We describe a fundamentally different approach to MRI referred to as SWIFT (sweep imaging with Fourier transformation). SWIFT exploits time-shared RF excitation and signal acquisition, allowing capture of signal from spins with extremely short transverse relaxation time, T2*. The MR signal is acquired in gaps inserted into a broadband frequency-swept excitation pulse, which results in acquisition delays of only 1-2 microseconds. In SWIFT, 3D k-space is sampled in a radial manner, whereby one projection of the object is acquired in the gaps of each frequency-swept pulse, allowing a repetition time (TR) on the order of the pulse length (typically 1 - 3 milliseconds). Since the orientation of consecutive projections varies in a smooth manner (i.e., only small increments in the values of the x, y, z gradients occur from view to view), SWIFT scanning is close to inaudible and is insensitive to gradient timing errors and eddy currents. SWIFT images can be acquired in scan times similar to and sometimes faster than conventional 3D gradient echo techniques. With its ability to capture signals from ultrashort T2* spins, SWIFT promises to expand the role of MRI in areas of research where MRI previously played no or negligible role. In this article, we show wood and tooth images obtained with SWIFT as examples of materials with ultrashort T 2*. Early experience suggests SWIFT can play a role in materials science and porous media research.