Limited Field Of View Research Articles

Cascaded Residual-Based Progressive-Refinement Generative Adversarial Network for Multi-Modal Cross-View Image Translation

Cross-view image-to-image translation refers to generating what the scenery looks like from different views at the same location, which involves simultaneous semantic and appearance translation. Previous works mainly focused on semantic translation that used a semantic map as additional information to guide the network to learn a good semantic mapping across different views. However, the appearance translation between two different views remains ambiguous due to the huge differences in distance and view angle, not to mention multi-modal translation. In this paper, we propose a novel end-to-end network called Cascaded Residual-based Progressive-refinement GAN (CRP-GAN). Specifically, an aerial image and a semantic map are used to generate multi-modal refined panoramas progressively. There are three novelties in the CRP-GAN. Firstly, it is able to generate ground-level panoramas of a wide field of view rather than images with a limited field of view, by fully exploiting the spatial information of the aerial image. Secondly, the proposed model is capable of generating multi-modal cross-view images, which is different from previous one-to-one image translation. Thirdly, a novel cascaded image refinement strategy is proposed to synthesize images with more details and less blurs at each stage of refinement. We conducted extensive experiments on CVUSA and Dayton datasets for cross-view image-to-image translation.

200 mm optical synthetic aperture imaging over 120 meters distance via macroscopic Fourier ptychography

Fourier ptychography (FP) imaging, drawing on the idea of synthetic aperture, has been demonstrated as a potential approach for remote sub-diffraction-limited imaging. Nevertheless, the farthest imaging distance is still limited to around 10 m, even though there has been a significant improvement in macroscopic FP. The most severe issue in increasing the imaging distance is the field of view (FoV) limitation caused by far-field conditions for diffraction. Here, we propose to modify the Fourier far-field condition for rough reflective objects, aiming to overcome the small FoV limitation by using a divergent beam to illuminate objects. A joint optimization of pupil function and target image is utilized to attain the aberration-free image while estimating the pupil function simultaneously. Benefiting from the optimized reconstruction algorithm, which effectively expands the camera’s effective aperture, we experimentally implement several FP systems suited for imaging distances of 12 m, 65 m, and 120 m with the maximum synthetic aperture of 200 mm. The maximum synthetic aperture is thus improved by more than one order of magnitude of the state-of-the-art works from the furthest distance, with an over fourfold improvement in the resolution compared to a single aperture. Our findings demonstrate significant potential for advancing the field of macroscopic FP, propelling it into a new stage of development.

A Near‐Zero Index Metamaterial Lens for Reduced Complexity and High‐Performance Active Electronically Scanned Arrays

AbstractA lens consisting of an anisotropic near‐zero index metamaterial (NZIM) is introduced for improving the far‐field performance of active electronically scanned arrays (AESA). Several simulation studies demonstrate how the NZIM lens (metalens) can be functionalized to transform the embedded element pattern of an array from a typical cosinusoidal shape to a flat‐topped pattern, dramatically reducing the gain at wider angles. This corresponds to reductions in scan loss and suppression of grating lobes in the desired field of view (FOV), especially for arrays with large element spacing (i.e., sparse or thinned arrays). The metalens concept is demonstrated through several simulation studies illustrating the beam shaping capability of NZIM materials. A fabricated metalens demonstrates full suppression of grating lobes and minimal scan loss with a ±10° FOV, which is ideally suited for limited FOV applications such as geosynchronous satellite communications.

CGLight: An effective indoor illumination estimation method based on improved convmixer and GauGAN

Illumination consistency is a key factor for seamlessly integrating virtual objects with real scenes in augmented reality (AR) systems. High dynamic range (HDR) panoramic images are widely used to estimate scene lighting accurately. However, generating environment maps requires complex deep network architectures, which cannot operate on devices with limited memory space. To address this issue, this paper proposes CGLight, an effective illumination estimation method that predicts HDR panoramic environment maps from a single limited field-of-view (LFOV) image. We first design a CMAtten encoder to extract features from input images, which learns the spherical harmonic (SH) lighting representation with fewer model parameters. Guided by the lighting parameters, we train a generative adversarial network (GAN) to generate HDR environment maps. In addition, to enrich lighting details and reduce training time, we specifically introduce the color consistency loss and independent discriminator, considering the impact of color properties on the lighting estimation task while improving computational efficiency. Furthermore, the effectiveness of CGLight is verified by relighting virtual objects using the predicted environment maps, and the root mean square error and angular error are 0.0494 and 4.0607 in the gray diffuse sphere, respectively. Extensive experiments and analyses demonstrate that CGLight achieves a balance between indoor illumination estimation accuracy and resource efficiency, attaining higher accuracy with nearly 4 times fewer model parameters than the ViT-B16 model.

