Counting people in crowded scenes is a highly challenging and important problem in computer vision and video surveillance. According to the statistics111https://en.wikipedia.org/wiki/List_of_human_stampedes, there are at least serious human stampedes happened in 2017, causing at least
people to die, and thousands of people were injured. Such tragedy can be prevented if we can estimate the crowd density and take preventive measures in time.
Precisely detecting or recognizing each person is intractable in crowed scenes due to the heavy occlusions and cluttered visual patterns, as shown in Figure 1. Therefore, most existing methods [28, 11, 26, 25, 22, 2, 21, 24, 17, 27]
choose to estimate the density map, of which the integral is equal to the number of people in a given image. With the recent breakthrough of deep learning in visual related tasks, most cutting-edge counting methods are based on the Convolutional Neural Network (CNN), e.g., MCNN and Hydra .
However, our study reveals an interesting phenomenon that, in the current all “going deeper” era, the network architectures in most counting methods are rather shallow. For example, MCNN only uses convolutional layers with a limited number of filters. The same situation exists even in many multi-subnet networks [2, 21, 22], e.g., the core component of Switch-CNN 
for generating density maps is exactly the same as in MCNN. Yet this is not simply because of hardware limitation, e.g., the auxiliary component in Switch-CNN, a density-level classifier, uses the very deep VGG-16 structure.
In fact, learning a Deep Neural Network (DNN) to estimate density map is not so straightforward and even frustrating. In both  and , the difficulty of optimizing their own DNN has been mentioned. Besides, the earlier research in  also points out that learning a DNN with the -loss, which is the major objective in most density map based methods, is vulnerable to outliers.
Based on the above considerations, we conduct in-depth study to dig out the main reason behind the disadvantages of learning DNNs for density map estimation. After delving into the learning process of existing methods, we attribute the main reason to the inhomogeneous density distribution
in crowd images. By inhomogeneous density distribution, we refer to the various intra-image and inter-image density levels. As we will see in the later sections, it will not only introduce outliers, but also result in other problems, such as the dying ReLU phenomenon, in the optimization process.
Fortunately, with the comprehensive analysis of the inhomogeneous density distribution problem, we are able to find an intuitive and natural solution for it: we simply transform our learning labels from 2D density maps to structured (3D) density maps. Such solution can reduce the effects of outliers sufficiently, and with a few modifications on the VGG-16 network, we successfully train a single Density-Aware Network (DAN), which obtains comparable performance with the much more complicated multi-subnet networks.
In summary, this paper focuses on the inhomogeneous density distribution problem in the high-density crowd counting task. The major contributions of this paper are dual: (a) We present the detailed analysis of how the inhomogeneous density distribution affects the density map based methods. We also point out the reasons of several abnormal phenomena in the optimization process, and provide solutions for them. We hope this can motivate other researchers to handle similar problems in their optimization process. (b) We propose the structured inhomogeneous density learning method as a practical solution for crowd counting, which is implemented by our DAN model. Extensive experiments on several datasets show that our simple method can obtain the state-of-the-art performance.
Ii Related Work
Earlier research on crowd counting mainly aims at low density scenes, and can be roughly divided into detection-based methods [9, 7] and regression-based methods [6, 15, 5, 4, 1]. In recent years, as high-density datasets [12, 27, 28] are built, density map-based methods, which is firstly introduced in , have gained more attention. Due to the page limitation, we focus on density map-based methods.
Most of the state-of-the-art density map-based methods are implemented using CNNs and its variants. A comprehensive survey of CNN-based counting models can be found in . Based on the network architecture, we can divide existing methods into (a) single-column networks, (b) multi-column networks and (c) multi-subnet networks, as shown in Figure 4:
Single-Column Networks: we consider that networks [17, 25] with the straight structure similar to the Fully Convolutional Networks  as the single-column networks. A typical example of this kind of networks is the Counting CNN (CCNN, ), which is simply trained by minimizing the -loss between its outputs and the ground-truth density maps. In , the CNN is combined with the LSTM to capture spatial and temporal dependencies. The advantage of this kind of networks is the simplicity of their architectures.
