Single pixel imaging opened the door to a cheaper camera architecture able to operate in a wide spectral range. Compressive sensing has been used with such an optical setup to reconstruct an image using ℓ1-minimization. To avoid this type of reconstruction, we consider an adaptive approach leading to a direct restoration of an image and for which we propose a new acquisition strategy. Our technique allows one to acquire an image in the wavelet domain with a progressive non-linear acquisition strategy. This scheme is based on the non-linear approximation of the wavelet transform which takes advantage of the transformation's sparsity. This approximation is applied in a multiresolution way and is shown to offer high compression performance on simulated data. One application of the single pixel camera concerns time-resolved acquisition to observe fluorescence lifetime images of biological structures.