State-of-the-art crowd counting algorithms [69, 71, 44, 49, 63, 54, 50, 38, 27, 48, 53, 34, 20, 47, 4] rely on Deep Networks to regress a crowd density, which is then integrated to estimate the number of people in the image. Their application can have important societal consequences, for example when they are used to assess how many people attended a demonstration or a political event.
In the “Fake News” era, it is therefore to be feared that hackers might launch adversarial attacks to bias the output of these models for political gain. While such attacks have been well studied for classification networks [15, 23, 41, 43, 42, 6], they remain largely unexplored territory for people counting and even for regression at large. The only related approach we know of  is very recent and specific to attacking optical flow networks, leaving the pixel-wise detection of attacks untouched.
In this paper, our goal is to blaze a trail in that direction. Our main insight is that if a two-stream network is trained to regress both the people density and the scene depth, it becomes very difficult to affect one without affecting the other. In other words, pixels that have been modified to alter the density estimate will also produce incorrect depths, which can be detected by estimating depth by unrelated means. When a ground-truth depth-map can be kept safe from the attacker, this is easily done. When this is not an option, we will show that using statistics from depth-maps acquired earlier suffices to detect tampering at a later date.
Fig. 1 depicts our approach. We will show that it is robust to both black-box attacks, in which the adversary does not know the existence of adversarial detector, and white-box attacks, in which the adversary can access not only the density regression network but also our adversarial detector. In other words, even when our approach is exposed, a hacker cannot mount an effective attack while avoiding detection. This is because in surveillance videos recorded by a fixed camera, untampered depth map pixels tend to change less over time than RGB image pixels. In a crowded scene, depth measurements are affected by the appearance and disappearance of people but the perturbations remain small whereas those of RBG pixels can be much larger due to illumination and appearance changes. Therefore, even if the attacker has access to the depth maps, it remains difficult to guarantee that they will be altered in a consistent and undetectable way. We will refer to this as the Density and Depth (DaD) model.
Our contribution therefore is an effective approach to foiling adversarial attacks on people counting algorithms that rely on deep learning. Its principle is very generic and could be naturally extended to other image regression tasks. Our experiments on several benchmark datasets demonstrate that it outperforms heuristic-based and uncertainty-based attack detection strategies.
2 Related Work
Early crowd counting methods [61, 60, 29] tended to rely on counting-by-detection, that is, explicitly detecting individual heads or bodies and then counting them. Unfortunately, in very crowded scenes, occlusions make detection difficult, and these approaches have been largely displaced by counting-by-density-estimation ones. They rely on training a regressor to estimate people density in various parts of the image and then integrating. This trend began in [8, 25, 12]
, using either Gaussian Process or Random Forests regressors. Even though approaches relying on low-level features[9, 7, 3, 45, 8, 19] can yield good results, they have now mostly been superseded by CNN-based methods [69, 63, 34, 54, 65, 44, 50, 71, 49, 4, 59, 35, 39, 51, 37, 36, 21, 72, 70, 58, 28, 31, 66, 40, 33, 64, 65, 52, 55, 10, 57, 67, 68], a survey of which can be found in . In this paper, we therefore focus on attacks against these.
Defense against Adversarial Attacks.
Deep networks trained to solve classification problems are vulnerable to adversarial attacks [15, 23, 41, 43, 42, 6]. Existing attack strategies can be roughly split in two categories, optimization-based and gradient-based. The former [43, 42, 6]
involve terms related to the class probabilities, which makes the latter[15, 23, 41] better candidates for attacks against deep regression models. The very recent work of  is the only one we know of that examines adversarial attacks for a regression task, that is, optical flow estimation. However, it does not propose defense mechanisms, which by contrast is the focus of this paper.
In the context of classification, one popular defense is adversarial training , which augments the training data with adversarial examples and has been shown to outperform many competing methods [16, 14, 18, 1, 26, 11]
. However, it needs access to adversarial examples, which are often not available ahead of time and expensive to generate during training. As a consequence, several alternative approaches have been proposed. This includes training auxiliary classifiers, ranging from simple linear regressors to complex neural networks, to predict whether a sample is adversarial or not. However, as shown in, such detection mechanisms can easily be defeated by an adversary targeting them directly.
In any event, none of these methods are designed to detect attacks at the pixel-level. Even the few researchers who have studied adversarial attacks for semantic segmentation [62, 30], which is a pixel-level prediction task, do not go beyond detection at the global image level.
A seemingly natural approach to performing pixel-level attack detection would be to rely on prediction uncertainty. In , the authors argue that Bayesian uncertainty  is a reliable way to detect the adversarial examples because the perturbed pixels generally have much higher uncertainty values. Uncertainty can be computed using dropout, as in , learned from data , or estimated using the negative log-likelihood of each prediction . In our experiments, we will extend this strategy to pixel-wise adversarial attack detection and show that our approach significantly outperforms it.
