1. An image pickup apparatus comprising: an image pickup device configured to pick up an optical image formed by an objective optical system, and to generate an image pickup signal based on the optical image that is picked up; and one or more processors comprising hardware, wherein the one or more processors are configured to: set a zoom magnification; when setting a position and size of a cut-out area, which is a part of an image represented by the image pickup signal, set a center of the cut-out area at a position deviated from a field-of-view center of the objective optical system in a case where the zoom magnification is a first magnification, and set the center of the cut-out area so that a distance between the center of the cut-out area and the field-of-view center decreases as the zoom magnification increases from the first magnification; and cut out the cut-out area, perform enlargement processing or reduction processing according to the zoom magnification and generate a zoomed image.