#### General description

In this family, the single-dish information is heavily processed before merging with the interferometric information. The basic idea is to produce from the single-dish observations pseudo-visibilities similar to the ones that would be produced by the interferometer if they were not filtered out.

1. The Single-Dish measurements are re-gridded and then FFTed into the uv plane.
2. The data are deconvolved of the single-dish beam ( ) convolution by division by its Fourier Transform (truncated to the antenna diameter).
3. The data are FFTed back to the image plane and multiplied by the interferometer primary beam, .
4. The result is FFTed again in the uv plane where the visibilities are sampled on a regular grid.
5. In the case of a mosaic, the two last operations are performed for each pointing center.

Using the properties of the Fourier transform, we can rewrite the measurement equation of an interferometer as
(16)
This equation means that the visibility measured by an interferometer at the spatial frequency is the convolution of the Fourier transform of the source intensity distribution by the Fourier transform of the primary beam. Hence, to get pseudo-visibilities truly consistent with interferometric visibilities, we must be able to reliably compute the convolution by the Fourier transform of the primary beam. This implies that we can compute pseudo-visibilities only for spatial frequencies lower than D-d. The use of the IRAM-30m to produce the short-spacing information of the NOEMA is thus ideal as it enables to recover pseudo-visibilities up to 15 m (=30 m-15 m). Once the pseudo-visibilities have been computed, they are merged with the interferometric visibilities and standard imaging and deconvolution are then applied to the merged data set.