Optimal Positioning Strategy for Multi-Camera Zooming Drones

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by IEEE: M. Vargas, C. Vivas and T. Alamo, "Optimal Positioning Strategy for Multi- Camera Zooming Drones," in IEEE/CAA Journal of Automatica Sinica, vol. 11, no. 8, pp. 1802-1818, August 2024, doi: 10.1109/JAS.2024.124455. “© 2024 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.” 1 Optimal positioning strategy for multi-camera, zooming drones Manuel Vargas, Carlos Vivas and Teodoro Alamo, Member, IEEE Abstract—In the context of multiple-target tracking and surveillance applications, this paper investigates the challenge of determining the optimal positioning of a single autonomous aerial vehicle or agent equipped with multiple independently-steerable zooming cameras to effectively monitor a set of targets of interest. Each camera is dedicated to tracking a specific target or cluster of targets. The key innovation of this study, in comparison to existing approaches, lies in incorporating the zooming factor for the onboard cameras into the optimization problem. This enhancement offers greater flexibility during mission execution by allowing the autonomous agent to adjust the focal lengths of the on-board cameras, in exchange for varying real-world distances to the corresponding targets, thereby providing additional degrees of freedom to the optimization problem. The proposed optimization framework aims to strike a balance among various factors, including distance to the targets, verticality of viewpoints, and the required focal length for each camera. The primary focus of this paper is to establish the theoretical groundwork for addressing the non-convex nature of the optimization problem arising from these considerations. To this end, we develop an original convex approximation strategy. The paper also includes simulations of diverse scenarios, featuring varying numbers of onboard tracking cameras and target motion profiles, to validate the effectiveness of the proposed approach. Index Terms—Projective transformation, visual surveillance, visual object tracking, unmanned aerial vehicle, convex optimization. I. INTRODUCTION The combination of a remotely piloted aircraft (RPAS) or unmanned aerial vehicle (UAV) with an on-board steerable camera, as a compound electromechanical device that can put “eyes in the sky” [1], has been extensively studied and successfully employed in a wide range of applications. This range from early intelligence-gathering military applications, to the more recent widespread use in entertainment activities, covering infrastructure inspection [2], [3], [4], search and rescue missions [5], traffic surveillance [6], [7], cooperative or uncooperative target tracking [8], [9], and a myriad of other applications. The on-board camera is typically mounted on an actuated gimbal, which serves a dual function. First, it acts as a line-of- sight stabilizer, counteracting the turns and/or tilts experienced by the aerial platform. Second, it functions as a visual tracking device, aimed at the desired target of interest [10], [11], [12], [13], [14]. For a better understanding of the impressive features and performance of contemporary aerial gimbaled zooming systems, such as the one mounted on the DJI Zenmuse H20©, M. Vargas, C. Vivas and T. Alamo are with the Department of System Engineering and Automation, University of Seville, Seville, Spain. E-mail: {mvargas,vivas,talamo}@us.es. please refer to the following video: https://www.youtube.com/ embed/G6VRhckcMeo?start=53&end=82. When the objective is to simultaneously keep track of several targets that might be distant from each other, while observing each one of them with sufficient visual detail, there are typically two approaches. The first approach is straightforward: deploying several vehicles with similar capabilities, each one devoted to a single target or a group of nearby targets [15], [16]. The second approach, on the other hand, focuses on shared attention, where a single vehicle periodically changes its camera’s aiming direction among different targets to avoid losing track of any of them permanently [17], [18], [19]. This second category, which has received less attention from the research community, is nevertheless quite attractive as it aims to accomplish the task with minimal resource deployment. This approach is highly beneficial in terms of implementation cost and energy demands during missions. Additionally, it is less intrusive, a feature that can be very relevant in certain areas of application such as wildlife monitoring or concealed surveillance operations. In terms of downsides, the shared-attention approach faces inherent constraints in the observation period of each target, inversely correlated with the total number of targets. Another challenge is the occurrence of ”blind lapses” during the targetswitching action, with no target within the field of view of the involved camera. This can become a compromising factor, especially in the presence of a large number of scattered targets with rather unpredictable trajectories. Furthermore, while intermittent observation may suffice for estimating target trajectories, in applications where closely monitoring and documenting activities or behaviors is paramount, such gaps in observation can pose a serious handicap. In a recent work [20], the concept of flying chameleons was introduced as a more flexible alternative in the context of aerial multi-target tracking and surveillance. Under this paradigm, one single aerial vehicle was equipped with several independently steerable cameras to visually track several targets in sufficient detail simultaneously. A convenient optimization problem was formulated, and several strategies for its resolution were proposed. However, in this previous work, the