real-time applications, despite an additional extraction process. Indeed, we show that our descriptors KSC-NAE, KSC-NPMSE-l 1 and KSC-NPMSE-l 2 are faster in terms of computational latency with a maximum of only 0.092s of MET on MSRAction3D dataset, 0.10 on UTKinect dataset, 0.137 on MSRC12 dataset and 0.092 on Multiview3D dataset. Also, Table 2, which presents the execution time per descriptor of each process on the four benchmarks. Since some state-of-the-art methods do not provide their codes, we added an asterisk in front of their name to indicate that the accuracy has been directly re-covered from the associated papers (the experimental conditions can be different than ours). For this reason, we could not provide the MET required for these methods.

However, in [START_REF] Rahmani | Histogram of oriented principal components for cross-view action recognition[END_REF], the time of calculation required for each frame is reported. The authors mention that the calculation of their descriptor takes about 2 seconds per frame (between 30 seconds and 100 seconds for a whole video of the MSRAction3D dataset).

Knowing that the machine used for their experiments [START_REF] Papadopoulos | Real-time skeleton-trackingbased human action recognition using kinect data[END_REF][START_REF] Ghorbel | 3D real-time human action recognition using a spline interpolation approach[END_REF] GHz with 24 GB RAM) is largely more powerful than our laptop, one can say that the descriptors presented in our paper require less execution time.

Thus, with these results, we can assume that the constraint of low computational latency is respected. However, it is important to notice that the computational latency also depends from a lot of parameters such as the image resolution (from the number of joints, in this case) as well as the number of classes (20 for MSRAction3D, 10 for UTKinect, 12 for MSRC12 and 12 for Multiview3D datasets) and Table 1, Table 3, Table 4 and Table 5 aim just at comparing execution time of different techniques applied to a similar situation.

Good accuracy and Robustness

Computational latency is an important criterion of evaluation, but is not sufficient.

In fact, the accuracy of recognition must be acceptable. On the MSRAction3D, our method allows us to recognize 89.64% of the actions correctly using KSC-NPMSEl 2 descriptor, which is a better score than other skeleton representations. On the other hand, even if depth-based descriptors give better results in terms of accuracy of recognition, their MET per descriptor remain too high and seem to be unsuitable for real-time applications (6.44s for HOG2, 27.333s for HON4D, 146.57s for SNV versus 0,092s for KSC). Table 3, Table 9 and Table 4 respectively report the accuracy of recognition of our method on UTKinect, MSRC12 and Multiview3D datasets and compare it to state-of-the-art methods. While on UTKinect the accuracy of recognition using KSC-NPMSE-l 2 descriptor is 96% (1% less than the highest score), the accuracy of our method on MSRC12 dataset outperforms recent state-of-the-art method with 94.27%

. On Multiview3D dataset, the accuracy of recognition given by the descriptor KSC-NPMSE-l 2 is also acceptable. When the training and testing are carried out using data 2 shows that computational latency remains relatively low on the four datasets using our method. Thus, we can conclude that the performance of our descriptor is acceptable on the four benchmarks and that our method is robust to the dataset changing.

NAE vs NPMSE

Thanks to the experiments, we can conclude that the NMPSE function gives better results than the NAE function on the four datasets. Indeed, we can notice a difference of around 5% on MSRAction3D dataset, 12% of UTKinect dataset, 10% on MSRC12 dataset and 4% on Multiview3D dataset between the descriptors KSC-NAE and KSC-NPMSE-l 2 , while the execution time per descriptor remains the same. Also, compared to the distance l 1 , the distance l 2 gives slightly more accurate results. In future work, it could be interesting to compare a wider range of more sophisticated distances. By observing the shape of the two curves (NAE and NPMSE-l 2 ) as shown by Figure 6, it can be noticed that the NMPSE is smoother than NAE and that NMPSE increases importantly only in the presence of key poses, while NAE increases more uniformly.

The superiority of NMPSE as a TVRF function could be explained by the fact that the notion of key frames is more exploited with the use of NMPSE. In the rest of experiments, only the KSC-NPMSE-l 2 will be taken into account since it gives the best results.

Benefits of Spatial Normalization (SN)

Table 9 reports the effect of the different processes of normalization. Each column of the table presents the obtained accuracy of recognition, after deleting one of the proposed processes contributing to spatial and temporal normalizations. We can observe that SN contributes considerably to increasing the accuracy on the three datasets. We 490 observe an increase of around 5% for MSRAction3D, 9% for UTKinect and approximately 0.67% and around 2 % for Multiview3D (for respectively SV and DV tests) and more than 1% for MSRC12, when SN is used. To explain the use of unitary Euclidean normalization instead of the use of an average skeleton as in [START_REF] Zanfir | The moving pose: An efficient 3D kinematics descriptor for low-latency action recognition and detection[END_REF], we report the results of our method combined with an average skeleton-based normalization on the four These results confirm the ability of our algorithm to cope with execution rate variability. This algorithm is very useful for pre-segmented actions, since it is fast and efficient.

However, to work efficiently, the video should start with a skeleton in a rest state. Indeed, the calculation of the NPMSE term, which quantifies the important phase of the motion, is based on this assumption.

Benefits of kinematic features

In Table 7, we evaluate the importance of each kinematic term. This table shows that position is generally the most discriminative value followed by velocity and acceleration. This may be due to the increase of the approximation error caused by the derivations. Nevertheless, the combined information of the three kinematic values give us the best amount of accuracy on the four datasets.

Parameter s influence

The parameter s, which represents the number of samples affects the accuracy of recognition. As shown in the information. On the other hand, if s is too high, the accuracy also decreases since the influence of the interpolation errors increases. Indeed, the original data is numerical and has been interpolated. Nevertheless, is important to notice that the choice of s does not affect the results considerably. For the values tested, we observe a decrease of up to 3% compared with the highest score of accuracy for a number of samples varying between 5 and 50.

Performance on a large-scale dataset: NTU + RGB-D dataset

In this part, we propose to test our method on the recent large-scale action recognition benchmark: NTU-RGB+D dataset. Because of the important amount of data, we use for this experimentation another laptop: an i7 macbook pro with 16 Go of RAM.

