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The low luminance levels of light-emitting diodes (LEDs) compared to arc lamps make it difficult to design high-brightness LED-based projectors. Besides, the specificities of LEDs do not always allow using the same design schemes as with arc lamp-based projection displays. This paper performs a taxonomy of the techniques that can be used to increase the brightness of LED-based projection displays. We show that, in etendue-limited systems, the perceived brightness depends on the system etendue limit, the efficiency of the light engine, and the source luminance. The ability to improve each of these parameters depends on the design constraints. The system etendue limit can be increased at the expense of bulkier, more complex, and more expensive designs. The light engine efficiency can be increased by using free-form shape components adapted to the shapes and the emission patterns of the considered LEDs. The apparent source luminance can be increased at the expense of the flux by either recycling light or restricting the light collection to a smaller etendue with higher average luminance. Luminance can also be increased by using multiple color primaries (spatial multiplexing) or pulsed LEDs (temporal multiplexing). Finally, we review how light recycling can be implemented to convert polarization without increasing etendue.