Safe shutdown/restart Raspberry Pi due to a power loss
I think this is bad idea, because:


You will be unable to use all capacitor energy. Capacitors will discharge from 5 to 4.5V or maybe 4V and your Pi will reset. In best case - 20% of capacity will be used.
Even if you change your circuit by adding some DC/DC converter - 10F 5V may be not enough. Raspberry Pi (model B) needs at least 2W. Thats 400mA at 5V.


You should consider conventional rechargeable or alkaline batteries. There are many designs with LiIon 18650 cells. One cell is 8-11Wh / watt hours. That means 3-4 hours on single cell with average efficiency DC/DC converter.