A position of a three-dimensional (3D) object relative to a display surface of an interactive display system is detected based upon the intensity of infrared (IR) light reflected from the object and received by an IR video camera disposed under the display surface. As the object approaches the display surface, a "hover" connected component is defined by pixels in the image produced by the IR video camera that have an intensity greater than a predefined hover threshold and are immediately adjacent to another pixel also having an intensity greater than the hover threshold. When the object contacts the display surface, a "touch" connected component is defined by pixels in the image having an intensity greater than a touch threshold, which is greater than the hover threshold. Connected components determined for an object at different heights above the surface are associated with a common label if their bounding areas overlap.