Hands can express a lot of information thanks to the many gestures that their fingers can compose. There are different types of gestures depending on the kind of information that they intend to transmit. Based on the researches of Kendon  and Quek et al. , a taxonomy of possible gesture categories is proposed as follows:
Deictic Gestures are the gestures that involve pointing to establish the identity or spatial location of an object within the context of the application domain.
Manipulative Gestures are usually performed by freehand movements to mimic manipulations of physical objects as in virtual reality interfaces.
Semaphoric Gestures are particular gestures that define a set of symbols to communicate with machines.
Gesticulation is one of the most natural forms of gesturing and it is commonly used in combination with conversational speech interfaces. These gestures are often unpredictable and difficult to analyse.
Language Gestures are the gestures used for sign languages. They are performed by using a series of gestures that combine to form grammatical structures for conversational style interfaces. In case of finger spelling, these gesture can be considered as semaphoric.
Hand gesture recognition provides a means to decode the information expressed by the reported categories which are always more used to interact with innovative applications, such as interactive games [3, 4], serious games [5, 6], sign language recognition [7, 8, 9, 10], emotional expression identification [11, 12], remote control in robotics [13, 14] or alternative computer interfaces [15, 16, 17, 18].
In general, the approaches used in hand gesture recognition can be divided into two main classes: 3D model-based  and appearance-based . The first uses key elements of the body parts to acquire relevant 3D information, while the second one uses images or video sequences to acquire key features.
In the past, several RGB cameras were necessary to obtain a 3D model of body parts, including hands. Recent works, supported by advanced devices, e.g., Microsoft Kinect , LMC , and novel modelling algorithms based on depth map concept  have enabled the use of 3D models within everyday application domains.
In this paper, a language hand gesture recognition solution using 3D model-based approaches is presented. Specifically, the proposed method uses skeletal-based modelling, where a virtual representation of skeleton hands (or other parts of the body) is mapped to specific segments. This technique uses joint angle parameters along with segment lengths, instead of intensive processing of all 3D model parameters. Then, it measures the variations over time of the skeleton joints whose spatial coordinates are acquired by a LMC. In particular, the angles formed by a specific subset of joints that involve distal, intermediate, and proximal phalanges for the index, middle, ring, and pinky, as well as the metacarpal for the thumb, can be considered highly discriminating to recognize many kind of hand gesture as confirmed by our tests. Moreover, these features have been selected as easy and quick to be extracted. Spatial information about the fingertips are also considered by the method in order to manage not articulated movements of the hands. In order to obtain a more accurate classifier, the information of the intra-finger angles and the spatial information of the palm of the hand are also considered. During the design of the proposed method, two challenges were fixed:
the search of a robust solution able to recognize also gestures that are similar to each other;
the achievement of the highest accuracy level compared with works of the current literature.
These goals have been obtained by using a stack of RNNs 
with Long Short Term Memory (LSTM) architecture, a particular type of Deep Neural Network (DNN) where connections between its units form a directed cycle. The RNNs, unlike the common DNNs, can model long term contextual information of temporal sequences thus obtaining excellent results in the sound analysis and speak recognition, as reported in 
. The LSTM is an architecture where a RNN uses special units instead of common activation function. LSTM units help to propagate and preserve the error through time and layers. This aspect of the LSTMs allows the net to learn continuously over many time steps, thereby opening a channel to link causes and effects remotely. An architecture formed by two or more stacked LSTM RNNs is defined as Deep LSTM (DLSTM). Such an architecture allows to learn at different time scales over the input sequences. In the experiment section, we focused on a challenging subset of gestures defined by the ASL . This ASL is chosen because it is composed of a considerable number of types of gesture with several degrees of complexity. The method proposed in this paper provides the following contributions:
the selection of a simple set of features based on the joint angles that are highly discriminative for the recognition of any kind of hand gesture, especially for language gestures;
the capability of analyzing and recognizing large number of gestures in the field of language gesture classification. The study of static and dynamic gestures, belonging to the ASL, provides the prerequisites for achieving a wider recognition system for sign language;
the DLSTM in combination with the skeleton extracted by the LMC has never been used before to recognize hand gestures. In addition, the LMC has been used in the design and collecting of a dataset, composed of a high number of gestures, that guarantees a high precision in the estimation of the positions of the joints of the hand.
