3D shape classification is a fundamental problem in the field of computer graphics and computer vision. It finds applications from traditional computer aided design and medical imaging to cutting-edge mixed reality and robot navigation. The challenge of 3D shape classification stems from the difficulty of characterizing 3D surface geometry, the variety of 3D transformation and deformation, and the imperfection of geometry and/or topology, etc. Hundreds of hand-crafted 3D shape descriptors have been proposed, either from 2D rendered views[1, 2, 3, 4, 5] or directly on 3D models [6, 7, 8, 9]. They are, however, often carefully designed to characterize only one or a few aspects of 3D shapes, making them hard to generalize well.
. By leveraging massive image databases such as ImageNet to pre-train the CNN and learn image descriptors for general vision tasks, MVCNN has significantly advanced the state-of-the-art of 3D shape classification. The method renders a 3D shape to RGB or depth images from different viewpoints, uses advanced CNN architecture to extract features for each view, and then aggregates the multi-view features to form the final feature representation based on max or average pooling. Albeit being simple yet effective, its best performance is achieved only when all views are used. In some practical scenarios, such as robot-operated active recognition, it is desirable to achieve object recognition with as-few-as-possible views, to minimize the robot movement cost.
Our key observation is that human is able to recognize a 3D shape without processing all views. Given the first view observation of a 3D shape, a human tends to first form hypotheses about which categories the shape may fall in, and then switch to the next viewpoint purposely, by moving himself around the shape or rotating the shape, to quick narrow down the uncertainty and refine hypotheses. This process is repeated for a few times until sufficient evidence are collected to minimize the uncertainty. There are two most prominent characters of the above procedure. First, the informative view is quite sparsely selected, far from being exhaustive. Second, both the next view selection and the predication making are deduced from the combined information from the previous observations over time.
Following the above intuition and drawing inspiration from visual attention model based on recurrent neural networks (RNN) [18, 19], we present View-Enhanced Recurrent Attention Model (VERAM) capable of automatically selecting an as short as possible sequence of views to classify 3D shapes. As shown in Fig. 1, our model is formulated as a goal-directed agent interacting with a 3D shape by using a camera sensor (Fig. 1(a)). At each time step, the agent actively selects the next viewpoint for the camera sensor, according to the current internal state . Next, the camera sensor captures the 3D shape from the new viewpoint to obtain a 2D image and pass it to the agent. The agent then extracts features for the new input image and updates its internal state. Such process is repeated for a few steps ( time steps in Fig. 1(b)). Finally, based on the aggregated information over time steps, the agent emits the predication as the probabilities of shape categories (Fig. 1(c)).
On the technical side, our method has two advantages over MVCNN. First, when pooling across all views, MVCNN discards the location information of each view. However, recent works reveal that viewpoint location plays an import role in enhancing the performance of the task of 3D shape classification [16, 17]. Second, processing all views involves high computational cost both at training and testing, although not every view is essential to recognition.
There have been a number of works using RNN-based visual attention model for image classification [18, 19, 20, 21], image captioning , and even for 3D object retrieval . However, a common issue with these RNN-based attention models is the unbalanced training of the subnetworks corresponding to next view estimation and shape classification. The classification subnetwork is usually easier to train than the view estimation one. Consequently, the model training can be easily trapped in a local minima: the classification subnetwork is overfitted while the view estimation one poorly trained, which in turn leads to degraded classification accuracy. In this work, we propose three key technical contributions with VERAM, aiming to enhance the view learning and achieve a significant performance boost to multi-view 3D shape classification.
We introduce three schemes to improve the information flow of gradient backpropagation from the view estimation subnetwork to the hidden units, achieving a balance with the training of the classification subnetwork. This overcomes the issue that the estimated views may get stuck at the boundary in the view parameterization space, which is usually encountered in previous works .
During the reinforcement learning of our attention model, we integrate the classification confidence of the current view into the gradient computation of the reward against the view. This leads to a highly efficient guidance to the next view estimation.
In VERAM, a novel loss function is proposed with a regularization term that enforces the estimated view to be distant to any of the previous ones so as to avoid view duplication.
The hybrid architecture of VERAM is trained for the subnetworks of shape classification and view estimation jointly, with the former using SGD  and the latter taking REINFORCE . We empirically evaluate our model on the ModelNet benchmark . Taking rendered gray-scale image as input and AlexNet as CNN architecture, without applying any data augmentation or network ensemble strategy, VERAM with views achieves average instance-level and class-level accuracy of and on ModelNet10, and on ModelNet40. The high accuracy and efficiency make our model scalable to large datasets and applicable to many online applications.
2 Related work
3D shape classification via hand-craft descriptors. There is a long history of work in 3D shape analysis and a large variety of hand-craft shape descriptors have been presented. The representative view-based descriptors cover Light Field descriptor , Elevation descriptor , Aspect Graph based descriptor [3, 4] and DFT/DTW panoramic descriptor , etc. Popular shape descriptors include Shape Histogram descriptor , Spherical Harmonic descriptor , 3D SURF , Heat Kernel Signatures 
, etc. These descriptors are largely hand-engineered and usually do not have enough generalization ability to adapt to the diversity of numerous 3D shapes in different categories. As a result, their performance has an obvious gap compared with the current dominant methods based on deep learning technology, which have achieved state-of-the-art performance in many tasks of computer graphics and computer vision. Our method falls into the deep learning class.
3D shape classification via deep CNN. A number of methods based on deep CNN have achieved state-of-the-art performance in 3D shape classification on public benchmarks. There are two categories. Shape-based methods [13, 26, 27, 28, 29, 30, 31, 32], perform convolutions with 3D filters on the voxels or point clouds in continuous 3D space, and the volumetric representation makes them have the ability of exploiting complete structure information. View-based methods [12, 37, 13, 14, 16, 17, 34, 35, 36] first render the 3D shape into 2D images from different viewpoints, and then apply 2D filters to carry out convolution for each view.
Compared with shape-based methods, the advantage of view-based methods is that the massive image databases can be used to pre-train the deep neural network and advanced network architectures succeeded in image recognition tasks can be employed. Partly for these reasons, to data, view-based methods shows better or comparable performance to shape-based methods. Moreover, convolution on 2D images is more efficient than on 3D volumetric space. View-based methods naturally need to fuse clues from different 2D views, and max or average pooling is the most common strategy to perform the task [12, 13],  , which lacks a view selection mechanism. Our method is view-based and also employs deep CNN to extract the descriptors for the rendered views. However, we adopt RNN-based visual attention model to learn attention policy of adaptively selecting a few number discriminative views, which is more effective and efficient.
