This invention teaches a quality assurance system for additive manufacturing. This invention teaches a multi-sensor, real-time quality system including sensors, affiliated hardware, and data processing algorithms that are Lagrangian-Eulerian with respect to the reference frames of its associated input measurements. The quality system for Additive Manufacturing is capable of measuring true in-process state variables associated with an additive manufacturing process, i.e., those in-process variables that define a feasible process space within which the process is deemed nominal. The in-process state variables can also be correlated to the part structure or microstructure and can then be useful in identifying particular locations within the part likely to include defects.