The rest of the paper is structured as follows. In Section II, the state-of-the-art of the gesture recognition is provided. The proposed method is described in Section III. Extensive experimental results are presented and discussed in Section IV. Finally, conclusions are drawn in Section V.
Ii Related Work
or Convolutional Neural Networks (CNNs)[38, 26]) are used to perform a classification phase. A reference work is reported in , where SVM is used with Histogram of Oriented Gradients (HOG) as feature vectors. Wang et al.  and Suryanarayan et al.  used a SVM with volumetric shape descriptors. Using same classifier, Marin et al. 
applied a combination of features extracted by Kinect and LMC sensors. Other interesting solutions are based on Hidden Markov Models (HMMs): Zun et al. propose a robust hand tracking to recognize hand signed digit gestures.
Different well-known techniques are extended and customized to reach increasingly better results. An example is shown in , where a semi-Markov conditional model to perform finger-spelling gesture recognition on video sequences is reported. The Hidden Conditional Random Field (HCRF) method proposed in Wang et al.  is instead used to recognize different human gestures. Lu et al.  use an extension of the HCRF to recognize dynamic hand gestures driven by depth data. Regarding the hand pose estimation, the solution proposed in Li et al. 
shows excellent results by applying a Randomized Decision Tree (RDT).
Another common solution is based on the use of Dynamic Time Warping (DTW). Although DTW does not belong to the class of machine learning techniques, it is often used in time series classification. In Vikram et al. , a DTW to support a handwriting recognition process based on the trajectory of the fingers extracted by a LMC is presented. In , the DTW with a novel error metric to match patterns, combined with a statistical classifier, is used to perform a tool to aid the study of basic music conducting gestures. In Sohn et al. 
, a pattern matching method by the combination of a DTW and a simple K-Nearest Neighbor (K-NN) classifier is used. Recently, the great performance of the deep neural networks has motivated the use of Convolutional Neural Networks (CNNs) in different application domains, including the gesture recognition as proposed in. In addition, analysing the behaviour of these nets in other fields [24, 25, 26], we have understood that the RNNs can suitably support the classification of temporal data sequences. Based on these observations, the proposed method was designed starting from two works that achieve outstanding results in the current literature: the first, proposed by Du et al. , where an hierarchical RNN for skeleton action recognition is used, and the second, proposed by Graves et al. , using a Deep Bidirectional LSTM for the speech recognition.
Let us consider, each hand gesture acquired by a user is represented by a set of feature vectors, where indicates the maximum number of time instants, inside a time interval , in which the features are extracted by a LMC. LMC is chosen as reference device for the acquisitions because it is optimized for the hands and the obtained skeleton model provides very accurate dynamic information about finger bones . A DLSTM is applied to model these sequences of data, where a time series of feature vectors (one vector for each time instant) is converted into a series of output probability vectors . Each indicates the class probability of the gesture carried out at time , with
. Finally, the classification of the gestures is performed by a softmax layer using classes, where is the set of the considered gesture classes of the ASL. The logical architecture of the proposed method is shown in Fig. 1.
Iii-a Feature Extraction
Each gesture can be considered as the composition of different poses, where each pose is characterized by particular angles. Such a concept has already been applied in several works to recognize human actions, using the angles formed by the body joints [45, 46, 47]. So, each feature vector , with , is mainly composed by (Fig. 2):
the internal angles , , , and of the joints between distal phalanges and intermediate phalanges. The internal angle considered for the thumb is computed between distal phalanx and proximal phalanx;
the internal angles , , , and of the joints between intermediate phalanges and proximal phalanges. The internal angle considered for the thumb is computed between proximal phalanx and metacarpal;
Each finger can be seen as a set of segments, where is the distal phalanx, is the intermediate phalanx (with the exception of the thumb, where is the proximal phalanx), and is the proximal phalanx (with the exception of the thumb, where is the metacarpal). The angles are calculated as follows:
with . Since the information provided by the angles is not sufficient to manage all types of existing gestures, especially dynamic gestures that perform movements in 3D space, additional information is used by considering the following features:
3D displacements of the position of the central point of the palm of the hand. These features are considered to manage hand translation on the 3D space;
3D displacements of the fingertip positions, with . These features are considered to manage hand rotation in 3D space;
the intra finger angles and , i.e., angles between two consecutive fingers. The fingers considered are the pointer finger, the middle finger, the ring finger and the pink finger. These features are used to handle special cases of static gestures that differ from each other only in intra finger angles, as shown in Fig. 3.