3D shape recognition via active view selection. Our method fits into the realm of active recognition. Indeed, active recognition through next view planning has been studied for quite a long time in computer vision . For 3D object recognition and pose estimation, next-best-view selection based on information rich model was proposed in . In each step, the next-best-view is selected as the voxels of which has the highest number of matches that have not been detected before. In 3DShapeNets framework , next-best-view for 2.5D recognition is selected according to which can maximize the mutual information to reduce the potential uncertainty. In contrast to these local optimum next-best-view selection methods, an approximately global optimum approach was proposed by using undirected graph search . However, the next-best-view selection is isolated from the neural network, and it is not a completely global optimum method. By contrast, the next-best-view predication of our method is embedded in deep RNN and is a global optimum approach. Recently, MV-RNN  combines RNN-based visual attention model with MVCNN  for 3D object retrieval. The view confidence and view location constrains are implicitly handled in the layer of feature representation. In contrast, VERAM does not combine MVCNN, and explicitly integrates view confidence and view location constrains into reward gradient and classification loss.
Visual attention model based on RNN. We draw inspiration from recent approaches that used RNN-based visual attention model to learn task-specific policies in various applications. The vision tasks include image classification [18, 19, 20, 21] , image caption generation , action with its boundary detection , and 3D object retrieval 
. The attention model is also used in non-visual task, such as learning policies for a Neural Turing machine. Our method builds on these directions and learns policies addressing the task for 3D shape classification. It extends the RNN-based visual attention models to be more robust to the common issue of the unbalanced training of the subnetworks, and provides a paradigm of how RNN-based attention model to support learning with view confidence and view location constrains integrally.
The proposed VERAM is a RNN-based visual attention model for 3D shape classification, and is formulated as a goal-directed agent interacting with a 3D shape. A graphical representation of VERAM is shown in Fig. 2.
The model sequentially processes a 3D shape within time steps. At each step , based on the internal state , the model actively selects a viewpoint and obtains an observation of 2D image rendered by the camera sensor from , and then, the model uses with to updates its internal state. This process is repeated until the predication is emitted at step . The architecture of the model will be described in subsection 3.1 and later in subsection 3.2, we explain how to use a combination of SGD and REINFORCE to train the model in end-to-end fashion.
The architecture of VERAM can be broken down into a number of sub-components including camera sensor, observation subnetwork, recurrent subnetwork, view estimation subnetwork and classification subnetwork. Each component maps the input into a matrix or vector output.
3.1.1 Camera sensor
As shown in Fig. 2, the input to camera sensor is a 3D shape and viewpoint location , the output of the camera sensor is the 2D image of the rendered view. 3D shape represented as polygon mesh is located at the center of the viewing sphere, as shown in Fig. 3 (a). The camera sensor can move on the surface of the viewing sphere and its location is indicated by the latitude and longitude. The camera sensor always points towards the centroid of the shape, and its upright vector is the tangent line of the latitude along clockwise direction. Phong reflection model is used to render the 3D shape into 2D images (). Under a perspective projection, the pixel color is determined by the reflected intensity of the polygon vertices. Example rendered images are shown in Fig. 3 (b).
Data preparation is necessary for efficient training, and we sample discrete views at every degrees both in latitude and longitude. If all views of a shape are rendered, they can be arranged in grid, as shown in Fig. 4
. The ordinal numberto in the grid along vertical and horizontal direction is corresponding to the latitude and longitude of the sampled viewpoint location, which defines the view parameterization space for VERAM.
There are two advantages of our sample strategy. First, the entire sphere is uniformly sampled both on latitude and longitude. Second, along the horizontal or vertical direction, the images are consecutively connected and form a circle. As a result, it provides a contiguous space for the agent to deploy the camera sensor in an absolute or relative coordinate. In the following sections, the absolute viewpoint location in the view parameterization space is represented as , and both , are in the range of . The magnified images of , and in Fig. 4 are shown in Fig. 3 (b). The function of the camera sensor can be formulated as
3.1.2 Observation subnetwork
The job of the observation subnetwork is to encode the information about both where the observation is taken as well as what has been seen. As Fig. 2 illustrates, at each time step , the observation subnetwork takes the rendered image and location tuple as input and output a vector .
We use to denote the output from function that takes image as input and is parameterized by weights . typically maps with a sequence of convolutional, pooling, and fully connected layers, and the output of which is high level features . Advanced network architectures succeeded in image recognition task can be used for , such as AlexNet , VGG-16 , ResNet , etc. The location tuple is mapped into embedding by a fully connected layer
. In our practice, a fully connected layer is corresponding to a rectified linear unit.
We concatenate the low bandwidth location with the high bandwidth view information by a fully connected layer and output the final observation feature vector . The observation subnetwork can be represented as
where , corresponding to the whole parameters of the observation subnetwork.
3.1.3 Recurrent subnetwork
The agent maintains an internal state which encodes the agent’s knowledge of the environment and summarizes information extracted from the history of past observations. It is instrumental to deciding how to act and where to deploy the camera sensor. In VERAM, this internal state is formed by the hidden units of the recurrent neural network and updated over each time step with the feature vector from the observation subnetwork, as shown in Fig. 2. The recurrent subnetwork is defined as
3.1.4 View estimation subnetwork
The view estimation subnetwork acts as a controller that directs attention based on the current internal state. In Fig. 2, the view estimation subnetwork takes the hidden units of recurrent subnetwork as input, and outputs to make a prediction on where to deploy the camera sensor to render the next view. The view estimation subnetwork consists of and . maps the hidden units into a two-dimensional coordinate tuple , formally defined as
is usually implemented by a layer followed by a specific transfer function. In the testing phase, is directly used as the next viewpoint location . The procedure of obtaining the image from location is non-differentiable, so in the training phase, a stochastic module needs to be used. It samples
stochastically from a Gaussian distribution with a mean
and a fixed variancefor reinforcement learning, defined as
The detail of view estimation subnetwork will be discussed in subsection 3.2.
3.1.5 Classification subnetwork
The classification subnetwork makes a classification and outputs the category probabilities of the input 3D shape based on the final internal state , which integrates the information of the interaction history between the agent and input 3D shape. In VERAM, the classification subnetwork has a fully connected layer followed by output layer, namely
Given the category label of shape , we can formulate learning as a supervised classification problem. However, because the architecture of VERAM is hybrid, training it involves challenges of handling the non-differentiable component, of keeping balance between learning subnetworks, while struggling with the overfitting problem caused by millions of parameters. In this section, we will describe how VERAM combines SGD  and REINFORCE  to solve these problems.
