Supervision of the physical layer of optical networks is an extremely relevant subject. To detect fiber faults, single-ended solutions such as the Optical Time Domain Reflectometer (OTDR) allow for precise measurements of fault profiles. Combining the OTDR with a signal processing approach for high-dimensional sparse parameter estimation allows for automated and reliable results in reduced time. In this work, a measurement system composed of a Photon-Counting OTDR data acquisition unit and a processing unit based on a Linearized Bregman Iterations algorithm for automatic fault finding is proposed. An in-depth comparative study of the proposed algorithm's fault-finding prowess in the presence of noise is presented. Characteristics such as sensitivity, specificity, processing time, and complexity, are analysed in simulated environments. Real-life measurements that are conducted using the Photon-Counting OTDR subsystem for data acquisition and the Linearized Bregman-based processing unit for automated data analysis demonstrated accurate results. It is concluded that the proposed measurement system is particularly well suited to the task of fault finding. The natural characteristic of the algorithm fosters embedding the solution in digital hardware, allowing for reduced costs and processing time.