Recently, a special type of data poisoning (DP) attack against deep neural network (DNN) classifiers, known as a backdoor, was proposed. These attacks do not seek to degrade classification accuracy, but rather to have the classifier learn to classify to a target class whenever the backdoor pattern is present in a test example. Here, we address the challenging post-training detection of backdoor attacks in DNN image classifiers, wherein the defender does not have access to the poisoned training set, but only to the trained classifier itself, as well as to clean (unpoisoned) examples from the classification domain. We propose a defense against imperceptible backdoor attacks based on perturbation optimization and novel, robust detection inference. Our method detects whether the trained DNN has been backdoor-attacked and infers the source and target classes involved in an attack. It outperforms alternative defenses for several backdoor patterns, data sets, and attack settings.