In this talk we discuss recovery guarantees for the reconstruction of wavelet coefficients from Walsh measurements, which appear in fluorescence microscopy, lensless cameras and other analogue measurement advices with an “on-off” behaviour. We consider both linear and non-linear reconstruction methods. For the analysis we discuss the typical structure of signals under the wavelet transform and the properties of the change of basis matrix. The results offer a guideline on the choice of sampling pattern and an insight to the relationship between Walsh functions and wavelets.