Since this dataset contains mutual actions involving more than one human, only the skeleton consuming more kinetic energy is considered. On the other hand, some joints are sometimes confused and have exactly the same coordinates. To overcome that, a small Gaussian noise is added to one of the two joints. We follow the same experimental protocol used in [START_REF] Shahroudy | NTU RGB+D: A large scale dataset for 3D human activity analysis[END_REF] and we propose to report the accuracy in Table 9 obtained for the two settings: cross-subject and cross-view. Compared to hand-crafted methods, our approach reaches the best score of accuracy while realizing the cross-subject test (with 51.12 %) and the second best score while performing cross-view test (with 51.39 %, slightly inferior to the accuracy obtained for the Lie Group [8] representation 52.76 %).

We specify that the parameter s has been fixed to 20. Despite the promising results, it can be noted in Table 9 that methods based on learned features remain more efficient on this dataset in terms of accuracy.

Conclusion and future work

In this work, we have presented a novel descriptor for fast action recognition. It is based on the cubic spline interpolation of kinematic values of joints, more precisely, position, velocity and acceleration. To make our descriptor invariant to anthropometric variability and execution rate variation, we perform a skeleton normalization as well as a temporal normalization. For this reason, a novel method of temporal normalization is proposed called TVR. The proposed method have shown its efficiency in terms of accuracy and computational latency on four different datasets.

However, actions are assumed to be already segmented. In future work, the issue of temporal segmentation will be studied. Some techniques developed for dynamical switched models can be used to detect different modes in an unsegmented sequence, allowing the detection of particular points of transition from an action to another or from an action to the rest state. First attempts have been made to adapt these methods in [START_REF] Boukharouba | Temporal video segmentation using a switched affine models identification technique[END_REF][START_REF] Ofli | Sequence of the most infor-730 mative joints (SMIJ): A new representation for human skeletal action recognition[END_REF][START_REF] Vidal | Applications of hybrid system identification in computer vision[END_REF][START_REF] Liu | 3D-based deep convolutional neural network for action recognition with depth sequences[END_REF].

Figure 1 :

 1 Figure 1: An overview of our approach: this figure describes the different steps to compute Kinematic Spline Curves (KSC). The first step represents the skeleton normalization which re-size skeleton to make it invariant to anthropometric variability. Based on the information of normalized joint positions, the joint velocities and joint accelerations are computed. Therefore, the three values are concatenated to obtain KF. To palliate execution rate variability, t is replaced by the NAT variable by using a Time Variable Replacement Function (TVRF). This step represents a novel temporal normalization method, referred as TVR. Then, each component KF is interpolated using a cubic spline algorithm in order to obtain M functions KF c . Finally, The uniform sampling of KF c allows us to build the final descriptor KSC.

Figure 2 :

 2 Figure 2: Skeleton modality extracted using KinectSDK

Figure 3 :

 3 Figure 3: Example of the effect of temporal normalization on a joint component trajectory f (t). In this example, f (t) refers to the x-coordinates of a joint. We consider two instances of the same type of action (Instance1 and Instance2). Top: the trajectories are plotted as functions of time. We can observe that the time slices of the two trajectories are different ( t N1 = t N2 ). Bottom: After the substitution of time by a NAT, we can observe that the trajectories vary in the same range [a,b]. An important remark can also be made: the two trajectories representing the same action type are more similar.

4. 3 . 1 .

 31 Normalized Accumulated kinetic Energy of the skeleton (NAE) as a Time Replacement Variable Function (TVRF)

Figure 4 :

 4 Figure 4: Let Instance 1 , Instance 2 and Instance 3 be three skeleton sequences with a temporal length respectively equal to N 1 , N 2 and N 3 . This figure shows how the TVR allows the expression of different instances with different temporal range in a same fixed range [a, b]

Figure 5 :

 5 Figure 5: Uniform sampling Interest: In this figure, we propose the visualization of a component curve sampling, after the temporal normalization. We consider two instances of the same type of action on the top and two instances containing different actions on the bottom. These samples are used to design the final descriptor. The more the curves are similar, the more the distances d i and d i between their samples are smaller, for i = 0...5 (because the number of samples s is equal to 6 in this example). The classification will be based on this distance since an SVM classifier is used.

  325

MSRAction3D dataset:

  This dataset represents one of the most famous benchmarks captured by a RGB-D camera and contains segmented human actions. This benchmark is composed of 20 actions, performed by 10 different subjects, 2 or 3 times. It provides 2 modalities: skeleton joints and depth maps. The challenge of this dataset resides in its very similar actions. UTKinect dataset: The UTKinect is also a well-known benchmark often used for the testing of RGB-D based human action recognition methods. It contains 3 modalities: RGB images, depth images and skeleton joints. 10 actions are performed twice by 10 subjects. This dataset is very challenging especially due to a significant intra-class variation. MSRC12 dataset: This dataset is generally used for skeleton-based action detection. It contains 594 sequences including 12 different types of gestures performed by 30 subjects. In total, there are 6244 annotated gesture instances. MSRC12 only provides skeleton joints and action points which correspond to the beginning of an action. Despite the fact that MSRC12 dataset has been initially proposed for action detection, it could be very useful for the action recognition task because of its large volume. Multiview3D Action 1 dataset: In opposition to the two other benchmarks, Multi-view3D is a very recent dataset. It also contains the three modalities captured by an RGB-D sensor. The dataset is composed of 12 actions performed by 10 actors in 3 different orientations (-30 • , 0 • , 30 • ). The challenge of this dataset is its wide body orientation variation. NTU RGB+D dataset: The NTU RGB+D dataset is a very large action recognition dataset containing human skeletons. It contains a total of 56880 sequences. This dataset is composed of 60 classes performed by 40 different subjects. Each skeleton is composed of 25 joints. It contains also 80 different viewpoints.

  The first one composed of depth-based descriptors, includes the square Histogram of Oriented Gradient (HOG2), the Histogram of Oriented Normals 4D (HON4D) and the Super Normal Vector (SNV) representations. The second one composed of skeleton-based descriptors, includes Joint Positions (JP), Relative Joint Positions (RJP), Quaternions (Q) and finally Lie Algebra Relative Pairs (LARP). These descriptors are tested on four datasets by respecting the same protocol of evaluation.

Figure 6 :

 6 Figure 6: The visualization of the NAE (green) and the NPMSE (blue) varying over time. Skeletons at some instants t k are also visualized.