: multi-column networks, or multi-scale networks are introduced to tackle the scale variance and distorted perspective map problem[28, 17, 24, 26]. This type of networks usually use multiple columns with the same functions to capture features at different resolutions. For example, MCNN  uses three columns of filters of different sizes; Hydra  uses a pyramid of input patches; and in , a multi-scale blob with different kernel sizes is introduced. Multi-column networks are usually more robust than the single column networks, since they can integrate features with different resolutions. However, as we will point out later, multi-column networks actually are global models, they will still be affected by the outliers in the training data.
Multi-Subnet Networks: multi-subnet networks [22, 11, 2, 21, 24], or multi-task networks are networks consisting of multiple subnets, and the main difference between multi-subnet networks and multi-column networks is that each subnet in multi-subnet networks has its own objective function. For example, Switch-CNN  is composed of three local regressors and one density-level classifier. The three regressors are trained to generate the their own density maps, while the density-level classifier is trained to select the optimal density map. A more complicate example is the CP-CNN  model, which has a global-context classifier, a local-context classifier, a density map regressor, and a fusion network to combine these subnets together. Indeed, among the three kinds of networks, multi-subnet networks can obtain the most impressive performance. However, learning a multi-subnet network is difficult, since it requires individually tuning for its each component.
Iii Crowd Counting with Density Map
In this section, we first briefly overview density map based counting methods in Section III-A, and then present the detailed analysis of how the inhomogeneous density distribution problem affects existing methods in Section III-B, so that we can solve it accordingly in our method.
Iii-a Problem Formulation
Given an image belonging to the image domain , the primary goal of this paper is to learn a function to estimate the number of people in , i.e., . To this end, we are given a training set consisting of images and their corresponding ground-truth head based annotations, i.e., , where , , and denote the height and width of the image, respectively, and if the center of a target appears at position , otherwise , , . Yet the information provided only by the count is limited, therefore we are also interested in inferring the density maps of the images. Specifically, given the annotation , the corresponding ground-truth density map is generated as follows:
where denotes the set of non-zero points in , and denotes a 2D normalized Gaussian function centered at point with smoothing factor 222For the compactness of expression, we have omitted the spatial transformation process of .. The size of depends on the specific feature representation, and we consider it as , where is the scaling factor. Calculating the count based on the density map can be done easily by summing over elements in , i.e.,
. In this way, most density map based methods can be summarized as learning a regression model by minimizing the following loss function:
where denotes the -loss, and can be any differential regularizer. Like in other deep learning models, gradient descent or its variants can be adopted to minimize .
Iii-B The Inhomogeneous Density Distribution
From the experimental results in previous research [28, 24, 2, 22], we observe three interesting phenomena: (a) Compared with global model, local model shows superior performance. This is observed from the fact that the three columns of CNNs in Switch-CNN  and those in MCNN  are the very same, and the major difference between these two networks is Switch-CNN discriminatingly selects one column for predicting, while MCNN merges all three columns into a global regressor. (b) Compared with shallow networks, quite surprisingly, DNNs perform poorly, especially on datasets with various density levels . (c) The mean absolute error (MAE) of current methods [28, 22] is proportional to the relative density level. If we divide images into groups based on their density levels, i.e., extremely low to extremely high, and consider the group with medium density level as the baseline, then groups far from the baseline suffer higher MAE than those near the baseline. All these phenomena indicate that blindly increasing the capacity of networks cannot help to solve our task. In order to increase the robustness of our solution, we must dig out the main reason behind the above phenomena, which, in fact, is the inhomogeneous density distribution in our images.
To understand how the inhomogeneous density distribution problem affects the performance of current methods, note that in the loss function defined in Eq.(2), the major objective is to minimize the -loss term, which is easily affected by outliers. Without loss of generality, we take MCNN as an example. For the convenience of discussion, we only consider its last layer, which contains a filter group of size
, and ReLU is used as the activation function. We assume the input of the last layer is noise-free. Letand denote the weights and bias of the filters, denote the input of the last convolutional layer, and be short for the residual that
, based on the chain rule we can calculate the partial derivative of-loss term with respect to and as
where is the indicator function. From Eq.(3) we can see that, for all , the gradient is dominated by . Therefore, if outliers exist, e.g., is affected by additive noise , then is equally affected by the bias , and consequently disrupts the optimization process.
