Suppose I have a mirror, or a picture on the wall, and I want to run a second video into this.
How could that be achieved? And even in such a way as to correct for perspective?
For example, like in this video.
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If you can use a tool like Apple's Motion or FCPX, or Adobe's After Effects, they all have the ability to put the 4 corners of a layer anywhere you want. With Motion or After Effects, you can even track the 4 corners and keep the "TV" image pinned in the right place.
In Motion, you can select a clip and open the Inspector. In the Inspector, press the "Properties" tab. There's a section in the properties called "Four Corner". If you twirl that open, it allows you to set where the 4 corners of the video go. You can put them wherever you like.
FCPX is similar - select the clip, and open the inspector. There should be a section in the inspector named "Distort". This will also let you set the 4 corners of the clip.
In After Effects you can select a layer and apply the built-in "Corner Pin" effect.
This can easily be accomplished using the "picture-in-picture" feature of most editing software. I've used Adobe Premiere and iMovie, and both include that feature. Premiere has the advantage that you can "mask" a layer, so it is not quite rectangular, in order to fit the shape of the mirror.
Make sure the camera for the main scene is on a tripod and does not move at all. If that camera moves, you will need to animate your picture-in-picture video and mask so it appears to stay inside the mirror's frame.