Hitfilm has a thing called mocha tracking that would be nice for that project ....
firstly you render 1 frame of the image with the stuffed dog, cut everything out of the picture except the dog - you do that with a photoediting software ( ie - photoshop ) - once you make that image - you take that and rig / animate it in CTA -
...and as cricky said, you need to do this on a green background ....( greenscreen, but with a greenbackground in CT or CTA )
once the animation is done and rendered .... you load both the original footage and the new animated footage to HITFILM ( www.hitfilm.com ) - first thing you do is apply chroma key to remove the green ....2nd thing you do ....is mocha tracking ....
basically since it's video....unless the video is on a tripod where the camera is perfectly still...the dog in the chair will be changing positions on screen based on whomever is holding the camera and how they are moving around ....this is where mocha tracking will lock the new animated footage of the dog and lock it on the screen over the original filmed dog.....
The mocha tracking step is really the hard part if you don't have a software doing it for you. it can be very tedious to manually match the constant shifting of the camera manually. what mocha can do in a few minutes with computerized motion tracking - will take many hours, possibly days to match the motions manually frame by frame to look natural.. ( known as video keyframing to video editors ).
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