1. An image processing apparatus for adjusting a gradation range of an input image, comprising: a luminance-color difference separation element which separates the input image into a luminance signal and a color difference signal; a gradation correction element which performs gradation correction on the luminance signal to adjust the gradation range to a predetermined gradation range; a color difference correction element which corrects the color difference signal based on the luminance signal obtained before the gradation correction and output from the luminance-color difference separation element, a luminance signal obtained after the gradation correction and output from the gradation correction element, and a theoretical limit characteristic of color reproduction; and a luminance-color difference synthesis element which synthesizes the luminance signal obtained after the gradation correction and a color difference signal obtained after the correction, into an image signal.