A general expression describing nondiffracting pulses whose transverse profile is a one-dimensional image is presented. The pulse turns out to be expressed as a superposition of two fields, possessing a purely translational dynamics, whose profiles are related to the field distribution on the the waist plane through an Hilbert transformation. The space-time structure of the generally X-shaped pulse is investigated and a simple relation connecting its transverse and the longitudinal widths is established. Specific analytical examples are considered and, in particular, the fundamental one-dimensional X waves are deduced and compared to their two-dimensional counterparts.