The quality of chest radiographs is a practical issue because deviations from quality standards cost radiologists' time, may lead to misdiagnosis and hold legal risks. Automatic and reproducible assessment of the most important quality figures on every acquisition can enable a radiology department to measure, maintain, and improve quality rates on an everyday basis. A method is proposed here to automatically quantify the quality according to the aspects of (i) collimation, (ii) patient rotation, and (iii) inhalation state of a chest PA radiograph by localizing a number of anatomical features and calculating some quality figures in accordance with international standards. The anatomical features related to these quality aspects are robustly detected by a combination of three convolutional neural networks and two probabilistic anatomical atlases. An error analysis demonstrates the accuracy and robustness of the method. The implementation proposed here works in real time (less than a second) on a CPU without any GPU support.