3.2.1 REINFORCE-based learning
In subsection 3.1.1, the camera sensor is formulated as to map 3D shape with to 2D image . However, is not a continuous function. As shown in Fig. 5, the gradient displayed as green arrow lines from observation subnetwork cannot back propagate to the view estimation subnetwork via the camera sensor. As a result, the view estimation subnetwork cannot be trained with standard back propagation.
REINFORCE  is adopted to solve this problem. Given a space of action sequences , is a distribution over and parameterized by , we wish to learn network parameters that maximize the expected reward of action sequences. The gradient of the objective is
Here is a reward assigned to each possible action sequence. Obviously, due to the high-dimensional space, this is a non-trivial optimization problem. REINFORCE addresses this by learning network parameters using Monte Carlo sampling. After running an agent’s current policy in its environment and obtaining interaction sequences of length , the approximation to the gradient equation is
Here, policy maps the history of past interactions with the environment to a distribution over actions for the current time step . is the cumulative future reward from the current time step to time step , is a baseline reward to reduce the variance of the gradient estimation. For a detail RNN-based REINFORCE, we refer to  and .
In our case, is to predict the next location , is summarized in the state of the hidden units . if the shape is classified correctly after steps and otherwise. Policy is a fully connected hidden layer that maps to as formally defined in formula (4), and is corresponding to . Gaussian distribution is adopt to implement the reinforcement algorithm. Assuming is the density function of the Gaussian distribution as defined in formula (5), is the sampled location, the gradient of w.r.t. is
This gradient can easily back propagate to , so can then be calculated. The flow direction of this gradient is shown as red arrow lines in Fig 5. REINFORCE learns model parameters according to this approximate gradient. It increases the log-probability of an action with a larger than expected cumulative reward, and decreases the probability if the obtained cumulative reward is smaller.
3.2.2 Enhancing the information flow of gradient
As shown in Fig. 5, observation subnetwork, recurrent subnetwork, and classification subnetwork are with standard deterministic neural network connections, and can be trained directly by back propagating gradients from the classification loss, namely, by SGD. By contrast, view estimation subnetwork containing stochastic module needs to be trained using REINFORCE described in subsection 3.2.1.
Giving deep insight into the architecture as shown in Fig. 2 and Fig. 5, we can find that there exists two ways starting from the hidden units and ending at classification subnetwork. Analogous to other RNN-based visual attention models, VERAM is essentially a parallel neural network, and suffers from the common issue of the unbalanced training of the subnetworks corresponding to next view estimation and shape classification. Specifically, as pointed in , the full model does not have enough time to learn a good attention policy before the classification subnetwork overfits to the data. In our practice, we find regardless of different initial parameters and different 3D shapes, the estimated view locations tend to get stuck at the boundary of the view parameterization space, namely or . Based on the visual attention model presented in , we propose three critical schemes to solve this issue, as shown in Fig. 5. It can be summarized as follows:
1) As mentioned in subsection 3.1.4, is implemented by a layer followed by a specific transfer function to force the estimated location into the target range. as well as are commonly used for their continuous interval can be easily mapped to view parameterization space. However, we find the problem of gradient disappearance caused by these transfer functions  is very serious, which means in reinforcement learning, the gradient from reward cannot back propagate to hidden units . VERAM adopts 
as this specific transfer function, which can effectively alleviate the vanishing gradient problem via the identity for positive values.
2) Similar to , we also need a function for to force the output of to fall into the view parameterization space. As shown in Fig. 5, the second operation is performed in  for this purpose. However, the gradient of reward bypasses it and directly applies to function. If the output of is out of the range, the gradient calculated by formula (9) will be not right for is not the actual viewpoint location. To solve this problem, VERAM performs in itself to simulate the output of is always in the range of view parameterization space.
3) In the backward process, formula (9) indicates that if the shape is misclassified, the learning process will encourage move away from . In the boundary, this also needs to be refreshed with sign as
Besides, if the shape is classified correctly, sign always sets to . Sign will multiplies to the result of formula (9). It means, in boundary, we still need to move to if is not in the range of [1/12, 1], although the shape is misclassified.
These three critical schemes enhance the information flow of gradient backpropagation from the view estimation subnetwork to the hidden units, and ensure each module be sequentially trained without break. As a result, it can effectively overcome the issue that the estimated views getting stuck at the boundary of the view parameterization space.
3.2.3 Learning with view confidence
The schemes for enhancing the information flow of gradient provides a basis for keeping a balance between the subnetworks. However, the classification subnetwork is still easier to train than the view estimation one, making the learned attention policy be easily trapped in local optimization. In this subsection, we propose a method of learning with view confidence for REINFORCE to solve this problem.
As mentioned in subsection 3.1.2, advanced network architectures succeeded in image recognition tasks can be used for to extract features for image . Besides, for all training shapes, we also extract the confidence of image . Concretely, first, we extract features for each image of all views of the training shapes. Second, each image is also labeled with the same category of the 3D shape. By this means, for each 3D shape in the training set, we can obtain pairs of . The collection of of all the 3D shapes in the training set will be taken as the input to the following simple network
We use the negative log likelihood criterion to train this network. After convergence, the output probability of corresponding to its category is extracted as the confidence of image . Fig. 6 presents the confidence of each 2D image shown in Fig. 4. The whiter the viewpoint, the more confidence the image has. We can see the image in almost has no confidence for its own category . According to the output of , the highest category probability of this image is .
If a shape is classified correctly, according to formula (9), reinforcement learning will encourage all to move to . On the contrary, it will encourage all to move away from . Intuitively, if a shape is classified correctly, but in step , the confidence of image located at is not high, we should weaken its effects of encouraging to move to . On the other hand, if a shape is misclassified, but the confidence of image located at is high, we should weaken its effects of encouraging to move away from . Based on this inspiration, combining formula (10), we refresh formula (9) as
If a shape is classified correctly, the first line of formula (12) is used to calculate the gradient, otherwise, the second line of formula (12) is used. VERAM incorporates view confidence into reinforcement learning, which leads to a highly efficient guidance to the next view estimation, and can effectively prevent the view estimation subnetwork from trapping into local optimization.
Note that confidence extraction is the preprocess procedure. The trained image classification network is only used to extract the confidence of image, and it will not be used anymore.
