The adoption of Artificial Intelligence methods within the instrumentation and measurements field is nowadays an attractive research area. On the one hand, making machines learn from data how to perform an activity, rather than hard code sequential instructions, is a convenient and effective solution in many modern research areas. On the other hand, AI allows for the compensation of inaccurate or not complete models of specific phenomena or systems. In this context, this paper investigates the possibility to exploit suitable Machine Learning techniques in a vision-based ophthalmic instrument to perform automatic Anterior Chamber Angle (ACA) measurements. In particular, two CNN-based networks have been identified to automatically classify acquired images and select the ones suitable for the Van-Herick procedure. Extensive clinical trials have been conducted by clinicians, from which a realistic and heterogeneous image dataset has been collected. The measurement accuracy of the proposed instrument is derived by extracting measures from the images of the aforementioned dataset, as well as the system performances have been assessed with respect to differences in patients' eye color. Currently, the ACA measurement procedure is performed manually by appropriately trained medical personnel. For this reason, Machine Learning and Vision-Based techniques may greatly improve both test objectiveness and diagnostic accessibility, by enabling an automatic measurement procedure.