We introduce a method for imaging heterogeneity at and near interfaces in Earth's lowermost mantle with broadband, three-component seismograms from global seismograph networks. Our approach is based on inverse scattering and allows the extraction of pertinent signal from large data sets and requires few a priori assumptions about the heterogeneity under study, which makes it complementary to the forward modeling of selected waveforms. Here we construct a generalized radon transform (for heterogeneous, anisotropic elastic media) to map broadband seismogram windows, comprising main arrivals with their coda and precursors, into multiple images of a target structure. The "common image point gathers" thus produced reveal multiscale variations in elastic properties near deep interfaces. The GRT can be applied to narrow and wide-angle data, and the (automated) extraction of signal from data over a wide range of epicentral distances enables exploration of core-mantle boundary (CMB) regions that cannot, with present-day data coverage, be imaged with the triplicated waveforms used in forward modeling studies. Tests with synthetic data, produced both with idealized and actual source-receiver distributions, illustrate pertinent aspects of the theory and show that (multiple) weak interfaces can be detected and located correctly, with a radial resolution of ~15 km for the frequencies used, even in the presence of (random) noise that would prohibit visual inspection and modeling of the subtle signals. We transformed the transverse component ScS waveforms into image gathers of a CMB patch beneath Central America. Juxtaposition of stacks of these gathers, involving ~35,000 seismograms, produces a two-dimensional (2-D) image profile revealing contrasts in elastic properties near the target depth of the CMB and ~280--340 km above it. The latter may mark the top of the so called D″ region. The images also reveal a richness of structures in between these depths. Combined with a statistical analysis of these singularities (described in paper 2), the approach to imaging presented here paves the way to large-scale seismic exploration of the lowermost mantle.