  Algorithm 1: Skeleton normalization at an instant t Input : (p a i (t), p b i (t)) 1≤i≤C represents the segment extremities ordered and C represents the number of connections with a i the root extremity and b i the other extremity of the segment i

		Output: (p norm a i	(t), p norm b i	(t)) 1≤i≤C with a i , b i ∈ 1, n
	1 p norm a1	(t) := p a1 (t)(p a1 (t) = p hip (t) represents the position of the hip joint)
	5	p norm b i	(t) := s i + p norm a i	(t)

2 for i ← 1 to C (C:Number of segments) do 3 S i := p a i (t) -p b i (t) 4 s i := Si/distance(p a i (t), p b i (t))

6 end joint accelerations A. These values are concatenated to obtain what we call Kinematic

Table 1 :

 1 Accuracy of recognition and execution time per descriptor(s) on MSRAction3D: AS1, AS2 and

	Descriptor	AS1(%) AS2(%) AS3(%) Overall(%) Time(s)
	HOG2 [6]	90.47	84.82	98.20	91.16	6.44
	HON4D [7]	94.28	91.71	98.20	94.47	27.33
	SNV [9]	95.25	94.69	96.43	95.46	146.57
	JP [8]	82.86	68.75	83.73	78.44	0.58
	RJP [8]	81.90	71.43	88.29	80.53	2.15
	Q [8]	66.67	59.82	71.48	67.99	1.33
	LARP [8]	83.81	84.82	92.73	87.14	17.61
	HOPC* [35]	-	-	-	86.5	-
	KSC-NAE (ours)	86.92	72.32	94.59	84.61	0.092
	KSC-NPMSE-l 1 (ours)	85.05	85.71	96.4	89.05	0.091
	KSC-NPMSE-l 2 (ours)	84.11	87.50	97.30	89.64	0.092

Table 2 :

 2 Execution time (s) of each process per KSC-NPMSE-l 1 descriptor on the four benchmarks

	Process	MSRAction3D UTKinect Multiview3D MSRC12
	Spatial Normalization	0.022	0.016	0.016	0.06
	Descriptor computing	0.07	0.064	0.073	0.077
	Classification	0.008	0.002	0.010	0.015
	Total	0.1	0.082	0.099	0.152
	Descriptor		Accuracy (%) MET (s)
	HOG2 [6]		74.15	5.03
	HON4D [7]	90.92	25.39
	SNV [9]		79.80	1365.33
	JP [8]		100	0.43
	RJP [8]		97.98	1.91
	Q [8]		88.89	1.40
	LARP [8]		97.08	42.00
	Random Forrest* [56]	87.90	-
	ST-LSTM* [21]	95	-
	KSC-NAE (ours)	84.00	0.08
	KSC-NPMSE-l 1 (ours)	94.00	0.09
	KSC-NPMSE-l 2 (ours)	96.00	0.10

Table 3 :

 3 Accuracy of recognition and MET per descriptor on UTKinect dataset. *The values have been

	recovered from the state-of-the-art
	Descriptor	Accuracy (%) MET (s)
	Logistic Regression* [12]	91.2	-
	Covariance descriptor* [55]	91.7	-
	KSC-NAE (ours)	84.00	0.137
	KSC-NPMSE-l 1 (ours)	93.22	0.134
	KSC-NPMSE-l 2 (ours)	94.27	0.134

Table 4 :

 4 Accuracy of recognition and MET per descriptor on MSRC12 dataset. *The values have been

	recovered from the state-of-the-art

Table 5 :

 5 Accuracy of recognition and MET per descriptor on Multiview3D dataset. *The values have been

	recovered from the state-of-the-art
	5.2. Results and discussion
	5.2.1. Low computational latency

Table 1 ,

 1 Table

Table 6 :

 6 Effect of each process on the accuracy of recognition using KSC-NPMSE-l 2 with the Same Viewpoint (SV), the score of accuracy is equal to 91.67% (around 4% percent less than the highest score and 4% percent better than Actionlet). And when the training and testing use data with Different Viewpoints (DV), the recognition accuracy is equal to 86.11% (versus 88.1% for LARP and 69.7% for Actionlet). As mentioned before, Table

Table 7 :

 7 Effect of each kinematic component on the accuracy of recognition using KSC-NPMSE-l 25.2.5. Benefits of Temporal Normalization (TN): the TVR algorithmIn Table5, the removal of the TN process (TVR algorithm) shows its important role in our method. Indeed, without TN, the accuracy of recognition decreases from 89.64%

	to 76.73% on MSRAction 3D, from 96% to 87% on UTKinect, from 91.67%(SV)-
	86.11%(DV) to 81.60%(SV)-68.40%(DV) on Multiview3D and from 94.17% to 81.43%.

Table 8 ,

 8 with varying the parameter s, the accuracy of recognition will also vary. The parameter s is therefore fixed according to the best score obtained ( s = 15 for MSRAction3D, UTKinect and Multiview3D and s = 40 for MSRC12). If s is fixed too small, the accuracy decreases because of a loss of numberof sampless 5 (%) 10 (%) 15 (%) 20 (%) 25 (%) 30 (%) 35 (%) 40 (%) 45 (%) 50 (%)

	MSRAction3D	86.03 86.90	89.64	88.75	88.46	88.14	87.55	87.86	86.96	86.94
	UTKinect	94.00 95.00	96.00	93.00	95.00	95.00	95.00	94.00	94.00	94.00
	Multiview3D (SV)	88.54 90.28	91.67	90.45	89.58	89.76	89.76	90.10	90.10	90.10
	Multiview3D (DV)	83.51 85.07	86.11	84.11	84.20	84.72	84.38	84.55	84.37	84.46
	MSC12	92.20 92.96	93.91	93.92	93.69	94.04	94.11	94.17	94.08	93.98

Table 8 :

 8 Effect of s the number of samples s on the accuracy of recognition using KSC-NPMSE-l 2

Table 9 :

 9 Accuracy of recognition of different methods on NTU-RGB+D

	Class of method Method	Cross-subject (%) Cross-view (%)
	hand-crafted	HOG2 [6]	32.24	22.27
		SNV [9]	31.82	13.61
		HON4D [7]	30.56	7.26
		Lie Group [8]	50.08	52.76
		Skeletal Quads [57]	38.62	41.36
		KSC-NMPSE-l 2 (ours)	51.12	51.39
	learned features HBRNN-L [58]	59.07	63.97
		ST-LSTM [21]	69.2	77.7
		SkeletonNet [22]	75.94	81.16
		Rahmani et al. [59]	75.2	83.1
		Clips + CNN + MTLN [20]	79.57	84.83


