**Abstract**

Radio astronomy imaging has been primarily focused on a planar approximation to a portion of the observed sphere, producing images of a fixed resolution. Historically, Fourier analysis played a pivotal role, with algorithmic modifications made to fit that paradigm. It was thought that spherical calculation was computationally impractical, and that inherent numerical instability meant only a dirty image (a very rough least-squares approximation) could be made. The computational and energy demands of instruments such as the planned Square Kilometre Array (SKA) have made a new approach imperative.

Here we present an efficient algorithm called Bluebild, that reconstructs directly on the celestial sphere, producing, for the first time, a true least-square estimate of the sky. Wide-field and flexible beamformed imaging follow naturally. It produces a continuous image description that may be stored independently of resolution, and sampled up to the fundamental telescope limit. A multi-scale sky decomposition becomes an intrinsic part of the process, and algorithmic linearity permits uncertainty assessment across the chain. The algorithm is fast, far simpler and more intuitive than previous methods. We show sky images produced are more accurate, and can be analysed in much more depth. Results with real LOFAR data will be presented.