New non-volatile memory (NVM) technologies enable direct, durable storage of data in an application's heap. Durable, randomly accessible memory facilitates the construction of applications that do not lose data at system shutdown or power failure. Existing NVM programming frameworks provide mechanisms to consistently capture a running application's state. They do not, however, fully support object-oriented languages or ensure that the persistent heap is consistent with the environment when the application is restarted. In this paper, we propose a new NVM language extension and runtime system that supports object-oriented NVM programming and avoids the pitfalls of prior approaches. At the heart of our technique is \emph{object reconstruction}, which transparently restores and reconstructs a persistent object's state during program restart. It is implemented in NVMReconstruction, a Clang/LLVM extension and runtime library that provides: (i) transient fields in persistent objects, (ii) support for virtual functions and function pointers, (iii) direct representation of persistent pointers as virtual addresses, and (iv) type-specific reconstruction of a persistent object during program restart. In addition, NVMReconstruction supports updating an application's code, even if this causes objects to expand, by providing object migration. NVMReconstruction also can compact the persistent heap to reduce fragmentation. In experiments, we demonstrate the versatility and usability of object reconstruction and its low runtime performance cost.