Sylvain Bertaina
email: sylvain.bertana@im2np.fr

Nandita Abhyankar

Maylis Orio

Naresh S Dalal
email: dalal@chem.fsu.edu

Measuring Motional Dynamics of (CH 3 ) 2 NH 2 + in the Perovskite-like Metal-Organic Framework [(CH 3 ) 2 NH 2 ][Zn(HCOO) 3 ]: The Value of Low-Frequency EPR

    Measuring Motional Dynamics of [(CH 3 ) 2 NH 2 ] + in the Perovskite-Like Metal-Organic Framework [(CH 3 ) 2 NH 2 ][Zn(HCOO) 3 ]: The Value of Low-Frequency Electron Paramagnetic Resonance

-INTRODUCTION

During the last decade or so, there has been considerable research interest in the family of metal-organic framework (MOF) compounds with the general perovskite formula ABX3, because of their structural versatility and potential applications [START_REF] Eerenstein | Multiferroic and Magnetoelectric Materials[END_REF][START_REF] Cheetham | There's Room in the Middle[END_REF][3][START_REF] Zhang | Ferroelectric Metal-Organic Frameworks[END_REF][START_REF] Li | Chemically Diverse and Multifunctional Hybrid Organic-Inorganic Perovskites[END_REF][START_REF] Asadi | Ferroelectricity in Metal-Organic Frameworks: Characterization and Mechanisms[END_REF] . A particularly popular class of such MOFs consists of formate frameworks [START_REF] Li | Chemically Diverse and Multifunctional Hybrid Organic-Inorganic Perovskites[END_REF][START_REF] Asadi | Ferroelectricity in Metal-Organic Frameworks: Characterization and Mechanisms[END_REF][START_REF] Jain | Order-Disorder Antiferroelectric Phase Transition in a Hybrid Inorganic-Organic Framework with the Perovskite Architecture[END_REF][START_REF] Jain | Multiferroic Behavior Associated with an Order-Disorder Hydrogen Bonding Transition in Metal-Organic Frameworks (MOFs) with the Perovskite ABX 3 Architecture[END_REF][START_REF] Mączka | Order-Disorder Transition and Weak Ferromagnetism in the Perovskite Metal Formate Frameworks of[END_REF]

with the general formula [A][M(HCOO)3].

Here, A + is a molecular cation such as the dimethylammonium ([(CH3)2NH2 + ]) or DMA + cation, and M 2+ is a 3d transition metal cation. Like their inorganic counterparts, such as SrTiO3, these MOFs exhibit two primary, spontaneous, and switchable internal domain alignments in the same phase: the alignment of electric polarization, as seen in ferroelectrics, and the alignment of magnetization, as seen in ferromagnets. More importantly, their electric polarization can be reversibly switched by an external magnetic field and the magnetization altered reversibly by an external electric field [START_REF] Eerenstein | Multiferroic and Magnetoelectric Materials[END_REF] . It has been emphasized in earlier reviews [START_REF] Eerenstein | Multiferroic and Magnetoelectric Materials[END_REF][START_REF] Cheetham | There's Room in the Middle[END_REF][3][START_REF] Zhang | Ferroelectric Metal-Organic Frameworks[END_REF] that in order to optimize their device potential, it is important to obtain a deeper understanding of the molecular interactions that underlie their ferroelectric and ferro(antiferro)magnetic phase transitions. In this regard, it is known that for these MOFs, the motional dynamics of the A + cation seem to play a pivotal role in driving the structural, ferroelectric transition [START_REF] Asadi | Ferroelectricity in Metal-Organic Frameworks: Characterization and Mechanisms[END_REF][START_REF] Jain | Order-Disorder Antiferroelectric Phase Transition in a Hybrid Inorganic-Organic Framework with the Perovskite Architecture[END_REF][START_REF] Jain | Multiferroic Behavior Associated with an Order-Disorder Hydrogen Bonding Transition in Metal-Organic Frameworks (MOFs) with the Perovskite ABX 3 Architecture[END_REF][START_REF] Mączka | Order-Disorder Transition and Weak Ferromagnetism in the Perovskite Metal Formate Frameworks of[END_REF][START_REF] Sánchez-Andújar | First-Order Structural Transition in the Multiferroic Perovskite-like Formate [(CH3)2NH2][Mn(HCOO)[END_REF] . In the high-temperature, paraelectric phase, the A + molecular unit, which possesses an axis of three-fold symmetry, appears to be dynamically disordered over three possible orientations [START_REF] Li | Chemically Diverse and Multifunctional Hybrid Organic-Inorganic Perovskites[END_REF][START_REF] Asadi | Ferroelectricity in Metal-Organic Frameworks: Characterization and Mechanisms[END_REF][START_REF] Jain | Order-Disorder Antiferroelectric Phase Transition in a Hybrid Inorganic-Organic Framework with the Perovskite Architecture[END_REF][START_REF] Jain | Multiferroic Behavior Associated with an Order-Disorder Hydrogen Bonding Transition in Metal-Organic Frameworks (MOFs) with the Perovskite ABX 3 Architecture[END_REF] . The phase transition coincides with the loss of this three-fold symmetry, as suggested by X-ray diffraction [START_REF] Li | Chemically Diverse and Multifunctional Hybrid Organic-Inorganic Perovskites[END_REF][START_REF] Asadi | Ferroelectricity in Metal-Organic Frameworks: Characterization and Mechanisms[END_REF][START_REF] Jain | Order-Disorder Antiferroelectric Phase Transition in a Hybrid Inorganic-Organic Framework with the Perovskite Architecture[END_REF] , neutron scattering [START_REF] Duncan | Local Structure of the Metal-Organic Perovskite Dimethylammonium Manganese( <scp>ii</Scp> ) Formate[END_REF] , and IR [START_REF] Clune | Magnetic Field-Temperature Phase Diagram of Multiferroic [(CH3)2NH2]Mn(HCOO)3[END_REF] , Raman [START_REF] Clune | Magnetic Field-Temperature Phase Diagram of Multiferroic [(CH3)2NH2]Mn(HCOO)3[END_REF][START_REF] Hughey | Phonon Mode Links Ferroicities in Multiferroic [(CH3)2NH2]Mn(HCOO)3[END_REF] , EPR [START_REF] Abhyankar | On Mn 2+ EPR Probing of the Ferroelectric Transition and Absence of Magnetoelectric Coupling in Dimethylammonium Manganese Formate (CH 3 ) 2 NH 2 Mn(HCOO) 3 , a Metal-Organic Complex with the Pb-Free Perovskite Framework[END_REF][15][START_REF] Šimėnas | Single Crystal Electron Paramagnetic Resonance of Dimethylammonium and Ammonium Hybrid Formate Frameworks: Influence of External Electric Field[END_REF][17] and NMR [START_REF] Abhyankar | Understanding Ferroelectricity in the Pb-Free Perovskite-Like Metal-Organic Framework [(CH 3 ) 2 NH 2 ]Zn(HCOO) 3 : Dielectric, 2D NMR, and Theoretical Studies[END_REF][START_REF] Besara | Mechanism of the Order-Disorder Phase Transition, and Glassy Behavior in the Metal-Organic Framework [(CH3)2NH2]Zn(HCOO)3[END_REF] spectroscopies. However, none of these techniques has provided a quantitative estimate of the correlation time or frequency of motion of the DMA + cation, a much sought-after datum in the phase transition model of these MOFs. This report presents a variable frequency EPR study aimed at measuring the timescale of motional dynamics of the A + cation [(CH3)2NH2] + in dimethylammonium zinc formate, [(CH3)2NH2]Zn(HCOO)3, henceforth referred to as DMAZnF. This sample was chosen because in this family of MOFs, [START_REF] Jain | Order-Disorder Antiferroelectric Phase Transition in a Hybrid Inorganic-Organic Framework with the Perovskite Architecture[END_REF][START_REF] Jain | Multiferroic Behavior Associated with an Order-Disorder Hydrogen Bonding Transition in Metal-Organic Frameworks (MOFs) with the Perovskite ABX 3 Architecture[END_REF] DMAZnF was the first compound shown to exhibit a dielectric transition. It has now been confirmed to be a ferroelectric below the dielectric phase transition [START_REF] Jain | Multiferroic Behavior Associated with an Order-Disorder Hydrogen Bonding Transition in Metal-Organic Frameworks (MOFs) with the Perovskite ABX 3 Architecture[END_REF][START_REF] Mączka | Order-Disorder Transition and Weak Ferromagnetism in the Perovskite Metal Formate Frameworks of[END_REF][START_REF] Sánchez-Andújar | Characterization of the Order-Disorder Dielectric Transition in the Hybrid Organic-Inorganic Perovskite-Like Formate Mn(HCOO) 3 [(CH 3 ) 2 NH 2[END_REF] .

