Robust Multi-Modal Sensor Fusion: An Adversarial Approach

TL;DR

This paper presents a robust multi-modal sensor fusion method using a generative network to learn a latent space, improving target detection and classification even with damaged or noisy sensors.

Contribution

It introduces a data-driven fusion approach that enhances robustness by exploiting an estimated latent space conditioned on sensor modalities, allowing flexibility and damage detection.

Findings

Achieves robustness against noisy and damaged sensors.

Improves detection and classification performance.

Demonstrates effectiveness in unconstrained surveillance settings.

Abstract

In recent years, multi-modal fusion has attracted a lot of research interest, both in academia, and in industry. Multimodal fusion entails the combination of information from a set of different types of sensors. Exploiting complementary information from different sensors, we show that target detection and classification problems can greatly benefit from this fusion approach and result in a performance increase. To achieve this gain, the information fusion from various sensors is shown to require some principled strategy to ensure that additional information is constructively used, and has a positive impact on performance. We subsequently demonstrate the viability of the proposed fusion approach by weakening the strong dependence on the functionality of all sensors, hence introducing additional flexibility in our solution and lifting the severe limitation in unconstrained surveillance settings with potential environmental impact. Our proposed data driven approach to multimodal fusion, exploits selected optimal features from an estimated latent space of data across all modalities. This hidden space is learned via a generative network conditioned on individual sensor modalities. The hidden space, as an intrinsic structure, is then exploited in detecting damaged sensors, and in subsequently safeguarding the performance of the fused sensor system. Experimental results show that such an approach can achieve automatic system robustness against noisy/damaged sensors.

Tables

Table 1. TABLE I: Performance Comparison for both Datasets

Method	Accuracy (Dataset-1)	Accuracy (Dataset-2)
Seismic Sensor	93.62 %	-
Acoustic Sensor	68.71 %	-
Imaging Sensor	90.33 %	-
Radar Sensor 1	-	89.33 %
Radar Sensor 2	-	86.73 %
Telescopic Imaging Sensor	-	83.57 %
Feature Concatenation	88.13 %	86.45 %
Dissimilar Sensor Fusion	91.61 %	-
Similar + Dissimilar Sensor Fusion	-	89.63 %
Dempster-Shafer Fusion	88.77 %	87.30 %
Event Driven Fusion (Without using GAN structure) [9, 10]	92.04 %	90.36 %
Canonical Correlation Analysis [28]	86.64 %	85.69 %
Discriminant Analysis	91.33 %	88.21 %
Dictionary Learning [29]	94.46 %	93.77 %
Hidden Space Generated by GAN	95.79 %	91.13 %
Event Driven Fusion + GAN	97.79 %	95.34 %

Equations

$$\begin{array}{r}{P}_{\Omega }({\gamma }_1^1,...,{\gamma }_k^l,...,{\gamma }_{K_L}^L)=\rho .P_{{\Omega }_{\text{MAXMI}}}({\gamma }_1^1,...,{\gamma }_k^l,...,{\gamma }_{K_L}^L)+\\ (1-\rho ).P_{{\Omega }_{\text{MINMI}}}({\gamma }_1^1,...,{\gamma }_k^l,...,{\gamma }_{K_L}^L),\end{array}$$

$$\begin{array}{rl}& \forall l\in \{1,...,L\},\\ & {\mathbf{F}}_{d\times N}^{\mathbf{l}}={{\mathbf{S}}^{\mathbf{l}}}_{d\times d_H}{\mathbf{H}}_{d_H\times N}.\end{array}$$

$$\forall l\in \{1,...,L\},{{\hat{\mathbf{H}}}^{\mathbf{l}}}_{d_H\times N}={\mathbf{G}}^{\mathbf{l}}({{\mathbf{X}}^{\mathbf{l}}}_{d_l\times N})\approx {\mathbf{H}}_{d_H\times N}$$

$$\begin{array}{rl}& \forall l\in \{1,...,L\},\\ & {{\mathbf{F}}^{\mathbf{l}}}_{d\times N}={{\mathbf{S}}^{\mathbf{l}}}_{d\times d_H}[({\mathbf{G}}^{\mathbf{l}}({\mathbf{X}}^{\mathbf{l}}){)}_{d_H\times N}].\end{array}$$

$$P^l(o_i^l)=\frac{exp(w_i^{l^T}{\mathbf{F}}^{\mathbf{l}}+b_i^l)}{{\sum }_{m=1}^{I}exp(w_m^{l^T}{\mathbf{F}}^{\mathbf{l}}+b_m^l)}.$$

$$\begin{array}{rl}& \underset{{\mathbf{G}}^{\mathbf{l}}}{\min }\underset{\mathbf{D}}{\max }\sum _{l=1}^{L}V({\mathbf{G}}^{\mathbf{l}},\mathbf{D}),\\ & V({\mathbf{G}}^{\mathbf{l}},\mathbf{D})=\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^L\{{\mathrm{I}\mathrm{E}}_{{\mathbf{G}}^{\mathbf{m}}(x^m)\text{texttildelow}{\mathrm{I}\mathrm{P}}_{\mathbf{H}}}[D^m({\mathbf{G}}^{\mathbf{m}}(x^m))]\\ & -{\mathrm{I}\mathrm{E}}_{x^l\text{texttildelow}{\mathrm{I}\mathrm{P}}_{{\mathbf{X}}^{\mathbf{l}}}}[D^m({\mathbf{G}}^{\mathbf{l}}(x^l))]\}.\end{array}$$

$$[\mathbf{A},\mathbf{B}]=\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}=0.$$

$$\begin{array}{r}\underset{{\mathbf{G}}^{\mathbf{l}},{\mathbf{Z}}^{\mathbf{l}}}{\min }\underset{\mathbf{D}}{\max }\sum _{l=1}^L\{V({\mathbf{G}}^{\mathbf{l}},\mathbf{D})+{\gamma }_2\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^L||\left[{\mathbf{Z}}^{\mathbf{l}},{\mathbf{Z}}^{\mathbf{m}}\right]|{|}^2\}.\end{array}$$

$$||{\mathbf{S}}^{\mathbf{l}}|{|}_{\infty ,1}=\sum _j\underset{i}{\max }(|s_{ij}^l|).$$

