Extract information from video through HDMI input processing
This is not a trivial exercise, but would be feasible with modern components.

I am not sure what you eventual project goals are, but you could bring up a prototype using an FPGA. The Xilinx Spartan-6 family, for example, has HDMI input and output capabilities. The Digilent Atlys Development Board seems like it could support your requirements. The board has the HDMI input and output connectors.

As far as processing, I believe that the Xilinx MicroBlaze core (which would run on the FPGA) can support Linux, if you feel an OS is necessary.

A few caveats:

If the HDMI signal is encrypted (HDCP), the task becomes harder.

If your eventual goal is a low cost consumer solution, using an HDMI capable FPGA may be cost prohibitive. An alternative would be to use a dedicated HDMI transmitter and receiver chips and a uC/DSP processor.