Re-use smartphone/tablets parts like a camera
The connector on the flex circuit from that camera was made by Hirose (it has an HRS logo on it), and might be in the BM10, BM14, or BM20 series. You can measure the pin spacing, stacking height, etc. to check which series it's in. For example, the Hirose BM14 series of FPC to board connectors has pins at 0.4 mm Pitch and 0.8 stacking height.
Mouser.com carries a 24-pin BM14 connector that might mate or match with the pictured connector. For other photos of Hirose FPC connectors, see google images for hirose fpc.
While you probably can find a matching connector (via careful measurement and perhaps some trial and error) and build an interface, it may be difficult to use the camera once it's hooked up. Most Arduino models have too little RAM to support image processing. A Yun might do it, a Tre, or some other micro entirely, eg Beagle or Raspberry.
(summary of comment thread)
I strongly recommend against trying to use any of these smartphone camera replacement parts. While they are cheap, this is because the signal they output is nigh impossible for you to use, even if you knew its specifications, which is unlikely given proprietary parts often have proprietary, non-public protocols. Even if you did have it, you'd likely have to implement a linux kernel extension for Raspberry Pi yourself to support it. Far more work than it's worth.
If you actually want to do image processing, there is a clear recommendation: the Raspberry Pi camera interface. RasPis have a special flex socket on the board for a camera, and you can buy cameras which work with the raspivid etc. commands included in raspbian with no extra setup. This allows you to use OpenCV or similar image processing libraries, or write your own, without having to worry about bitbanging obscure camera protocols.
No-name camera boards can be bought from eBay or similar sites for less than US$20, and they support 30fps 5 MP image/video capture to the Raspberry Pi natively and easily. Just search for "Raspberry Pi camera".
However...
If you're really sure you're up to the challenge of hacking a proprietary camera module, you'll want to do thorough research finding the exact protocol specifications for these camera chips first, and try to find ones that are the most common, for the highest chance of compatibility, and likelihood of others having made progress on using them. Good luck!