DMAZnF displays a first-order structural phase transition at about TC=170K. X-ray diffraction reveals that at high temperature (HT), the compound exists in the trigonal R3c phase while at low temperature (LT), its structure belongs to the monoclinic noncentrosymmetric Cc space group [START_REF] Sánchez-Andújar | First-Order Structural Transition in the Multiferroic Perovskite-like Formate [(CH3)2NH2][Mn(HCOO)[END_REF] . Despite this structural phase transition, the global structure does not change significantly. Figure 1 shows the crystal structure of DMAZnF in the HT phase. The Zn(HCOO)3 -subsystem, in which the Zn 2+ ions are connected through the formate groups, forms a cage in which the DMA + cations are located. The symmetry of the HT phase allows the nitrogen atom to occupy one of three equivalent positions. In Figure 1, we have differentiated these positions by three colors: yellow, green, and blue. The nitrogen continuously moves between these three positions (as represented by the round arrow). The position of any DMA + ion is independent of the positions of the neighboring DMA + ions in the paraelastic/paraelectric phase. The symmetry of the LT phase (Figure SI1) allows only one position for the N atom. In the LT ferroelastic/ferroelectric phase, DMA + cations are ordered in a checkerboard arrangement, with alternate DMA + cations pointing in slightly different directions.

Electron paramagnetic resonance (EPR) spectroscopy is the tool of choice to probe the anisotropy and crystal field effects acting on paramagnetic ions. In particular cases, it is also possible to probe the fluctuation of the crystal field in time. In this paper, we focus only on the HT phase (T>170K), in which the DMA + cations are thought to undergo hindered rotational reorientations between three equivalent sites within the zinc formate cage. In order to perform EPR experiments, we have doped the DMAZnF (diamagnetic) with a small (~ 0.1 mole %) amount of Mn 2+ , which acts as a paramagnetic probe of crystal-field fluctuations due to the motion of the DMA + cations around it. Thus, the timescale of the motion of DMA + can be measured via the Mn 2+ spin probe.

We show that the lack of measurements of the timescale of motional dynamics of DMA + can be ascribed to the fact that this motional timescale falls well below the timescales of optical-IR spectroscopy and neutron scattering, which are techniques used in earlier studies of these materials. As detailed below, this timescale falls in the range of the rarely used low-frequency (~ 4 GHz) S-band EPR technique. The obtained data, supplemented by theoretical modeling (DFT calculations), provide strong support for the dynamic behavior of the DMA + cation and its role in the ferroelectric transition.

2-EXPERIMENTAL DETAILS:

Synthesis: Single crystals of DMAZnF were grown by solvothermal methods reported previously. A reaction mixture containing 1 mmol anhydrous ZnCl2, 6 mL DMF, and 6 mL deionized H2O was spiked with trace amounts of MnCl2. The exact amount of Mn in the final product was not determined but by comparison with the EPR-standard free radical diphenylpicrylhydrazyl (DPPH), the amount was assessed as less than 1 mole % in the final product. The mixture was heated overnight in a Teflon-lined autoclave. The solution was gradually cooled over a few hours, and the supernatant was set aside to crystallize. Over a period of a few days, block-like single crystals with dimensions of 1-3 mm were obtained.

Single crystal diffraction: Single crystal X-ray diffraction was employed to confirm the crystal structure and the relationship between the external morphology and unit-cell and crystal axes.

Figure 1 shows a schematic of the pseudocubic crystal structure of DMAZnF, with eight Zn ions at the corners of the cube and the DMA + cation at the center. As is known, DMAZnF undergoes a paraelectric-to-ferroelectric phase transition at TC = 166 K, with a hysteresis of less than 1 K, in agreement with our heat capacity studies [START_REF] Jain | Order-Disorder Antiferroelectric Phase Transition in a Hybrid Inorganic-Organic Framework with the Perovskite Architecture[END_REF] . In the currently accepted model of the ferroelectric transition, at T > TC, the DMA + cations are dynamically disordered between three positions, with 1/3 rd site occupation at each position, as depicted in Fig. 1. At TC, the dynamic disordering ceases and the DMA + cations become ordered. As mentioned in the Introduction, despite several studies using neutron scattering and IR, Raman, EPR and NMR spectroscopies, it has not been possible to pin down the actual timescale or frequency of the order-disorder dynamics of the DMA + cation. This issue provided the main motivation for the present undertaking.