problem was significantly curtailed by the assumption of fixed focal lengths for the onboard cameras. As a result, capturing a closer view of a specific target required physically moving the aerial vehicle closer to it, potentially compromising the observation of the other targets. The present paper extends the previous works by incorporating the zooming effect on the on-board cameras, which can be automatically adjusted in real-time according to an optimal 2 criterion. This new feature introduces a great deal of flexibility to the conceptual system, significantly enhancing its capabilities by adding new degrees of freedom to the optimization problem. However, this enhancement comes with a significant toll on the theoretical development, as it breaks away from the convex nature of the original problem. To address this challenge, we introduce an entirely new formulation in this work that not only circumvents this downside but also ensures the feasibility of the resulting optimization problem. The superior mission capabilities enabled by the introduction of the zooming effect have been studied in previous works for the case of single-camera setups [21], [22]. However, a cautionary note must be observed here. The application of high magnification factors can make fine image stabilization more challenging, typically demanding the confluence of highaccuracy gimbal control and electronic image stabilization [23], [24]. Bearing this latter consideration into account, the present work is intended to take advantage of those enhanced capabilities in the context of multi-camera, multitarget tracking. The proposed framework is well-suited for applications across various domains. In high seas search and rescue missions, where people and small boats may be adrift after shipwrecks or disasters, continuous tracking is crucial for prioritizing rescue efforts [25], [26], [27]. Similarly, in surveillance missions involving suspicious vessels [28], such as those used by smugglers or illegal migration groups, simultaneous tracking of multiple vessels might be essential for gathering visual evidence and strengthening border control [29], [30]. This approach is also valuable for covert military operations [31], perimeter defense [32], [33], convoy protection [34], as well as for homeland security missions and other security applications [35]. Moreover, in sporting events [36], [37], cinematography [38], [39], wildlife monitoring [40], [41], and traffic surveillance [42], [43], the framework would allow for simultaneous tracking and filming of multiple targets while minimizing intrusion and maximizing coverage. The three main contributions of the present paper can be summarized as follows: •A significant enhancement of the existing concept for multi-camera, multi-target tracking agents, achieved by integrating the zooming capability into the on-board tracking cameras. •Introduction of a novel optimization approach tailored for this enhanced framework, addressing the inherent non-convexity of the problem through convex relaxation techniques while ensuring its feasibility. •Extension of prior research to enable the alternative of tracking clusters rather than individual targets, broadening the applicability and efficacy of the proposed framework. It is important to emphasize that this is a theoretical work, focused on highlighting the potential applications of the proposed concept in visual tracking and surveillance of multiple moving targets. The proposed optimal guidance law is designed solely to provide the autonomous vehicle and the set of on-board cameras with optimal setpoints for position and focal lengths, based on the estimated locations of targets at each instant. This work does not address any specific low-level path planning strategy in the 3D environment [44], [45]. Such strategies should account for the maneuverability and dynamic restrictions of the vehicle itself, which can vary significantly depending on whether dealing with holonomic vehicles like VTOL (vertical take-off and landing) multirotor drones or vehicles with strong motion constraints such as fixed-wing aircrafts. These considerations are beyond the scope of this paper and will be addressed in future works, where target observation and vehicle guidance can be integrated in a unified manner. Table Iintroduces the notation used throughout the paper. As a general rule, roman typeface is reserved for vectors and matrices. The rest of the paper is organized as follows. Section II introduces the working scenario, including the assumptions made on the different elements considered and the principles used to formulate the optimization problem. Section III is dedicated to presenting the formulation of the optimization problem. Section IV describes the convex relaxation of the stated problem and provides an intuitive geometric interpretation of this approximation. It also outlines the proposed optimization algorithm. Section V offers some simulation experiments demonstrating the performance of the proposed strategy. Finally, Section VI presents the conclusions of the work and outlines future lines of research. II. FRAMEWORK In this section, we describe the operational framework, proposing a single multi-camera, zooming drone, referred to as the agent, as a suitable concept in the context of aerial surveillance and multi-target tracking. First, we outline the specific operational environment and the underlying assumptions. Next, we elaborate on the foundational concepts behind the optimal reference-generation strategy for single agents, which will be proposed in the following section. A. Working Scenario As mentioned earlier, the concept of integrating multicamera setups onto autonomous aerial vehicles can be applied across a wide range of platforms, spanning from light multirotor systems