This study proposes a novel non-negative matrix factorisation (NMF) variant L1/2-NMF after visualisation and analysis of the process of target recognition via NMF for synthetic aperture radar (SAR) images. NMF has been applied to obtain pattern feature in SAR images. This study considers the intrinsic character and the physical meaning of NMF feature when applied for SAR automatic target recognition. At the base of obtaining the linear relationship between the sample to be recognised and the train samples, the whole recognition process is detailed and vividly visualised. Meanwhile, lots of researches have been done to improve NMF methods by enforcing sparse constraint with L1-norm, such as non-negative sparse coding (NNSC), local NMF and sparse NMF. Compared with L1-norm, L1/2-norm has been shown to have a more natural sparseness. In this study, a novel variant of NMF with L1/2 constraint, called L1/2-NMF is proposed, and is carried out a thorough study by applying it in SAR target recognition. Experimental results on MSTAR public database show that both the basis and coding matrices obtained by L1/2-NMF have higher sparseness than those obtained by NMF, NNSC and NMF with sparseness constraints (NMFsc). The recognition results demonstrate that the L1/2-NMF outperforms NNSC, NMFsc and non-smooth NMF.