EPR spectroscopy:

EPR experiments were performed using three conventional Bruker spectrometers operating at S-band (E600 -4 GHz), X-band (EMX -9.6 GHz), and Q-band (E600 -34 GHz), respectively. The sample was mounted on a suprasil rod and its orientation was measured using a goniometer. All measurements were performed with H//[102] axis. The temperature was varied from 300K to 170K (the phase transition temperature), using an Oxford ITC temperature controller. 10 min were allowed for stabilization at each temperature. Magnetic field modulation associated with lock-in detection was employed, resulting in the derivative of the signal. All measurements were carried out with the static field H perpendicular to the (012) face of the crystal. It is noted that the sensitivity of the S-band spectrometer is about an order of magnitude lower compared to those at X and Q-band. Therefore, the signal-to-noise ratio is correspondingly worse in the S-band spectra.

3-COMPUTATIONAL DETAILS

All theoretical calculations were based on the Density Functional Theory (DFT) and were performed with the ORCA program package [START_REF] Neese | The ORCA Program System[END_REF] . To facilitate comparisons between theory and experiments, all DFT models were obtained from the experimental X-ray crystal structures, and were optimized while constraining the positions of all heavy atoms to their experimentally derived coordinates. Only the positions of the hydrogen atoms were relaxed because these are not reliably determined from the X-ray structure. Geometry optimizations were undertaken using the GGA functional BP86 [START_REF] Perdew | Density-Functional Approximation for the Correlation Energy of the Inhomogeneous Electron Gas[END_REF][START_REF] Perdew | Erratum: Density-Functional Approximation for the Correlation Energy of the Inhomogeneous Electron Gas[END_REF][START_REF] Becke | Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior[END_REF] in combination with the TZV/P [START_REF] Schäfer | Fully Optimized Contracted Gaussian Basis Sets of Triple Zeta Valence Quality for Atoms Li to Kr[END_REF] basis set for all atoms and by exploiting the resolution of the identity (RI) approximation in the Split-RI-J variant [START_REF] Neese | Metal and Ligand Hyperfine Couplings in Transition Metal Complexes: The Effect of Spin-Orbit Coupling as Studied by Coupled Perturbed Kohn-Sham Theory[END_REF] with the appropriate Coulomb fitting sets [START_REF] Weigend | Accurate Coulomb-Fitting Basis Sets for H to Rn[END_REF] . Increased integration grids (Grid4 and GridX4 in ORCA convention) and tight SCF convergence criteria were used. The zero-field splitting (ZFS) parameters were obtained from single-point calculations using the BP functional. Scalar relativistic effects accounted for the metal center, and were included with ZORA paired with the SARC def2-TZVP(-f) basis sets [START_REF] Pantazis | All-Electron Scalar Relativistic Basis Sets for Third-Row Transition Metal Atoms[END_REF] and the decontracted def2-TZVP/J Coulomb fitting basis sets for all atoms. Increased integration grids (Grid4 and GridX4 in the ORCA convention) and tight SCF convergence criteria were used in the calculation. The spin-spin contribution to ZFS was calculated on the basis of the UNO determinant [START_REF] Sinnecker | Spin-Spin Contributions to the Zero-Field Splitting Tensor in Organic Triplets, Carbenes and BiradicalsA Density Functional and Ab Initio Study[END_REF] .

RESULTS AND DISCUSSION

In the rigid model: EPR spectra of DMAZnF:Mn 2+ have been reported recently, [START_REF] Šimėnas | Single Crystal Electron Paramagnetic Resonance of Dimethylammonium and Ammonium Hybrid Formate Frameworks: Influence of External Electric Field[END_REF] with particular attention paid to the LT phase. However, the dynamics of interest occur in the HT phase. We therefore focused mainly on the dynamics in the HT phase. Figure 2 shows an example of a spectrum recorded at X-band and T=170K (just above the ferroelectric phase transition temperature). In general, the spectrum of a Mn 2+ ion (S=5/2, I=5/2) is composed of six sets of 5 allowed transitions (DmS=±1, DmI =0), giving 30 allowed lines. In our case, the spectrum is composed of only 6 allowed DmS=±1 (mI=-1/2 to 1/2) transitions. The (mS=±5/2 to ±3/2, and mS=±3/2 to ±1/2) transitions are invisible because while they do occur, they are too broad to be resolved. Between the allowed transitions, we observed 10 weaker forbidden (DmS=±1, DmI=±1) transitions [START_REF] Wolga | Forbidden Transitions in the Paramagnetic-Resonance Spectrum of Mn2+ in Cubic MgO+[END_REF] . The presence of the forbidden transitions proves that there is a non-negligible crystal field anisotropy. Multiple-quantum transitions, observed in weak anisotropy, [START_REF] Sorokin | Multiple Quantum Transitions in Paramagnetic Resonance[END_REF][START_REF] Bertaina | Tunable Multiphoton Rabi Oscillations in an Electronic Spin System[END_REF] are not seen here.

To simulate the EPR spectra, we used the following Hamiltonian:

Here, the first term represents the Zeeman interaction, the second the hyperfine interaction, while the last term represents the crystal field interaction.

In solid-state EPR spectroscopy, the signal is usually described by the rigid model, in which the motion of atoms is small compared to the timescale (~1/f, f = frequency) of measurement.

We used the Matlab module Easyspin [START_REF] Stoll | a Comprehensive Software Package for Spectral Simulation and Analysis in EPR[END_REF] to simulate the spectra.

The green line shows the spectrum simulated using the standard rigid model and the following parameters: g=2.01, A=-265MHz and D=B0 2 /3=250(±50)MHz. To take into account the large anisotropy strain, we used a Gaussian distribution of D, DD~150MHz. The simulation describes the experimental data accurately, and the fit parameters are in agreement with those reported in other EPR studies 15 . Due to the very large line broadening, it was not possible to extract other crystal field parameters from the data.

We are conscious that the physical meaning of this very large value of D-strain is not obvious, but we present a plausible hypothesis based on our theoretical calculations (vide infra).

