I spent this summer doing exactly that for an internship. I set up a xilinx Zynq chip to run a basic video capture and streaming program (based on gstreamer), w

I spent this summer doing exactly that for an internship. I set up a xilinx Zynq chip to run a basic video capture and streaming program (based on gstreamer), with the encoding portion handled by the fpga. This chip has a baked-in encoder supporting AVC and HEVC, but I suppose you could implement or buy your own IP block for other codecs. Xilinx makes multiple boards based off the same chip with support for video codecs and SDR.