In the inverse problems of Fourier synthesis encountered in aperture synthesis, the data are complex quantities corresponding to an irregular sampling of the Fourier transform of some non-negative object function. CLEAN is a deconvolution method which is intensively used in this context. This series of two papers is meant to provide, as an alternative to CLEAN, a well-formulated deterministic method, in which the propagation of errors is under control. The method introduced in part I is referred to as FIRST or WIPE: FIRST for the principles (Fourier Interpolation and Reconstruction via Shannon-type Techniques), and WIPE for the corresponding deconvolution method (WIPE is reminiscent of CLEAN). The main purpose of part II is to complete the general framework presented in part I. In particular, we present a new method for computing the dusty map and the dirty beam. The related techniques of fast Fourier sampling and gridding are thoroughly analysed. We then explicitly describe the imaging kernel of WIPE by presenting the basic algorithm, the role of the Krylov spaces in the control of its robustness and the calculation of the image reconstruction error bars. Among the regularization parameters appearing in the definition of the objective functional of FIRST, those related to the choice of the object representation space play an essential role. This is why an entire section is devoted to the methods that can be devised and implemented for constructing this space, in a global manner or step by step, interactively or automatically. All the technical developments presented in this paper are illustrated with the aid of appropriate simulations, so that the main aspects of FIRST and WIPE can be understood in a concrete manner. The methodology underlying FIRST can be applied to more complex experimental situations such as those encountered, for example, in mosaic image reconstruction, multifrequency Fourier synthesis and in multiresolution aperture synthesis. This is outlined in the concluding comments.