Frequency and clinical implications of findings on true whole-body 18F-FDG PET in the assessment of breast cancer.

In the assessment of breast cancer using 18-F FDG PET/CT, the incremental clinical benefit in performing a true whole-body PET/CT (with a field of view (FOV) from the vertex to the toes) over a limited whole-body PET/CT (with a FOV from the base of skull to the mid-thighs) is uncertain. Two hundred and one studies of 120 patients who underwent staging or restaging true whole body 18F-FDG PET/CT for breast cancer were retrospectively identified. Any abnormal hypermetabolic or structural focus outside the limited FOV was recorded and characterised, and referenced with the patient's known disease status and any symptomatology. A total of 18 (9.0%) studies had FDG avid and/or structural abnormalities detected outside the limited whole-body FOV which were identified as malignant. Seventeen out of 18 (94.4%) were skeletal and of these, 15/17 (88.2%) were located within the lower limbs. In three cases, there were de novo findings but identified in the presence of interval progression of other metastases within the limited whole-body FOV. None of these additional findings is known to have resulted in a change to staging or clinical management. In the assessment of breast cancer, a true whole-body PET/CT can reveal metastases outside the limited whole-body FOV, but these are unlikely to be encountered in isolation and therefore may have little bearing on clinical stage or management. Ultimately, while the choice of FOV should still be based on the individual patient situation, routine utilisation of the true whole-body FOV in the asymptomatic patient may not be necessary.

Distributed Multi-Sensor Fusion for Multi-Group/Extended Target Tracking with Different Limited Fields of View

The identification of sensing group targets and extended targets is of paramount importance in the context of vehicle tracking and early warning detection. As the scope of target monitoring and tracking extends, conventional single-sensor-based tracking techniques are proving to be inadequate in meeting the practical demands of the field. Consequently, multi-sensor fusion tracking technology has emerged as a viable alternative. However, the use of multiple sensors is constrained by their limited fields of view (FOVs), which leads to issues such as the loss of target detection and the introduction of false targets after fusion. Hence, this study proposes a combination of weighted geometric averaging (WGA) and weighted arithmetic averaging (WAA) methods to solve distributed multi-group/extended target tracking with different fields of view. Specifically, a local Poisson multi-Bernoulli mixture (PMBM) filter was first used on individual sensors. Subsequently, we combined a sequential fusion technique and the proposed fusion approach to fuse the PMBM filter densities. The efficacy and superiority of this approach were demonstrated through simulations.

AI-based open-source software for cephalometric analysis from limited FOV radiographs

BackgroundArtificial Intelligence (AI) in dental diagnostics is evolving, offering innovative approaches for conducting cephalometric analysis with less manual input and overcoming the limitations of traditional imaging methods. To enhance the diagnostic processes in dentistry, an open-source software that utilises AI to improve the extraction of cephalometric values from limited field of view (FOV) images was created. Material and MethodsReduced FOV images lack several vital cephalometric landmarks, prompting the creation of predictive models to estimate missing values. The GridSearchCV algorithm and other algorithms were used to construct predictive models using software and to select the best models. The software was validated by comparing the predicted values with the actual measurements and calculating the mean squared error using Excel. Further validation involved a randomly selected cohort of 25 untreated orthodontic cases. ResultsEvaluation of the software showed that it was effective in accurately predicting key cephalometric measurements, suggesting that it could be a reliable tool for clinical use. However, some variations were noted in its predictive accuracy across different measurements, indicating areas that could benefit from further development. The software could align closely with the actual cephalometric measurements through detailed statistical analysis. ConclusionsThe integration of AI into cephalometric analysis with the software could represent progress, potentially leading to more efficient dental diagnostics and a reduction in the need for additional X-rays. This study aimed to advance the integration and refinement of AI in healthcare, focusing on minimising bias and understanding its impact on clinical decisions. In future studies, the application of AI in dental practice should be expanded to address these challenges. Clinical Significance StatementThis software integrates AI into clinical practice to enhance the diagnostic and therapeutic phases for patient benefit. It enables precise and comprehensive cephalometric analyses using data previously considered insufficient, thereby reducing the need for X-rays and improving patient care.