to heavy fixed-wing UAVs. This setup typically comprises one central wide or ultra-wide angle camera, referred to as C0, surrounded by ntracking cameras denoted as Ci, where i=1, . . . , n. The central camera’s role is to provide a perfectly zenithal, low-resolution overview of the scene below, while the tracking cameras, equipped with motorized zoom lenses, are able to acquire images of distant targets with the desired level of visual detail. Fig. 1illustrates this concept for a multirotor vehicle with n=2 tracking cameras. Fig. 1also illustrates the coordinate frames involved in the setup. First, {W}represents the world coordinate frame defined, for instance, following the ENU (east–north–up) configuration. Its origin is typically located at a convenient stationary point on the ground surface. The next reference frame, 3 TABLE I NOTATION Symbol Description nNumber of tracking cameras. mNumber of nested intervals present in the optimization index. {W}World coordinate frame. {Ci}i-th camera frame (i={0,...,n},i= 0 refers to central cam.). C jzi3D position of i-th target with respect to {Cj}. zi3D position of i-th target with respect to {W}. riEquivalent radius of i-th target in the scene. x3D position of the vehicle with respect to {W}. x∗Optimal value of x. x∗ cConvex approximation to the optimal value of x. ˆx Value of xwhere the convex approximation is made. ℓi(ˆx) Linear ray in R3, starting in ziand with direction (ˆx−zi). pi,r Vehicle’s ideal reference position with respect to i-th target. x, y, h Components of x, that is: x=[x, y, h] ⊤. XConvex feasibility set constraining the solution. hmin Minimum limit of the allowed altitude range. hmax Maximum limit of the allowed altitude range. diEuclidean distance to the i-th target: di:= ∥x−zi∥2. dScalar dummy variable. ηiRelative to i-th target, unitary vector used in convex relaxation. fiActual focal length of the i-th camera. fi,r Reference value for fi. fi,min Minimum allowed value for fi. fi,max Maximum allowed value for fi. siEquivalent radius of i-th target’s projection on the image plane. si,r Reference value for si. si,min Minimum allowed value for si. si,max Maximum allowed value for si. si,pix Equivalent to si, but expressed in pixels. si,r,pix Equivalent to si,r, but expressed in pixels. si,min,pix Equivalent to si,min, but expressed in pixels. si,max,pix Equivalent to si,max, but expressed in pixels. Li,j j-th lower limit imposed on some measure of the i-th cameratarget pair. Ui,j j-th upper limit imposed on some measure of the i-th cameratarget pair. µi,νiWeighting coefficients present in the optimization index, relative to the i-th camera-target pair. τi,j , σi,j Weighting coefficients, relative to the i-th camera-target pair and the j-th nested interval, present in the optimization index. βiExponent in the proposed optimization index, linked to the i-th camera-target pair. αi,j Exponent in the proposed optimization index, linked to the i-th camera-target pair and to the j-th nested interval. ρwiEffective pixel width of the i-th image sensor, in metric units per pixel. ρhiEffective pixel height of the i-th image sensor, in metric units per pixel. ρiRegularized pixel length, to account for sensors with nonsquare pixels ρi:= √ρwiρhi. e1,e2,e3Vectors defining the canonical basis in R3. J(x) Initial optimization index. J◦(x) Simplification of J(x), by removing the non-convex terms. Jc(x,ˆx) Convex approximation of J(x) around ˆx. Ψi,Λi,Γ iMain terms composing the optimization index for the i-th target. lDistance between centers of targets or clusters. Fig. 1. Illustration of the concept using a multirotor platform, a central wide-angle camera on a 2-DOF gimbal, and two tracking cameras with their respective 3-DOF gimbals. Adapted from [20]. denoted as {B}, is rigidly attached to the vehicle’s body. Each camera has its own coordinate frame, denoted as {Ci}. Each gimbal adjusts the inertial orientation angles necessary for the corresponding camera by defining the orientation of {Ci}with respect to {W}. For a detailed description of the kinematics involved, please refer to [20]. We consider multi-target tracking as a two-level task. At the bottom level, wherever the flying platform is positioned, the target aiming problem must be solved. At the top level, assuming that the targets are being correctly aimed to, a steering strategy must be designed so that the vehicle can receive an optimal positioning set-point, as a function of the current target distribution, according to a given criterion. In this work, we strictly focus on the second stage. B. General assumptions and simplifications Here, we outline the a priori conditions that are assumed in the subsequent description. First, it is assumed that an estimate of the vehicle’s altitude and global position is available at all times, provided by the navigation sensory system. Additionally, information about the inertial orientation for each on-board camera is assumed to be available. Furthermore, it will be assumed that the navigation control system (in the case of the aerial platform) or the corresponding orientation control systems (in the case of each gimbal-camera set) are in proper working order, fulfilling the provided attitude, position, or aiming setpoints as efficiently as possible. In order to simplify subsequent geometrical descriptions, it is also postulated that the camera is attached to the corresponding gimbal in such a way as to ensure that any gimbalinduced rotation occurs around the camera nodal point, thus eliminating noticeable parallax effects. In practice, cameras are typically mounted to minimize gravitational inertia, making that particular setup unlikely. However, due to the expected significant proximity between the theoretical nodal point and the center of gravity relative to the distances to the objects 