In the recently proposed technique Sweep Imaging with Fourier Transform (SWIFT), frequency-modulated radiofrequency pulses are used in concert with simultaneous acquisition to facilitate MRI of samples with very short transverse relaxation time. In the present work, sweep MRI is reviewed from a reconstruction perspective and several extensions and modifications of the current methodology are proposed. An algorithm for algebraic image reconstruction is derived from a comprehensive description of signal formation, including interleaved radiofrequency transmission and acquisition of arbitrary timing as well as the relevant filtering and decimation steps along the receiver chain. The new reconstruction approach readily permits several measures of optimising the signal sampling strategy as demonstrated in simulations and imaging experiments. Employing a variety of radiofrequency pulse envelopes, water and rubber phantoms as well as bone samples with transverse relaxation time in the order of 500 μsec were imaged at signal bandwidths of up to 96 kHz.