$$\begin{array}{r}\underset{{\mathbf{G}}^{\mathbf{l}},{\mathbf{S}}^{\mathbf{l}},{\mathbf{Z}}^{\mathbf{l}}}{\min }\underset{\mathbf{D}}{\max }\sum _{l=1}^L\{V({\mathbf{G}}^{\mathbf{l}},\mathbf{D})+{\gamma }_1||{\mathbf{S}}^{\mathbf{l}}|{|}_{\infty ,1}\\ +{\gamma }_2\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^L||\left[{\mathbf{Z}}^{\mathbf{l}},{\mathbf{Z}}^{\mathbf{m}}\right]|{|}^2\}\end{array}$$

$$C_{LOSS}^l({\mathbf{F}}^{\mathbf{l}})=\sum _{n=1}^N\sum _{i=1}^I-y_{n_i}\log \sigma (c_i^l({\mathbf{f}}_{\mathbf{n}}^{\mathbf{l}})),$$

$$\begin{array}{rl}& \underset{{\mathbf{G}}^{\mathbf{l}},{\mathbf{Z}}^{\mathbf{l}},{\mathbf{S}}^{\mathbf{l}},{\mathbf{C}}^{\mathbf{l}}}{\min }\underset{\mathbf{D}}{\max }\mathcal{L}({\mathbf{G}}^{\mathbf{l}},\mathbf{D},{\mathbf{Z}}^{\mathbf{l}},{\mathbf{S}}^{\mathbf{l}},{\mathbf{C}}^{\mathbf{l}})\\ & \mathcal{L}({\mathbf{G}}^{\mathbf{l}},\mathbf{D},{\mathbf{Z}}^{\mathbf{l}},{\mathbf{S}}^{\mathbf{l}},{\mathbf{C}}^{\mathbf{l}})=\sum _{l=1}^L\{V({\mathbf{G}}^{\mathbf{l}},\mathbf{D})+{\gamma }_1||{\mathbf{S}}^{\mathbf{l}}|{|}_{\infty ,1}\\ & +{\gamma }_2\sum _{\begin{array}{c}m=1\\ m\ne l\end{array}}^L||\left[{\mathbf{Z}}^{\mathbf{l}},{\mathbf{Z}}^{\mathbf{m}}\right]|{|}^2+{\gamma }_3{C}_{LOSS}^l({\mathbf{F}}^{\mathbf{l}}),\end{array}$$

$$\begin{array}{r}{P}_t^f(o_i)=P_t(o_i^1,o_i^2,...,o_i^L)=\rho .P_{\text{MAX MI}}(o_i^1,o_i^2,...,o_i^L)\\ +(1-\rho ).P_{\text{MIN MI}}(o_i^1,o_i^2,...,o_i^L),\end{array}$$

$$R_t^{l^{\prime }}=\{P_t^{l^{\prime }}(o_1^l),P_t^{l^{\prime }}(o_2^l),...,P_t^{l^{\prime }}(o_I^l),P_t^{l^{\prime }}(un{c}^l)\},$$

$$\begin{array}{r}{P}_t(a^1,a^2,...,a^L)=\rho .P_{\text{MAX MI}}(a^1,a^2,...,a^L)\\ +(1-\rho ).P_{\text{MIN MI}}(a^1,a^2,...,a^L),\end{array}$$

$$\begin{array}{r}{P}_t^f(o_i)=P_t((\underset{l=1}{\overset{L}{\bigwedge }}{o}_i^l)\underset{m_1=1}{\overset{L}{\bigvee }}(un{c}^{m_1}\underset{\begin{array}{c}l=1\\ l\ne m_1\end{array}}{\overset{L}{\bigwedge }}{o}_i^l)\\ \underset{m_1,m_2=1}{\overset{L}{\bigvee }}(un{c}^{m_1}\wedge un{c}^{m_2}\underset{\begin{array}{c}l=1\\ l\ne m_1\\ l\ne m_2\end{array}}{\overset{L}{\bigwedge }}{o}_i^l)\\ ...\underset{\begin{array}{c}{m}_1,m_2\\ ...m_{L-1}=1\end{array}}{\overset{L}{\bigvee }}(\underset{j=1}{\overset{L-1}{\bigwedge }}un{c}^{m_j}\underset{\begin{array}{c}l=1\\ \forall j,l\ne m_j\end{array}}{\overset{L}{\bigwedge }}{o}_i^l)),\end{array}$$

$$P_t^f(o_i)=P_t((o_i^1\wedge o_i^2)\vee (un{c}^1\wedge o_i^2)\vee (o_i^1\wedge un{c}^2)),$$

$$Do{C}_t^l=(1-p_D({\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{l}})).Ac{c}_{train}^l,$$

$$\begin{array}{rl}& \forall j,l\ne m,\\ & \sum _{\begin{array}{c}l=1\\ l\ne m\end{array}}^L\mathcal{I}(||{\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{m}}-{\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{l}}|{|}^2>T)\ge \{\begin{array}{l}\frac{L-1}{2},\text{ if }L\text{ is odd},\\ \frac{L}{2}-1,\text{ if }L\text{ is even},\end{array}\\ & \text{And, }\sum _{\begin{array}{c}j,l=1\\ j,l\ne m\end{array}}^L\mathcal{I}(||{\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{j}}-{\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{l}}|{|}^2<T)\ge \{\begin{array}{l}\frac{L-1}{2},\text{ if }L\text{ is odd},\\ \frac{L}{2}-1,\text{ if }L\text{ is even},\end{array}\end{array}$$

$$p_D({\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{l}})=\frac{d_{lev}}{{\max }_v{d}_v},$$

$$lev=arg\min v$$

$$s.t.\exists j\in \{1,...,J_{lev}\},{\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{l}}\in Z_{lev}^j,$$

$$p_D({\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{l}})>T,$$

$${\hat{\mathbf{f}}}_{\mathbf{t}}^{\mathbf{l}}={\mathbf{S}}^{\mathbf{l}}\left(\frac{{\sum }_{m\in \Gamma }Do{C}_t^m.{\hat{\mathbf{h}}}_{\mathbf{t}}^{\mathbf{m}}}{{\sum }_{m\in \Gamma }Do{C}_t^m}\right),$$

