I considered posting this on other SE sites such as Audio-Video Production and Photography, but I didn't feel I'd get the definitive, fact-based (rather than experience-based) answer I'm seeking.
Background
I used a GoPro HD Hero (which has a 170-degree view) to record a track day (driving a car) and a snowboarding trip. Playing back the videos after each event, I sensed immediately that they seemed slow. However, since clocks and speech seemed to progress at a normal rate, I dismissed my senses, figuring adrenaline and wind must've made me perceive real life faster than it actually was.
And there's probably some truth to this adrenaline-and-wind explanation, but watching the videos again today (more than a year later), I sensed the slowness as I always did. I came up with a potential, partial explanation, but still lacking; I was hoping someone might be able to shed some light as to whether this "slow video" phenomena is merely psychological, or an expected effect of optics.
Partial Explanation
I think there may be an effect, due to depth perception being altered by wide-angle lenses.
Suppose we were to record a skateboarder from the side, i.e. so the skateboarder would skate from left to right in our camera's field of view. In this case, because the axis of movement (the skateboarder's "line") would be perpendicular† to the axis of distortion (the camera's "line", or depth), there would be no distortion of perceived speed: the skateboarder moves relative to an undistorted frame of reference (the ground).
However, if we were to mount the camera on the skateboarder's helmet, i.e. record in a first-person view, then the axes of movement and distortion would be one and the same. From here though, I'm not sure how to explain why this translates to slower-seeming playback.
Question
In layman's terms, my problem is, "The ground and the trees pass by me slower," when playing back a wide-angle video.
- Is this psychological, or is there a physical explanation?
- Is there a mathematical formula that can tell by how much one must speed up playback in order for "the ground and the trees" to pass by as they seemed to in real life?
Footnotes
† Technically, I guess "perpendicular" would be a curve, i.e. staying equidistant from the camera, but for simplicity, suppose the skateboarder is far enough from the lens that we may consider the curve a "line".
Best Answer
I believe the explanation is quite simple. When you take a wide angle video, you are mapping 170 degrees of azimuth onto the film. When you play that back, you play it back on a screen that takes up roughly 40-60 degrees of azimuth. The edges are compressed relative to the center when recording the image ($r=f\sin\theta$), but not when playing back the image. Because larger physical distances are being recorded in a smaller film space, the velocity of objects at the edges appears slower (and the speed at which objects depart the field of view is what most strongly influences your sense of motion). The apparent velocity in the middle of the image is being recorded quite accurately, but we have difficulty judging the velocity of things which are moving toward or away from us; we judge velocity best when it is nearly tangential (at the edge of the frame, with a fisheye lens).
This is not present in all types of lenses. Standard lenses map the environment to a focal plane using the perspective mapping $r=f\tan(\theta)$, where $r$ is the distance across the imaging plane and $\theta$ is the angular position in the scene. The angular derivative of this function is $dr/d\theta=f\sec^2(\theta)$, which means that the change in pixel position with respect to change in scene angle increases strongly as theta increases.
The GoPro lens is approximately an equisolid fisheye lens (there are many possible fisheye mappings, but this is among the most common). The mapping for this type of lens is $r=2f\sin(\theta/2)$, which has angular derivative $dr/d\theta=f\cos(\theta/2)$. This means that changes in scene angular position correspond to smaller and smaller changes across the imaging plane as $\theta$ increases.
Since playback of recorded images is typically not reprojected with the same mapping with which it was recorded, this results in distortions when watching video playback. For standard lenses, this over-expands the edges, while for fisheye lenses, the edges of the image are under-expanded.