1. A color gamut conversion device comprising: a saturation function generation unit that generates a saturation of an outer circumference of a color gamut in a color space of an input image as a function for lightness and hue and generates a saturation of an outer circumference of a color gamut in a color space of an output image as a function for lightness and hue; a saturation ratio calculation unit that calculates a saturation ratio of the saturation of the outer circumference of the color gamut of the input image to the saturation of the outer circumference of the color gamut of the output image; an input image acquisition unit that acquires an input image to be subjected to color gamut conversion; a lightness/chromaticity acquisition unit that acquires, in a pixel unit, lightness and chromaticity of a predetermined scene from a measurement device capable of measuring lightness and chromaticity; an out-of-gamut pixel selection unit that selects a pixel whose acquired chromaticity exists outside the color gamut of the color space of the input image; a hue calculation unit that calculates a hue of each pixel selected as the pixel outside the color gamut, from the lightness and the chromaticity of each selected pixel; a histogram generation unit that generates a histogram in which a plurality of pixels selected as the pixels outside the color gamut are counted for each pair of lightness and hue; a magnification determination unit that determines, within a range of not less than 1 and not more than the saturation ratio, a magnification by which the chromaticity of the input image to be subjected to the color gamut conversion is to be multiplied for each pair of lightness and hue based on the histogram; a color space conversion unit that calculates lightness and hue of each selected pixel of the input image to be subjected to the color gamut conversion; a chromaticity expansion unit that multiplies the chromaticity of the target pixel of the input image to be subjected to the color gamut conversion by a magnification in accordance with the pair of the lightness and the hue of the target pixel; and a color space inverse conversion unit that converts the image whose color gamut was expanded into an image with the original color space.