The high fidelity reconstruction of compressed and low-resolution magnetic resonance (MR) data is essential for simultaneously improving patient care, accuracy in diagnosis and quality in clinical research. Sponsored by the Royal Society through the Newton Mobility Grant Scheme, we held a half-day workshop on reconstruction schemes for MR data on the 17th of August 2016 to discuss new ideas from related research fields that could be useful to overcome the shortcomings of the conventional reconstruction methods that have been evaluated up to date. Participants were 21 university students, computer scientists, image analysts, engineers and physicists from institutions from 6 different countries. The discussion evolved around exploring new avenues to achieve high resolution, high quality and fast acquisition of MR imaging. In this article, we summarise the topics covered throughout the workshop and make recommendations for ongoing and future works.