3ds Max/Unity

Autodesk 3ds Max [34] is a 3D content creation suite used to create 3D models for games and animation. Unity3D [35] is a real-time cross-platform initially released by Unity Technologies in 2005 with the goal of playing 2D and 3D games and interactive simulations. Unity enables users to immerse themselves in a scene using intuitive tools, such as asset tracking, rendering, and scripting. In recent years, the use of 3ds Max and Unity software for the development of 3D models and architectural visualizations in critical infrastructure protection has increased [36]. The 3D models and scenes generated in 3ds Max can be exported directly to Unity 3D to create games and interactive experiences. In PPS design, tools for skinning, texturing, rigging, and animating are used to create 3D models of the infrastructure. The detailed design of the PPS elements, including CCTV cameras, perimeter barriers, characters, etc. can also be incorporated into the 3ds Max/Unity for the iterative evaluation process.

EASI Model

With the integration of 3D models, a physical security simulation was conducted using an A^* heuristic pathfinding algorithm for vulnerability assessment. The cost function used in the 3D A^* heuristic pathfinding was calculated based on the EASI model to estimate the probability of interruption (P_I) along the adversary path. The EASI model, developed by Sandia National Laboratories to evaluate the effectiveness of the PPS for nuclear facility security systems under threat from outsiders in the 1970s [37]. Since then, the EASI model has been widely used owing to its simplicity and ease of use [38]. The effectiveness of a physical security system was evaluated using the EASI model along a specific adversary intrusion path in a probabilistic analysis. It uses the performance measures for a PPS function that include the probability of detection (P_D), probability of alarm communication to the response force, probability of deployment of the response force to the adversarys location, time to deploy the response force to the adversary's location, and time to complete the remaining adversarial attacks after detection to determine the probability of interruption (P_I) as follows: $\begin{array}{c}{P}_{\text{I}}=P_{\text{D}}^1\cdot P_{\text{C}}^1\cdot P\left(R|A_1\right)\\ +{\displaystyle \sum _{n=2}^N}{P}_{\text{D}}^n\cdot P_{\text{C}}^n\cdot P\left(R|A_n\right)\\ \times {\displaystyle \prod _{i=1}^{n-1}}\left(1-P_{\text{D}}^i\right)\end{array}$ (1) where $P_{\text{D}}^n$ (n=1,2,...,N) represents the detection probability for each detection point. $P_{\text{C}}^n$ is the probability of a successful alarm communication with response forces after identifying an adversarial attack at the n^th detection point, which is typically set as a constant. P(R|A_n) is the probability that the response forces successfully interrupt an adversarial attack after receiving an alarm message at the nth detection point. The mean and standard deviations of the response time (RFT) and adversary task time (TR) were used to calculate the value of P(R|A). Assuming that RFT and TR are mutually independent and normally distributed, the random variable X(X=TR-RTF≥0) for the timely detection characterization follows a normal distribution with mean ${\mu }_X$ and variance ${\sigma }_X^2$. The probability of P(R|A) can be calculated as: $\begin{array}{c}P\left(R|A\right)=P\left(X\ge 0\right)\\ ={\displaystyle {\int }_0^{\infty }}\frac{1}{\sqrt{2\pi {\sigma }_X^2}}\\ \exp \left[-\frac{{\left(X-{\mu }_X\right)}^2}{2{\sigma }_X^2}\right]\text{d}X\mathrm{,}\end{array}$ (2) where TR and RFT can be obtained from the shortest path search or the most vulnerable path search with the lowest PI value. Detailed explanations of how to calculate the TR and RFT values can be found in Ref. [22].

A^* Heuristic Pathfinding Algorithm in 3D Environment

A^* algorithm [39] is the best-first search algorithm that has been widely used for map traversal to determine the shortest path between the initial and final points. A^*-algorithm searches for the most promising path through the state space using a heuristic function, where an estimate of the cost from the current state to the target is considered as informed information to guide the path search process more efficiently. Heuristic function H(n)) is given by $H\left(n\right)\le H^{*}\left(n\right)\mathrm{,}$ (3) where H(n) is the heuristic cost and $H^{*}\left(n\right)$ is the estimated cost. The heuristic cost should be less than or equal to the estimated cost to guarantee admissibility and consistency with respect to the heuristic function used in the A^* search algorithm for optimal path identification [40].