Compared to the nominal value of D=250MHz, the value of DD is not at first understandable since DD, which is caused by small local strains in the lattice, should be a perturbation. Here we found a strain of about 60%, which should destroy the crystal structure. However, another scenario can explain this large value. We recall that the Mn 2+ spin is surrounded by eight nearest-neighbor DMA + cations. In the HT phase, each DMA + can occupy one of three equivalent positions, and is constantly moving between these positions. However, if the timescale of the measurement is fast enough (faster than the time of motion between two positions), the DMA + cations appear frozen in one configuration. We can estimate the number of configurations of eight DMA + cations surrounding one Mn 2+ spin. With eight DMA + cations, each capable of occupying one of three equivalent positions, we have 3 [START_REF] Jain | Multiferroic Behavior Associated with an Order-Disorder Hydrogen Bonding Transition in Metal-Organic Frameworks (MOFs) with the Perovskite ABX 3 Architecture[END_REF] = 6561 possible configurations. The sample contains many billions of Mn 2+ spins, each experiencing a crystal field corresponding to one of these 6561 configurations. Thus, the EPR signal recorded is like a "snapshot" of all the configurations at the same time. All configurations are equally probable and each gives a different EPR spectrum. The spectrum shown in Fig. 2 is actually the sum of all 6561 spectra. We simulated this effect using a Gaussian distribution of D.

In order to confirm this model, we used DFT. We have estimated the ZFS of selected configurations. We chose to work with a minimal model consisting of one Mn 2+ ion bound to six formate anions and surrounded by eight DMA + cations (see figure SI2). The resulting metal cluster displays a quasi-octahedral coordination geometry. Based on the HT X-ray structure that identified three equivalent positions of the nitrogen atom in each DMA + , we considered several configurations in which the Mn-N distances can vary from 4.61 to 5.78 Angstroms. This provided a random sampling of the possible configurations (see Table SI3). The distribution of the ZFS parameter was found to be DDDFT~125 MHz, in rather good agreement with the experimentally estimated DD ~150 MHz .

The results of DFT calculations lead to several conclusions. Firstly, the crystal field effect on the Mn 2+ spin has previously been considered to originate solely from the surrounding octahedral oxygens. Here we prove that the DMA + cations also play an important role in determining the crystal field of the M 2+ ion. The reason is that although the oxygens are in firstneighbor positions, they form an octahedron with a nearly cubic symmetry and consequently, a weak anisotropy. On the contrary, although the DMA + cations are farther away than the oxygens, they create a low-symmetry crystal field around Mn 2+ .

Secondly, in the HT phase, the DMA + cations are considered to be undergoing hindered rotation between their three equivalent positions [START_REF] Abhyankar | Understanding Ferroelectricity in the Pb-Free Perovskite-Like Metal-Organic Framework [(CH 3 ) 2 NH 2 ]Zn(HCOO) 3 : Dielectric, 2D NMR, and Theoretical Studies[END_REF][START_REF] Besara | Mechanism of the Order-Disorder Phase Transition, and Glassy Behavior in the Metal-Organic Framework [(CH3)2NH2]Zn(HCOO)3[END_REF] but if the timescale of the measurement is small enough, the cation appears frozen. This is the rigid limit, which is the case for X-band measurements at low temperature (170K<T<200K) and Q-band measurements at all temperatures.

In the slow-motion regime: Intriguingly, at higher temperatures (>200K) in X-band measurements, the lineshape is different compared to the one in Fig. 2. Figure 3 shows the X-band EPR spectrum of DMAZF:Mn 2+ at room temperature. The forbidden transitions disappear and the linewidths are smaller for the outer lines (mI=±5/2) than for the central lines (mI=±1/2). The rigid-model simulation did not yield the experimental lineshape at room temperature, for any chosen values of fit parameters. This is because the rigid limit assumes that the crystal field is constant in time (i.e. the atoms are fixed) and the linewidth is included phenomenologically, as a distribution of crystal fields, in the Hamiltonian. On the contrary, in the HT phase, the DMA + cation can occupy one of three equivalent positions and is almost free to move between these positions. Depending on the rate, the motion of DMA + must be taken into account and the rigid limit is no longer valid. The snapshot described before is now blurred by the motion of the cation, which is comparable or fast compared to the timescale of the measurement. This model is called the "slow-motion" model and has been used in the past to describe the EPR spectra of viscous solutions [START_REF] Mclachlan | Line Widths of Electron Resonance Spectra in Solution[END_REF][START_REF] Goldammer | Electron-Paramagnetic-Resonance Study of Manganese Ions Bound to Concanavalin A[END_REF][START_REF] Von Goldammer | Electron Paramagnetic Relaxation and EPR-Line Shapes of Manganese Ions in a Sulfonated Polystyrene Ion-Exchange Resin at Various Degrees of Hydration[END_REF] . In these papers, the Mn 2+ ions were moving in a viscous solution, causing fluctuation of the anisotropic interaction with the solvent as a function of time. In our case, the Mn 2+ ion is fixed but the motion of the DMA + cations leads to the same effect as that observed for Mn 2+ spins surrounded by a viscous solution.

As is known [START_REF] Mclachlan | Line Widths of Electron Resonance Spectra in Solution[END_REF][START_REF] Goldammer | Electron-Paramagnetic-Resonance Study of Manganese Ions Bound to Concanavalin A[END_REF][START_REF] Von Goldammer | Electron Paramagnetic Relaxation and EPR-Line Shapes of Manganese Ions in a Sulfonated Polystyrene Ion-Exchange Resin at Various Degrees of Hydration[END_REF] , the choice of model is based on the correlation time (tc) of the process modulating the ZFS. Depending on the value of tc relative to the timescale of measurement (inverse of the microwave frequency), the DMA + is seen as either rigid or moving. If tc< 1/f, then during the timescale of the measurement, the DMA + appears to be moving between the three possible positions: this is the slow-motion regime. If tc> 1/f, then the DMA + appears to be frozen in one of the three positions during the measurement: this is the rigid limit mentioned above. Figure 3 shows the simulation in the slow-motion regime, using D=250MHz, A=265MHz and tc =2.10 -9 s. This model describes the room-temperature spectrum accurately.

In order to support our model, we acquired EPR spectra of DMAZF:Mn 2+ at a lower frequency (S-band -4GHz), where the effect of motional broadening is expected to be enhanced.

