The subject of this paper is a new two-step method of optical imagery. In a first step the object is illuminated with a coherent monochromatic wave, and the diffraction pattern resulting from the interference of the coherent secondary wave issuing from the object with the strong, coherent background is recorded on a photographic plate. If the photographic plate, suitably processed, is replaced in the original position and illuminated with the coherent background alone, an image of the object will appear behind it, in the original position. It is shown that this process reconstructs the coherent secondary wave, together with an equally strong ‘twin wave’ which has the same amplitude, but opposite phase shifts relative to the background. The illuminating wave itself can be used for producing the coherent background. The simplest case is illumination by a point source. In this case the two twin waves are shown to correspond to two ‘twin objects’, one of which is the original, while the other is its mirror image with respect to the illuminating centre. A physical aperture can be used as a point source, or the image of an aperture produced by a condenser system . If this system has aberrations, such as astigmatism or spherical aberration, the twin image will be no longer sharp but will appear blurred, as if viewed through a system with twice the aberrations of the condenser. In either case the correct image of the object can be effectively isolated from its twin, and separately observed. Three-dimensional objects can be reconstructed, as well as two-dimensional. The wave used in the reconstruction need not be the original, it can be, for example, a light-optical imitation of the electron wave with which the diffraction diagram was taken. Thus it becomes possible to extend the idea of Sir Lawrence Bragg’s ‘X -ray microscope’ to arbitrary objects, and use the new method for improvements in electron microscopy. The apparatus will consist of two parts, an electronic device in which a diffraction pattern is taken with electrons diverging from a fine focus, and an optical synthetizer, which imitates the essential data of the electronic device on a much enlarged scale. The theory of the analysis-synthesis cycle is developed, with a discussion of the impurities arising in the reconstruction, and their avoidance. The limitations of the new method are due chiefly to the small intensities which are available in coherent beams, but it appears perfectly feasible to achieve a resolution limit of 1 Å, ultimately perhaps even better.