In this study, we demonstrate a sparsity-regularized, complex, blind deconvolution method for removing sidelobe artefacts and stochastic noise from optical coherence tomography (OCT) images. Our method estimates the complex scattering amplitude of tissue on a line-by-line basis by estimating and deconvolving the complex, one-dimensional axial point spread function (PSF) from measured OCT A-line data. We also present a strategy for employing a sparsity weighting mask to mitigate the loss of speckle brightness within tissue-containing regions caused by the sparse deconvolution. Qualitative and quantitative analyses show that this approach suppresses sidelobe artefacts and background noise better than traditional spectral reshaping techniques, with negligible loss of tissue structure. The technique is particularly useful for emerging OCT applications where OCT images contain strong specular reflections at air-tissue boundaries that create large sidelobe artefacts.