The present invention discloses a method to improve the image resolution of a microscope. This improvement is based on the mathematical processing of the complex field computed from the measurements with a microscope of the wave emitted or scattered by the specimen. This wave is, in a preferred embodiment, electromagnetic or optical for an optical microscope, but can be also of different kind like acoustical or matter waves. The disclosed invention makes use of the quantitative phase microscopy techniques known in the sate of the art or to be invented. In a preferred embodiment, the complex field provided by Digital Holographic Microscopy (DHM), but any kind of microscopy derived from quantitative phase microscopy: modified DIC, Shack- Hartmann wavefront analyzer or any analyzer derived from a similar principle, such as multi-level lateral shearing interferometers or common-path interferometers, or devices that convert stacks of intensity images (transport if intensity techniques: TIT) into quantitative phase image can be used, provided that they deliver a comprehensive measure of the complex scattered wavefield. The hereby-disclosed method delivers superresolution microscopic images of the specimen, i.e. images with a resolution beyond the Rayleigh limit of the microscope. It is shown that the limit of resolution with coherent illumination can be improved by a factor of 6 at least. It is taught that the gain in resolution arises from the mathematical digital processing of the phase as well as of the amplitude of the complex field scattered by the observed specimen. In a first embodiment, the invention teaches how the experimental observation of systematically occurring phase singularities in phase imaging of sub-Rayleigh distanced objects can be exploited to relate the locus of the phase singularities to the sub-Rayleigh distance of point sources, not resolved in usual diffraction limited microscopy. In a second, preferred imbodiment, the disclosed method teaches how the image resolution is improved by complex deconvolution. Accessing the object's scattered complex field - containing the information coded in the phase - and deconvolving it with the reconstructed complex transfer function (CTF) is at the basis of the disclosed method. In a third, preferred imbodiment, it is taught how the concept of "Synthetic Coherent Transfer Function" (SCTF), based on Debye scalar or Vector model includes experimental parameters of MO and how the experimental Amplitude Point Spread Functions (APSF) are used for the SCTF determination. It is also taught how to derive APSF from the measurement of the complex field scattered by a nanohole in a metallic film. In a fourth imbodiment, the invention teaches how the limit of resolution can be extended to a limit of ?/6 or smaller based angular scanning. In a fifth imbodiment, the invention teaches how the presented method can generalized to a tomographic approach that ultimately results in super-resolved 3D refractive index reconstruction.