This paper presents a method for target detection and classification of improvised explosive devices (IEDs), based on a joint low-rank and sparse decomposition of ground penetrating radar (GPR) signals. First the acquired GPR signals are decomposed into a low-rank component, containing the background clutter and the ground surface reflections, and a set of convolutional sparse codes, representing the target signals. Then, features are extracted from each reconstructed signal and classified using support vector machines. Experiments are conducted with real data acquired in the wild from 18 types of IEDs. Experimental results are presented which show that individual GPR traces can be classified with 73.8% accuracy. Furthermore, the IED type can be identified with 84.2% accuracy by combining individual signal classifications.