$$a_1^v:0\le v\le 10m/s,a_2^v:15m/s\le v\le 35m/s,$$

$$a_1^r:0<r\le 300m,a_2^r:300m<r,$$

$$a_1^{cs}:0<cs\le 20m^2,a_2^{cs}:15m^2\le cs\le 50m^2.$$

$$a_1^d:0\le d\le 60pixels,a_2^d:90pixels\le d\le 210pixels,$$

$$a_1^{ar}:0<AR\le 1.5,a_2^{ar}:1.5<AR.$$

$$o_1:\{a_1^r\wedge [(a_2^v\wedge a_2^d)\vee (a_2^{cs}\vee a_2^{ar})]\}$$

$$o_2:\{a_1^v\wedge a_1^d\wedge a_2^r\wedge a_1^{cs}\wedge a_1^{ar}\}$$

$$\begin{array}{r}{\theta }_d^{r^{(j)}}\leftarrow {\theta }_d^{r^{(j-1)}}+{\mu }_D\left\{\frac{d\mathcal{L}({\mathbf{G}}^{\mathbf{l}},\mathbf{D},{\mathbf{Z}}^{\mathbf{l}},{\mathbf{S}}^{\mathbf{l}},{\mathbf{C}}^{\mathbf{l}})}{d{\mathbf{z}}^{\mathbf{R}}}\frac{d{\mathbf{z}}^{\mathbf{R}}}{d{\mathbf{z}}^{\mathbf{R}-1}}...\frac{d{\mathbf{z}}^{\mathbf{r}+1}}{d{\theta }_d^r}\right\}\end{array}$$

$${\mathbf{C}}^{{\mathbf{l}}^{(\mathbf{j})}}\leftarrow {\mathbf{C}}^{{\mathbf{l}}^{(\mathbf{j}-1)}}-{\mu }_G\left\{\frac{d\mathcal{L}({\mathbf{G}}^{\mathbf{l}},\mathbf{D},{\mathbf{Z}}^{\mathbf{l}},{\mathbf{S}}^{\mathbf{l}},{\mathbf{C}}^{\mathbf{l}})}{d{\mathbf{C}}^{\mathbf{l}}}\right\}$$

Full text

lot of research interest, both in academia, and in industry. Multimodal fusion entails the combination of information from a set of different types of sensors. Exploiting complementary information from different sensors, we show that target detection and classification problems can greatly benefit from this fusion approach and result in a performance increase. To achieve this gain, the information fusion from various sensors is shown to require some principled strategy to ensure that additional information is constructively used, and has a positive impact on performance. We subsequently demonstrate the viability of the proposed fusion approach by weakening the strong dependence on the functionality of all sensors, hence introducing additional flexibility in our solution and lifting the severe limitation in unconstrained surveillance settings with potential environmental impact. Our proposed data driven approach to multimodal fusion, exploits selected optimal features from an estimated latent space of data across all modalities. This hidden space is learned via a generative network conditioned on individual sensor modalities. The hidden space, as an intrinsic structure, is then exploited in detecting damaged sensors, and in subsequently safeguarding the performance of the fused sensor system. Experimental results show that such an approach can achieve automatic system robustness against noisy/damaged sensors.

I Introduction

Sensor fusion is known to broadly classify fusion techniques into three classes. Data level fusion is used when combining diverse raw data, to subsequently proceed with inference. Feature level fusion first extracts information from raw data, and then merges these features to make coherent decisions. Fusion at the decision level allows each sensor to reach its own individual decision (on the target identity), prior to an optimal combination of these decisions. Decision level fusion has received a lot of attention when exploiting heterogeneous modalities, as it allows each modality to have an independent feature representation, thus providing additional information. A review of classical fusion approaches can be found in [1, 2], and a more recent alternative data-driven perspective is provided by [3].

The viability of many fusion approaches strongly hinges on the functionality of all the sensors, making it a fairly restrictive solution. The severity of this limitation is even more pronounced in unconstrained surveillance settings, where the environmental conditions have a direct impact on the sensors, and close manual monitoring is difficult or even impractical. Partial sensor failure can hence cause a major drop in performance in a fusion system if timely failure detection is not performed. Furthermore, even if these sensors are successfully detected, the common adopted solution is to ignore the damaged sensor, with a potential negative impact on the overall resulting performance (i.e. relative to when all sensors are functional). We consider exploiting the prior information about the relationship between these sensor modalities, typically available from past observations during training of the fusion system, so that the latter can safeguard a high performance. There has recently been an interest in such a transfer of knowledge. In [11], the authors introduce hallucination networks to address a missing modality at test time by distilling knowledge into the network during training. Here, a teacher-student network is implemented using an $L^{2}$ loss term (hallucination loss) to train the student network. This idea has been further extended in [12, 13, 14]. In [14] where a Conditional Generative Adversarial Network (CGAN) was used to generate representative features for the missing modality. Furthermore, [15] shows that considering interactions between modalities can often lead to a better feature representation. Recent work in Domain Adaptation [16, 17, 18] addresses differences between source and target domains. In these works, the authors attempt to learn intermediate domains (represented as points on the Grassman manifold in [16, 17] and by dictionaries in [18]) between the source domain and the target domain.

Our Contributions: In this paper, we propose a robust sensor fusion algorithm, that can detect damaged sensors on the fly, and take the required steps to safeguard detection performance. We use a special case of the Event Driven Fusion technique recently proposed in [9, 10], and modify it in order to include reliability of individual sensors. This reliability measure is adaptive, and accounts for the sensor condition during implementation. We also propose a data driven approach for learning the features of interest which were handcrafted in [9]. In addition, we learn a hidden latent space between the sensor modalities, and the optimal features for classification are driven by the existence of this hidden space. Furthermore, it also provides robustness against damaged sensors. This hidden space is learned via a generative network conditioned on individual sensor modalities. In contrast to [14], we do not require any target feature space in order to learn the optimal hidden space estimate. The hidden space is structured so that it can accommodate both shared and private features of sensor modalities. We forego the use of a-priori knowledge about sensor damage adopted in [10], and can hence detect damaged sensors based on deviations in the generated hidden space.

