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We solve the general problem of determining, through imaging, thethree-dimensional positions of $N$ weak incoherent point-like emitters in anarbitrary spatial configuration. We show that a structured measurement strategyin which a linear interferometer feeds into an array of photo-detectors isalways optimal for this estimation problem, in the sense that it saturates thequantum Cram\'er-Rao bound. We provide a method for the explicit constructionof the optimal interferometer. Further explicit results for the quantum Fisherinformation and the optimal interferometer design that attains it are obtainedfor the case of one and two incoherent emitters in the paraxial regime. Thiswork provides insights into the phenomenon of super-resolution throughincoherent imaging that has attracted much attention recently. Our results willfind a wide range of applications over a broad spectrum of frequencies, fromfluorescence microscopy to stellar interferometry.

The quantum limit to incoherent imaging is achieved by linear interferometry