We develop and test a new two-dimensional model for binocular combination of the two eyes' luminance profiles. For first-order stimuli, the model assumes that one eye's luminance profile first goes through a luminance compressor, receives gain-control and gain-enhancement from the other eye, and then linearly combines the other eye's output profile. For second-order stimuli, rectification is added in the signal path of the model before the binocular combination site. Both the total contrast and luminance energies, weighted sums over both the space and spatial-frequency domains, were used in the interocular gain-control, while only the total contrast energy was used in the interocular gain-enhancement. To challenge the model, we performed a binocular brightness matching experiment over a large range of background and target luminances. The target stimulus was a dichoptic disc with a sharp edge that has an increment or decrement luminance from its background. The disk's interocular luminance ratio varied from trial to trial. To refine the model we tested three luminance compressors, five nested binocular combination models (including the Ding-Sperling and the DSKL models), and examined the presence or absence of total luminance energy in the model. We found that (1) installing a luminance compressor, either a logarithmic luminance function or luminance gain-control, (2) including both contrast and luminance energies, and (3) adding interocular gain-enhancement (the DSKL model) to a combined model significantly improved its performance. The combined model provides a systematic account of binocular luminance summation over a large range of luminance input levels. It gives a unified explanation of Fechner's paradox observed on a dark background, and a winner-take-all phenomenon observed on a light background. To further test the model, we conducted two additional experiments: luminance summation of discs with asymmetric contour information (Experiment 2), similar to Levelt (1965) and binocular combination of second-order contrast-modulated gratings (Experiment 3). We used the model obtained in Experiment 1 to predict the results of Experiments 2 and 3 and the results of our previous studies. Model simulations further refined the contrast space weight and contrast sensitivity functions that are installed in the model, and provide a reasonable account for rebalancing of imbalanced binocular vision by reducing the mean luminance in the dominant eye.