3 Density and Depth Model
As discussed in Section 2, most state-of-the-art crowd counting algorithms rely on a deep network regressor , that takes an image as input and returns , an estimated density map, which should be as close as possible to a ground-truth one in norm terms. Here, stands for the network’s weights, which have been optimized for this purpose.
An adversarial attack then involves generating a perturbation , where maps the input image and ground-truth densities to in such a way that is visually indistinguishable from while yielding a crowd density estimate that is as different as possible from the ground-truth one.
We will review the best known ways to generate such attacks in Section 4.1. Here, our concern is to define so as to defeat them by ensuring that they are easily detected. To this end, we leverage an auxiliary task, depth estimation, as follows.
|1 - 2||3364 conv-1||1||33512 conv-2|
|3 - 4||33128 conv-1||3||33512 conv-2|
|2 2 max pooling||4||33256 conv-2|
|5 - 7||33256 conv-1||5||33128 conv-2|
|2 2 max pooling||6||3364 conv-2|
|8 - 10||33512 conv-1||7||111 conv-1|
|11||context-aware features |
Instead of training a single regressor that predicts only people density, we train a two-stream network where one stream predicts people density and the other depth at each pixel. We write
where and are the estimated densities and depths while and are two regressors parameterized by the weigths . The network that implements and comprises a single encoder and two decoders, one that produces the densities and the other the depths. It is depicted by Fig. 2 and we provide details about its architecture in Table 1. Note that some of the weights in are specific to the first decoder and others to the second.
As the two decoders use the same set of features as input, it is difficult to tamper with the results of one without affecting that of the other, as we will see in Section 4.4. More specifically, if pixel is perturbed to change the local density estimate, the local depth estimate is likely to be affected as well. We therefore take the relative error in depth estimation with respect to the ground-truth depth map
to be an indicator of a potential disturbance. In practice, we label a pixel as potentially tampered with if this difference is larger than the top 5% of this difference in the training dataset, and we will evaluate the influence of this hyper-parameter in Section 4.6. In test sequences for which the ground-truth depth map can also be tampered with by the attackers, we can use the statistics of the training depth maps to also detect such tampering, as will be shown in Section 4.5.
Given a set of training images with corresponding ground-truth density maps and ground-truth depth maps , we learn the weights of the two regressors by minimizing the loss
where is the batch size and is a hyper-parameter that balances the contributions of the two losses. We found empirically that yields the best overall performance, as will be shown in section 4.4.
To obtain the ground-truth density maps , we rely on the same strategy as previous work [27, 49, 71, 48, 32]. In each training image , we annotate a set of 2D points that denote the position of each human head in the scene. The corresponding ground-truth density map is obtained by convolving an image containing ones at these locations and zeroes elsewhere with a Gaussian kernel of mean
In this section, we first introduce the existing adversarial attack methods that can be used against a deep regressor and describe the evaluation metric and the benchmark datasets we used to assess their performance. We then use them against our approach to demonstrate its robustness, and conclude with an ablation study that demonstrates that our approach is robust to the hyper-parameter setting and works well when used in conjunction with several recent crowd density regressors[71, 27, 37] .
4.1 Attacking a Deep Regressor
While there exist many adversarial attackers [15, 23, 41, 43, 42, 6], their effectiveness have been proven mostly against classifiers but far more rarely against regressors . As discussed in Section 2, the gradient-based methods [15, 23, 41] are the most suitable ones to attack regressors and we focus on the so-called Fast Gradient Sign Methods (FSGMs), which are the most successful and widely used ones. We will distinguish between black-box attacks in which the attacker does not know that we use depth for verification purposes and white-box attacks in which they do.
If the attacker is unaware that we use the depth map for verification purposes, they will only try to affect the density map. They might then use one of the following variants of FSGM.
It generates adversarial examples designed to increase the network loss as much as possible for the correct answer, thereby preventing the networks from predicting it. Given an input image , the ground-truth density , and the regressor of Eq. 1 parametrized by , the attack is performed by iterating
times. The adversarial example is then taken to be and we will refer to this as FSGM-U(n). It is a single-step or multiple-step attack without target and guarantees that the resulting perturbation is bounded by . For consistency with earlier work [15, 23], when , we reformulate this attack as
Unless otherwise specified we use , , and , as recommended in earlier work . These numbers are chosen to substantially increase the crowd counting error while keeping the perturbation almost imperceptible to the human eye. We will analyze the sensitivity of our approach to these values in Section 4.6. An example of this attack is shown in Fig. 3. By comparing Fig. 3(d) and (e), we can see that the attack made some people “disappear”.
Targeted FSGM (FSGM-T(n)) .