3.2.4 Learning with view location constrains
In practice, we find the visited view of each time step may overlap. To solve this problem and make better use of the complementary capacities of different views. A novel regular term for view location constrains is supplemented to loss function and learned at the same time to make the distribution of the visited view locations much more reasonable and mutually complementary.
The regular term for the view location constrains is depend on the prior knowledge of the applications. For 3D shape classification, there are total view locations for the camera sensor, but VERAM only needs a few time steps to form the judgment, so the visited views at least should be separated from each other. This weak regular term is adopted by VERAM.
Particularly, after the agent has moved through steps, we add pairwise distance layer for each two view locations and the agent has visited. We train this part of the model with the loss of to force the visited views to separate from each other. The loss is formally defined as
By integrating this loss function into the learning framework of recurrent attention model, VERAM can explicitly circumvents view duplication and the performance is further improved.
4 Experiments and Discussions
4.1 Dataset, criteria and implementation details
Dataset. We evaluate VERAM along with current state-of-the-arts on ModelNet10 and MondelNet40 . ModelNet10 contains categories with unique 3D shapes, and ModelNet40 contains categories with unique 3D shapes. The training set and testing set have been split on the website.
Criteria. We report classification accuracy with two level criteria. Instance-level accuracy is the ratio of the number of shapes that are classified correctly to the number of the total shapes. Class-level accuracy is the average of instance-level accuracy among all categories. Class-level accuracy is more objective for there are big difference among different categories in the number of testing shapes, from 20 to 100, but instance-level is more intuitive for it directly reflects how many shapes are misclassified.
Implementation details. VERAM is implemented by Torch
on the platform with NVIDIA GeForce TITAN X. It needs about
VERAM is implemented by Torch on the platform with NVIDIA GeForce TITAN X. It needs aboutepochs for training. The learning rate is set to in the first epochs, then decreases linearly to minimum at epoch . Momentum is set to . Based on grid search, the fixed variance of Gaussian distribution for REINFORCE is set to .
CNN architecture and recurrent subnetwork are two main components that affect the performance of VERAM. CNN is used to encode the rendered image. For CNN architecture, AlexNet  and ResNet  are used in our experiments. For recurrent subnetwork, linear mapping and LSTM  are adopted to update the hidden state in each time step. We will give the detail settings in each part of the evaluation.
Theoretically, the parameters of CNN in the observation subnetwork can be fine-tuned with the recurrent network at the same time. However, the training progress will be very slow because the forward and backward propagation need to proceed at each time step. For efficient training, the parameters of AlexNet and ResNet pre-trained on ImageNet are fixed and without further fine-tuning process on ModelNet. However, we can select more previous layer of CNN to counteract this influence.
The source code as well as the trained model of VERAM will be released at our project page: www.kevinkaixu.net/projects/veram.html.
4.2 Comparison with state-of-the-arts
In this subsection, we will compare the performance of VERAM with state-of-the-art deep neural-based methods, especially with the view-based methods. The performance of VERAM is achieved by taking AlexNet as CNN architecture for fair comparison. LSTM is adopted for recurrent subnetwork.
Comparison with deep neural-based methods. The performance of VERAM is compared with state-of-the-art deep neural-based methods on 3D shape classification, as summarized in table I. These methods can be roughly grouped into two categories: shape-based and view-based. Shape-based methods cover 3DShapeNets , VoxNet , SubVolSup , AniProbing , VRN & VRN-Ensemble , FPNN , PointNet , PointNet++ , O-CNN  and So-Net. The view-based methods include MVCNN , DeepPano , PANORAMA-NN, PANORAMA-ENN, MVCNN-Alex , MVCNN-MultiRes , Pairwise , DomSetClust , RotationNet and the proposed VERAM. Besides, FusionNet  exploits both volumetric representation and projective pixel representation.
Among shape-based methods, VRN-Ensemble  achieves the best instance-level accuracy, on ModelNet10 and on ModelNet40. This result is by summing predictions from separately trained five VRN models and one Voxception model. When comparing VERAM with the single VRN model, VERAM gets the better performance, vs. on ModelNet10, and vs. on ModelNet40.
Among view-based methods, DeepPano , PANORAMA-NN and PANORAMA-ENN are based on the panoramic image of 3D shape. PANORAMA-ENN has a clear advantage over VERAM, vs. on ModelNet10, vs. on ModelNet 40. PANORAMA-ENN needs to obtain three panoramas from different principle axes and each panorama needs to extract SDM, NDM and magnitude of gradient image of NDM, while VERAM only takes grayscale image as input. Moreover, these three panoramic methods can be regarded as based on continuous views, while VERAM and other view-based methods are based on discrete views.
Table II gives the detail comparison against state-of-the-art discrete view-based methods with their different processing strategies. Apparently, RotationNet, DomSetClust and MVCNN-MultiRes get the better performance. By augmenting the classification task with pose estimation, RotationNet gets instance-level accuracy on ModelNet10 and on ModelNet40, both are state-of-the-art performance. They are achieved by alerting 11 different camera system orientations. According to table 7 of , with AlexNet, On ModelNet 40, there are 8 of total 11 camera system orientations (except th, th, th) under which the performance of RotionalNet is less than , which is inferior to VERAM (). Considering VERAM only uses a single camera system orientation, there is potential for VERAM to further reduce the gap.
DomSetClust ever gets instance-level accuracy and class-level accuracy on ModelNet40, and both better than the performance of VERAM. However, to obtain such performance, DomSetClust needs to take grayscale, depth and surface normals image as input and fine-tune the CNN network. When DomSetClust and VERAM both use grayscale image as input, VERAM has an clear advantage, vs. instance-level accuracy, and vs. class-level accuracy.
The comparison of VERAM and MVCNN-Alex is more fair for they both taking grayscale image as input, AlexNet as CNN, and with single resolution. VERAM has an obvious advantage over MVCNN-Alex, vs. instance-level accuracy and vs. class-level accuracy. The performances of VERAM is matchable with MVCNN-MultiRes, but the latter needs to implement two times of voxelization and three times of rendering.
Based on the above study, it can be concluded that under the equivalent conditions of render representation, resolution, CNN architecture and number of views, VERAM outperforms all state-of-the-art view-based methods.
|Method||Input||Reso- lution||CNN||Fine tune||ModelNet10||ModelNet40|
Comparison with alternative view-selection methods. VERAM falls into the category of active view selection. Among the methods in table I, both 3DShapeNets  and Pairwise  also adopt active view selection for 3D shape classification. Besides, MV-RNN  extends RNN-based visual attention model by integrating MVCNN  into the architecture for 3D object retrieval. To give a comprehensive comparison, we implemented MV-RNN for 3D shape classification and the input to MV-RNN is vector for each rendered image extracted from ResNet152. Both VERAM and MV-RNN take grayscale image as input, and 3DShapeNets takes depth image as input. In contrast, Pairwise exploits more information and takes both grayscale and depth image as input.