But where do the outliers come from? Remember that is generated by convolving the head annotation map with the 2D Gaussian kernel (Eq.(1)), of which the essential idea is, the sum of density values in the area of one head is equal to . Therefore, in ideal situation, we should generate the high and compact Gaussian response for people far from the camera (high-density area), while the low and flat response for those near the camera (low-density area). However, due to the lack of annotation, current methods have to use a “moderate” Gaussian template to generate the density map. In this way, for high-density / low-density areas, the density values of the real training template are lower / higher than those of the ideal one, and consequently the learned model will underestimate / overestimate the counts, which is exactly in accordance with the experimental results in .
With the inhomogeneous density distribution, the aforementioned phenomena can be explained well now: (a) Since Switch-CNN selects the training images for each of its regressors individually, it actually removes the outliers in a certain extent. On the other hand, the geometry-adaptive kernel in MCNN does help to reduce the bias, but the outliers remain in the training process. (b) With deeper architectures, networks are more likely to over-fit the outliers, degrading their generalization ability. (c) The real training template is quite different from the ideal response in areas with extreme density levels, therefore the trained models are more affected by the outliers in these areas.
Besides, by analyzing the gradient of the -loss term, we can also infer the following problems will occur if we train a deep neural network to minimize the -loss without deliberate consideration:
Dying ReLU: For any , i.e., when the network overestimates the density, if the value of or the learning rate for gradient descent is too large, then will decrease dramatically, causing the activation value for in a wide range of values. Since the gradient can only pass through a ReLU when the activation value is larger than , it is hard to update in this case, and consequently the output of the ReLU will always be .
Exploding Gradients: Similar to Eq.(3), we have , so if the value of is too large, e.g., , it may cause the gradient to explode. In fact, Daniel et al.  mention that their CCNN cannot converge with the Xavier Initialization . Although they didn’t explain the reason behind it, we can see clearly that it is caused by the term.
Saddle Points: The gradient is dominated by , and all in contribute equally. Hence, if there are a few with unbounded, extremely high values, they may cancel out the partial derivative of the other elements in , namely reaching the critical points. Furthermore,  points out that if the critical points are with much larger than the global minimum, then they are exponentially likely to be the saddle points and will slow than the learning process333For the relationship between critical points and saddle points, interested readers can refer to  for more details.
Iv Density-Aware Network
Based on the aforementioned analysis of the inhomogeneous density distribution problem, we can see that the ways of tackling it are quite straightforward: we can either (a) minimizing the difference between ideal templates and real training templates, or (b) reduce the effects of outliers, or, as we will show in Section IV-A, we can even combine and accomplish both goals simultaneously within a unified structured learning framework, and implement the framework easily via the proposed Density-Aware Network (DAN) in Section III-A.
Iv-a Separating Inhomogeneous Density Distribution as Structured Learning
Our solution for the inhomogeneous density distribution problem is intuitive and natural: we extend the 2D density maps to the structured density maps (3D), of which the last dimension implicitly indicates the density levels, and we use an individual Gaussian kernel on each density level.
Formally, our goal now is to learn a structured regressor , where denotes the number of predefined density levels. In order to do so, we need to convert the original head annotation into the new structured label , ,
. Here we propose a heuristic soft mapping rule to determine the value offor :
where is the normalization term to make sure , is the empirical smoothing parameter, and denotes the index of the optimal density level for . To determine , notice that the average distances between annotated points in the high-density areas are usually much smaller than those in the low-density areas, therefore, we simply define a strictly increasing thresholds , and calculate the average distance from point to its Top-5 nearest annotated points, and consider as the index of the minimum threshold that large than or equal to the average distance. Similar to Eq.(1), we generate the structured density map on level , , by convolving with a normalized Gaussian filter with the empirical smoothing parameter .
Remark: Figure 4 shows a toy example of the generating process of the structured density maps. The benefits of learning such structured density maps are trial: (a) Compared with the single Gaussian template solution, apparently multiple templates are less biased, and hence are more suitable to model the inhomogeneous density distribution; (b) Each level learns to generate its own density maps resembling the local model, thus can help to reduce the effects of outliers. (c) The outputs of all levels are generated simultaneously, which naturally serves as the regularizer, e.g., it is unlikely for a position to have high response values on both the high-density and the low-density levels.