4 of interest, this approximation remains valid. Moreover, it is reasonable to assume that the real-time operation of the visualfeedback gimbal control loop will address any misalignment issues. On the other hand, the effect of lens distortion will be ignored in our description. This will not entail any loss of generality in the proposal, provided that, in the presence of significant lens distortion, images will undergo a rectification process before further processing. With regard to target position estimation and aiming, we are only interested in the position and apparent size on the image of each target or cluster of targets, regardless of their orientation. In the case where a camera is tracking a single target, the referred position will be defined by the apparent center of its silhouette in the image, while in the case of tracking clusters, a suitable aiming center will be defined as a point of interest. On the other hand, as mentioned above, the description of any low-level attitude control of the aerial platform or any lowlevel gimbal control is out of the scope of the paper, as these can be considered issues largely solved from a theoretical and technical point of view [46], [47], [7], [48]. For the sake of simplicity in the description, it is assumed that the distances between the position of any camera frame {Ci}and the own vehicle’s body frame {B}are negligible, compared to the distances from the aerial platform to any one of the targets. In fact, for the intended practical applications, it is expected that this difference comes in several orders or magnitude. It will be assumed that if a particular target is within the field of view of one or more onboard cameras, its absolute position with respect to {W}can be estimated at any time. This estimation may be achieved by adhering to the flatearth assumption, as demonstrated in [20], which eliminates the need for stereo, elevation maps, or any other rangegathering mechanism. This simplification is approximately valid in certain contexts and has been previously exploited by numerous other authors, as evidenced in [17], [49], [12], [50], [28], [13], [19]. Notably, offshore search and rescue (SAR) operations [25], [26], [27] represent a clear example where the flat-earth assumption can be effectively leveraged. For our purposes, if this assumption holds, and with knowledge of the absolute position of the aerial vehicle and the inertial orientation of the observing camera(s), the absolute position of the target can be derived (please refer, for instance, to the description provided in [20]). Similarly, an estimation of the target’s size in the real world could be geometrically derived. In the description that follows, however, we will not necessarily stick to the flat-earth assumption, as any other depth estimation mechanism would be equally valid for our purposes. During initialization, it is anticipated that the downwardfacing, wide-angle camera (C0) will have all the targets of interest within its field of view, enabling the estimation of the position of each target with respect to this camera, denoted as C0zi, where i= 1, . . . , n. Using this vector and the known kinematics of the camera arrangement, it becomes feasible to estimate the position of the target with respect to the tracking camera that will be assigned to it1,Ci. Subsequently, the corresponding camera is oriented towards the target in a manner that positions it approximately at the center of its field of view (please refer again to [20] for the detailed computational description). Finally, the visual servoing module of that gimbal-camera set could be initiated to address any misalignments in the initial aiming towards the target. When employing a clustering strategy, it can be directly implemented within the two-dimensional space of the central camera’s view. Moreover, the positions corresponding to the aiming center of each cluster should be converted from image coordinates to 3D positions. As it can be understood from the description in the previous paragraph, when clustering is utilized, it is just a matter of replacing the point of interest of a single target by the aiming center of the appropriate cluster. C. General guidelines for optimal positioning This section delineates the guidelines for configuring the strategy of the optimal setpoint provision for the multi-camera, multi-target tracking system under investigation. As we place special emphasis on target observation, the objective is not merely to keep the vehicle as close as possible to the set of targets, but to do so while promoting a certain degree of verticality of the viewpoint with respect to each one of them. Simultaneously, we aim to ensure an optimal balance between the distance to each target and the focal length of the respective camera. Essentially, the optimization problem involves quantitatively harmonizing these three aspects. As stated in [20], enforcing verticality may be of particular interest for several reasons: •The absence of any constraint or concern regarding verticality would grant unrestricted flexibility to obtain images of the target from fully zenithal to fully horizontal viewpoints. The subsequent variability in the target’s appearance could pose large challenges for the image-based tracking algorithms, potentially making the tracking task more complex. •In a scenario where multiple targets of unspecified height are in motion, enabling tracking of such targets without imposing minimum verticality and altitude constraints can potentially result in an increased likelihood of occlusions. These occlusions may be due to uninteresting objects obstructing the targets, interference between the targets themselves, or even the risk of one onboard camera obstructing the line of sight of another. •In cases where the goal is to survey targets with minimal intrusion, even with the intention of going unnoticed by the targets themselves, it is advantageous to apply a minimum altitude threshold while optimizing vertical alignment relative to the target set. •When working with clusters instead of isolated targets, we will observe that our isotropic metric for assessing the observation quality of a given cluster becomes more accurate as the perspective of the viewpoints becomes less prominent. 