In terms of vulnerability analysis in the design and evaluation of a physical protection system, the probability of interruption (P_I) is used as a cost function for the most optimal pathfinding. The calculation of the total cost function F(n) is divided into two parts: i) G(n), which indicates the real cost of the path from the starting point to the current point n and ii) H(n), which represents the estimated heuristic cost from the current point n to target point N_g. In most cases, A^* algorithm does not know the actual distance or cost until it determines a path. The A^*-algorithm is equivalent to the Dijkstra algorithm when the heuristic function A^* becomes zero. As the estimated heuristic cost is the exact cost of reaching the destination node from the current node n, the lowest-cost path can be found using algorithm A^* with the fastest speed [41]. However, it is generally impractical to determine an optimal heuristic function that always matches the exact cost because the paths to be searched are unknown.

In this study, an innovative algorithm A^* is proposed to determine the most vulnerable adversary path in a 3D security environment. The 3D A^* search algorithm was adapted from one of our previous studies [28] which was carried out to simulate physical security by visual backtracking search on a 2D-graph model of the PPS. As depicted in Fig. 2, the heuristic function $H^{*}\left(n\right)$ to reach the destination node (N_s) from the current node n can be estimated as: $H^{*}\left(n\right)=\text{Δ}{P}_{\text{I}}^n=P_{\text{I}}^s-P_{\text{I}}^n$ (4) where $P_{\text{I}}^n$ and $P_{\text{I}}^s$ represent the cumulative probabilities of interruptions at the current node n and source node N_s, respectively. Assuming that m potential detection opportunities exist along the adversary's path starting from the current node n to the source node (N_s), $P_{\text{I}}^s$ can be obtained recursively using Eq. (5): $\begin{array}{ll}{P}_{\text{I}}^s\hfill & =P_{\text{D}}^s\cdot P_{\text{C}}\cdot P\left(R|A_s\right)+\left(1-P_{\text{D}}^s\right)\cdot P_{\text{I}}^{s-1}\hfill \\ \hfill & \begin{array}{l}=P_{\text{D}}^s\cdot P_{\text{C}}\cdot P\left(R|A_s\right)+\left(1-P_{\text{D}}^s\right)\\ \cdot \left[P_{\text{D}}^{s-1}\cdot P_{\text{C}}\cdot P\left(R|A_{s-1}\right)+\left(1-P_{\text{D}}^{s-1}\right)\cdot P_{\text{I}}^{s-2}\right]\end{array}\hfill \\ \hfill & \begin{array}{l}=P_{\text{D}}^s\cdot P_{\text{C}}\cdot P\left(R|A_s\right)+\left(1-P_{\text{D}}^s\right)\\ \cdot \{P_{\text{D}}^{s-1}\cdot P_{\text{C}}\cdot P\left(R|A_{s-1}\right)+\left(1-P_{\text{D}}^{s-1}\right)\end{array}\hfill \\ \hfill & \cdot [P_{\text{D}}^{s-2}\cdot P_{\text{C}}\cdot P\left(R|A_{s-2}\right)+\left(1-P_{\text{D}}^{s-2}\right)\cdot P_{\text{I}}^{s-3}]\}\hfill \\ \hfill & \begin{array}{l}=P_{\text{D}}^s\cdot P_{\text{C}}\cdot P\left(R|A_s\right)+\left(1-P_{\text{D}}^s\right)\\ \cdot \{\cdots [P_{\text{D}}^{n+1}\cdot P_{\text{C}}\cdot P\left(R|A_{n+1}\right)+\left(1-P_{\text{D}}^{n+1}\right)P_{\text{I}}^n]\}\end{array}\hfill \\ \hfill & =f^{[s-n]}\left(P_{\text{D}}^{n+1}\mathrm{,}P\left(R|A_{n+1}\right)\mathrm{,}{P}_{\text{I}}^n\right)\hfill \end{array}$ (5) where $P_{\text{I}}^n$ is the cumulative probability of an interruption at current node n. P(R|A_s) is the probability that the response forces successfully interrupt an adversarial attack after receiving an alarm alert at the n^th detection point. $P_{\text{I}}^n$ can also be expressed in the following recursive form: $P_{\text{I}}^n=P_{\text{D}}^n\cdot P_{\text{C}}\cdot P\left(R|A_n\right)+\left(1-P_{\text{D}}^n\right)\cdot P_{\text{I}}^{n-1}$ (6) To obtain a heuristic function