From February 2003 to September 2005 the Solar Mass Ejection Imager on the Coriolis spacecraft detected 207 interplanetary coronal mass ejections (ICME) in the inner heliosphere. We have examined the data from the Large Angle Spectroscopic Coronagraph (LASCO) on the SOHO spacecraft for evidence of coronal transient activity that might have been the solar progenitor of the Solar Mass Ejection Imager (SMEI) events, taking into account the projected speed of the SMEI event and its position angle in the plane of the sky. We found a significant number of SMEI events where there is either only a weak or unlikely coronal mass ejection (CME) detected by LASCO or no event at all. A discussion of the effects of projection across large distances on the ICME measurements is made, along with a new technique called the Cube-Fit procedure that was designed to model the ICME trajectory more accurately than simple linear fits to elongation-time plots. Of the 207 SMEI events, 189 occurred during periods of full LASCO data coverage. Of these, 32 or 17% were found to have a weak or unlikely LASCO counterpart, and 14 or 7% had no apparent LASCO transient association. Using solar X-ray, EUV and Ha data we investigated three main physical possibilities for ICME occurrence with no LASCO counterpart: (1) Corotating interaction regions (CIRs), (2) erupting magnetic structures (EMS), and (3) flare blast waves. We find that only one event may possibly be a CIR and that flare blast waves can be ruled out. The most likely phenomenon is investigated and discussed, that of EMS. Here, the transient erupts in the same manner as a typical CME, except that they do not have sufficient mass to be detected by LASCO. As the structure moves outward, it accumulates and concentrates solar wind material until it is bright enough to be detected by SMEI.