1Without loss of generality, each target (or cluster) is alloted a number that matches the number of the tracking camera to which it is assigned. 5 On the other hand, regarding the optimal trade-off between distance to a given target and the focal length featured by the corresponding camera, one might simply believe that trying to exploit, as much as possible and without any penalty, the available zooming range of the camera is, from an economic or energetic point of view, the best approach. However, this might result in excessive pressure on the part of the aiming device in terms of its accuracy or stability specifications. Alternatively, trying to find a convenient balance between avoidance of unnecessary vehicle displacements and pushing the zooming mechanism to consistently work on its top limit, could be a wise policy. III. PROBLEM FORMULATION This section introduces the proposed strategy for providing references to the multi-camera agent, comprising a suitable combination of the vehicle’s position relative to the set of targets and the specific values of the focal lengths. The main idea is that the proposed optimization index will consist of a term for each target/tracking camera pair. Each of these terms will include several subterms, each imposing penalties for violating specific conditions, which will be described in detail. We denote by x∈R3the main decision variable, which represents the absolute position of the aerial vehicle with respect to {W}. According to one of the assumptions made in Section II-B, it can also be used as an approximation of the position of any onboard camera. We assume the presence of n targets, with positions zi,i= 1, . . . , n, also referred to {W}. Each tracking camera is in charge of tracking one particular target and each target is monitored by a single tracking camera. The scalar rirepresents the equivalent radius of the i-th target, and is used as a convenient measure of the size of each target in the real world. On the other hand, didenotes the Euclidean distance to the i-th target, di=∥x−zi∥2. The focal length of the i-th camera is denoted by fi. It will be assumed that a preferred reference value of this parameter has been a priori chosen within the working range fi,r ∈[fi,min, fi,max]. Starting from the perspective projection principle, we can express the following equivalence of ratios: di ri =fi si , i = 1, . . . , n. (1) where sirepresents the equivalent radius of the projected image of the object in metric units. This distance can be more conveniently obtained from its equivalent expressed in pixels, si,pix:si≈ρisi,pix, where ρirepresents the regularized pixel length of the corresponding camera. This approximation becomes exact for image sensors with square pixels. Suppose the focal length is set to the reference value fi,r. The distance di,r, at which the radius of the target in the image equals a predefined desired value, si,r, is obtained from the following expression: di,r ri =fi,r si,r , i = 1, . . . , n. We conclude that di,r =rifi,r si,r , i = 1, . . . , n represents the most convenient distance to each target, as it ensures the desired size of the target in the image when using the reference focal length. According to our discussion in Subsection II-C, vertical views are often preferred, so the ideal position pi,r for the camera with regard to the i-th target would be pi,r = zi+  0 0 1  di,r = zi+ e3di,r, i = 1, . . . , n. As a starting point, we could obtain the position xby solving the optimization problem min n X i=1 µi∥x−pi,r∥2 2,(2) The weighting factors µi∈R≥0serve to potentially prioritize some targets over others. This unconstrained, strictly convex optimization problem has an explicit solution. By differentiating with respect to x, we obtain that the solution x∗satisfies the following equation: n X i=1 2µi(x∗−pi,r)=0. That is, x∗= n P i=1 µipi,r n P i=1 µi .(3) The problem statement can be improved in different ways. For example, constraints could be included in the vehicle’s position with respect to {W}, i.e., min x∈X n X i=1 µi∥x−pi,r∥2 2. For instance, a restricted altitude range could be specified: X=x∈R3:hmin ≤x⊤e3≤hmax.(4) Another improvement has to do with the verticality of the viewpoints. It should be noted that the altitude provided by (3) does not account for the dispersion of the targets. To prioritize altitude in cases where the targets are widely scattered, a new term could be introduced, such as the following: n X i=1 νi(∥x−zi∥2−(x −zi)⊤e3)2.(5) As we can see, for any particular target, this term penalizes the disparity between the distance to the vehicle, given by the norm of (x−zi), and the projection of this same vector onto the vertical direction. 