that is sufficiently close to the exact cost and never overestimates the value, we use the conservative value of P(R|A_s) to replace the nested iterations of P(R|A_s) (n<i≤s) in an unknown estimation of $P_{\text{I}}^s$. P(R|A_i) is a monotonically increasing function that satisfies P(R|A_i)≤P (R|A_i+1) under the condition of timely detection, with X=TR-RFT≥0. where the time delay remaining for the completion of the adversarial task (TR) also increases monotonically ($T{R}_i\le T{R}_{i+1}$) as the adversaries move away from the target in backward pathfinding. Consequently, the following inequality holds for $H\left(n\right)\le H^{*}\left(n\right)$. $\begin{array}{ll}{H}^{*}\left(n\right)\hfill & =P_{\text{I}}^s-P_{\text{I}}^n\hfill \\ \hfill & \begin{array}{l}=P_{\text{D}}^s\cdot P_{\text{C}}\cdot P\left(R|A_s\right)+\left(1-P_{\text{D}}^s\right)\\ \cdot \{\cdots [P_{\text{D}}^{n+1}\cdot P_{\text{C}}\cdot P\left(R|A_{n+1}\right)\\ +\left(1-P_{\text{D}}^{n+1}\right)P_{\text{I}}^n]\}-P_{\text{I}}^n\end{array}\hfill \\ \hfill & \begin{array}{l}\ge P_{\text{D}}^{\min }\cdot P_{\text{C}}\cdot P\left(R|A_n\right)+\left(1-P_{\text{D}}^{\min }\right)\\ \cdot \{\cdots [P_{\text{D}}^{\min }\cdot P_{\text{C}}\cdot P\left(R|A_n\right)\\ +\left(1-P_{\text{D}}^{\min }\right)P_{\text{I}}^n]\}-P_{\text{I}}^n\end{array}\hfill \end{array}$ (7) Thus, the heuristic function $H\left(n\right)$ can be rewritten as $\begin{array}{c}H(n)=P_{\text{D}}^{\min }\cdot P_{\text{C}}\cdot P\left(R|A_n\right)+\left(1-P_{\text{D}}^{\min }\right)\\ \cdot \left\{\cdots \left[P_{\text{D}}^{\min }\cdot P_{\text{C}}\cdot P\left(R|A_n\right)+\left(1-P_{\text{D}}^{\min }\right)\cdot P_{\text{I}}^n\right]\right\}\\ -P_{\text{I}}^n\end{array}$ (8) where $p_{\text{D}}^{\text{min}}$ returns the minimum value of the map of the non-zero detection field. Although admissibility can be guaranteed with the above heuristic function, we still do not know which path the adversaries will take to attack. It is also unknown how many detection opportunities and delay elements are reserved for a segment of nondetermined path from the current position n to starting point N_s. To obtain a heuristic as close to the estimated cost as possible, the probability that the adversary will be detected by the intrusion detection system is considered as a random variable $\Phi$. The unknown number of opportunities (m) remaining for adversarial detection along the path from the current node n to the source node N_s is estimated as the expected value of the random variable $\Phi$, which is denoted as $E(\Phi )$. $m=E(\text{Φ})=p\cdot N_{\text{gids}}$ (9) where p denotes the possibility of a sensing element being located in a node cell. Consequently, (1- p) represents the probability that a cell area is not covered by the detection field. The probability of a node falling into the detection area can be approximated as the ratio of the detection area to the total site area. m remains in the integer realm after removing the fractional part of the floating-point numbers to count the detection opportunities. N_grids denotes the number of cells in which adversaries must cross the protected areas to reach the target. N_grids can be approximately calculated using Eq. (10). $N_{\text{grids}}=\frac{D}{d_{\text{cell}}\cdot {\upsilon }_{\text{A}}}$ (10) where d_cell represents the diagonal length of a two-dimensional square unit cell or three-dimensional cube unit cell to avoid overestimating the number of unit cells through which the shortest path must pass. υ_A denotes the mean velocity of the adversarial attacks. D is the distance between the current node n and destination node N_s.