3DShapeNets provides the baseline performance on both datasets among all methods. With the deep neural networks pre-trained on ImageNet, Pairwise makes a great progress on the performance. With 12 views, it gets and instance-level accuracy on ModelNet10 and ModelNet40 respectively. MV-RNN adopts CNN2 of max pooling to perform classification and the accuracy increases with the number of views. The performance of MV-RNN outperforms that of Pairwise with
instance-level accuracy on ModelNet10 and ModelNet40 respectively. MV-RNN adopts CNN2 of max pooling to perform classification and the accuracy increases with the number of views. The performance of MV-RNN outperforms that of Pairwise withviews. VERAM outperforms 3DShapeNets, Pairwise and MV-RNN on both datasets in every view. Although VERAM also uses the pre-trained model on ImageNet. However, the next view selection of VERAM is embedded seamlessly in RNN with view confidence and view location constrains learning.
4.3 Evaluation of the enhancement of VERAM
In this subsection, we will evaluate the three key technical components of VERAM on conducting to improve the performance step by step. ResNet is used to encode the rendered image for it is more compact. We use linear mapping for recurrent network instead of LSTM to stress the ability of three key components to select more discriminative views to enhance the view learning. In the end of this subsection, we will present the results when using LSTM as recurrent network. We trained models for each network setting with the specified super parameters, and use the average class-level accuracy for comparison.
Enhancing the information flow of gradient. RNN-based visual attention model suffers from the problem of unbalanced training of the subnetworks, and the estimated view locations tend to get stuck at the boundary in the view parameterization space. Fig. 8 (left) shows the heat map of view location frequency of each time step by applying the visual attention model presented in , denoted as ClassicalRAM, to predict all in the testing set of ModelNet10, time steps =. It can be seen that starting from the view located at , the views of the next three steps almost all locate at the boundary . To solve this issue, three schemes are proposed for VERAM as described in subsection 3.2.2 to enhance information flow of gradient backpropagation. For simplicity, we call it BoundaryRAM. Fig. 8 (right) shows the heat map of view location frequency of each time step by applying a trained BoundaryRAM model to the same dataset. There are significant difference among the heat maps of different time steps.
The comparison of the average class-level accuracy between ClassicalRAM  and BoundaryRAM on ModelNet10 and ModelNet40 is shown in Fig. 9. The horizontal axis is time steps from to . Note that is a super parameter, and the accuracy of each is the average value from trained models with the same network setting. Leaving out =, i.e., =, the accuracy of BoundaryRAM goes up from min to max on ModelNet10, and from min to max on ModelNet40. From Fig. 8 and Fig. 9, it can be seen that the three critical schemes presented in subsection 3.2.2 can effectively overcome the issue that the estimated views getting stuck at the boundary, and achieve a apparent performance enhancement.
Learning with view confidence. The classification subnetwork of VERAM is easier to train than the view estimation one, and learned attention policy can be easily trapped in local optimization. In practice, with the same network, we find the accuracy of each category from different trained BoundaryRAM model fluctuates widely. With the number of time steps increases, the model trained by BoundaryRAM will be more unstable. To alleviate this problem, based on BoundaryRAM, in subsection 3.2.3, we propose a method of learning with view confidence for REINFORCE to provide effective guidance to agent on where to deploy the model’s attention, here called ConfRAM. The comparison of the average class-level accuracy among ClassicalRAM , BoundaryRAM and ConfRAM on ModelNet10 and ModelNet40 is shown in Fig. 10.
From Fig. 10, it can be seen that, the performance of ClassicalRAM drops after the number of time steps =, while BoundaryRAM drops after = on ModelNet10 and after = on ModelNet40. By contrast, ConfRAM can exploit more views and converge at about = with significant higher performance. Compared with BoundaryRAM, leaving out =, i.e., =, the accuracy of ConfRAM goes up from min to max on ModelNet10, and from min to max on ModelNet40.
Learning with view location constrains. ConfRAM can achieve a stable and fairly good performance. However, the visited view location of each time step may overlap. In subsection 3.2.4, based on BoundaryRAM and ConfRAM, a weak regular term is adopted by VERAM to keep the visited views separated from each other, here called LocRAM, equally to the whole VERAM. The comparison of the average class-level accuracy among ClassicalRAM , BoundaryRAM, ConfRAM and LocRAM on ModelNet10 and ModelNet40 is shown in Fig. 11.
Fig. 11 shows LocRAM learning with the weak regular term of view location constrains can obtain a clear performance improvement. Compared with ConfRAM, leaving out =, i.e., =, the accuracy of LocRAM goes up from min to max on ModelNet10, and from min to max on ModelNet40. It should be pointed out that the regular term of VERAM is only a weak constrain.
The above experiments of this subsection use linear mapping as recurrent subnetwork, and the previous step will have less impact when the interval between which and the last step increases. When replacing linear mapping with LSTM, the comparison of the average class-level accuracy among ClassicalRAM , BoundaryRAM, ConfRAM and LocRAM on ModelNet10 and ModelNet40 is shown in Fig. 12. Since LSTM is effective at capturing long-term temporal dependencies, the already selected discriminative view prevents the performance from decreasing obviously. However, VERAM with three key components still achieves a significant performance boost over ClassicalRAM. When =, the accuracy is improved by on ModelNet10 ( vs. ), and by on ModelNet40 ( vs. ).
4.4 Affecting factors on the performance of VERAM
Effect of different recurrent subnetworks. The final predication is based on the hidden state of recurrent subnetwork which is updated over each time step. Linear mapping and LSTM are mostly used for this purpose. When using linear mapping as recurrent subnetwork, formula is specified as = . For LSTM, the total units is set to and each of which is composed of cell, input gate, output gate and forget gate. Fig. 13 gives the comparison of the performance of VERAM with linear mapping and with LSTM on ModelNet40 when =. The left uses AlexNet as CNN while the right uses ResNet as CNN.
