The RHESSI spacecraft images hard X-ray emission from solar flares with an angular resolution down to ~2'' and an energy resolution of 1 keV. For such a Rotating Modulation Collimator (RMC) instrument, imaging information is gathered not as a set of spatial images, but rather as a set of (energy-dependent) spatial Fourier components (termed visibilities). We report here on a novel technique which uses these spatial Fourier components in count space to derive, via a regularized spectral inversion process, the corresponding spatial Fourier components for the electron distribution, in such a way that the resulting electron visibilities, and so the images that are constructed from them, vary smoothly with electron energy E. ``Stacking'' such images then results in smooth, physically plausible, electron spectra for prominent features in the flare. Application of visibility-based analysis techniques has also permitted an assessment of the density and volume of the electron acceleration region, and so the number of particles it contains. This, plus information on the rate of particle acceleration to hard-X-ray-producing energies [obtained directly from the hard X-ray spectrum I(ε)] allows us to deduce the specific acceleration rate (particles s-1 per particle). The values of this key quantity are compared with the predictions of various electron acceleration scenarios.