Fig. 2Full-screen View

Detection opportunities for cost estimation

To compute the length of the shortest path in a 3D scene at a lower computational cost, we introduced waypoint navigation on the coarse grid map to compress the path representation. A waypoint is a point along a path that can be added manually or automatically to accelerate path finding [42]. Waypoints are generally designated at the must-pass entrances of adversarial pathways to construct an exact heuristic for guiding the shortest path between any pair of coarse grid locations. As illustrated in Fig. 3, the 3D space can be divided into several layers according to the number of coarse grids. For example, a pair of waypoints can be added at the corners of stairs or elevators that connect two parallel grid planes of building stories. Thus, the calculation of three-dimensional space distance can be transformed into the sum of the lengths of the pairs of waypoints across the three two-dimensional connecting planes. The final heuristic function is calculated as follows: $H(n)=H\left(n\mathrm{,}{w}_1\right)+cost\left(w_1\mathrm{,}{w}_2\right)+H\left(w_2\mathrm{,}\text{ goal}\right)$ (11) The exact heuristic can be pre-computed for the shortest path between any pair of waypoints $cost\left(w_1\mathrm{,}{w}_2\right)$. The heuristic costs with pairs of waypoints ($w_1\mathrm{,}{w}_2$) that are close to the current and goal nodes can also be evaluated using $H\left(n\mathrm{,}{w}_1\right)$ and H(w₂, goal), respectively. The distance heuristic can be similarly estimated. $D=D\left(n\mathrm{,}{w}_1\right)+D\left(w_1\mathrm{,}{w}_2\right)+D\left(w_2\mathrm{,}\text{ goal}\right)$ (12) The heuristic function of each path segment can be estimated using the diagonal distance defined in Eq. (13): $\begin{array}{c}\begin{array}{c}D\left(n\mathrm{,}{w}_1\right)=\left(\left|x_n-x_{w_1}\right|+\left|y_n-y_{w_1}\right|+\left|z_n-z_{w_1}\right|\right)\\ +\left(\sqrt{2}-2\right)\min \left(\left|x_n-x_{w_1}\right|\mathrm{,}\left|y_n-y_{w_1}\right|\right)\end{array}\\ \begin{array}{c}D\left(w_1\mathrm{,}{w}_2\right)=\left(\left|x_{w_1}-x_{w_2}\right|+\left|y_{w_1}-y_{w_2}\right|+\left|z_{w_1}-z_{w_2}\right|\right)\\ +\left(\sqrt{2}-2\right)\min \left(\left|x_{w_1}-x_{w_2}\right|\mathrm{,}\left|y_{w_1}-y_{w_2}\right|\right)\end{array}\\ \begin{array}{c}D\left(w_2\mathrm{,}\text{ goal}\right)=\left(\left|x_{w_2}-x_{\text{goal}}\right|+\left|y_{w_2}-y_{\text{goal}}\right|+\left|z_{w_2}-z_{\text{goal}}\right|\right)\\ +\left(\sqrt{2}-2\right)\min \left(\left|x_{w_2}-x_{\text{goal}}\right|\mathrm{,}\left|y_{w_2}-y_{\text{goal}}\right|\right)\end{array}\end{array}$ (13) For 3D path planning, spatial navigation was created for object positioning and guidance. As shown in Fig. 4, the special navigation enables 26-direction navigation by considering the three pairs of orthogonal axes. The 3D calibration board for the characterization of agents' movements consisted of three parallel planes. Each plane can be considered as a vertical space with a fixed step-size input ξ to describe the enemy's ability to move up and down. The navigation arrow points to adjacent cells in 3D space with a visual representation of the three colored arrows. The agent can move in eight directions (blue arrows): forward (F), backward (B), right (R), left (L), right forward (R-F), left forward (L-F), right backward (R-B), and left backward (L-B), when the diagonal distance is used for the heuristic estimation. The upward and downward arrows are distinguished by red and green, respectively, to show the upward (U) and downward (D) motions. Reachable adjacent vertices can be recognized well and efficiently with a reduced-space search using the Raycast function in Unity. The air cell spaces were identified as unreachable considering that the enemy could only walk up the stairs to the second floor. Therefore, the selection of the search path can be significantly reduced using the designated stairs.