II Related Work

II-A Model Based Sensor Fusion

A model-based sensor fusion approach was proposed in [6, 7, 8].In this work, two different sensor fusion models were explored, namely, Similar Sensor Fusion (SSF), and Dissimilar Sensor Fusion (DSF). The SSF model assumes that the independent sensors in the network are similar to each other (eg. 5 radars looking at the same object). Additional sensors in this case do not provide any new information, but can be used to confirm information from other sensors. This model attempts to find a fusion result which is most consistent with all the individual sensor reports. On the other hand, the DSF model assumes that all sensors observe dissimilar characteristics of the target, and hence each sensor provides novel information about the target. An additional sensor in this case generates increased clarity on the target identity. These two models should be viewed as “extreme cases” of decision level fusion. There are many practical cases in which the sensors are neither completely similar nor completely dissimilar. Event Driven Fusion [9, 10] addresses this limitation and is discussed below.

II-B Event Driven Fusion

Event Driven Fusion [9, 10] looks at fusion under a different light, by combining occurrences of events to reach a probability measure for a target identity. Each sensor is said to make a decision on the occurrence of certain events that it observes, rather than making a decision on the target identity. This technique also explores the extent of dependence between features being observed by the sensors, and hence generates more informed probability distributions over the events.

Consider the set of objects/targets, $O=\{o_{1},o_{2},...,o_{I}\}$. Let the $k^{th}$ feature from the $l^{th}$ sensor be $F_{k}^{l}$. Then, a mutually exclusive set of events, $\Omega_{k}^{l}=\{a_{kj}^{l}\}_{j=1,...,J_{kl}}$, is defined over the feature $F_{k}^{l}$. Here, $a_{kj}^{l}$ is the $j^{th}$ event on $F_{k}^{l}$ and is described as $a_{kj}^{l}:F_{k}^{l}\in[u_{j},v_{j})$, $u_{j}\in{\rm I\!R^{+}}$, $v_{j}\in{\rm I\!R^{+}}$, and $v_{j}>u_{j}$. A probability report is generated by the $l^{th}$ sensor for each of its features, $R_{k}^{l}=\{\Omega_{k}^{l},\sigma_{B}(\Omega_{k}^{l}),P_{k}^{l}\}$, where, $\sigma_{B}(\Omega_{k}^{l})$ is the Borel sigma algebra of $\Omega_{k}^{l}$, and $P_{k}^{l}$ is the set of probabilities over all the events in $\sigma_{B}(\Omega_{k}^{l})$. An object/target is then defined as $o_{i}\in\sigma_{B}(\Omega)$, where, $\Omega=\Omega_{1}^{1}\times...\times\Omega_{k}^{l}\times...\times\Omega_{K_{L}}^{L}$. The joint probability in the product space is determined as a convex combination of a distribution with minimal mutual information and one with maximal mutual information. For events $\gamma_{k}^{l}\in\sigma_{B}(\Omega_{k}^{l})$

[TABLE]

where $\rho\in[0,1]$ is a pseudo-measure of correlation between the features. $\rho\approx 1$ when the features are highly correlated, and $\rho=0$ when they are independent. For any object defined as a combination of events in the product space $\Omega$, $o_{i}\in\sigma_{B}(\Omega)$, rules of probability can then be used to determine the object probability. For instance, in a 2-D scenario, an object may be defined as a combination of events $\gamma_{1}\in\sigma_{B}(\Omega_{1})$, and $\gamma_{2}\in\sigma_{B}(\Omega_{2})$. The combination defined in the product space, $\Omega=\Omega_{1}\times\Omega_{2}$, may be of the form $o:\{\gamma_{1}\wedge\gamma_{2}\}$ or $o:\{\gamma_{1}\vee\gamma_{2}\}$. Given the joint probability, $P_{\Omega}$, rules of probability can be used to determine the fused object probability as follows:

• •

$o:\{\gamma_{1}\wedge\gamma_{2}\}:P^{f}(o)=P_{\Omega}(\gamma_{1},\gamma_{2})$

• •

$o:\{\gamma_{1}\vee\gamma_{2}\}:P^{f}(o)=P_{1}(\gamma_{1})+P_{2}(\gamma_{2})-P_{\Omega}(\gamma_{1},\gamma_{2})$

Where, $P_{1}(\gamma_{1})$ and $P_{2}(\gamma_{2})$ are the marginal probabilities for detection of the events $\gamma_{1}$ and $\gamma_{2}$ as seen by sensors $1$ and $2$.

II-C Generative Adversarial Networks

The Adversarial Network was first introduced by Goodfellow et al. [19] in 2014. In this framework, a generative model is pitted against an adversary: the discriminator. The generator aims to deceive the discriminator by synthesizing realistic samples from some underlying distribution. The discriminator on the other hand attempts to discriminate between a real data sample and that from the generator. Both these models are approximated by neural networks. When trained alternatively, the generator learns to produce random samples from the data distribution which are very close to the real data samples. Following this, Conditional Generative Adversarial Networks (CGANs) were proposed in [20]. These networks were trained to generate realistic samples from a class conditional distribution, by replacing the random noise input to the generator by some useful information (see Figure 1). Hence, the generator now aims to generate realistic data samples, given the conditional information. CGANs have been used to generate random faces, given facial attributes [21] and to also produce relevant images given text descriptions [22].

III Problem Formulation

Consider $L$ sensors surveilling an area of interest. We wish to detect/recognize targets, $O=\{o_{1},o_{2},...,o_{I}\}$ given the data collected by the sensors. Let the $n^{th}$ observation from the $l^{th}$ sensor be denoted by, $\bm{x^{l}_{n}}=\{\bm{x^{l}}_{\bm{n}_{q}}\}_{q=1...d_{l}}$.Let $\bm{X^{l}}_{d_{l}\times N}=\{\bm{x_{n}^{l}}\}_{n=1...N}$ be the set of $N$ observations, each of dimension $d_{l}$, made by the $l^{th}$ sensor.