6 For convenience, the new optimization problem can be rewritten by grouping both previous set of terms together as follows: min x∈X n X i=1 Γi(x,zi)(6) with Γi(x,zi) = µi∥x−pi,r∥2 S+νi(∥x−zi∥2−(x −zi)⊤e3)βi. To provide more generality to the approach, a weighted Euclidean norm has been introduced to penalize the discrepancy (x−pi,r). We thus assume that S∈R3×3is positive definite and that ∥x−pi,r∥2 S= (x −pi,r)⊤S (x −pi,r). Additionally, the square form in (5) has been generalized by allowing any βi≥1as the exponent. Property 1: Suppose that µi≥0, νi≥0, βi≥1,i= 1, . . . , n and Sis a positive definite matrix. Then the function n X i=1 Γi(x,zi)is convex. Proof: First, we notice that µi∥x−pi,r∥2 Sis, by construction, a strictly convex term. Regarding the second term in Γi, since the Euclidean norm is convex and we are subtracting from it a linear form of the variable, this very difference is also convex. Additionally, we observe that this difference is always non-negative. According to a property of convex functions (see, for instance, Example 3.13 of [51]), if g(x) is a convex and nonnegative function of xand βi≥1, then g(x)βi is convex. Therefore, n X i=1 Γi(x,zi)is convex as it results from the non-negative weighted sum of convex functions. Another set of relevant constraints arises when considering bounds on the focal distances. Suppose that one desires to impose that each focal length filies within its operational range [fi,min, fi,max]: fi,min ≤fi≤fi,max, i = 1, . . . , n. (7) As discussed earlier, in accordance with the desired radius of the target projection on the image, si,r, there exists an optimal ratio between the distance to the target and the focal length: di fi =ri si,r , i = 1, . . . , n. (8) We infer that, in order to guarantee the optimal ratio di fi , while preserving the constraints (7), it is advisable to impose the following conditions: ri si,r fi,min ≤di≤ri si,r fi,max, i = 1, . . . , n. By gradually relaxing the equality constraint imposed by (8), one could also consider imposing limits on the radius siin the image, that is si,min ≤si≤si,max. Again, from the pin-hole camera model (1), and the constraints si∈[si,min, si,max], fi∈[fi,min, fi,max]we obtain ri si,max fi,min ≤di≤ri si,min fi,max, i = 1, . . . , n. Violating the previous inequalities indicates that the constraint si∈[si,min, si,max]cannot be satisfied even if the focal length is adjusted to its operational limits (either the target is too close or too distant). To simplify the expressions in the paper, we denote by Li,j and Ui,j various lower and upper bounds on the distance di= ∥x−zi∥2to the i-th target. Let us denote Li,1=ri si,r fi,min, Ui,1=ri si,r fi,max , i=1, . . . , n. Li,2=ri si,max fi,min, Ui,2=ri si,min fi,max , i=1, . . . , n. We notice that, by construction, di,r ∈[Li,1, Ui,1]⊆[Li,2, Ui,2], i = 1, . . . , n. We now enumerate various situations concerning each target, ranging from optimal positioning to non-acceptable positioning: •x = pi,r: This represents optimal vertical alignment with respect to a specific target, as well as the optimal focal length and projected image size. •di=di,r: Optimal focal length and image size. •di∈[Li,1, Ui,1]: Focal length within the operational range and optimal image size. •di∈[Li,2, Ui,2]: Both focal length and image size are within their operational ranges. •di∈ [Li,2, Ui,2]: The size in the image will not meet the constraints, even when setting the focal length to the limits of the operational range. A. Cost functions Ψiand Λi When the number of targets is greater than 3, it is generally not possible to choose xin such a way that the vehicle is placed at the optimal distance with respect to every target. We now define a general class of functions that could be used to penalize the vehicle not being placed at the optimal distance with respect to the i-th target. Consider, as in the previous subsection, that for each target, we have a nested set of intervals that include the desired distance di,r. That is, di,r ∈[Li,0, Ui,0]⊆[Li,1, Ui,1]⊆[Li,2, Ui,2]⊆. . .⊆[Li,m, Ui,m]. According to our previous description, the first interval could be defined as: Li,0=Ui,0=di,r. This constitutes a mere gambit to allow a more general and well-rounded description of the nested-intervals strategy. We propose penalizing functions Ψi(·)and Λi(·)that assign a positive cost when di=∥x−zi∥2is not included in the nested set of intervals. For any scalar variable, d, these functions will be defined as: Ψi(d) = m X j=0 τi,j (max{0, d −Ui,j})αi,j (9) Λi(d) = m X j=0 σi,j (max{0, Li,j −d})αi,j ,(10) where 7 •The scalars τi,j ≥0and σi,j ≥0serve to potentially assign different weights to each target (i.e. prioritizing some targets). They also allow us to penalize some violations more than others. •The positive integers αi,j ≥1are exponents, which will typically be chosen equal to 1 or 2. Example: Suppose that m= 2,τi,0=σi,0= 1,τi,1=σi,1= 2,τi,2=σi,2= 10. Suppose also that one makes Li,0=di,r , Ui,0=di,r , Li,1=ri si,r fi,min , Ui,1=ri si,r fi,max , Li,2=ri si,max fi,min , Ui,2=ri si,min fi,max. Then Ψi(di)+Λi(di)=|di −di,r|+ 2 X j=1 τi,jmax{di −Ui,j , Li,j −di,0}αi,j . The first term penalizes the deviation between di=∥x−zi∥2 and di,r, the second term penalizes deviations from the optimal choice of focal length for di, and the third term penalizes distances where the target’s specifications for image size cannot be met within the operational range of the focal length. We notice that Λi(∥x−zi∥2)is a non-increasing function of ∥x−zi∥2, since it penalizes being too close to the i-th target. The level sets of the function are non-convex, because the inequality Λi(∥x−zi∥2)≤ρtranslates into a constraint of the form ∥x−zi∥2≥rρ, which is non-convex. The opposite happens with functions Ψi(∥x−zi∥2), which, as stated in the following property, are convex with respect to x. This convexity will be relevant when considering numerical optimization methods to obtain the optimal value for x. Property 2: Suppose that τi,j ≥0,αi,j ≥1,i= 1, . . . , n, j= 1, . . . , m. Define function Ψ : R≥0→R≥0as Ψi(d) = m X j=0 τi,j (max{0, d −Ui,j})αi,j , i = 1, . . . , n. Then Ψi(∥x−zi∥2)is convex with respect to x. Proof: By definition, Ψi(∥x−zi∥2) = m X j=0 τi,j (max{0,∥x−zi∥2−Ui,j})αi,j . We notice that Ψi(∥x−zi∥2)consists of the non-negative weighted sum of terms max{0,ΥUi,j (x)}αi,j , where ΥUi,j (x) = ∥x−zi∥2−Ui,j , i = 1, . . . , n, j = 1, . . . , m. (11) Since the Euclidean norm is convex with respect to its argument (see, e.g., subsection 3.1.5 of [51]), we can infer that the functions ΥUi,j (x)are convex. Furthermore, due to the convexity of ΥUi,j (x)and the inequality αi,j ≥1, we can conclude that max 0,ΥUi,j (x)αi,j is also convex (the proof of this claim is presented in Appendix A). Therefore, Ψi(|x−zi|2)is convex since it results from the non-negative weighted sum of convex functions. B. Proposed optimization problem Define the cost function J(x) = n X i=1 Ψi(∥x−zi∥2)+Λi(∥x−zi∥2)+Γi(x,zi).