All the listed features are independent by the reference. Thus, the input vector assigned to the DLSTM at time is:
Iii-B Sampling Process
As each person can perform the same gesture with different speeds, and as we want to analyze the sequences having all the same number of samples, we have implemented a sampling process able to select the most significant feature values within the entire time interval of the gesture sequence. This means that data are acquired only in the most significant time instants, where an instant of time is defined as significant when the joint angles and the hand central point position vary substantially between and (as explained below). Let and , with and , be the functions that represent the value of and angles at time , and let be the function that represents the value of (i.e., the displacement of the centre of the hand with respect to the previous position at time ) at time . For each function , with and , the Savitzky-Golay filter  is applied. The Savitzky-Golay filter is a digital filter able to smooth a set of digital data in order to increase the signal-to-noise ratio without greatly distorting the signal. Now, the significant variations on the considered features are identified through the relative maximum and minimum of each . All the time instants associated with at least one relative minimum or relative maximum of a feature are used to create a new set , which represents a set of possible important time instants to sample. In Fig. 4, an example of this sampling phase is shown, where the behaviour of the function (the angle of the distal phalange of the index finger) for an instance of the gesture ”milk” is considered. The signal in Fig. 4 is cleaned of any noise, caused by the acquisition device or tremors of the hand, by applying the Savitzky-Golay filter. Then, the maximum and minimum relative points are identified and sampled. In the example, only the procedure for the feature is shown, but this step is performed for each feature . Now, depending on the cardinality of the set of the sampled time instants, the following cases must be considered:
if , then the remaining time instants to be sampled are randomly selected in ;
if , then, only some significant time instants are sampled for each feature. Let be the set of the samples in obtained from the relative maximum and minimum of the feature (), we want to know the number of time instants that can be sampled for each such that . Each is obtained thought the following proportion . Then, from each set, we will randomly take samples.
After the sampling step, each acquisition instance is composed by a sequence of feature vectors. The proposed sampling procedure is dynamically based on the value of the features.
Iii-C Deep Last Short Term Memory network
A fundamental component in the proposed work is the network used in the classification of the hand gesture. This network is based on multiple LSTMs, which unlike other types of NN, is able to efficiently analyze time sequences of data. Several factors, such as the error blowing up problem  and the vanishing gradient , do not allow the use of common activation functions (e.g., tanh or sigmoid) to suitably train a network composed by multiple RNNs. This problem can be tackled with the LSTM units (Fig. 5).