From Fig. 13, it can be seen that LSTM achieves the better performance than linear mapping. When using AlexNet to encode the visual representation, VERAM with LSTM has significant advantage over with linear mapping. For example, the instance-level accuracy increases from to by LSTM when =. When using ResNet as CNN, LSTM still achieves the better performance than linear mapping. However, the improvement is slight. When =, it only gets instance-level accuracy increment from to . Besides, for both AlexNet and ResNet, the performance of linear mapping when = is slightly lower than when =, but the performance of LSTM is quite the opposite.
Effect of different CNN architectures. As described in subsection 3.1.2, VERAM needs to encode the visual representation for the rendered image at each time step. A number of advanced CNN architectures can be adopted for implementation. Among them, AlexNet is the first notably successful CNN architecture on ImageNet while ResNet is the recently proposed representing the advanced level, and they are compared in this subsection. For each 2D image of 3D shape, we extract features from layer of the AlexNet pre-trained on ImageNet. Analogously, we extract features from layer of the ResNet152 pre-trained on ImageNet 11K, which is available on MXNet . For efficient training, we didn’t fine-tune AlexNet or ResNet model on ModelNet. Fig. 14 presents the performance of VERAM with AlexNet and with ResNet on ModelNet40 when =. The left uses linear mapping as recurrent network while the right uses LSTM.
Fig. 14 indicates that when using linear mapping as RNN, the performance of ResNet soars about percent compared with that of AlexNet and has a clear advantage. However, when using LSTM as RNN, the performance of AlexNet and ResNet are comparable to each other. At =, the best instance-level accuracy of VERAM with AlexNet is , and with ResNet, it is . Theoretically, the feature extracted from ResNet has more capacity of discrimination than feature extracted from AlexNet. However, the features extracted from CNN in each time step are further merged by LSTM, and acoording to Fig.
. Theoretically, the feature extracted from ResNet has more capacity of discrimination than feature extracted from AlexNet. However, the features extracted from CNN in each time step are further merged by LSTM, and acoording to Fig.13, the improvement of LSTM on VERAM with AlexNet is much more significant than on VERAM with ResNet. As a result, the performance gap between these two CNNs is reduced. We noticed the performance difference among different CNN architectures in recent work  on 3D shape recognitions is about .
Effect of time steps T. Time steps in VERAM is a super parameter and it determines how many images does VERAM need to render and observe before emit the classification predication. Fig. 15 shows the best instance-level and class-lever accuracy VERAM achieved on ModelNet40 with different time steps from to . AlexNet is used to encode the visual representation and LSTM is adopt as recurrent subnetwork. For clarity, we only append and its accuracy for . In Fig. 15, the rate of accuracy growth is fast in the first few steps, then becomes slower and slower with the increase of time step and converges at =, where VERAM achieves the best performance, instance-level accuracy and class-level accuracy. After that, the performance even slightly decreases. From Fig. 15 it can be concluded as follows:
1) The performance of VERAM can quickly converge within a few time steps. On ModelNet40, VERAM can get instance-level accuracy with only 3 views, which is as high as of the best performance ().
2) Increasing would do little to improve the performance after the convergence. VERAM uses the same parameters and network for each time step, so that the total number of parameters will not expand with the increase of . Although increasing means VERAM can obtain more information, but the larger means VERAM needing to cope with more views with the same number of parameters.
Effect of shape alignment. As Pairwise  and RotationNet , VERAM needs a unified coordinate system to render shapes and each shape should be rendered from pre-defined viewpoints, so shape alignment is important to the proposed method. On the website of ModelNet , all shapes in ModelNet10 are manually cleaned and their orientations are well aligned. The shapes in ModelNet40 are also cleaned, but a small part of shapes are needed to align in the pre-processing stage. The reported performance of VERAM is based on such aligned orientation.
To quantify the effect of the alignment on the performance, we conduced an experiment with aligned ModelNet40 newly released on the website of ModelNet. Taking AlexNet as CNN and LSTM as RNN, we trained three VERAM models (=, = and =) with the aligned training dataset and tested the classification on both aligned and unaligned (i.e., randomly rotated) shapes. The results are shown in Fig. 16. The instance-level accuracy of unaligned shapes only got , and with , and views. Moreover, it only increases slightly with the increment number of views. These indicate that VERAM is rather sensitive to the pre-defined viewpoints. We notice that this limitation is generally encountered in several view-based methods. For instance, RotationNet achieves on ShapeNetCore55 dataset. In future, we plan to merge alignment network into VERAM to alleviate this problem.
Complexity. With the extracted features from AlexNet and using LSTM as RNN, the total number of parameters of VERAM is about M, which is lesser than that of VGG16 (M) and ResNet152 (M). In our single TITAN X GPU platform and in the testing phase, it takes less than second to extract features for an image by AlexNet. Besides feature extraction, rendering and predication run very fast, and it only needs about second to emit the decision. As shown in Fig. 15, the performance of VERAM is very stable with different after . In summary, if we set =, which can already obtain high accuracy, it takes about second to classify a 3D shape (not including the time spent on moving the camera).
4.5 Detailed statistics
VERAM achieves the best average class-level accuracy on ModelNet10 (with ResNet) and on Modelnet40 (with AlexNet). The number of correctly classified shapes over the total number of shapes of each category with these two best models are reported in table III.
On ModelNet10, and are the two worst categories and they achieve and class-level accuracy respectively. out of in the testing set are misclassified. One each of them is misclassified as and , and the other are misclassified as . A total of are misclassified. One each of them is misclassified as and and the others are misclassified as . Two misclassified shapes of and are shown in the first row of Fig. 17.
On ModelNet40, the classification accuracy of is and is the worst among all categories. gets accuracy and holds the second worst position. Total among 20 are misclassified. One each of them is misclassified as , and , and the other are misclassified as . The testing set of ModelNet40 contains and are misclassified. Among them, shapes are misclassified as and the other are spread to categories including , , , , . Two misclassified shapes of and are shown in the second row of Fig. 17.
According to our intuition, the shapes shown in Fig. 17 are hard to be correctly classified by only using the visual representations. At least partly, the mistake dues to a shape may has diversified functions. We suspect that more structure information in 3D space is needed to be fully exploited to meet such challenges.
We have presented VERAM, a recurrent attention model capable of actively selecting a sequence of views for highly accurate 3D shape classification. To address the problem commonly found in existing RNN-based attention models, i.e., the unbalanced training of the subnetworks, we propose to 1) enhance the information flow of gradient backpropagation for the view estimation subnetwork, 2) design a highly informative reward function for the reinforcement training of view estimation, and 3) formulate a novel loss function that explicitly circumvents view duplication.
