Anomaly detection is a challenging problem in machine learning, and is even more so when dealing with instances that are captured in low-level, raw data representations without a well-behaved set of engineered features. The Radial Basis Function Data Descriptor (RBFDD) network is an effective solution for anomaly detection, however, it is a shallow model that does not deal effectively with raw data representations. This paper investigates approaches to modifying the RBFDD network to transform it into a deep one-class classifier suitable for anomaly detection problems with low-level raw data representations. We show that approaches based on transfer learning are not effective and our results suggest that this is because the latent representations learned by generic classification models are not suitable for anomaly detection. Instead we show that an approach that adds multiple convolutional layers before the RBF layer, to form a Deep Radial Basis Function Data Descriptor (D-RBFDD) network, is very effective. This is shown in a set of evaluation experiments using multiple anomaly detection scenarios created from publicly available image classification datasets, and a real-world anomaly detection dataset in which different types of arrhythmia are detected in electrocardiogram (ECG) data. Our experiments show that the D-RBFDD network out-performs state-of-the-art anomaly detection methods including the Deep Support Vector Data Descriptor (Deep SVDD), One-Class SVM, and Isolation Forest on the image datasets, and produces competitive results for the ECG dataset.