Fig. 3Full-screen View

(Color online) Waypoints for path navigation

Fig. 4Full-screen View

Moving directions in 3D environment

Following backward pathfinding, the heuristic time delay remaining after detection $T_{\text{TR}}^{\text{H}}$ can be estimated by the following Eq. (14): $T_{\text{TR}}^H=T_{\text{TR}}^n+\frac{D}{v_{\text{A}}}+T_{\text{TD}}^{\overrightarrow{n}s}$ (14) where $T_{\text{TR}}^n$ denotes the adversary task time defined for the current node n, which can be obtained by iterative calculation from its parent node n-1 to the first detection opportunity. $T_{\text{TD}}^{\overrightarrow{n}s}$ is the cumulative time delay along the path segment from the current node n to the initial point of an adversarial attack (N_s) in reverse traversal. D represents the diagonal distance between the current node n and initial point of the adversarial attack (N_s), which is determined by Eq. (13).

3D Modeling

In this section, a hypothetical laboratory facility is considered as an example of a PPS for demonstration. The entire facility site was 236 m long and 183 m wide. 3D Building Information Modeling (BIM) of the hypothetical laboratory facility was implemented based on 3ds Max/Unity3D platform. As shown in Fig. 5, the laboratory site has three main buildings within the protected area, which is surrounded by a perimeter barrier. The two gates were located at the front and right sides of the laboratory facility. The asset is located in Room #207 of Building #3 and is used as the target enclosure to protect it from theft, sabotage, and other malevolent attacks. Building #3 is a two-story quadrangular courtyard building with only one entrance facing the right. The dimensions of Building #3 is 71 m × 54 m × 21 m. Figure 6 presents the 3D BIM model with both bird's eye view and head-up displays. To obtain a better display of the pathways under planning, we perceived Building #3 from multiple perspectives using 3ds Max/Unity. The interior view of the building was also presented with motion perception. The architectural style of Building #2 was similar to that of Building #3, but with two gateways. Building #1 is a modern building constructed using concrete. The outer walls of the three buildings were mounted with infrared detectors to prevent enemies from climbing walls or breaking windows.

Fig. 5Full-screen View

(Color online) The layout of laboratory facility

Fig. 6Full-screen View

(Color online) 3D bird's eye and motion preception of the Building #3

The 2D plane layout of Building #3 is shown in Fig. 7. There is a corridor and walkway that connect rows 0. f rooms on each floor. Three stairs (stairwells #1, #2, and #3) are configured in the building to connect the ground floor to the second floor. For each staircase, an omnidirectional camera (360°) was installed to monitor illegal break-ins. In addition to the video surveillance of the staircases, the internal and external aspects of Building #3 were monitored. Eight CCTV cameras were distributed in the interior corners of the building and each floor was equipped with four cameras. The interior and exterior surveillance cameras are referred to as {CCTV #2-1, CCTV #2-2, CCTV #2-3, CCTV #2-4} and {CCTV #3-1, CCTV #3-2, CCTV #3-3, CCTV #3-4}, respectively. As demonstrated in the 3D displays in Fig. 6, the agents can choose any of the three staircases on the second floor after passing through gate #3 of building #3. Apparently, the 3D models of buildings allow professionals to gain insights into a structure when starting to play animation. The room-numbering sequence on the second floor is shown in Fig. 7 to quickly locate and track the agent's position. A built-in green garden was located in the central part of Building #3. The walkways, corridors, and aisle stairs between the green garden and the rooms are marked in pink. The red lines on the periphery of the black exterior walls represent the pairwise infrared detectors. The violet-orange circles represent the omnidirectional cameras blocked by stairwells or walls.