Automated Pipeline for Multi-LiDAR Extrinsic Calibration: Experimental Evaluation and Performance Analysis

Abstract. In recent years, 3D LiDAR (Light Detection and Ranging) has become a crucial sensor in various applications such as autonomous vehicles, robotics, object detection, precision forestry, and agriculture. However, specific LiDAR sensors, such as Velodyne VLP16, exhibit some drawbacks, such as a limited field of view and sparse data density, making them inadequate for certain specific applications. Hence, this research proposes a method for calibrating two VLP16 LiDAR sensors to improve coverage and reduce blind spots. The pipeline for performing the calibration begins with mounting LiDARs correctly in a rod at a specific orientation and distance, followed by the selection of multiple sites for data collection, then performing a registration algorithm for estimating calibration parameters, and then an accuracy assessment of the calibrated point clouds. The registration algorithm used here is a modified version of ICP (Iterative Closest Point), which overcomes the need for initialization and eliminates manual intervention in installing targets or retro-reflectors. Finally, we evaluated the accuracy of the fused point cloud collected in an open environment using two calibrated Velodyne VLP16 sensors. For accuracy assessment, we used the PCA eigenvalue and RMSE value to observe how tightly point clouds are fused. As a calibration result, we got the orientation and translation parameters, which are used to achieve the common coordinate system, and accomplished calibration accuracy up to single-digit precision from all the experimental sites. Now the system of two calibrated VLP16 sensors will provide higher coverage and increased data density and might be useful for forest applications.

Lensless Mueller holographic microscopy with robust noise reduction for multiplane polarization imaging

Lensless holographic microscopy has emerged as a powerful and cost-effective tool for computational imaging, offering high resolution over a large field of view, beneficial for various biological applications. However, conventional approaches can struggle with contrast and accurate visualization of diverse components over the samples, which can directly affect the diagnostic precision of the techniques. Mueller imaging, while offering detailed, stain-free observations of polarized light responses in samples, often has a limited field of view and single plane information. This is due to the use of high NA microscope objectives and generally complex hardware setups, thus narrowing its practical effectiveness. This work introduces a Lensless Mueller Holographic Microscopy (LMHM) system that overcomes these limitations, enabling large field of view, volumetric multi-layer imaging, and Mueller matrix computation using in-line lens-free holography setup. The proposed system provides precision visualization of polarization information in samples, offering high-quality features due to the incorporation of a numerical multi-height Gerchberg-Saxton reconstruction algorithm with additional complex field filtering and a physical rotating diffuser. The proposed LMHM framework is validated with a calibrated USAF 1951 birefringent test target. A multiplane sample containing cloth fiber is utilized to study the LMHM capabilities of imaging volumetric samples. Finally, the LMHM is used to analyze two mice’s brain slices, effectively showcasing this organ’s anatomy. Among other structures in the brain, the proposed method easily allows the visualization of, e.g., the corpus callosum. These results constitute a proof-of-concept evaluation for bioimaging applications.

PERF: Panoramic Neural Radiance Field From a Single Panorama.

Neural Radiance Field (NeRF) has achieved substantial progress in novel view synthesis given multi-view images. Recently, some works have attempted to train a NeRF from a single image with 3D priors. They mainly focus on a limited field of view with a few occlusions, which greatly limits their scalability to real-world 360-degree panoramic scenarios with large-size occlusions. In this paper, we present PERF, a 360-degree novel view synthesis framework that trains a panoramic neural radiance field from a single panorama. Notably, PERF allows 3D roaming in a complex scene without expensive and tedious image collection. To achieve this goal, we propose a novel collaborative RGBD inpainting method and a progressive inpainting-and-erasing method to lift up a 360-degree 2D scene to a 3D scene. Specifically, we first predict a panoramic depth map as initialization given a single panorama and reconstruct visible 3D regions with volume rendering. Then we introduce a collaborative RGBD inpainting approach into a NeRF for completing RGB images and depth maps from random views, which is derived from an RGB Stable Diffusion model and a monocular depth estimator. Finally, we introduce an inpainting-and-erasing strategy to avoid inconsistent geometry between a newly-sampled view and reference views. The two components are integrated into the learning of NeRFs in a unified optimization framework and achieve promising results. Extensive experiments on Replica and a new dataset PERF-in-the-wild demonstrate the superiority of our PERF over state-of-the-art methods. Our PERF can be widely used for real-world applications, such as panorama-to-3D, text-to-3D, and 3D scene stylization applications.

