[Detecting Thrown Ball] Which camera to use and which arduino board to use?

The usual answer is the Arduino is too slow and too memory limited to be able to do this processing in real time.

I suspect the Arm embedded processors (Teensy 3.0, Due, Mbed, etc.) are more optimized for low power situations. They are presumably faster than the Arduino (particularly at processing 16/32-bit data), but they may not be fast enough or have enough memory to hold a complete image in memory. Perhaps something like a Beaglebone (500-700Mhz depending on power issues) or Rasberry Pi (700Mhz) would give you the additional speed and memory to do what you want.