Reconstructing infinite-dimensional signals from a limited amount of linear measurements is a key problem in many applications such as medical imaging, single-pixel and lensless cameras, fluorescence microscopy etc. Efficient techniques for such a problem include generalized sampling and its compressed versions, as well as methods based on data assimilation. All of these methods have in common that the reconstruction quality depends highly on the subspace angle between the sampling and the reconstruction space. In this paper we consider the case of binary measurements, which, after a standard subtraction trick, can be converted to a 1 and -1 setup. These measurements are modelled with Walsh functions, which form the kernel for the Hadamard transform. For the reconstruction we use wavelets. We show that the relation between the amount of data sampled and the coefficients reconstructed has to be only linear to ensure that the angle is bounded from below and hence the reconstruction is accurate and stable.