We first seek to discover a hidden common space111In [3], such a space was referred to as an ‘information subspace’ $\bm{H}_{d_{H}\times N}$, of features captured by and hence present in the $L$ sensors, where $d_{H}$ is the dimension of the common subspace. This structure, unknown a priori, may include both shared and non-shared features across the sensors. We refer to the non-shared features of specific sensors, as private. Given a hidden space, this thus amounts to being able to select from each sensor the optimal set of features for classification via a selection matrix, $\bm{S}^{\bm{l}}$,

[TABLE]

Since the hidden space represents the information shared by the sensors, there must also exist a mapping, $\bm{G^{l}}:\bm{X^{l}}\to\bm{H}$ such that,

[TABLE]

From Equations III and 2 we have,

[TABLE]

The existence of such a hidden space makes it possible to detect damaged sensors, and safeguard the system performance against them. When the $l^{th}$ sensor is damaged, representative features for that sensor are reconstructed from the hidden space via the selection operator, $\bm{F^{l}=S^{l}H}$. Following the feature extraction, a linear classifier, $c_{i}^{l}(\bm{F^{l}})=w_{i}^{l^{T}}\bm{F^{l}}+b_{i}^{l}$, is used to determine the classification score for the $i^{th}$ object as seen by the $l^{th}$ sensor. The probability of occurrence of this object is then determined as,

[TABLE]

Finally, given these probability reports, $R^{l}=\{P^{l}(o_{i}^{l})\}$, the objective is to determine the fused probability report $R^{f}=\{P^{f}(o_{i})\}$, which is achieved using a special case of Event Driven Fusion [9, 10].

IV Proposed Approach

As discussed in the previous section, the hidden space, $\bm{H}$, can be estimated from the $l^{th}$ sensor observations as, $\bm{\hat{H}^{l}}=\bm{G^{l}}(\bm{X^{l}})$. The mapping $\bm{G^{l}}$ is approximated by a neural network and is realized as the generator of a Conditional Generative Adversarial Network (CGAN), that generates the estimate of the hidden space, $\bm{\hat{H}^{l}}$, while conditioned on the observations of the $l^{th}$ sensor. Hence, we will have $L$ generators that generate $L$ estimates of the hidden space.

The desired output of these generators is to generate the estimate, $\bm{\hat{H}^{l}}$, such that, $\forall l\in\{1,...,L\},$ $\bm{\hat{H}^{l}}=\bm{G^{l}(X^{l})}\approx\bm{H}$. On the other hand, the discriminator attempts to correctly identify the source modality of the estimated hidden space. This results in a score assignment $D^{l}(\hat{h})$ for an estimate $\hat{h}$ generated by $G^{l}$. When updating the parameters for the $l^{th}$ generator, the hidden space estimates generated by all the other generators are assumed to be the target space, and the $l^{th}$ generator attempts to replicate these spaces, while at the same time attempting to generate an estimate that confuses the discriminator.

The standard formulation for the Generative Adversarial Network [19, 20] is known to have some instability issues [32]. Specifically, if the supports of the estimated hidden spaces are disjoint, which is highly likely when the inputs to the generators are coming from different modalities, a perfect discriminator is easily learned, and gradients for updating the generator may vanish. This issue was addressed in [32], and solved by using a Wasserstein GAN. So, using the sensor observations as the conditional information in the WGAN formulation [32], we have,

[TABLE]

The discriminator $\bm{D}$, in the above formulation is required to be compact and $K$-Lipschitz. This is done by clamping the weights of the discriminator to a fixed box (eg. $\theta_{d}\in[-0.01,0.01]$) [32]. The discriminator is updated once after every generator update. That is, for $L$ sensors, the discriminator is updated $L$ times for one update to the $l^{th}$ generator (see Algorithm 1). While this causes the estimates to be close to each other, it does not guarantee that the generated hidden spaces share the same basis.

A similar idea toward finding a common representation between multiple modalities was very recently and independently proposed in [31]. In this paper the authors attempt to find a common latent space between text and image modalities in order to perform image/text retrieval. Given a query from the image modality, the image is transformed into the latent common space, and the text with closest representation is retrieved.

In our work, we additionally ensure that the hidden space estimates share a common basis, enforcing as best can be done, a common subspace. These hidden estimates are later exploited in detecting damaged sensors on the basis of their similar operation, thus making the common subspace constraint important. To that end, we exploit an important property of commutation between the operators that are responsible for transforming the data into the common subspace. Linear operators $\bm{A}\in{\rm I\!R^{nxn}}$ and $\bm{B}\in{\rm I\!R^{nxn}}$ are said to commute if,

[TABLE]

Theorem 1**.**

[24, 25]** If $\bm{A}\in{\rm I\!R^{nxn}}$ and $\bm{B}\in{\rm I\!R^{nxn}}$ are commuting linear operators, they share common eigenvectors.

Theorem 2**.**

[24, 25]** If $\bm{A}\in{\rm I\!R^{nxn}}$ and $\bm{B}\in{\rm I\!R^{nxn}}$ are commuting operators that are also individually diagonalizable, they share a common eigenbasis.

If the operators $\bm{Z^{1}},...,\bm{Z^{L}}$ commute, they will share a common eigenbasis [24, 25]. Furthermore, since the transformation, $\bm{Z^{l}}[\bm{M^{l}}(\bm{X^{l}})]$ lies in the range space of the operator, $\bm{Z^{l}}$, $\forall l,\bm{\hat{H}^{l}}=\bm{Z^{l}}[\bm{M^{l}}(\bm{X^{l}})]$ lie in a common subspace, due to the shared basis. While exact commutation cannot be guaranteed, we include a penalty term in the optimization to encourage the operators to commute, hence leading to operators that are ‘almost commuting’. Note that $\bm{Z^{l}}$ must be a square matrix for validly enforcing the commutation cost as per Equation 4. By including this with the GAN loss, Equation IV becomes,

[TABLE]

While the CGAN network is expected to eventually generate samples that lie in the same subspace as the target data, even in the absence of this auxiliary loss, we observe that its inclusion nevertheless improves the generator performance tremendously, in spite of the fact that the target space is ill defined. Experiments show that including the commutation term speeds up convergence, and also yields better hidden space estimates.

