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I'm looking to do an automated focus stacking using a manual macro lens, and I would like to find a solution that avoid focus breathing (i.e. a change in the FOV while changing the focus).

I was thinking about using a motorized macro bellow to allow me to adjust the focus while keeping the camera stationary (the lens would be moving closer to the subject).

If that would not work, is there any other suggestion that would minimize breathing? Would it be possible to also move the camera away from the subject while the lens extend to compensate for the change in FOV that might happen?

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A bellows that moves the camera body while keeping the lens (and subject) stationary is the best option. By moving the camera body you shift the sensor through the DOFocus at the image plane, which keeps the lens' perspective constant. This should result in the minimum of image shifts assuming the stack isn't really deep... but such a setup is uncommon.

However, all methods of shifting the focus result in some image/magnification shift. If you use a bellows it introduces the bellows factor which changes the relative size of the image circle, which changes the light density/exposure, and the apparent magnification. If you shift the lens/camera together it changes the perspective which changes the relative distances/sizes. And if you shift the focus (ring) it changes the magnification.

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