(12) Suppose also that the decision variable x∈R3is constrained to belong to a given non-empty convex set X, e.g. (4). The subsequent optimization problem corresponds to the constrained minimization of the proposed index P(x) : x∗= arg min x∈X J(x).(13) One of the virtues of the stated optimization problem is that it is always feasible, which is of special relevance in this kind of applications. The cost function J(x) consists of three different types of terms, which are analyzed in the following: (i) Ψ(∥x−zi∥2): This term, in view of Property 2, is convex with respect to xand non-decreasing with ∥x−zi∥2. (ii) Λi(∥x−zi∥2): As mentioned in the previous subsection, this term is non-convex and non-increasing with ∥x− zi∥2. (iii) Γi(x,zi): According to Property 1, this term is also convex on x. Since Xis assumed to be convex, the only source of nonconvexity in the proposed optimization problem are the terms Λi(∥x−zi∥2),i= 1, . . . , n. We detail in the following section how to build convex upper approximations to Λi(∥x−zi∥2). These are then used to formulate a convex approximation of the entire original problem. IV. CONVEX RELAXATION As commented before, the functions Λi(∥x−zi∥2) = m X j=0 σi,j (max{0, Li,j − ∥x−zi∥2})αi,j , for i= 1, . . . , n, are not convex with respect to x. We are, nevertheless, interested in attaining a convenient convex approximation of such functions. In general terms, convexification offers an attractive alternative from a practical point of view, as it will allow a more efficient implementation of future practical applications. By approximating non-convex optimization problems with convex problems, convexification reduces computational complexity and facilitates the development of scalable and robust solutions. Specifically, this section will demonstrate how to derive upper convex bounds for each function Λi(|x−zi|2). The resulting approximate convex optimization algorithm will be evaluated under various scenarios in Section V. By construction, Λi:R≥0→R≥0is a non-increasing function on its argument. That is, da≤dbimplies Λi(da)≥Λi(db), i = 1, . . . , n. (14) Suppose that ηi∈R3, satisfies ∥ηi∥2≤1. Then, η⊤ i(x −zi)≤ ∥ηi∥2∥x−zi∥2≤ ∥x−zi∥2. 8 From this and the non-increasing nature of Λi:R≥0→R≥0 with respect to its argument, we obtain Λi(η⊤(x −zi)) ≥Λi(∥x−zi∥2),(15) ∀η∈ { η∈R3:∥η∥2≤1},∀x∈R3. The following property shows how to obtain η1,η2,...,ηnin such a way that the upper bound Λi(η⊤ i(x −zi)) coincides with the original function Λi(∥x−zi∥2)at a given point ˆx. Property 3: Define the functions ηi:R3→R3,i= 1, . . . , n, as ηi(ˆx) =    ˆx −zi ∥ˆx −zi∥2 if ˆx = zi 0otherwise. Then, Λi((x −zi)⊤ηi(ˆx)) ≥Λi(∥x−zi∥2),∀x∈R3, i = 1, . . . , n. Moreover, the equality holds along the ray ℓi(ˆx) = {zi+ ληi(ˆx) : λ∈R≥0}. That is, given i∈ {1, . . . , n}, Λi((x −zi)⊤ηi(ˆx)) = Λi(∥x−zi∥2),∀x∈ℓi(ˆx). Proof: We first prove the first claim. Given i∈ {1, . . . , n}, we consider two cases, ˆx = zi, and ˆx =zi: (i) ˆx = zi: In this case ηi(ˆx) = 0 ∈R3. From ηi(ˆx) = 0 and the non-increasing nature of Λi:R≥0→R≥0, see (14), we obtain Λi((x −zi)⊤ηi(ˆx)) = Λi(0) ≥Λi(∥x−zi∥2),∀x∈R3. (ii) ˆx = zi: We have that ˆx = ziimplies ∥ηi(ˆx)∥2= 1. Thus, it follows from inequality (15) that Λi((x −zi)⊤ηi(ˆx)) ≥Λi(∥x−zi∥2),∀x∈R3. We now prove the second claim. As before, we consider two cases, ˆx = zi, and ˆx =zi: (i) ˆx = zi: In this case ηi(ˆx) = 0 and the ray ℓi(ˆx) collapses to a single point, ℓi(ˆx)={ˆx}. Thus, the equality Λi(∥x− zi∥2) = Λi((x −zi)⊤ηi(ˆx)) = Λi(0) holds trivially for all x∈ℓi(ˆx). (ii) ˆx = zi: For every x = zi+ληi(ˆx) ∈ℓi(ˆx), we have ∥x−zi∥2=∥zi+ληi(ˆx) −zi∥2 =|λ| ∥ηi(ˆx)∥2=|λ|. (x −zi)⊤ηi(ˆx) = (zi+ληi(ˆx) −zi)⊤ηi(ˆx) =ληi(ˆx)⊤ηi(ˆx) = λ=|λ|, where the last equality holds because, according to the definition of the ray ℓi(ˆx),λ≥0. Hence, for every x∈ ℓi(ˆx), we have (x −zi)⊤ηi(ˆx) = ∥x−zi∥2. and from this, we finally conclude Λi((ˆx −zi)⊤ηi(ˆx)) = Λi(∥ˆx −zi∥2),∀x∈ℓi(ˆx). The previous property states that an upper bound for index J(x) introduced in (12) can be obtained by replacing Λi(∥x− zi∥2)with Λi((x −zi)⊤ηi(ˆx)). As it is claimed in the next property, the obtained upper bound is convex on x. Property 4: Given ˆx ∈R3, define the functions ηi:R3→ R3,i= 1, . . . , n, as ηi(ˆx) :=    ˆx −zi ∥ˆx −zi∥2 if ˆx = zi 0otherwise. Then, Jc(x,ˆx) := n X i=1 Ψi(∥x−zi∥2)+Λi((x−zi)⊤ηi(ˆx))+Γ(x,zi), (16) satisfies (i) Jc(ˆx,ˆx) = J(ˆx),∀ˆx ∈R3. (ii) Jc(x,ˆx) ≥J(x),∀x∈R3,∀ˆx ∈R3. (iii) Jc(x,ˆx) is a strictly convex function with respect to x, for all ˆx ∈R3. Proof: By construction, we have (ˆx −zi)⊤ηi(ˆx) = ∥ˆx − zi∥2,i= 1, . . . , n. Thus, n X i=1 Λi((ˆx −zi)⊤ηi(ˆx)) = n X i=1 Λi(∥ˆx −zi∥2),∀ˆx ∈R3. From this equality, we directly infer the first claim of the property. Additionally, in view of Property 3, we have Λi((x −zi)⊤ηi(ˆx)) ≥Λi(∥x−zi∥2),∀x∈R3, i = 1, . . . , n. which proves the veracity of the second claim. In order to finish the proof, we first notice that, as stated in Property 2, the terms Ψi(∥x−zi∥2)are convex on x. We now proceed similarly to the proof of Property 2to show that the terms Λi((x −zi)⊤ηi(ˆx)) are also convex on x. Note that Λi((x−zi)⊤ηi(ˆx)) consists of the non-negative weighted sum of terms of the form max{0,ΥLi,j (x,ˆx)}αi,j , where ΥLi,j (x,ˆx) = Li,j −(x −zi)⊤ηi(ˆx) (17) i= 1, . . . , n, j = 1, . . . , m. are