IV-A Structure of the Hidden Space/Selection Operators

In addition to ensuring that the hidden space estimates generated by different modalities lie in a common subspace, it is also important to structure the hidden space in a way that it not only contains information shared between the sensor modalities, but also contains private information of individual modalities. This is especially important for heterogeneous sensors as they may provide additional information along with some common information. This conforms with the notion that no two sensors are completely similar or dissimilar. This is equivalent to structuring $\bm{S^{l}}$ such that it selects features that are relevant to the $l^{th}$ modality, while ignoring others. Such a structure can be achieved by encouraging the columns of the selection matrix to be close to zero. If the $m^{th}$ column of $\bm{S^{l}}$ is zeroed out, then the $m^{th}$ feature in the hidden space, $\bm{H}$, does not contribute toward the $l^{th}$ modality. This allows us to find the latent space, $\bm{H}$, that naturally separates information shared between different modalities from which is private to each modality. We implement this by minimizing the $L_{\infty,1}$ norm, i.e., $\min_{\bm{S^{l}}}||\bm{S}^{\bm{l}}||_{\infty,1}$, where,

[TABLE]

This minimizes the sum of the maximum of each column in $\bm{S^{l}}$. Upon adding this term to the generator cost functional, we get,

[TABLE]

Notice that the above formulation has a trivial solution of setting $\bm{S^{l}}=\bm{0}$, and, $\bm{Z^{l}}=0$ for all $l$. In order to avoid this solution, we also optimize the classification based on the selected features, $\bm{F^{l}}$, for each modality, via the minimization of the cross-entropy loss. This ensures that the learned features, $\bm{F^{l}}$, are optimal for object detection/recognition, which is only possible if $\bm{S^{l}}\neq 0$, and, $\bm{Z^{l}}\neq 0$. Given the sensor observations, $\bm{X^{l}}$, and the classifier $\bm{C^{l}}=\{c_{i}^{l}\}$, the cross-entropy loss is computed as,

[TABLE]

where, $Y_{n}=\{y_{n_{i}}\}$ is the ground truth for the $n^{th}$ sample, $\sigma$ is the soft-max function, and, $\bm{f_{n}^{l}}=\bm{S^{l}}[\bm{G^{l}}(\bm{x_{n}^{l}})]$.

Finally, the optimization task is,

[TABLE]

where, $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ are hyper-parameters that control the contribution of different terms toward the optimization.

The necessary updates to train these networks are summarized in the Appendix (Algorithm 1). This setup learns the optimal features for classification $\bm{F^{l}}$, while driven by the existence of the hidden space $\bm{H}$, such that $\bm{F^{l}}=\bm{S^{l}H}$. Due to the inclusion of the selection matrix $\bm{S^{l}}$, followed by the Classification Layer, note that hand-crafting of features as in [9, 10], is no longer required. The features of interest are learned and automatically selected by the optimization during the training phase. The effects of the different terms in Equation IV-A on the determined hidden spaces are shown in Section V-B in Figures 9, 10, and 11. The block diagram for the proposed approach is summarized in Figure 2.

IV-B A Special Case of Event Driven Fusion

In order to fuse the individual decisions from the sensors we use a special case of Event Driven Fusion. Instead of defining feature events for each object as in [9], an event in this case is the occurrence of the $i^{th}$ object as seen by the $l^{th}$ sensor, $o_{i}^{l}$. The $l^{th}$ classifier, $\bm{C^{l}}=\{c_{i}^{l}\}$, provides the corresponding probability given the test sample, $\bm{x_{t}^{l}}$, $P_{t}^{l}(o_{i}^{l})=\frac{exp(c_{i}^{l}(\bm{f_{t}^{l}}))}{\sum_{m=1}^{I}exp(c_{m}^{l}(\bm{f_{t}^{l}}))}$, as previously discussed in Section III. Each sensor report is now represented as, $R_{t}^{l}=\{P_{t}^{l}(o_{i}^{l})\}$, and the fused probability of occurrence of the $i^{th}$ object as per the rules of Event Driven Fusion [9] is determined as,

[TABLE]

where $\rho$ is a pseudo-measure of correlation between the sensor modalities. For making an informed decision in favor of an object, this formulation assumes all the modalities to be equally reliable. This is not always true in practice, as certain sensors may provide more discriminative information than others, and are hence more reliable. It is thus important to weigh the various sensor decisions by a Degree of Confidence (DoC). The DoC in the decisions made by the $l^{th}$ sensor given the test sample, $\bm{x_{t}^{l}}$, is denoted by, $DoC_{t}^{l}\in[0,1]$, where $DoC_{t}^{l}=0$ implies that the sensor observations do not provide any useful information about the target identity, and $DoC_{t}^{l}=1$ implies that information provided by the sensor is highly discriminative with respect to target classification. The individual sensor reports, $R_{t}^{l}=\{P_{t}^{l}(o_{i}^{l})\}$, are now redefined as,

[TABLE]

where $P_{t}^{l^{\prime}}(o_{i}^{l})=DoC_{t}^{l}*P_{t}^{l}(o_{i}^{l})$, and $P_{t}^{l^{\prime}}(unc^{l})=1-DoC_{t}^{l}$ is the probability that the $l^{th}$ sensor is uncertain about the target identity. The joint probability distribution for the new sensor reports is now rewritten as,

[TABLE]

where, $a^{l}\in\{o_{1}^{l},o_{2}^{l}...,o_{I}^{l},unc^{l}\}$, and the probability of occurrence of the $i^{th}$ target is computed as,

[TABLE]

with a degree of confidence, $DoC_{t}^{f}=1-P_{t}(\bigwedge_{l=1}^{L}unc^{l})$, in the fused decision.

For a 2D scenario (i.e. two sensors), the probability of occurrence of the $i^{th}$ target is,

[TABLE]

with a degree of confidence, $DoC^{f}=1-P(unc^{1}\wedge unc^{2})$, in the fused decision. This formulation takes into account the potential uncertainties in the decisions made by individual sensors. Figure 3 shows the joint distribution to be determined in a 2D case.

The degree of confidence in the $l^{th}$ sensor for classification of the test sample, $\bm{x_{t}^{l}}$, is determined as,

[TABLE]

where, $p_{D}(\bm{\hat{h}_{t}^{l}})$ is the probability of damage for the $l^{th}$ sensor given the test sample, $\bm{x_{t}^{l}}$ (discussed in detail in the next subsection), and $Acc_{train}^{l}$ is the training accuracy of the $l^{th}$ sensor. The training accuracy in Equation 14 represents prior information about the discrimination power of the sensor, while $(1-p_{D}(\bm{\hat{h}_{t}^{l}}))$ represents the sensor condition at the time of operation/testing.

