1. An image processing apparatus having a color conversion unit configured to perform color conversion using a look-up table (LUT) for defining conversion from an input color to an output color, the apparatus comprising: an input unit configured to input an image that is to be a target of color conversion; an accepting unit configured to accept designation of a specific color; and a generation unit configured to register an output value corresponding to the specific color at an unused grid point in the LUT, and generate a specific color-associated LUT, wherein the LUT is a three-dimensional LUT in which N grid points are arranged in each axial direction, the generation unit secures M unused grid points in each axial direction by rearranging a conversion table such that a three-dimensional space which exists within the LUT and in which (N-M) grid points are arranged in each axial direction shows an entire color space shown by the LUT, and reducing the number of grid points to be used, and if the specific color is used in an image input by the input unit, the color conversion unit performs color conversion using the specific color-associated LUT generated by the generation unit.