Fig. 7Full-screen View

2D Plane map for the second floor of Building #3

Generation of 3D Detection Field

In this study, 3D models of the PPS elements were developed to project the detection field. Two types of detection elements were considered for the detection field projection. One is a gate access control system, such as a keycard, fab, facial recognition, or fingerprint scanner, etc. with credentials to enter. The detection probability of such sensors was approximated as a point estimate, as summarized in Table 1. Another type of detection element is the CCTV cameras mounted in the internal or external corners of buildings. The detection zone of the CCTV camera is shown in Fig. 8b. The detection space of the CCTV camera has a conical search light shape. The probability distribution of the camera's projected region on the ground was approximated by a linear decay function defined in the following Eq. (15): $P_{\text{D}}^i=P_{\text{D}-\text{center}}\cdot \left(1-\frac{D_i}{R}\right)\mathrm{,}$ (15) where $P_{\text{D}}^i$ denotes the probability of detection at the i-th cell node covered by the spotlight of the CCTV camera. $P_{\text{D}-\text{center}}$ refers to the probability of detection in the center of the spotlight area. Di is the distance between the i-th cell node and the center point of the spotlight area. R is the radius of the spotlighted circle.

Fig. 8Full-screen View

(Color online) Modeling of CCTV camera

The entire detection field generated for a hypothetical laboratory facility is illustrated in Fig. 9. The detection field is presented in the form of thermography, in which the sensitive detection area is marked with a highlighted foreground color display. The pixel colors of the detection circles gradually decayed to pale yellow and even to a gray background, which represented the non-detection areas. In addition, the critical detection area (TR<RFT) is highlighted with a light pink background on the map to provide insight into the potential vulnerabilities. The internal structures of buildings #1 and #2 are not shown here because of the lower security risk levels identified for evaluating the effectiveness of the PPS.

Fig. 9Full-screen View

(Color online) Detection field projected for the hypothetical laboratory facility

Scene Setting for the Case Demonstration

As pinpointed in Fig. 7, the protected target is located in Room #207 of Building #3. The adversaries are assumed to start their attacks on the right side of the laboratory site. The surveillance monitoring system and response forces are located in the left corner of Building #2. The response forces must reach the target room prior to the arrival of the enemies to interrupt them once an alarm is triggered. Therefore, the shortest path was chosen based on the response forces to reach the target at the highest speed. A relatively small PPS response time (T_RFT=61 s) was obtained by traveling along the shortest path to the target enclosure and considering a good training exercise for the response forces. An adversary attack is conducted under the following assumptions:

• Obstacles such as walls, perimeter fences, etc. are impenetrable. That is, enemies do not have the ability to destroy objects.

• Any open space on the ground is accessible. The enemies can move freely in the open areas.

• The step length is set to ξ=0.3m by taking into account the optimal height of a staircase step.

• The detection probability (P_D) and time delays (T_TD) defined for the various PPS elements including the outer gate point (Gate #1, Gate #2), building interior gate (Gate #3), and room doors are summarized in Table 1.

• The average speed of enemies and response forces are assumed to be V_A=3 m/s and V_RF=2.5 m/s for T_TR and T_RFT values calculation.

• An adversarial attack will be successfully interrupted by the response forces as long as the response forces get into the target enclosure before the enemies do.

Optimal Pathfinding Using 3D A^* Heuristic Algorithm and Dijkstra Algorithm

Based on the model assumptions and model parameter inputs, the vulnerability of the heuristic pathfinding solution can be obtained using 3D A^*-algorithm. The most vulnerable adversary path identified using the measure P_I in the 3D A^* heuristic search is shown in Fig. 10.

