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I have two webcams, which are mounted close together. I plan to use them to find out the true depth of certain pixels of images. Suppose I use a standard multi camera calibration setup and find out the relative orientation of one with respect to the other camera, can it perform as well as a stereo camera setup for depth estimation? What are the challenges typically? How accurate must the calibration be?

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Yes you can use two camera the same as using single stereo camera for depth perception.
Step on calibration camera for two camera and stereo camera is same.

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The challenges you are going to face is more how to mount the cameras and find the relative positions between them (also the rotation). You can find a lot of designs for mounts for the cameras, also 3D printers' designs. The calibration is pretty similar to the one for one camera chess pattern etc. There are also a lot of tutorials explaining how you should proceed. Have fun!

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