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Advances in deep learning have revolutionized cyber-physical applications, including the development of Autonomous Vehicles. However, real-world collisions involving autonomous control of vehicles have raised significant safety concerns regarding the use of Deep Neural Networks (DNN) in safety-critical tasks, particularly Perception. The inherent unverifiability of DNNs poses a key challenge in ensuring their safe and reliable operation. In this work, we propose Perception Simplex (PS), a fault-tolerant application architecture designed for obstacle detection and collision avoidance. We analyze an existing LiDAR-based classical obstacle detection algorithm to establish strict bounds on its capabilities and limitations. Such analysis and verification have not been possible for deep learning-based perception systems yet. By employing verifiable obstacle detection algorithms, PS identifies obstacle existence detection faults in the output of unverifiable DNN-based object detectors. When faults with potential collision risks are detected, appropriate corrective actions are initiated. Through extensive analysis and software-in-the-loop simulations, we demonstrate that PS provides predictable and deterministic fault tolerance against obstacle existence detection faults, establishing a robust safety guarantee.