Fig. 10Full-screen View

(Color online) The most vulnerable adversary path searched by A^* algorithm

The search results were also compared with those of the Dijkstra algorithm, as summarized in Table 2. The complexity of the proposed heuristic scheme was estimated by counting the number of nodes traversed by the algorithm and the elapsed time required to complete the optimal path. Figure 11 shows the shortest path with the lowest P_I obtained by the Dijkstra algorithm. To provide a clearer path display, we made the building transparent except for the skeleton frame. In Unity3D environment, users can also follow the character model for path exploration in the animation mode. Thus, the most desirable benefits of 3D modeling and animation can be discovered in PPS design and simulations.

Fig. 11Full-screen View

(Color online) The most vulnerable adversary path searched by Dijkstra algorithm

This shows that a huge number of 134,298 cell nodes are traversed by the Dijkstra algorithm for the most vulnerable path identification, with an elapsed time of 2,299,362 ms. In contrast, the search efficiency was significantly improved by proposed algorithm A^*. Saving more than two-thirds of the computation time when applying algorithm A^* for optimal pathfinding in a hypothetical scene with the magnitude of discrete cell nodes N_total= 145,521. The most vulnerable adversarial paths identified by the Dijkstra and A^* algorithms have the same measures of P_I=0.466414838754809, TR=111.91 s, and cumulative detection probability $P_{\text{D}}^{\text{acc}}=0.145123303533361$.

From the bird-seye view shown in Fig. 10 and Fig. 11, we can see that the adversaries take the same path to the target. Starting from the initial point on the right side of the laboratory site, the adversaries enter the facility through gate #1 and then choose the shortest path to bypass the detection fields around buildings #1 and #2. As adversaries approach Building #3, their movements are monitored by a series of CCTV cameras, in which the searchlight covers most of the floor area. Therefore, adversaries tend to be more cautious in selecting their path directions to quickly approach the target without being detected by surveillance monitoring systems. Screenshots of the hidden routes inside Building # 3 are shown in Fig. 12. Figure 12 also presents the visual comparisons with different segments of the path between the Dijkstra algorithm and the A^* algorithm for the full validation of the path. The path varies slightly only in the nondetection areas (P_D=0), but the cost function P_I is not affected by the identification of the optimal global path.

Fig. 12Full-screen View

(Color online) Visual comparisons with different path segments

During the heuristic search analysis, the cost values, including [G(n), H(n), F(n)], were also reviewed for cross-validation of the optimal paths. As compared in Fig. 13, the heuristic cost H(n) shows a monotone decreasing change as we advance the path towards the origin of adversarial attack. However, it also exhibits local volatility in path guidance when the simultaneous contrast effects of the combined detection and delay functions are considered for the heuristic estimation of H(n). In other words, the simultaneous contrast effects resulting from the decreased cumulative probability of detection and increased probability of adversarial interruption (P(R|A)) can lead to unnecessary searches for local path optimization. Theoretically, the estimated cost H(n) will continue to decrease until zero because fewer detection grids need to be computed for the short-distance heuristic when the point moves close to the target. However, the monotonic increase in TR and P(R|A) has a countereffect on the heuristic estimates in backward pathfinding. There may even be a slight increase in the estimated value of H(n) during the path planning. This surprising phenomenon is highlighted in Fig. 13(a) and (b). Surprisingly, a sudden jump in the overall cost estimation F(n) resulting from a significant increase in the time delay at a detection point can be observed for a critical path element such as Gate #1. Gate #1 served as the main entrance to a limited area of the hypothetical laboratory facility. Because the delay elements in the open areas are not considered in the heuristic estimation, a sudden increase in the time delay caused by gate #1 immediately creates an obvious shoot-up in the overall cost estimation F(n) when compared to its neighboring nodes inside the boundary fence of the hypothetical laboratory facility. Gate nodes with relatively large values for the overall cost estimate F(n) are pushed down on OpenList. In such cases, the algorithm must prioritize the search in the free-walking area (Fig. 10 where the neighboring nodes are in fact fantastic with lower value of F(n), but not the optimal path extended to the off-site source node. This also leads to many unnecessary searches and the wastage of computational resources.

Fig. 13Full-screen View

(Color online) Optimality of cross-validation in scattered data approximation