Self-reset analog-to-digital converters (ADCs) allow for digitization of a signal with a high dynamic range. The reset action is equivalent to a modulo operation performed on the signal. We consider the problem of recovering the original signal from the measured modulo-operated signal. In our formulation, we assume that the underlying signal is Lipschitz continuous. The modulo-operated signal can be expressed as the sum of the original signal and a piecewise-constant signal that captures the transitions. The reconstruction requires estimating the piecewise-constant signal. We rely on local smoothness of the modulo-operated signal and employ wavelets with sufficient vanishing moments to suppress the polynomial component. We employ Daubechies wavelets, which are most compact for a given number of vanishing moments. The wavelet filtering provides a sequence consisting of a sum of scaled and shifted versions of a kernel derived from the wavelet filter. The transition locations are estimated from the sequence using a sparse recovery technique. We derive a sufficient condition on the sampling frequency for ensuring perfect reconstruction of the smooth signal. We validate our reconstruction technique on a signal consisting of sinusoids in both clean and noisy conditions and compare the reconstruction quality with the recently developed repeated finite-difference method.