IV-C Sensor Failure Detection

Since $\bm{H}$ is designed so that it be common for all the sensors, it may be used to detect a damaged sensor.

IV-C1 Cross-Sensor Tracking

The estimate $\bm{\hat{H}^{m}}$, based on erroneous observations, will significantly deviate from the estimates from normal observations (i.e. sensors that are not damaged), $\bm{\hat{H}^{l}},l\neq m$. This allows detection of damage in a sensor during the testing phase. The $m^{th}$ sensor is said to be damaged if,

[TABLE]

where, $\mathcal{I}(.)$ is the indicator function, $T$ is a threshold value determined from the training data, and $\bm{\hat{h}_{t}^{l}}$ is the estimated hidden space given the observation $\bm{x_{t}^{l}}$. This equation considers a sensor damaged when the distance between the hidden spaces of more than half the sensors in the network is less than some threshold $T$, while the distance of the hidden space of the sensor in question is more than $T$. The limitation with this approach is that we can only detect up to $\frac{L-1}{2}/\frac{L}{2}-1$ damaged sensors.

IV-C2 Hierarchical Clustering

A faulty sensor can also be detected by comparing the hidden space generated by the test sample with that generated from training data. One way to proceed is by clustering functional sensor observations (from training data), and verify whether the hidden space estimate generated by the test sample can be associated to any of these clusters. The concatenation of the training data, $\hat{\bm{H}}_{d\times LN}^{C}=\{\hat{\bm{H}}^{1},\hat{\bm{H}}^{2},...,\hat{\bm{H}}^{L}\}$, is first used to construct a clustering tree based on an Agglomerative approach (see Algorithm 2 in Appendix). The probability of damage of the $l^{th}$ sensor is then computed as,

[TABLE]

where, $d_{v}$ is the cut-off distance at clustering level $v$, and,

[TABLE]

$v=1,...,V$ are the clustering levels, $J_{v}$ is the number of clusters at level $v$, and $Z_{v}^{j}$ is the $j^{th}$ cluster at level $v$. This is a measure of how quickly the hidden space estimate, $\bm{\hat{h_{t}^{l}}}$, can be clustered with the training data.

The $l^{th}$ sensor is said to be damaged if,

[TABLE]

where, $T$, is a threshold value which will depend on the dataset, types of sensors, SNR, etc. In our evaluation, we compute the optimal thresholds at different SNRs for the training data, and these thresholds are later used during the testing phase in order to determine the state of a sensor. This probability measure is also used in Equation 14, in order to adapt the DoC based on the sensor condition in a functional mode. If the sensor is damaged, the representative features are generated using the selection matrix as,

[TABLE]

where, $\Gamma$ is the set of working sensors.

V Experiments and Results

We validate our proposed approach by running experiments on two different datasets.

The first dataset we use is pre-collected data from a network of seismic, acoustic, and imaging sensors deployed in a field, where people/vehicles were walking/driven around in specified patterns. Details about this sensor setup and experiments can be found in [26]. This dataset has been previously used for target detection in [9, 14, 27, 28]. Here, we use this dataset to classify between human targets, vehicular targets, and no targets. Some data samples from the sensors can be seen in Figure 5. This will be referred to as ‘Dataset-1’ in the following discussions.

For our second experiment, we select two sensors, namely a Radar sensor and a telescopic optical sensor, for one (latter) of which we have acquired real data. For technical reasons, our Radar measurements were never co-measured with the optical data, and so we use MATLAB Simulink in order to simulate the radar responses. Both sensors are ideally synchronized when observing a given target, which in our case, can be any object in outer space, such as satellite or space debris. Each generated radar signal over one second is correlated with two telescopic images. Samples for objects with different velocities, cross-sections, ranges, and aspect-ratios are then generated.

Object classes are defined in the same way as in [10], based on events on the feature values. Note that these events are no longer required for training purposes, and are only used for the purposes of labeling the data. For the radar, we use the features, velocity ($v$), cross-section ($cs$), and range ($r$), and the events are defined as,

[TABLE]

Note that the events $a_{i}^{v},a_{i}^{r},$ and $a_{i}^{cs}$ are defined in the same way as $a_{kj}^{l}$ in Section II-B. From the telescopic imaging sensor, the features, displacement ($d$) and aspect ratio ($AR$) define the following events,

[TABLE]

Furthermore, the objects for classification are defined in terms of these events as,

[TABLE]

This will be referred to as ‘Dataset-2’ in the following discussions.

V-A Implementation Details

Figure 7-(a) shows the detailed block diagram for implementation of the discussed approach on Dataset-1. As noted earlier we no longer require handcrafted features and event definitions as in [9, 10], since we let the generative structure (see Figure 7) guide the learning of features. Similarly, the block diagram for implementation of this system on Dataset-2 can be seen in Figure 7-(b). The output of the $l^{th}$ generator, $\bm{G^{l}}$, for a test sample, $\bm{x_{t}^{l}}$, is a $d_{H}$-dimensional estimate of the hidden space, $\bm{\hat{h}^{l}_{t}}$. This hidden space is also used to detect potential damages to the sensors deployed. The optimal features $\bm{f_{t}^{l}}$, are subsequently selected by each sensor for making decisions on target identities via the selection matrix $\bm{S^{l}}$. The decisions are then fed into the fusion system which synthesizes a decision on the basis of the rules discussed in Section IV-B.

The generator network uses 1-D convolutions in case of seismic/acoustic and radar modalities, and 2-D convolutions in case of the imaging sensors. We use a 6 layered Neural Network, with 2x2 max pooling layer after the $2^{nd}$ and $4^{th}$ layer. ReLU activation is applied after every convolutional layer. The first 4 layers are convolutional, while the last two are fully connected. The first two convolutional layers use a filter size of 5 and the next two use a filter size of 3. The first fully connected layer is used to transform the output of the convolutional layers into a $d_{H}$ dimensional representation. All the layers preceding the final fully connected layer approximate the mapping $\bm{M^{l}}$, while the final layer approximates the operator $\bm{Z^{l}}$, and transforms the data into a common subspace.