Iv-B Robust Regression
With the structured density maps, we are now ready to define our objective function. As we have pointed out in previous sections, directly learning the inhomogeneous density distribution with the loss will cause many problems, e.g., being sensitive to outliers. Therefore, we propose to minimizing the following cost-sensitive Huber loss:
where denote the residual on the -th density level, is the Huber loss (element-wise) that:
and is the weight defined as:
where , and
are hyperparameters and, and denote the predicted and ground-truth count, respectively. To see how optimizing can obtain a more robust model, one should notice that if , otherwise , so it clips the gradient to no matter how large the bias
is. This property reduces the effects of outliers and provides relief from the exploding gradient problem. Furthermore, the weightis introduced to emphasize hard, high-residual examples. The term is the relative absolute residual, and the term is used to prevent from dividing . It is easy to see that is inversely proportional to , so that examples with high estimation error can gain more attention in our learning process. The hyperparameter controls the range of that , and adjusts the slope of curve of , as demonstrated in Figure 5. We set throughout this paper. and together can ensure the gradient stay within a reasonable range, and hence reduce the difficulty of optimization.
Iv-C One Network, Two Goals
In order to realize the structured density map learning, we propose a simple yet effective Density-Aware Network (DAN) in this section. The intuition behind our design of DAN is straightforward: as to regression, since shallow neural networks can generate rather good density maps, DNNs shall be able to do so as well, due to their higher capacity; as to classification, DNNs are also capable of distinguishing various density levels, e.g., in cutting-edge multi-subnet networks [2, 22], the very deep VGG-16  is adopted as the classifier for density levels. Therefore, we believe that a single deep architecture is sufficient for learning the structured density maps. Besides, classifying density levels and learning the density maps are highly related tasks, therefore using a single network can let them share visual cues more effectively.
The proposed DAN is a multi-branch network and its architecture is shown in Figure 3: the “trunk” of DAN, namely the shared blocks before splitting into multiple branches, stems from the VGG-16 network. Specifically, we adopt the Conv1 to Conv5 blocks in VGG-16, and remain only the first and the second pooling layers in VGG-16, in this way, the downscaling factor of DAN is . We attach branches to the end of the “trunk”, and each branch consists of two convolutional blocks to generate its corresponding density maps. The specific number of and the sizes of filters in the branches can be adjusted based on the dataset, and we find and filters of perform well in most situations. Note that the whole DAN is fully convolutional, therefore it can handle images of arbitrary size.
Besides, in order to prevent the dying ReLU phenomenon, we also change all the activation functions in DAN to the leaky ReLU  that . As to the weight initialization, the weights of filters in Conv1 to Conv5
are initialized by the VGG-16 model pre-trained on the ImageNet. For the filters in the branches, as we have analyzed above, high initial value ofwill cause the exploding gradient problem easily. Therefore, after generating with the Xavier initialization method , we multiply by a small number , i.e., . In our experiments, we find that can prevent gradient explosion effectively.
Table I summarizes all major modifications in DAN, which can be implemented easily. The architecture of DAN is flexible and can be compatible with most current deep learning frameworks. Unlike other density map based methods, DAN can be trained in a totally end-to-end way, thus makes it a considerable solution for the crowd counting problem.
|Architecture:||VGG-16 multi-branch networks|
|Activation function:||ReLU leaky ReLU|
|UCF ||40 : 10||1279.48||960.13|
|WorldExpo||3380 : 600||48.29||38.22|
|ShanghaiTechA||300 : 182||500.29||456.56|
|ShanghaiTechB||400 : 316||123.59||94.52|
denote the average and standard deviation of counts, respectively.
This section provides the detailed analysis of our experiments on several dataset [12, 27, 28]. Our mainly concerns on three aspects: (a) The comparison between the proposed structured inhomogeneous density learning and other state-of-the-art methods (Section V-C), which can show that the proposed method is a simple yet considerable solution for crowd counting in high-density scenes. (b) The ablation study of the proposed Density-Aware Network (DAN) for implementing structured learning on the ShanghaiTech, Part A  (Section V-D), which validates our opinions on how the inhomogeneous density distribution affects the learning process. (c) Qualitative results of the proposed method (Section V-E), in which not only the high-quality results, but also those in the worst situations of the proposed methods are demonstrated.