Instead of simply preventing the network from finding the right answer , we can target a specific wrong answer . This is achieved using the slightly modified iterative scheme
Again, we take the adversarial example to be and use the same values as before for and . We will refer to this as FSGM-T(n). In our experiments, we take the targets to be the true value plus one, which creates an obvious error while yielding tampered images that are undistinguishable from the original ones.
If the attacker knows that we are using the depth maps and has access to both and , the two regressors of Eq. 1, their natural reaction will be to try to modify the density maps while leaving the depth maps as unchanged as possible. To this end, we propose the following exposed variations of the untargeted and targeted FSGM attacks described above.
Untargeted Exposed FSGM (FSGM-UE(n)).
The iterative scheme becomes
for consistency with earlier work. The additional term in the loss function aims to preserve the predicting power ofwhile compromising that of as much as possible. We again use the same values as before for and , and is the same balancing factor as in Eq. 3.
Targeted Exposed FSGT (FSGM-TE(n)).
4.2 Evaluation Datasets
We use three different datasets to evaluate our approach. The first two are RGB-D datasets with ground-truth depth value obtained from sensors. Since depth sensors may not always be available, we also evaluate our model with a third dataset that contains RGB images with an accurate perspective map. This perspective map is a depth map computed from the scene geometry instead of using depth sensors. As such, it only represents the scene, not the people in it. This will let us show that our approach not only works for RGB-D datasets but also achieves remarkable performance in RGB images if scene geometry is available.
This is a large-scale RGB-D dataset with 2,193 images and 144,512 annotated heads. The valid depth ranges from 0 to 20 meters due to the limitation in depth sensors. The lighting condition ranges from very bright to very dark in different scenarios. We use the same setting as in , with 1,193 images as training set and the remaining ones as test set, and normalize the depth values from [0,20] to [0,1] for both training and evaluation.
It is acquired by a fixed indoor surveillance camera. This dataset is divided into three video sequences, named FLOW, QUEUE and GROUPS. The crowd motion varies from sequence to sequence. In the FLOW sequence, people walk from point to point. In the QUEUE sequence, people walk in a line. In the GROUPS sequence, people move inside a controlled area. There are 1,260 frames in the FLOW sequence with 3,542 heads. The QUEUE sequence contains 5,031 heads in 918 frames, and the GROUPS sequence encompasses 1,180 frames with 9,057 heads. We follow the same setting as in , taking 20% of the images of each scene as training set and using the remaining ones as test set.
The above two RGB-D datasets contain depth information acquired by sensors. Such information is hard to obtain in outdoor environments, particularly if the scene is far from the camera. Therefore, we also evaluate our approach on the Venice dataset. This dataset contains RGB images and an accurate perspective map of each scene. It was obtained using the grid-like ground pattern, as shown in Fig. 4, and thus does not depend on any depth sensor. The dataset contains 4 different sequences for a total of 167 annotated frames with fixed 1,280 720 resolution. Our experimental setting follows that of [37, 36], with 80 images from a single long sequence as training data, and the images from the remaining 3 sequences for testing purposes.
4.3 Metrics and Baselines
In all our experiments, we partition the images into four parts and tamper with one while leaving the other three untouched. We then measure two things:
How well can we detect the pixels that have been tampered with? We measure this in terms of the mean Intersection over Union
where is the number of images, is 1 for the pixels in image predicted to have been tampered with according to Eq. 2 and is the ground-truth perturbation mask.
How well do modifications of the depth map correlate with modifications of the predicted density? As in many previous works [71, 69, 44, 49, 63, 54, 37, 36] we quantify these modifications in terms of the mean absolute error for densities and depths along with the root mean squared error for density. They are defined as
where is the number of test images, and denote the true number of people in the th image and depth value at pixel of the th image, and and are the estimated values. is the number of tampered pixels in the th image. In practice is obtained by integrating the predicted people densities.
In the absence of prior art on defenses against attacks of density estimation algorithms, we use the following baselines for comparison purposes.
RANDHALF and RANDQUARTER. We randomly label either half or a quarter of the pixels as being under attack, given that we know a priori that exactly a quarter are. We introduced RANDHALF to show that using a random rate other than the true one does not help.
. Since adversarial attacks is caused by modifying the input image, it can be seen as heteroscedastic aleatoric uncertainty, which assumes that observation noise vary with input. We threshold the uncertainty values to classify each pixel as perturbed or not and report the results obtained with the best threshold.
ENSEMBLES. We use the approach of  that relies on a scalable method for estimating predictive uncertainty estimates from deep nets using a proper scoring rule as the training criteria. The optional adversarial training process is not used as we do not know the potential attackers in advance. As before, we threshold the uncertainty values to obtain a pixel-wise classification map and report the best results.
