1. An interactive display comprising: one or more display layers that are at least partially transparent, the one or more display layers having a first side and a second opposing side; a first illuminator positioned at a first location behind the one or more display layers, such that at least some light emitted from the first illuminator passes from the first illuminator through the first side and the one or more display layers towards an object positioned on the second opposing side of the one or more display layers during illumination of the first illuminator; a second illuminator positioned at a second location behind the one or more display layers, such that at least some light emitted from the second illuminator passes from the first illuminator through the first side and the one or more display layers towards the object positioned on the second opposing side of the one or more display layers during illumination of the second illuminator; a camera positioned behind the one or more display layers with the first and second illuminators, the camera being configured to take images of at least a portion of the first side of the one or more display layers; a phase management mechanism configured to: control alternation of phases of the interactive display, the interactive display having a plurality of phases including at least a first phase, a second phase, and a third phase; when in the first phase, controlling the first illuminator to dominate over the second illuminator, and in the first phase causing the camera to take a first image; when in the second phase, controlling the second illuminator to dominate over the first illuminator, and in the second phase causing the camera to take a second image; and when in the third phase, controlling all illuminators, including the first illuminator and the second illuminator, to be turned off, and in the third phase causing the camera to take a third image; an image construction mechanism configured to construct an input image using at least at least a portion of the first image, at least a portion of the second image, and at least a portion of the third image.