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My goal is to put a lens or an array of lenses in front of my actioncam to use its high resolution for recording of undistorted normal-eyed footage, to avoid having to buy two cameras for the occasional video blog.

Now I looked into it and what I can choose from are frequency admission, focal length and diameter.

What do I know about my actioncam? I has a 175° - f/2.8 lens in it. I know its sensor has a ratio of 4/3.

How do I figure out, which lens(es) of what diameter(s) in which distance(s to each other) to use?

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    Why not use a cheap alternative such as a smartphone to shoot the video? Smartphones have a more 'natural' 35mm focal length equiv of between 28-35mm which should be enough. If you insist on human focal length of ~50mm, you can use some phone lens adapters which are probably easier to come by than a bespoke hack for your actioncam. e.g. tomsguide.com/us/best-iphone-lenses,review-2366.html – Florin Draghici Nov 13 '15 at 15:36
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If you use a single lens, or even just a few lenses, the image quality will be very bad. The actioncam has a very small sensor and thus very small pixels. The resulting demand on the lens to delivery resolution is very high. Simple lenses do not have adequate resolution for this and will give you a blurry image.

You would be much better off using a different camera. If you were so inclined, it would be more effective to spend the money you would spend on this project to purchase a low cost camera to record the video with.

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In the case you have things lying around, you can go this route:

http://petapixel.com/2013/12/10/diy-adapter-allows-gopro-cameras-use-interchangeable-lenses/

However, I agree, that it might come in more handy (and most likely cheaper and with better image quality) to use a smartphone or some other camera.

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