A general stitching measurement for stereo deflectometry

A slope stitching algorithm is proposed to fulfill the measurement of a large object under the limited field of view in a stereo deflectometry system. The tested object can be easily divided into three sub-apertures under a unique world coordinate system w.r.t. the valid regions of the two cameras. Then the sub-apertures are stitched via slopes instead of point cloud. And, to produce a much smoother slope in the common regions of two cameras, a weighted average method is used to guide the merging process of the slope. The method expands the measurement area into a binocular combination area without the need for predefined markers or moving any devices. Experimental results confirm the feasibility and efficiency of our proposed algorithm, highlighting its advantages in testing large-size optical elements with high accuracy and cost-effectiveness.

A Pressure Sensing Device to Assist in Colonoscopic Procedures to Prevent Perforation-A Case Study.

Colonoscopy has a limited field of view because it relies solely on a small camera attached to the end of the scope and a screen displayed on a monitor. Consequently, the quality and safety of diagnosis and treatment depend on the experience and skills of the gastroenterologist. When a novice attempts to insert the colonoscope during the procedure, excessive pressure can sometimes be applied to the colon wall. This pressure can cause a medical accident known as colonic perforation, which the physician should prevent. We propose an assisting device that senses the pressure applied to the colon wall, analyzes the risk of perforation, and warns the physician in real time. Flexible pressure sensors are attached to the surface of the colonoscope shaft. These sensors measure pressure signals during a colonoscopy procedure. A simple signal processor is used to collect and process the pressure signals. In the experiment, a colonoscope equipped with the proposed device was inserted into a simulated colon made from a colon extracted from a pig. The processed data were visually communicated to the gastroenterologist via displays and light-emitting diodes (LEDs). The device helps the physician continuously monitor and prevent excessive pressure on the colon wall. In this experiment, the device appropriately generated and delivered warnings to help the physicians prevent colonic perforation. In the future, the device is to be improved, and more experiments will be performed in live swine models or humans to confirm its efficacy and safety.

Solar Wind Structures from the Gaussianity of Magnetic Magnitude

The heliosphere is permeated with highly structured solar wind originating from the Sun. One of the primary science objectives of Parker Solar Probe (PSP) is to determine the structures and dynamics of the plasma and magnetic fields at the sources of the solar wind. However, establishing the connection between in situ measurements and structures and dynamics in the solar atmosphere is challenging: most of the magnetic footpoint mapping techniques have significant uncertainties in the source localization of a plasma parcel observed in situ, and the PSP plasma measurements suffer from a limited field of view. Therefore, it lacks a universal tool to self-contextualize the in situ measurements. Here we develop a novel time series visualization method named Gaussianity Scalogram. Utilizing this method, by analyzing the magnetic magnitude data from both PSP and Ulysses, we successfully identify in situ structures that are possible remnants of solar atmospheric and magnetic structures spanning more than 7 orders of magnitude, from years to seconds, including polar and midlatitude coronal holes, as well as structures compatible with supergranulation, “jetlets” and “picoflares.” Furthermore, computer simulations of Alfvénic turbulence successfully reproduce the Gaussianization of magnetic magnitude, supporting the observed distribution. Building upon these discoveries, the Gaussianity Scalogram can help future studies to reveal the fractal-like fine structures in the solar wind time series from both PSP and a decades-old data archive.

Regiomontanus angle maximization for catadioptric sensors with paraboloidal mirrors

Dioptric cameras with conventional perspective projection have well established analytical properties. However, they suffer from perspective distortions and only have a limited field of view. Catadioptric cameras offer panoramic imaging. Their extensive field of view together with projection specific image analysis, can simplify many computer vision tasks. Several properties of catadioptric projection for geometric primitives such as points and lines have been addressed and have also been used for calibration. However, higher order geometric properties are yet to be investigated. Such analysis is complicated by the specifics of the warping of the scene by catadioptric projection. One such property, that is the subject of this work, is the Regiomontanus angle maximization relative to the effective viewpoint of the sensor. This work considers catadioptric sensors with paraboloidal mirrors, that is, paracatadioptric sensors. Analytical ray tracing of a simplified 1D world object gives its projection in the image and an expression for its length. The optimization of the length of the projection results in a third degree equation for the Regiomontanus distance that can be solved explicitly. The Khayyam geometric solution of this equation provides the Regiomontanus distance of maximum subtended projection for these cameras. Applications of these results in various contexts are presented and discussed.