V-a Implementation Details
In order to minimizing the cost-sensitive Huber loss , we adopt the Adam optimizer  with a small learning rate, e.g., to . The model will converge in about K iterations. In each iteration, we select a training image and randomly crop a patch of the image size as the training example. We perform data augmentation by horizontally flipping images and adding Gaussian noisy to images.
As to the hyperparameters, the smoothing parameter for soft mapping, ; the density thresholds depend on the image sizes, and a customary choice is , and the smoothing parameters for Gaussian Filters, are set to ; the threshold in Huber loss, , is set to .
Our implementation of DAN is in MATLAB with the MatConvNet framework , source code will be released in the future.
To validate the effectiveness of our method, we perform extensive experiments on four publicly available datasets, including UCF , WorldExpo , ShanghaiTech A and B . The statistics of these datasets are summarized in Table II. The counts in our experiments cover a wide range from to , which can reduce the possibility of over-fitting. For the purpose of fair comparison, following most existing methods, we adopt the mean absolute error (MAE) and the mean square error (MSE) as the metrics for evaluation, which are defined as follows:
V-C Quantitative Analysis
ShanghaiTech: The first dataset we evaluate our method on is the ShanghaiTech dataset  including part A and B. Part A includes training images and images with counts ranging from to , while Part B includes training images and test images with counts ranging from to . We compare the proposed DAN networks with deep learning based state-of-the-art methods on this dataset, including Zhang et al. , MCNN , Cascaded-MTL , Switch-CNN , Huang et al. , MSCNN  and CP-CNN .
From the experimental results demonstrated in Table III, we can see that the performance of DAN is significant: it outperforms all state-of-the-art methods by a large margin in Part B, considering both MAE (13.2) and MSE (20.1). As to Part A, DAN only gets the second best performance with respect to MAE. However, note that CP-CNN is a multi-subnet network with higher capacity than DAN, therefore we also compare DAN with the fused network consisting of the global context model (VGG-16) and the density map regressor (MCNN) in CP-CNN (denoted as CP-CNN(G)), and find out that DAN can achieve better accuracy (MAE reduced by ). Besides, Switch-CNN  also employs the VGG-16 architecture to construct one of its subnets, and we can see that the proposed DAN outperforms Switch-CNN on both Part A and Part B. These results indicate that, with our method, a simply end-to-end trained deep network can achieve the state-of-the-art performance.
|Part A||Part B|
|Zhang et al. ||181.8||277.7||32.0||49.8|
|Huang et al. ||-||-||20.2||35.6|
|CP-CNN (G) ||89.9||127.9||-||-|
|Method||Scene 1||Scene 2||Scene 3||Scene 4||Scene 5||Avg|
|Chen et al.||red2.1||55.9||red9.6||11.3||red3.4||16.5|
|Zhang et al.||9.8||blue14.1||14.3||22.2||3.7||12.9|
|Huang et al. ||4.1||21.7||11.9||11.0||3.5||10.5|
|Switch-CNN with perspective map ||4.2||14.9||14.2||18.7||4.3||11.2|
WorldExpo: The second dataset for evaluation is WorldExpo , which contains test images from scenes. We compare DAN with state-of-the-art methods on this dataset, including Chen et al., Zhang et al., MCNN , ConvLSTM , Huang et al. , Switch-CNN  and CP-CNN .
As demonstrated in Table IV, the performance of all methods are very close (from to ). The major reasons behind this are dual: on the one hand, counts in this dataset range from to , with the smallest standard deviation value () among all the datasets, therefore it is hard to distinguish the performance of different methods based on the MAE metric; on the other hand, this dataset provides the ground-truth perspective maps for generating training templates, which obviously reduces the effects of outliers. Nevertheless, the proposed DAN still gets the lowest MAE in Scene 5, and the second lowest MAE considering the average MAE across all scenes. Furthermore, we do not use the ground-truth perspective maps in the training process of DAN, which indicates our method is robust to variant perspectives.