The LSTM can be seen as memory blocks that are one or more self-connected memory cells and three multiplicative units: the input, output and forget gates. These gates provide continuous analogues of write, read, and reset operations for the cells. Although LSTM allows to manage the problem of the vanishing gradient, the input time series often have a temporal hierarchy, with information that is spread out over multiple time scales which can not be adequately recognized by simple recurrent networks such as LSTMs. For this reason, Deep RNNs or LSTMs were introduced. In fact, by constructing recurring networks formed over multiple layers, a higher abstraction on the input data is reached . Increased input abstraction does not always bring benefits because the effectiveness of these networks depends on the task and the analysed input. In several works, such as [26, 51, 52], it was observed empirically that Deep LSTMs work better than shallower ones on speech recognition. The audio signals, for example analysed in speech-to-text task, can be elaborated on more abstractions ranging from the entire pronounced phrase to the syllables of each word, and each abstraction can be captured in different time scales within the period considered. Like in the case of audio sequences analysed in the speech recognition problem, gestures of the hand can be examined over multiple time scales. In fact, every gesture can be seen as the composition of so many small movements and sub-gesture of the hand, and its suitable for this type of network. Based on these considerations, the LSTM stack-based solution have been experimented and then compared to the performance of a single-level network. The first step is to define the activation functions of memory cell of the (the first layer of the proposed neural network), as well as the input, output, and forget gates computed by using iteratively the following equations (from ):
where , , , and denote the input gate, forget gate, output gate and cell activation vectors, respectively. These vectors have the same length of the hidden vector . Instead, , , , and are the weights of the input gate, forget gate, output gate and cell to the input. In addition, , , and are the diagonal weights for peep-hole connections. Finally, the terms , , , and
indicates the input, forget, cell and output bias vectors, respectively. We have that
is the logistic sigmoid function andis the element-wise product of the vectors. Once the activation functions for the first level have been defined, the next step is to define the upper level activation functions.
DLSTMs are architectures obtained by stacking multiple LSTM layers where the output sequence of one layer forms the input sequence for the next layer (Fig. 6). The memory cell of an at time , in addition to the classic and vectors, takes in input the , i.e., the hidden state at time of the below . So, the activations of the memory cells of of the network higher levels (i.e., ) are given by the following equations:
The output of the DLSTM network, at time , with -layers, is given by:
where is a bias vector, is the hidden vector of the last layer and is the weight from the hidden layer to output layer. The output
defines a probability distribution over thepossible gesture classes, where (the element of ) is the estimated probability of a specific class at time for the acquired gesture . Finally, all results are collected and normalized into the softmax layer, through the following equations:
for each , with . The classification of gesture will be given by the highest probability contained in .
Iii-D Network Training
Given a dataset composed of
train gesture sequences, the goal is to minimize the following maximun-likelihood loss function:
where , , is an input sequence of the training dataset , is the ground-truth label of and is the Kronecker delta or delta function. This formulation is referred to the cross-entropy error proposed in . The Back-Propagation Through Time (BPTT) algorithm 
is used to obtain the objective function derived with respect to all the weights and to compute the minimization based on the stochastic gradient descent.
Accuracy results on the proposed dataset by varying the number of stacked LSTMs in the network architecture. (a) Accuracy results using 800 epochs for each considered architecture and (b) accuracy results using 800 epochs for 1-LSTM, 2-LSTM, 3-LSTM and 4-LSTM; for the 5-LSTM and 6-LSTM are used 1600 and 1800 epochs, respectively. The x-axis indicates the number of the stacked LSTMs, while the y-axis indicates the accuracy values.
Iv Experimental Results
This section describes the experiments performed to evaluate the behavior of the proposed approach. All the experiments were performed by using a LMC on an Intel i5 3.2GHz, 16GB RAM, with a GeForce GTX 1050ti graphics card. The DLSTM network and the BPTT algorithm, used to compute the minimization based on the stochastic gradient descent, are implemented by using the Keras111https://keras.io/ framework. The main purposes of the experiments were the assessment of the joint angles as salient features for the recognition of the hand gestures, the overall robustness of the proposed method and its higher accuracy compared with the current literature. The experiments were performed by using a challenging subset of gestures defined by the ASL and described in Section IV-A. A discussion of the obtained results is presented in Section IV-D, and a comparison of the proposed method with key works of the current state-of-the-art is reported in Section IV-E.
Currently, there is no public ASL dataset with a large number of classes and with information on the hand joint. Based on these reasons, we acquired the data of 30 gestures to create a new dataset. This dataset consists of 12 dynamic gestures and 18 static gestures taken by the ASL. These additional gestures are chosen to represent much of the variations in joint angles and finger positions that occur when the hand perform a gesture. The static gestures are: 1, 2-V, 3, 4, 5, 6-W, 7, 8, 9, A, B, C, D, H, I, L, X and Y. indeed, the dynamic gesture are: bathroom, blue, finish, green, hungry, milk, past, pig, store and where. The dataset is composed of 1200 hand gesture sequences, coming from 20 different people. Each gesture was collected by 15 males and 5 females, aged 20 to 28 years. Each person performed the hand gestures twice, once for each hand. different gestures are considered. The sequences from 14 people are used to create the train set while sequences of the remaining 7 people were used to form the test set. So, the 7 people used in the tests are never taken into consideration during the training phase. As previously described in Section III-A, each sequence is acquired according to a sampling process, with and .