Automated extrinsic calibration of solid-state frame LiDAR sensors with non-overlapping field of view for monitoring indoor stockpile storage facilities

Several industrial and commercial bulk material management applications rely on accurate, current stockpile volume estimation. Proximal imaging and LiDAR sensing modalities can be used to derive stockpile volume estimates in outdoor and indoor storage facilities. Among available imaging and LiDAR sensing modalities, the latter is more advantageous for indoor storage facilities due to its ability to capture scans under poor lighting conditions. Evaluating volumes from such sensing modalities requires the pose (i.e., position and orientation) parameters of the used sensors relative to a common reference frame. For outdoor facilities, a Global Navigation Satellite System (GNSS) combined with an Inertial Navigation System (INS) can be used to derive the sensors’ pose relative to a global reference frame. For indoor facilities, GNSS signal outages will not allow for such capability. Prior research has developed strategies for establishing the sensor position and orientation for stockpile volume estimation while relying on multi-beam spinning LiDAR units. These approaches are feasible due to the large range and Field of View (FOV) of such systems that can capture the internal surfaces of indoor storage facilities.The mechanical movement of multi-beam spinning LiDAR units together with the harsh conditions within indoor facilities (e.g., excessive humidity, wide range of temperature variation, dust, and corrosive environment in deicing salt storage facilities) limit the use of such systems. With the increasing availability of solid-state LiDAR units, there is an interest in exploring their potential for stockpile volume estimation. Despite their higher robustness to harsh conditions, solid-state LiDAR units have shorter distance measurement range and limited FOV when compared with multi-beam spinning LiDAR. This research presents a strategy for the extrinsic calibration (i.e., estimating the relative pose parameters) of installed solid-state LiDAR units inside stockpile storage facilities. The extrinsic calibration is made possible using deployed spherical targets and a complete, reference scan of the facility from another LiDAR sensing modality. The proposed research introduces strategies for: 1) automated extraction of the spherical targets; 2) automated matching of these targets in the solid-state LiDAR and reference scans using invariant relationships among them; and 3) coarse-to-fine estimation of the calibration parameters. Experimental results in several facilities have shown the feasibility of using the proposed methodology to conduct the extrinsic calibration and volume evaluation with an error percentage less than 3.5% even with occlusion percentages reaching up to 50%.

Graph-based multi-agent reinforcement learning for collaborative search and tracking of multiple UAVs

This paper investigates the challenges associated with Unmanned Aerial Vehicle (UAV) collaborative search and target tracking in dynamic and unknown environments characterized by limited field of view. The primary objective is to explore the unknown environments to locate and track targets effectively. To address this problem, we propose a novel Multi-Agent Reinforcement Learning (MARL) method based on Graph Neural Network (GNN). Firstly, a method is introduced for encoding continuous-space multi-UAV problem data into spatial graphs which establish essential relationships among agents, obstacles, and targets. Secondly, a Graph AttenTion network (GAT) model is presented, which focuses exclusively on adjacent nodes, learns attention weights adaptively and allows agents to better process information in dynamic environments. Reward functions are specifically designed to tackle exploration challenges in environments with sparse rewards. By introducing a framework that integrates centralized training and distributed execution, the advancement of models is facilitated. Simulation results show that the proposed method outperforms the existing MARL method in search rate and tracking performance with less collisions. The experiments show that the proposed method can be extended to applications with a larger number of agents, which provides a potential solution to the challenging problem of multi-UAV autonomous tracking in dynamic unknown environments.

Robot-Assisted Augmented Reality (AR)-Guided Surgical Navigation for Periacetabular Osteotomy.