UCF: The UCF  is the most difficult dataset, because it has only images for training and has the highest standard deviation value (). Similar to other methods, we perform 5-fold cross validation on this dataset. We compare DAN with state-of-the-art methods on UCF, including Lempitsky et al. , Idrees et al. , Zhang et al. , MCNN , Huang et al. , CCNN , MSCNN , CNN Boosted , Hydra 2s , Cascaded-MTL , Switch-CNN  and CP-CNN .
The experimental results are shown in Table V. From this result we can see that DAN can achieve rather high accuracy, its MAE is , which is between that of Switch-CNN () and CP-CNN (). This again validates our opinion that a single deep network with proper learning settings can reduce the effects of outliers significantly.
|Lempitsky et al. ||493.4||487.1|
|Idrees et al. ||419.5||541.6|
|Zhang et al. ||467.0||498.5|
|Huang et al. ||409.5||563.7|
|CNN Boosted ||364.4||-|
|Hydra 2s ||333.73||425.26|
V-D Ablation Study
Remember that the major conclusion of this paper is that the inhomogeneous density distribution problem will result in the bias to outliers and other difficulties in the learning process. To validate this, we conduct an ablation study on the ShanghaiTech dataset with Part A. We compare the performance of DAN with different settings: training the VGG-16 part in DAN with -loss (denoted as VGG-16), DAN adopting Huber loss and only branch (denoted as DAN-1-H), and DAN with different numbers of branches (denoted as DAN-1 to DAN-4). All the variants of DAN are trained with the same settings.
The experimental results are shown in Table VI. Firstly, we have tried to train the VGG-16 with loss several times, but unfortunately, all end up with the exploded gradients. This confirms our opinion that outliers can cause difficulties in the learning process. With the Huber loss and other modifications in our network, we succeed in learning the deep networks (DAN-1-H), which also outperforms most current networks. Secondly, our weighting strategy in the loss function can increase the performance significantly, as the performance of DAN-1 is higher than that of DAN-1-H. Lastly, we test DAN with different number of branches, of which the density thresholds are set to be distributed as equally as possible, e.g., selected from . We find that with more branches, the performance of DAN is getting better. This can be understood as with more branches, the differences between training templates and ideal templates are smaller, and more local models are ensembled, and consequently the networks are more robust.
V-E Qualitative Analysis
We demonstrate representative examples of our experimental results on the ShanghaiTech for qualitative analysis in this section. Figure 6 and 7 shows images with their ground-truth structured density maps and predicted density maps on Part A and B, respectively. As in , we also calculate the Peak Signal to Noise Ratio (PSNR) as the metric to evaluate the quality of our predicted density maps.
From these results we can see that the proposed method perform well in most situations, not matter for extremely high-density scenes (Part A) or less crowd scenes (Part B). Besides, most PSNR values of our predicted density maps are above dB, which indicates DAN can generate high-quality density maps. More importantly, notice that our predicted structured density maps are divided into levels clearly, therefore validates our opinion that DAN can accomplish the regression task and the classification task at the same time.
Yet there are a few failure cases for the proposed method. To analyze this, we demonstrate the top-4 images ranked by the MAE in Figure 6 and 7. We notice that a common characteristic of these images is that in the extremely high-density areas of these images, due to the restricted resolutions, even human cannot precisely distinguish the targets. Especially for images in Part A, sizes of targets are even less than pixels, e.g., the last row in Figure 6. Therefore, for these images, the resolution of our network is not high enough to handle them. Fortunately, these images are rare in all datasets, hence we still can consider DAN as a practical solution for the crowd counting problem.
In this paper, we focus on the crowd counting problem in extremely high-density scene images and take in-depth analysis into the phenomena that existing deep learning-based methods do not work well while deeper networks are employed. Our comprehensive study reals that it can be heavily attributed to the inhomogeneous density distribution problem, and a feasible solution is provided by extending the density map from 2D to 3D, with a extra dimension implicitly indicating the density level. Based on this, we also present a single Density-Aware Network that is simple and easy to train. Extensive experiments demonstrate that it achieves the state-of-art performance on several challenging datasets.
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