Experimental determination of the concentration dependence of a liquid mixture diffusion coefficient usually involves numerous experimental runs, performed for the solutions with small concentration difference, prepared for each concentration value. This takes time, but allows using analytical solutions of the diffusion equation, obtained with constant diffusion coefficient, for interpretation of the obtained data. The concentration data can be obtained either by analysis of the taken samples, or (for transparent liquid mixtures) by nonintrusive optical measurements in the test cell. In the present study, new method is proposed, which enables one to obtain the whole concentration curve for the diffusion coefficient of transparent liquid mixture in a single experimental run. Quantitative refractometry using Background Oriented Schlieren (BOS) technique is used. First, instantaneous distributions of the apparent displacement, which is proportional to the refractive index gradient, are measured for one-dimensional diffusion process in a rectangular glass cell. Large or even maximal initial concentration differences are used, in contrast to conventional approach. The refractive angle is large, hence new variants of BOS technique are employed: BOS based on Fourier Transform Profilometry and M-array BOS. They are much more robust with respect to large displacement and large displacement gradient, typically observed in thin diffusion layers, formed by two miscible liquids. Also, additional information related to deformation of the dot images and formation of double and triple images of the same dot can be obtained in M-array BOS. Then, the obtained vertical profiles of the apparent displacement are used to determine the coefficients of the cosine expansion for the concentration vertical profile. Substitution of these coefficients and their temporal derivatives into the diffusion equation leads to overdetermined system of linear equations for the coefficients of concentration dependence of the diffusion coefficient, which is assumed linear or parabolic. Measurements are performed for the aqueous solutions of sodium chloride, ethanol and glycerol. Comparison of the obtained results with literature data shows that coefficients of the linear dependence can be accurately measured if contribution of the quadratic term is not important (sodium chloride). For diffusion in glycerol-water and ethanol-water mixtures, which exhibit strongly nonlinear dependence of the diffusion coefficient on concentration, the coefficients of parabolic approximation can be obtained. Measurements of the ethanol-water diffusion coefficient for the complete concentration range are not possible, because the refractive index concentration dependence for this mixture has maximum at ethanol mass fraction about 0.75, which leads to vanishing sensitivity of BOS measurement near this point. However, separate measurements of two branches of the diffusion coefficient concentration curve below and above the refractive index maximum are possible.