Periacetabular osteotomy (PAO) is an effective approach for the surgical treatment of developmental dysplasia of the hip (DDH). However, due to the complex anatomical structure around the hip joint and the limited field of view (FoV) during the surgery, it is challenging for surgeons to perform a PAO surgery. To solve this challenge, we propose a robot-assisted, augmented reality (AR)-guided surgical navigation system for PAO. The system mainly consists of a robot arm, an optical tracker, and a Microsoft HoloLens 2 headset, which is a state-of-the-art (SOTA) optical see-through (OST) head-mounted display (HMD). For AR guidance, we propose an optical marker-based AR registration method to estimate a transformation from the optical tracker coordinate system (COS) to the virtual space COS such that the virtual models can be superimposed on the corresponding physical counterparts. Furthermore, to guide the osteotomy, the developed system automatically aligns a bone saw with osteotomy planes planned in preoperative images. Then, it provides surgeons with not only virtual constraints to restrict movement of the bone saw but also AR guidance for visual feedback without sight diversion, leading to higher surgical accuracy and improved surgical safety. Comprehensive experiments were conducted to evaluate both the AR registration accuracy and osteotomy accuracy of the developed navigation system. The proposed AR registration method achieved an average mean absolute distance error (mADE) of 1.96 ± 0.43 mm. The robotic system achieved an average center translation error of 0.96 ± 0.23 mm, an average maximum distance of 1.31 ± 0.20 mm, and an average angular deviation of 3.77 ± 0.85°. Experimental results demonstrated both the AR registration accuracy and the osteotomy accuracy of the developed system.

Monocular Metasurface for Structured Light Generation and 3D Imaging with a Large Field-of-View.

Structured light three-dimensional (3D) imaging technology captures the geometric information on 3D objects by recording waves reflected from the objects' surface. The projection angle and point number of the laser dots directly determine the field-of-view (FOV) and the resolution of the reconstructed image. Conventionally, diffractive optical elements with micrometer-scale pixel size have been used to generate laser dot arrays, leading to limited FOV and point number within the projection optical path. Here, we theoretically put forward and experimentally demonstrate a monocular geometric phase metasurface composed of deep subwavelength meta-atoms to generate a 10 798 dot array within an FOV of 163°. Attributed to the vast number and high-density point cloud generated by the metasurface, the 3D reconstructed results showcase a maximum relative error in depth of 5.3 mm and a reconstruction error of 6.07%. Additionally, we propose a spin-multiplexed metasurface design method capable of doubling the number of lattice points. We demonstrate its application in the field of 3D imaging through experiments, where the 3D reconstructed results show a maximum relative depth error of 0.44 cm and a reconstruction error of 2.78%. Our proposed metasurface featuring advanced point cloud generation holds substantial potential for various applications such as facial recognition, autonomous driving, virtual reality, and beyond.

Aberration correction in epi-fluorescence microscopy using unknown speckle illumination.

Diffraction-limited imaging in epi-fluorescence microscopy remains a challenge when sample aberrations are present or when the region of interest rests deep within an inhomogeneous medium. Adaptive optics is an attractive solution, albeit with limited field of view and requiring relatively complicated systems. Alternatively, reconstruction algorithms have been developed over the years to correct for aberrations. Unfortunately, purely post-processing techniques tend to be ill-posed and provide only incremental improvements in image quality. Here, we report a computational optical approach using unknown speckle illumination and a matched reconstruction algorithm to correct for aberrations and reach or surpass diffraction limited resolution. The data acquisition is performed by shifting an unknown speckle pattern with respect to a fluorescent object. A key advantage is that the speckle statistics are preserved upon propagation through the aberrations, which avoids the double pass of information through the aberrating medium typical of epi-fluorescence microscopy. The method recovers simultaneously a high-resolution image, the point spread function of the system that contains the aberrations, the speckle illumination pattern, and the shift positions.

Time-optimal surveillance between two differential drive robots with a limited field of view

Keeping an intelligent agent under surveillance with an autonomous robot as it travels in the environment is one of the essential tasks in robotics. This work addresses a pursuit-evasion surveillance problem between two Differential Drive Robots moving on a plane without obstacles. One of them, the pursuer, is equipped with a limited field-of-view sensor and aims to maintain the other robot (evader) inside its detection region for as much time as possible. On the contrary, the evader wants to escape as soon as possible. Unlike prior studies that address the same problem, based on heuristic approaches, we frame it as a zero-sum differential game, and we use differential game theory to find a solution with theoretical guarantees. In particular, we compute the players' time-optimal motion strategies to achieve their goals near the end of the game and provide analytical expressions describing them. These strategies are in Nash equilibrium, meaning that any unilateral deviation by a player from these strategies does not provide such player benefit toward its goal. Our analysis exhibits the existence of a Transition Surface in which the evader switches its controls. We also determine the game's winner based on the players' initial configurations and velocities.