  Single-pixel imaging is an indirect imaging technique which utilizes
simplified optical hardware and advanced computational methods. It offers novel
solutions for hyper-spectral imaging, polarimetric imaging, three-dimensional
imaging, holographic imaging, optical encryption and imaging through scattering
media. The main limitations for its use come from relatively high measurement
and reconstruction times. In this paper we propose to reduce the required
signal acquisition time by using a novel sampling scheme based on a random
selection of Morlet wavelets convolved with white noise. While such functions
exhibit random properties, they are locally determined by Morlet wavelet
parameters. The proposed method is equivalent to random sampling of the
properly selected part of the feature space, which maps the measured images
accurately both in the spatial and spatial frequency domains. We compare both
numerically and experimentally the image quality obtained with our sampling
protocol against widely-used sampling with Walsh-Hadamard or noiselet
functions. The results show considerable improvement over the former methods,
enabling single-pixel imaging at low compression rates on the order of a few
percent.
