We present a method to characterize vertical cracks in a fast way using burst vibrothermography. In this technique the sample is excited by ultrasounds and, at the defect, rubbing of the contacting surfaces produces heat that can be detected as a temperature rise at the surface using an infrared camera. In this work, first we present the solution of the direct problem, i.e., the calculation of the surface temperature distribution produced by a vertical heat source representing a crack excited by an ultrasound burst, and we choose the information that will be used to characterize the crack, namely, one thermogram and one timing-graph. Next we address the inverse problem, consisting of finding the heat source distribution that is responsible for the observed surface temperature. This inverse problem is ill-posed, and a naïve inversion process is unstable. We propose to use three penalty terms, based on zero order Tikhonov and Total Variation functionals and the Lasso method, to stabilize the inversion. By inverting synthetic data, we analyze the performance of the algorithm as a function of the depth of the heat source and we study the effect of the burst duration and noise level in the data on the quality of the reconstructions. Finally, we invert experimental data taken in samples containing calibrated heat sources. The results show that it is possible to characterize vertical cracks down to depths of 6 mm in AISI 304 stainless steel.