When being applied to real scenarios, VERAM has several limitations which may spark future research:
Shape alignment. The experiment in subsection 4.4 shows that the result is sensitive to alignment and this issue is also reported in. To address this problem, the basic strategy is by deepening the network and augmenting the training data to force the network to cover different viewpoint variants. Inspired by the approaches of learning transform parameters as , we think the more promising approach is to merge the alignment network into VERAM.
Time steps. VERAM uses the fixed time steps for predication. To make the model capable of stopping observation once it has enough information, the reward of MV-RNN  contains the information gain and it terminates the process when entropy is less than the threshold, while  designed a subnetwork to learn a binary prediction indicator. Both of them provide the insight to extend our model with varying instead of fixed time steps.
Inaccessible views and occlusion. In real scanning scenario, there are cases where some predicted views are physically inaccessible. Varying time steps is necessary and the proposed method also needs to be extended to learn the relevance of each view to the task as TAGM . Occlusion also has severe adverse effect since the feature representation of each view is by convoluting the entire image. The key to this problem is to encode the part-based feature as in  to spot informative visible parts of the partially occluded 3d shape.
Moving cost and views passed through. The cost of moving scanner should be considered carefully in real scanning scenario. One feasible approach is to model the cost of moving scanner as the circle distance between adjacent selected views and add the cost to the reward for reinforce learning. Another issue is VERAM omits the continuous images obtained when moving the scanner from the current selected viewpoint to the next one. Although these views are not the best for next observation, but they also have the potential to help the classification to be more efficient.
Visual feature encoding. The pre-trained deep CNN is adopted by VERAM to extract visual features. For efficiency reason, such deep network is hard to fine-tuned with the view estimation subnetwork simultaneously, although what and where to observe are coupled for each other. It seems that visual attention model should exploit a more efficient network to learning what to observe.
This work was supported in part by the National Natural Science Foundation of China (61373135, 61672299, 61702281, 61532003, 61572507 and 61622212), and the Postdoctoral Science Foundation of Jiangsu Province of China (No.1701046A).
D. Y. Chen, X. P. Tian, Y. T. Shen, and M. Ouhyoung, ”On visual similarity based 3D model retrieval,” Comput. Graph. Forum, vol. 22, no. 3, pp. 223-232, 2003.
-  J. L. Shih, C. H. Lee, and J. T. Wang, ”A new 3D model retrieval approach based on the elevation descriptor,” Pattern Recogn, vol. 40, no. 1, pp. 283-295, 2007.
-  C. M. Cyr and B. B. Kimia, ”3D object recognition using shape similiarity-based aspect graph,” Proc. IEEE Conf. Comput. Vis., pp. 254-261, 2001
-  M. Ulrich, C. Wiedemann, and C. Steger, ”Combining scale-space and similarity-based aspect graphs for fast 3d object recognition,” IEEE trans. on Pattern Analysis and Machine Intelligence, vol. 34, no. 10, pp. 1902-1914, 2012
-  P. Papadakis, I. Pratikakis, T. Theoharis, and S. Perantonis, ”PANORAMA: A 3D shape descriptor based on panoramic views for unsupervised 3D object retrieval,” Int. J. Comput. Vision, vol. 89, no. 2, pp. 177-192, 2010.
-  M. Ankerst, G. Kastenm ller, H. P. Kriegel, and T. Seidl, ”Nearest neighbor classification in 3D protein databases,” Proc. the Seventh International Conference on Intelligent Systems for Molecular Biology, pp. 34-43, 1999.
-  M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz, ”Rotation invariant spherical harmonic representation of 3d shape descriptors,” Proc. Symp. Geometry Process, pp. 156-164, 2003.
-  J. Knopp, M. Prasad, G. Willems, R. Timofte and L. Van Gool, ”Hough transform and 3D SURF for robust three dimensional classification,” Proc. 11th Eur. Conf. Comput. Vis., pp. 589-602, 2010.
-  A. M. Bronstein, M. M. Bronstein, L. J. Guibas, and M. Ovsjanikov, ”Shape Google: Geometric Words and Expressions for Invariant Shape Retrieval,” ACM Trans. Graphics, vol. 30, no. 1, pp. 1-20, 2011.
-  A. Krizhevsky, I. Sutskever, and G. Hinton, ”ImageNet classification with deep convolutional neural networks,” Proc. Adv. Neural Inf. Process. Syst., pp. 1097-1105, 2012.
-  K. Simonyan and A. Zisserman, ”Very deep convolutional networks for large-scale image recognition,” arXiv:1409.1556, 2014.
-  H. Su, S. Maji, E. Kalogerakis, and E. G. Learned-Miller, ”Multi-view convolutional neural networks for 3D shape recognition,” Proc. Int. Conf. Comput. Vision, pp. 945-953, 2015.
C. R. Qi, H. Su, M. Niessner, A. Dai, M. Yan and L. J. Guibas, ”Volumetric and multi-view CNNs for object classification on 3D data,” Proc. 2016 IEEE Conf. Computer Vision and Pattern Recognition, pp. 5648-5656, 2016.
-  C. Wang, M. Pelillo, and K. Siddiqi, ”Dominant Set Clustering and Pooling for Multi-View 3D Object Recognition,” Proc. 2017 British Machine Vision Conference(BMVC), 2017
-  J. Deng, W. Dong, R. Socher, L. J. Li, K. Li and L. Fei-Fei, ”Imagenet: A Large-Scale Hierarchical Image Database,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 248-255, 2009.
-  E. Johns, S. Leutenegger, and A. J. Davison, ”Pairwise decomposition of image sequences for active multi-view recognition,” Proc. 2016 IEEE Conf. on Computer Vision and Pattern Recognition, pp. 3813-3822, 2016.
-  A. Kanezaki, Y. Matsushitay and Y. Nishida, ”RotationNet: Joint Object Categorization and Pose Estimation Using Multiviews from Unsupervised Viewpoints,” Proc. 2018 IEEE Conf. on Computer Vision and Pattern Recognition, pp. 5010-5019, 2018.
-  J. Ba, V. Mnih, and K. Kavukcuoglu, ”Multiple object recognition with visual attention,” arXiv:1412.7755, 2014.
-  V. Mnih, N. Heess, A. Graves, and K. Kavukcuoglu, ”Recurrent models of visual attention,” Proc. Adv. NIPS, pp. 1-9, 2014.
-  P. Sermanet, A. Frome, and E. Real, ”Attention for fine-grained categorization,” arXiv:1412.7054, 2014.