Iv-B Selection of the Optimal Number of Stacked LSTM
Several tests have been conducted to chose the optimal number of stacked LSTMs to use in the proposed architecture. The hidden units per LSTM are , i.e., the hidden units are equal to the number of input time instances considered for each gesture. In Fig. 6(a), it is shown as an architecture composed by 4 levels gives the best accuracy results using 800 epochs. In fact, although several levels of LSTM allow to analyze complex time sequences by dividing them into multiple time scales, the 5-LSTM and the 6-LSTM require more epochs to be trained. Increasing the number of epochs needed to train the 5-LSTM and 6-LSTM architectures (i.e., 1600 epochs for the 5-LSTM and 1800 for the 6-LSTM), the Fig. 6(b) shows how their results improves. We can notice how greater abstraction on input does not provide substantial benefits from a certain number of levels, and the accuracy gained by the network begins to converge to a precise value. In conclusion, 4 levels are appropriated for the proposed network and represent a good compromise between training times and system accuracy. The choice of the learnig rate influences the speed of the convergence of the cost function. If the learning rate is too small, the convergence is obtained slowly, while if the learning rate is too large, the cost function may not decrease in every iteration and therefore it could not converge. In the proposed method, the learning rate is set to through large empirical tests.
Iv-C Feature Effectiveness Analysis
In order to verify the effectiveness of the features in classifying the set of gestures taken by the ASL dataset, some tests have been carried out. The tests carry out the training of the network and the subsequent classification (using two different sets to train and test the network, respectively) of the ASL gestures, where the number of stacked LSTM is 4 (as explained in Section IV-B), using only subsets of as features. The results in Table I shown as the combination of and features is able to discern alone an high number of gestures of the ASL vocabulary and it reaches better classification results with respect to separate use of these two features. Although the single feature does not offer good performance, it greatly improves classification when used with and . Instead, the combination of features that relate to the movements of the hand () are unable by themselves to classify the gesture of the hand but, if combined with the features of the angles, allow the method to achieve high performance (as discussed to follow).
Iv-D Hand gesture recognition on the ASL dataset
To evaluate the method, we have used very popular metrics. First of all, we have used the accuracy as main metric. In addition, we have also introduced precision, recall and f1-score metrics can be considered a de facto standard to measure the quality of this class of algorithms . the obtained results are presented in Table II. To better analyse the proposed approach and according to the tests performed to recognize the different hand gestures, the confusion matrix is computed (Fig. 8). Each column of the matrix represents the instances in a predicted gesture, instead each row represents the instances in a current gesture. The main diagonal of the matrix represents the instances correctly classified by the DLSTM. The elements below the diagonal represent the false positives, i.e. the gestures that are incorrectly classified within a class of interest. The elements above the diagonal are the false negatives, i.e., the gestures incorrectly classified as not belonging to a class of interest. The distinction of some gestures is very hard, since they are very similar to other gestures in the dataset. Despite this, the proposed method does not suffer of ambiguity issues. The only exceptions are given by the gestures 6 with W (Fig. 8(a)) and 2 with V (Fig. 8(b)). The variations in their joint angles are minimal and difficult to see even to the human eye: moreover, the LMC device fails to capture these variations. For this reason, these gestures have been gropued in the same class. Tests performed without grouping these classes achieved 91.5178% of accuracy. In fact, the decrease in accuracy is caused precisely by some incorrect classifications regarding the 2,6,v and W classes.