convex functions since they are linear functions on x. The convexity of max{0,ΥLi,j (x,ˆx)}αi,j is proven in Appendix A. From the convex nature of these functions, we conclude that Λi((x −zi)⊤ηi(ˆx)) is convex on xsince it results from the non-negative weighted sum of convex functions. This, along with the convexity of the terms Γi(x,zi)and Ψi(·), established by Properties 1and 2respectively, proves the third claim. A. Graphical interpretation It should be clear by now that our problem unfolds in the Euclidean 3D space. However, in order to provide an intuitive graphical interpretation of the approximation provided by the convex relaxation, in this subsection, we will make a 2-dof reduction of the problem, assuming that all targets keep the same constant value of the second coordinate (let us assume that for all targets this component remains equal to zero) while they move around. By dropping this second component, we will be able to plot a 3D surface representing the original optimization index J(x) described by (12), as well as the 15 the actual image coverage, while the green circumferences in the bottom-right subplots reflect the reference values si,r. The file named testComplexScenario.mp4, included as supplementary material, provides the complete simulation of this experiment in video format. VI. CONCLUSIONS AND FUTURE WORK A novel approach to the problem of multi-target tracking and surveillance by means of a single autonomous aerial agent is presented. Based on the premise that such an agent is equipped with several independently-steerable zooming cameras, the objective is to infer the best reference position for the agent at each instant, ensuring optimal monitoring of a set of designated targets. By exploiting the flat-earth assumption and the knowledge of the agent’s altitude measurement, the position and equivalent radius of the targets or clusters can be obtained as a normalized planar footprint on the scene below. Ideal monitoring of a target or cluster is then formulated as the fit of this equivalent footprint projection to a reference circle centered on the image acquired by the tracking camera allocated to that target or cluster. Simultaneously, it is essential to ensure that the solution respects the allowed altitude band for the vehicle and that the resulting focal length of every camera remains within its operational range, as long as it is feasible. The resulting constrained optimization problem is found to be intrinsically non-convex and non-differentiable. An original approximation to the problem is developed through convex relaxation to cope with this non-convex nature while ensuring the feasibility of the problem. Simulated experiments demonstrate the greater flexibility of the proposed scheme to adapt to varied target scattering conditions and significant differences in target or cluster sizes. As part of future works, immediate interest is focused on integrating an optimal control law for effectively guiding the vehicle, considering its dynamic characteristics alongside the requirements for ideal target monitoring described in this paper. Another immediate future task is extending the problem to a collaborative multi-agent framework. Generalizing the solution to coordinate a team of multi-camera agents in scenarios with many widely scattered targets will enable visual surveillance most effectively. Additionally, there is interest in reformulating the problem to utilize pure visual information directly, without relying on any 3D reconstruction methods, which can be seen as the main limitation in the proposed approach. APPENDIX A PROOF OF THE CONVEXITY OF (max{0,Υ(x)})α Throughout the description given in this appendix, f(x), h(x) and Υ(x) represent generic scalar functions of vector variable, while g(·)represents a generic scalar function of scalar variable. The scalar αwill be used as a generic exponent, while µis used as weighting factor. Property 6: Suppose that Υ : Rnx→Ris convex. Then f(x) = (max{0,Υ(x)})αis convex in Rnxfor every α≥1. Proof: The proof relies on standard results from the composition of convex functions (see, e.g. subsection 3.2.4. of [51]), and it is included here for completeness. Denote h(x) = max{0,Υ(x)}. Since h(x) is the pointwise maximum of two convex functions, it is also convex (see, e.g. subsection 3.2.3. of [51]). From the convexity of h(x) we have h(µxa+ (1 −µ)xb)≤µh(xa) + (1 −µ)h(xb), ∀xa,∀xb,∀µ∈[0,1]. Defining g:R→Ras g(y) = yαwe obtain f(x) = (max{0,Υ(x)})α=hα(x) = g(h(x)), where, by construction, h(x) ∈R≥0,∀x. We analyze the first and second derivatives of g(y) = yαfor non-negative values of yand α≥1: g′(y) = αyα−1≥0,∀y≥0 g′′(y) = α(α−1)yα−2≥0,∀y≥0. From here, we infer that g(y) = yαis a nondecreasing and convex function in R≥0for every α≥1. Suppose now that xaand xbare two arbitrary vectors in Rnx,µ∈[0,1], and α≥1. Taking into account that h: Rnx→R≥0is convex in Rnx, and that, given α≥1,g(y) = yαis non decreasing and convex in R≥0, we obtain, f(µxa+ (1 −µ)xb)=(h(µxa+ (1 −µ)xb))α =g(h(µxa+ (1 −µ)xb)) ≤g(µh(xa) + (1 −µ)h(xb)) ≤µg(h(xa)) + (1 −µ)g(h(xb)) =µf(xa) + (1 −µ)f(xb). We notice that the first inequality is due to the convexity of h(·)and the non-decreasing nature of g(·)in R≥0. The second inequality is due to the convexity of g(·)in R≥0. Thus, we have proved that f(µxa+ (1 −µ)xb)) ≤µf(xa) + (1 −µ)f(xb),∀xa, ∀xb,∀µ∈[0,1]. Hence, f(x) is convex. ACKNOWLEDGMENT This work was supported by grants PID2022-142946NA- I00 and PID2022-141159OB-I00, funded by MICIU/AEI/ 10.13039/501100011033 and by ERDF/EU. REFERENCES [1] M. Schwager, B. J. Julian, M. Angermann, and D. Rus, “Eyes in the sky: Decentralized control for the deployment of robotic camera networks,” Proceedings of the IEEE, vol. 99, no. 9, pp. 1541–1561, 2011. 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