BAYESIAN. We further compare our model with Bayesian uncertainty , which uses dropout as Bayesian approximation of model uncertainty. Again, we threshold the uncertainty value and report the results for the best threshold.
The baseline models are trained with the same crowd density regression networks as our approach.
4.4 Comparative Performance
|Original image||Image under FSGM-U(1) attack||Region of interest||Ground truth||OURS|
Using the CAN  architecture.
CAN is an encoder-decoder crowd density estimation architecture, that delivers excellent performance. We use it to implement and duplicate its decoder to implement . Recall from Section 3 that we use the hyper-parameter of Eq. 3 to balance the people density estimation loss and the depth estimation loss while training and . In Table 3, we report the performance of the two regressors as a function of the value of . yields the best performance overall and we use regressors trained using this value in all our other experiments. Interestingly, training and jointly yields a better density regressor than training alone, which is what we do when we set to zero.
We report the counting and depth errors with/without attack in Table 2 for the 3 datasets. All the attacks cause large increase in crowd counting errors, which always comes with a substantial increase in depth estimation error. The exposed methods reduce slightly this increase but at the cost of also making the attack less effective.
|Attack||RANDHALF||RANDQUARTER||HETERO ||ENSEMBLES ||BAYESIAN ||OURS|
|Attack||RANDHALF||RANDQUARTER||HETERO ||ENSEMBLES ||BAYESIAN ||OURS|
|Attack||RANDHALF||RANDQUARTER||HETERO ||ENSEMBLES ||BAYESIAN ||OURS|
In Tables 4, 5, and 6, we report the pixel-wise adversarial detection accuracy for the ShanghaiTechRGBD, MICC and Venice datasets. Our approach outperforms all the baseline models by a large margin for all the attacks. In Fig. 5, we show a qualitative result.
|ShanghaiTechRGBD dataset||MICC dataset||Venice dataset|
4.5 Tampering with the Ground-Truth Depth Map
The results of Section 4.4 were obtained under the assumption that we have access to a ground-truth depth map that is safe from attack. In some scenarios, this might not be the case and the attacker might be able to tamper with the depth map. Fortunately, even if this were the case, the attack would still be detectable as follows. Given training depth maps recorded by a fixed depth sensor, we can record the min and max depth values for each pixel as
Given ground-truth depth maps that are exposed and can be tampered with, the tampered depth value at pixel of the ground-truth depth map can be written as
where is the mean depth value in the ground-truth depth maps in the test dataset, and is a scalar that represents the perturbation strength of a potential attack. We take the perturbation to be a function of because the ground-truth depth map is not an input to our network. If the attacker were able to choose an appropriate for each pixel of the ground-truth depth map, the tampering indicator of Eq. 2 could be compromised. Fortunately, such an attack is very likely to be detected using the following simple but effective approach. If or , we label the ground-truth pixel as potentially tampered with. As shown in Fig. 8, the pixel-wise detection accuracy is over 90% even for extremely small perturbations with and quickly increases from there. This makes the attacker’s task difficult indeed.
4.6 Sensitivity Analysis
We now quantify the influence of the two main hyper-parameters introduced in Section 4.1 that control the intensity of the attacks.
We change the value of in Eq. 4 and Eq. 5 from 1.0 to 35.0 for all attacks and plot the resulting mIoU in Fig. 7. Our model can detect very weak attacks with down to 1.0 and its performance quickly increases for larger values. In the supplementary material, we will exhibit the monotonous relationship between and the people density estimation error. When , there is already a small perturbation of the density estimates—around 6 in for ShanghaiTechRGBD—that then become much larger as increases. The number of iterations is set to as recommended in earlier work .
Strength of White Attacks.
To check the robustness of our model against white-box attacks, we evaluate different values in the loss term of Eq. 7, whose role is to keep the depth estimate as steady as possible in ShanghaiTechRGBD. We tested our approach for values of ranging from 0.01 to 100.0 and report the detection accuracy results along with the crowd counting error and depth error in Table 7. For larger values , both the crowd density error and the detection rate drop. In other words, increasing makes the attack harder to detect but also weaker. We show the same trend in the other datasets in the supplementary material.
5 Conclusion and Future Perspectives
In this paper, we have shown that estimating density and depth jointly in a two-stream network could be leveraged to detect adversarial attacks against crowd-counting models at the pixel level. Our experiments have demonstrated this to be the case even when the attacker knows our detection strategy, or has access to the ground-truth depth map. In essence, our approach is an instance of a broader idea: One can leverage multi-task learning to detect adversarial attacks. In the future, we will therefore study the use of this approach for other tasks, such as depth estimation, optical flow estimation, and semantic segmentation.
Acknowledgments This work was supported in part by the Swiss National Science Foundation. We also would like to thank Krzysztof Lis and Krishna Nakka for helpful discussions.
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