user2811: I was looking at alternatives to the GoPro (such as the Sony FDR-X1000V), but they all seem to only shoot wide-angle with significant perspective distortion (fisheye). Are there action cameras which do not introduce this sort of distortion?

user17194: Yi Technologies action cameras have lens distortion correction as a built-in feature. The lens does introduce distortion but the onboard processor compensates for it. 
I assume we'll see more of this type of compensation embedded in cameras in the future.

user10931: There are two major advantages to recording fisheye distortion rather than rectilinear distortion.  First, it is optically easier to do, which means one can make lenses that are better in other metrics (such as chromatic aberration, resolution, vignetting, etc) while remaining small enough and light enough to fit into the body of an action camera.  Second, the relative size of an object doesn't change radically as its moves from center to edge when the camera is not under tight control.  For that reason, it is possible to define a crop area that follows the subject as it bounces around the full frame and then use motion tracking to create a relatively stable view of the object, which can then be de-fished.  If it were rectilinear, the stabilization process might have to do some major zoom and tilt corrections as well, which would be very distracting.

2811: <p>I was looking at alternatives to the GoPro (such as the Sony FDR-X1000V), but they all seem to only shoot wide-angle with significant perspective distortion (fisheye). Are there action cameras which do <em>not</em> introduce this sort of distortion?</p>
 17194: <p>Yi Technologies action cameras have <a href="http://www.yitechnology.com/yi-4k-plus-action-camera-specs" rel="nofollow noreferrer">lens distortion correction as a built-in feature</a>. The lens does introduce distortion but the onboard processor compensates for it. </p>

<p>I assume we'll see more of this type of compensation embedded in cameras in the future.</p>
 10931: <p>There are two major advantages to recording fisheye distortion rather than rectilinear distortion.  First, it is optically easier to do, which means one can make lenses that are better in other metrics (such as chromatic aberration, resolution, vignetting, etc) while remaining small enough and light enough to fit into the body of an action camera.  Second, the relative size of an object doesn't change radically as its moves from center to edge when the camera is not under tight control.  For that reason, it is possible to define a crop area that follows the subject as it bounces around the full frame and then use motion tracking to create a relatively stable view of the object, which can then be de-fished.  If it were rectilinear, the stabilization process might have to do some major zoom and tilt corrections as well, which would be very distracting.</p>