The discriminator is implemented as a 3 layered fully connected network. Each layer preceeding the final layer transforms the $d-$dimensional input to $\frac{d}{2}$ and applies ReLU activation. The final layer transforms the features to a $L-$dimensional vector which represents the predictions from the discriminator network.

In determining the dimension of the hidden space, we search over values ranging from $d=50$ to $d=5000$. For Dataset-1 we find the best performance at $d=500$, while for Dataset-2, $d=700$. In both cases we observe that at best performance, $d<<d_{l}$.A search over $[10^{-5},10]$ was performed to select the optimal $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$. For Dataset-1 the set of parameters used were $\gamma_{1}=10^{-3},\gamma_{2}=10^{-4},\gamma_{3}=1$ and $\gamma_{1}=10^{-3},\gamma_{2}=10^{-3},\gamma_{3}=1$ for Dataset-2. As mentioned in Section IV-C, the value of the threshold $T$ will depend on the SNR of the signal as well as the sensor type. The optimal threshold for various sensors is computed based on the validation set during training. The SNR vs Threshold curve for the various sensors is shown in Fig. 8.

V-B Performance Analysis

Table I shows the performance of different techniques as compared with the proposed approach of using a Generative Adversarial Network to learn optimal features, which are then used for target classification. All sensors are assumed to be working normally in Table I. Comparisons are carried out with approaches that perform fusion at the decision level, as well as those that seek a hidden space for fusion. We find that using the individual sensor features (whose learning is driven by the existence of a structured hidden space) for classification and performing decision level fusion on these classifications yields better performance relative to those based solely on the hidden space. For the evaluation of Model Based Fusion approaches (see Section II-A), the dissimilar sensor setting is considered for Dataset-1 as all three sensors are different. On the other hand, for Dataset-2, the two radars are first combined using similar sensor fusion and the result of this is combined with the telescopic sensor using dissimilar sensor fusion.

We also compare the effects of different losses in our objective. First we implement a system which uses an Adversarial setup, along with the classification losses, i.e., $\gamma_{1}$, and $\gamma_{2}$ in Equation IV-A are set to [math]. The losses are seen in Figure 9-(a). The blue plot represents the negative of the discriminator loss for the GAN network (y-axis on left), while the orange plot corresponds to the commutation loss (y-axis on right). Figure 9-(b) shows the sum of pairwise distances between the hidden estimates generated by the three sensors, i.e. $\text{Magnitude}(k)=\sum_{l,m=1}^{L}(\bm{H^{l}}(k)-\bm{H^{m}}(k))^{2}$, where, $k\in\{1,...,d_{H}\}$, refers to the feature number in the $d_{H}$-dimensional hidden space. Figure 9-(c) shows the individual classification performance of the sensors. It is observed that the performance of Seismic and Acoustic Sensors improves as the model is updated, but the performance of the imaging sensor is very low.

Figure 10 shows the optimization losses when the commutation term is added to the objective, i.e. only $\gamma_{1}$ is set to zero in Equation IV-A. This amounts to minimizing of the pairwise commutation loss, hence forcing the hidden space estimates to lie in a common subspace. It is seen that including the commutation cost leads to closer hidden spaces (Figure 10-(b)).The performance of the imaging sensor, however, does not significantly increase.

Figure 11 shows the optimization loss when all the terms in Equation IV-A are active. The imaging sensor now starts giving better performance. Since the $L_{\infty,1}$ norm on the selection matrix allows representation of private/shared features in the hidden space, the heterogeneity of the imaging sensor is maintained, and selected features are more optimal for classification based on the imaging sensor. The effects of the $L_{\infty,1}$ norm can also be seen in the differences between the hidden estimates in Figure 11-(b), where some features now exhibit more distinguishing diversity compared to others, as they may correspond to private features.

V-C Robustness Analysis

The major advantage of learning a hidden space between the modalities is the ability to detect sensor damage in operation, and to generate representative features for that damaged sensor as seen in Equation 18. The contribution of the representative features towards the fused decision can also be controlled via the degree of confidence.

We also study how the performance of the system varies with different Signal to Noise Ratios. That is noise is selectively added to some sensors while another is normally functioning.

Figures 12 and 13 show the degradation of the fusion performance as the SNR decreases for Dataset-1 and Dataset-2, respectively. The plots with ‘asterisk’(*) markers represent approaches that search for a common subspace in order to fuse the different modalities, while those with ‘dot’(.) markers represent approaches that perform fusion at the decision level. Finally, the plots with ‘square’ markers show the performance of our proposed approach. The blue plot uses adaptive DoC $\left(DoC^{l}_{t}=(1-p_{D}(\hat{h}_{t}^{l})).Acc_{train}^{l}\right)$, while the orange plot only uses the prior information $\left(DoC^{l}_{t}=Acc_{train}^{l}\right)$, which is also the case for other plots using decision level fusion. It is observed that it is important to update the DoC during the implementation based on the sensor condition, rather than only depending on the prior information about the discriminative abilities of the sensor.

Furthermore, the above graphs show that the degradation has severe effects in the case where the seismic and imaging sensors are damaged. This is due to the fact that the discriminative power of the acoustic sensor is low, and in spite of adapting the DoC and generating representative features, the performance is limited by the information contained in the observations of the sensor. A similar trend is also observed for Dataset-2, where the telescopic imagery has lower discriminative ability compared to the radar sensors.

VI Conclusion

In this work, we proposed a data driven approach to learn a structured hidden space between sensors by way of a bank of Generative Adversarial Networks. The maps into the hidden space are forced to commute with each other, leading to the estimates from different modalities to lie in a common subspace. Enforcing commutation is also shown to lead to faster convergence and better hidden space estimates. The hidden space is subsequently used to learn the features of a target of interest for classification. The ‘private’ and ‘shared’ information in a hidden space was further enhanced by including a selection matrix that selects features of interest for each modality before classification. The hidden space serves a dual purpose of detecting noisy/damaged sensors and mitigating sensor losses by ensuring a graceful performance degradation. Experiments on multiple datasets show that the proposed approach secures a great degree of robustness to noisy/damaged sensors, and outperforms existing fusion algorithms.

Appendix

Algorithms