In Fig. 10, the Train/Test plots are shown. The first plot (Fig. 9(a)) shows the Train/Test accuracy over the iterations, instead the second plot (Fig. 9(b)) contains the loss curves that represent the sum of the errors provided for each training or test instance. In this work, the loss curves are calculated as maximum-likelihood loss function, described in Section III-D. Instead, the curves of accuracy represent the training or validation instances correctly recognized. After a certain number of iterations (), the test accuracy curve converges.
|Features||Accuracy 14 Gestures||Accuracy 28 Gestures|
|Skeleton-based Dynamic hand gesture recognition ||88.24%||81.90%|
|Key frames with convolutional neural network ||82.90%||71.90%|
|Joint Angles Similarities and HOG2 for Action Recognition ||83.85%||76.53%|
|HON4D: Histogram of Oriented 4D Normals for Activity Recognition from Depth Sequences ||78.53%||74.03%|
|3-D human action recognition by shape analysis of motion trajectories on riemannian manifold ||79.61%||62.00%|
We compared the proposed method with key works of the current state-of-the-art presented in [55, 58, 57, 46, 56] on the SHREC dataset . The SHREC dataset has been selected as: (a) it provides different types of data to allow comparisons between methods based on different acquisition sensors; (b) it allows the classification of hand gestures with different degrees of complexity; (c) it provides data that allow to extract all the features necessary for the proposed method. The SHREC dataset is also an excellent example of a semaphoric gesture to show the effectiveness of the proposed method on other categories of gesture. The SHREC dataset contains 14 dynamic gestures performed by 28 participants (all participants are right handed) and captured by the Intel RealSense short range depth camera. Each gesture is performed between 1 and 10 times by each participant in two way: using one finger and the whole hand. Therefore, the dataset is composed by 2800 sequences captured. The depth image, with a resolution of 640x480, and the coordinates of 22 joints (both in the 2D depth image space and in the 3D world space) are saved for each frame of each sequence in the dataset. For the proposed method we only needed the 3D coordinates of the joints from which we derived the features of our interest. The depth images and hand skeletons were captured at 30 frames per second and the length of sample gestures ranges from 20 to 170 frames. Since some sequences of the dataset are very short, in order to avoid sampling with a very low
value, we used the padding technique to increase the length of these sequences to an acceptable value of(i.e., ). As shown in Table III, the proposed method outperforms the accuracy values of the other works, both in the dataset divided into 14 classes and in the dataset divided into 28 classes. The confusion matrices obtained from the tests are shown in Fig. 11. By analyzing these matrices well we can see that the method can classify very well the gestures performed using only one finger, while in the version using the whole hand some mismatches occur. In details, the gesture 16 (SWIPE LEFT) is sometimes erroneously classified as gesture 26 (SWIPE V) and gesture 8 (PINCH), instead, the gesture 18 (SWIPE UP) is confused with the gesture 6 (EXPAND). By carefully analyzing the variations of the feature values, we notice that the angles obtained from these instances are similar and the movements of the hand in space are not substantial. Despite these isolated cases, the method achieves excellent performance. This result demonstrates how the DLSTM and the selected features are a very powerful solution in recognizing different types of hand gestures. Although our work is focused on language gestures, the tests carried out on this dataset have highlighted how our method can also handle semaphoric gestures, i.e., gestures that define a set of symbols to communicate with machines.
In this paper, a novel language hand gesture recognition approach based on a DLSTM is presented. A set of new discriminative features based on both joint angles and fingertip positions are used in combination with DLSTM for the first time to obtain high accuracy in the hand gesture recognition. In addition, a novel dataset based on a large subset of the ASL is created to train and test the proposed method. Moreover, this dataset is used to analyze the effectiveness of the extracted features and the behavior of the network by varying the number of stacked LSTMs. As a future development, we would like to create a public data set based on the ASL composed of a higher number of gestures. A possible improvement of the method could be obtained by integrating the RGB information, as well as the skeleton, in order to solve ambiguous cases (such as those between gesture 6 and gesture W).
This work was supported in part by the MIUR under grant ”Dipartimenti di eccellenza 2018-2022” of the Department of Computer Science of Sapienza University.
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