Unfortunately, the sensitivity of the S-band spectrometer is about 2 orders of magnitude lower than at X-band. As a consequence, the signal-to-noise ratio of the measurements is poor and the signals are noisy. Figure 4 shows a spectrum recorded at S-band at room temperature. The purpose of this measurement was to compare the two models describing the X-band results. The blue line is the simulated spectrum corresponding to the slow-motion regime, obtained using the correlation time extracted from X-band results (Fig. 3). The green line is the simulated spectrum corresponding to the rigid limit, obtained using the Hamiltonian parameters extracted from Fig. 2 (g=2.01, A=-265MHz and D=B0 2 /3=250MHz and DD~150MHz). Although the linewidth is slightly overestimated, the slow-motion regime describes the experimental data much more accurately than the rigid limit. At Q-band, we find the linewidth of 8 G to be nearly independent of temperature. At X-band, the linewidth starts at 15 G at room temperature and then decreases with temperature, finally reaching a minimal value of about 8 G. At S-band, the linewidth is about 40 G, and decreases with temperature to 30 G. Applying the slow-motion model [START_REF] Von Goldammer | Electron Paramagnetic Relaxation and EPR-Line Shapes of Manganese Ions in a Sulfonated Polystyrene Ion-Exchange Resin at Various Degrees of Hydration[END_REF] with tc =2.10 -9 s ( at T=300K), we obtained linewidths of 46 G at S-band, 15 G at X-band, and 1 G at Q-band. While the simulated linewidths at S-band and X-band are in good agreement with experimental results, those at Q-band are not. This is because the timescale of Q-band measurements is very fast compared to the correlation time of the motion of the DMA + ion. Thus, the system is best described by the rigid-limit model, in which inhomogeneous broadening of the linewidth has been estimated to be 8 G. This value is the threshold below which slow-motion broadening is hidden by the inhomogeneous broadening discussed previously. At X-band, both regimes occur. At high temperature, slow-motion broadening is dominant. Upon decreasing the temperature, the correlation time increases and below about 220K, the DMA + cation appears frozen at the measurement timescale and the rigid limit is reached.

The timescale of spectroscopy is an important parameter to take into account, especially if the motion inside the system is important. This is the case in the DMA formate family, in which the nature of the multiferroicity is directly connected to the motion of DMA + . In high-energy spectroscopy (X-ray, neutron), the timescale is extremely short and the DMA + ions appear frozen in the HT phase, giving the reported result of three equivalent positions of nitrogen [START_REF] Jain | Order-Disorder Antiferroelectric Phase Transition in a Hybrid Inorganic-Organic Framework with the Perovskite Architecture[END_REF] . On the other side, in low-energy spectroscopy, the DMA + appears to be moving, irrespective of the experimental temperature [START_REF] Besara | Mechanism of the Order-Disorder Phase Transition, and Glassy Behavior in the Metal-Organic Framework [(CH3)2NH2]Zn(HCOO)3[END_REF] . Dielectric susceptibility measurments [START_REF] Abhyankar | Understanding Ferroelectricity in the Pb-Free Perovskite-Like Metal-Organic Framework [(CH 3 ) 2 NH 2 ]Zn(HCOO) 3 : Dielectric, 2D NMR, and Theoretical Studies[END_REF] have shown that the motion of the dipoles, which is directly related to the motion of the DMA + ions, has a characteristic frequency f0=2.97x10 [START_REF] Hughey | Phonon Mode Links Ferroicities in Multiferroic [(CH3)2NH2]Mn(HCOO)3[END_REF] Hz and energy Ea= 3600K (0.31 eV) . These values lead to a correlation time of the dipole tc = 1/f0 exp(Ea/kT), of 5.10 -9 s at T=300K. This is in reasonably good agreement with the value of the EPR correlation time of 2.10 -9 s. It also agrees with the correlation time obtained by a recent 15 N NMR study [START_REF] Šimėnas | Spectroscopic Study of[END_REF] .

CONCLUSION:

This work extends earlier EPR studies of the phase transition of the metal-organic framework compound DMAZnF, using the Mn 2+ (S = 5/2) paramagnetic ion as a spin probe. Emphasis is laid on understanding the relative signal intensities of the hyperfine peaks in the HT, paraelectric phase of this compound; which is the parent of this class of MOFs, as detailed in the Introduction. In particular, Simenas and coworkers 17 have used pulsed EPR , ENDOR, and CW EPR to probe the motion of the DMA + cation , but these studies focused more on the order parameter than on the large spread in the value of the D parameter in the paraelectric phase.

The authors did perform an Arrhenius analysis of the motional effect based on temperature variation of the linewidth, but a quantitative analysis of the rate of motion was not described.

We argued that the timescale of the motion is slow enough that going to frequencies lower than X-band might provide fruitful information about this motion. Indeed, S-Band (4 GHz) EPR spectra yielded clear signatures of the slow motion of both the formate and the DMA + groups; the rate of hindered rotation of DMA + is well described by the correlation time of 2.10 -9 s at ambient temperature. Additionally, the unusual result that ΔD, i.e. the spread in D, is larger than D itself can be rationalized by our DFT calculations. We anticipate that our studies will spark new interest in lower-frequency EPR , which can be combined with DFT studies to understand the role of molecular motion in the mechanism of the ferroelectric phase transition in these hybrid MOFs.

SUPPORTING INFORMATIONS:

Crystallographic structure of the LT phase. Minimal model used for the DFT calculation and the results of the 14 configurations used.

Figure 1 :

 1 Figure 1: Schematic representation of Mn 2+ -doped DMAZnF in the HT phase. The nitrogen

Figure 2 :

 2 Figure 2: X-band EPR spectra of the Mn 2+ spin probe in DMAZnF, recorded just above Tc. The

Figure 3 :

 3 Figure 3: X-band EPR spectra of the Mn 2+ spin probe in DMAZnF, recorded at room

Figure 4 :

 4 Figure 4: S-band (4 GHz) EPR spectra of the Mn 2+ spin probe in DMAZnF, recorded at room

Figure 5 :

 5 Figure 5: Peak-to-peak linewidth of the rightmost EPR line, recorded between 300K and 170K

ACKNOWLEDGEMENTS

We thank CNRS' research infrastructure RENARD (FR3443) for EPR facilities, and the US National Science Foundation, via Award CHE-1464955. The NHMFL is supported by the Cooperative Agreement Grant No. DMR-1157490 and the State of Florida.