Projector and method for projecting an image

A projector having an illumination modulator to modulate the intensity of the light in a pixel-individual manner, a projection lens projecting the light being modulated in a pixel-individual manner onto an image modulator to generate the image to be projected such that each illumination pixel is associated with at least one image pixel. A controller which generates illumination control data and generates image control data for the image modulator. The illumination control data are generated such that they have an on-value for each illumination pixel that is associated with at least one image pixel intended to represent a brightness value in the image that exceeds a predetermined threshold value according to the image data, said on-value selected such that each illumination pixel being associated with at least one image pixel intended to represent a brightness value according to the image data that exceeds a predetermined threshold value and falls short of a predetermined maximum value, is also at least temporarily switched to the second state during periods when the at least one associated image pixel is switched to the second state.