1. A three-dimensional (3D) interaction apparatus for providing a user interaction service for a 3D image, the 3D interaction apparatus comprising: a memory including computer-executable instructions; a processor configured to read and execute the computer-executable instructions; an action calculator configured to calculate 3D action information corresponding to an action of a user; a mapping information calculator configured to calculate a manipulation position in a coordinate system of the 3D image matching each of coordinate values of the 3D action information, based on the 3D action information and 3D information of each of features of the 3D image; and an interaction controller configured to, when an arbitrary image processing request based on an action of a user is made for the calculated manipulation position, perform a corresponding image processing operation and display a 3D image obtained through the performed image processing operation.