-  S. Semeniuta and E. Barth, ”Image classification with recurrent attention models,” Proc. 2016 IEEE Symposium Series on Computational Intelligence (SSCI), pp. 1-7, 2016.
-  K. Xu, J. Ba, R. Kiros, K. Cho, A. Courville, R. Salakhudinov, R. Zemel, and Y. Bengio, ”Show attend and tell: Neural image caption generation with visual attention,” Proc. 32nd Int. Conf. Mach. Learn., pp. 2048-2057, 2015.
-  K. Xu, Y. Shi, L. Zheng, J. Zhang, M. Liu, H. Huang, H. Su, D. Cohen-Or and B. Chen, ”3d attention-driven depth acquisition for object identification,” ACM Trans. Graphics, vol. 35, no. 6, pp. 1-14, 2016.
-  D. E. Rumelhart, G. E. Hinton, and R. J. Williams, ”Learning representations by back-propagating errors,” Nature, vol. 323, no. 6088, pp. 533-536, 1986.
R. J. Williams, ”Simple statistical gradient-following algorithms for connectionist reinforcement learning,” Machine learning, vol. 8, no. 3-4, pp. 229-256, 1992.
-  Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang and J. Xiao, ”3d shapenets: A deep representation for volumetric shapes,” Proc. 2015 IEEE Conf. on Computer Vision and Pattern Recognition, pp. 1912-1920, 2015.
-  D. Maturana and S. Scherer, ”VoxNet: A 3D Convolutional Neural Network for real-time object recognition,” Proc. 2015 IEEE/RSJ International Conf. on Intelligent Robots and Systems (IROS), pp. 922-928, 2015.
-  A. Brock, T. Lim, J. Ritchie, and N. Weston, ”Generative and Discriminative Voxel Modeling with Convolutional Neural Networks,” arXiv:1608.04236, 2016.
-  Y. Li, S. Pirk, H. Su, C. R. Qi, and L. J. Guibas, ”FPNN: Field Probing Neural Networks for 3D Data,” arXiv:1605.06240, 2016.
-  C. R. Qi, H. Su, K. Mo, and L. J. Guibas, ”PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation,” arXiv:1612.00593, 2016.
-  C. R. Qi, L. Yi, H. Su, and L. J. Guibas, ”PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space,” arXiv:1706.02413, 2017.
-  P. S. Wang, Y. Liu, Y. X. Guo, C. Y. Sun, and X. Tong, ”O-CNN: Octree-based Convolutional Neural Networks for 3D Shape Analysis,” ACM Trans. Graphics, vol. 36, no. 4, pp. 1-11, 2017.
-  J. Li, B. Chen, and G. Lee, ”SO-Net: Self-Organizing Network for Point Cloud Analysis,” Proc. 2018 IEEE Conf. on Computer Vision and Pattern Recognition, pp. 9397-9406, 2018.
-  B. Shi, S. Bai, Z. Zhou, and X. Bai, ”DeepPano: Deep panoramic representation for 3-D shape recognition,” IEEE Signal Process. Lett., vol. 22, no. 12, pp. 2339-2343, 2015.
-  K. Sfikas, T. Theoharis and I. Pratikakis, ”Exploiting the PANORAMA representation for convolutional neural network classification and retrieval,” Proc. 2017 Eurographics Workshop on 3D Object Retrieval, 2017.
-  K. Sfikas, I. Pratikakis and T. Theoharis, ”Ensemble of PANORAMA-based convolutional neural networks for 3D model classification and retrieval,” Computers & Graphic, vol. 71, pp. 208-218, 2017.
-  Z. Xie, K. Xu, W. Shan, L. Liu, Y. Xiong and H. Huang, ”Projective Feature Learning for 3D Shapes with Multi-View Depth Images,” Computer Graphics Forum, vol. 34, num. 7, pp. 1-11, 2015.
-  S. D. Roy, S. Chaudhury, and S. Banerjee, ”Active recognition through next view planning: a survey,” Pattern Recognition, vol. 37, no. 3, pp. 429-446, 2004.
-  K. Wu, R. Ranasinghe, and G. Dissanayake, ”Active recognition and pose estimation of household objects in clutter,” Proc. 2015 IEEE International Conference on Robotics and Automation (ICRA), pp. 4230-4237, 2015.
-  S. Yeung, O. Russakovsky, G. Mori, and L. Fei-Fei, ”End-to-end learning of action detection from frame glimpses in videos,” Proc. 2016 IEEE Conf. on Computer Vision and Pattern Recognition, pp. 2678-2687, 2016.
-  W. Zaremba and I. Sutskever, ”Reinforcement learning neural Turing machines,” arXiv:1505.00521, 2015.
-  K. He, X. Zhang, S. Ren, and J. Sun, ”Deep residual learning for image recognition,” Proc. 2016 IEEE Conf. on Computer Vision and Pattern Recognition, pp. 770-778, 2016.
-  S. Hochreiter and J. Schmidhuber, ”Long short-term memory,” Neural computation, vol. 9, no. 8, pp. 1735-1780, 1997.
-  D. Wierstra, A. Foerster, J. Peters, and J. Schmidhuber, ”Solving deep memory POMDPs with recurrent policy gradients,” Proc. International Conference on Artificial Neural Networks, pp. 697-706, 2007.
-  S. Hochreiter, ”The vanishing gradient problem during learning recurrent neural nets and problem solutions,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 6, no. 2, pp. 107-116, 1998.
-  D. A. Clevert, T. Unterthiner, and S. Hochreiter, ”Fast and accurate deep network learning by exponential linear units (elus),” arXiv:1511.07289, 2015.
-  T. Chen, M. Li, Y. Li, M. Lin, N. Wang, M. Wang, T. Xiao, B. Xu, C. Zhang, and Z. Zhang, ”MXNet: A flexible and efficient machine learning library for heterogeneous distributed systems,” arXiv:1512.01274, 2015.
-  V. Hegde and R. Zadeh, ”Fusionnet: 3d object classification using multiple data representations,” arXiv:1607.05695, 2016.
M. Jaderberg, K. Simonyan, and A. Zisserman. ”Spatial transformer networks,” Proc Advances in neural information processing systems, pp. 2017-2025, 2015.
-  W. Pei, T. Baltrǔsaitis, D. M. Tax, and L. P. Morency, ”Temporal attention-gated model for robust sequence classification,” Proc. 2017 IEEE Conf. on Computer Vision and Pattern Recognition, pp. 820-829, 2017.