1. A ptychographic imaging system with convex relaxation, the system comprising: an LED array configured to provide coherent radiation to illuminate a specimen being imaged, the LED array configured to activate different individual LEDs of the LED array to sequentially illuminate the specimen from a plurality of incident angles; a lens configured to collect light issuing from the specimen; a digital radiation intensity detector configured to receive light propagated from the lens and to capture intensity distributions for a sequence of low resolution intensity images of the specimen from sequential activation of individual LEDs in the LED array, each low-resolution intensity image based on light issuing from the specimen while illuminated from one of the incidence angles; and a processor in communication with the digital radiation detector to receive the intensity distributions of the sequence of low resolution images, the processor configured to: generate a convex problem based on the sequence of low resolution images; and optimize the convex problem to reconstruct a high-resolution image of the specimen, wherein reconstructing the high-resolution image comprises recovering phase data from the sequence of the low-resolution intensity images.