1. An image processing device for performing edge-preserving smoothing, the image processing device comprising: a reducing unit which reduces an input image; a reduced image smoothing unit which performs a smoothing processing on a reduced image generated by the reducing unit with edges preserved; and an edge-preserving enlarging unit which enlarges an image generated by the reduced image smoothing unit with the edges preserved, wherein the edge-preserving enlarging unit performs a filtering processing after enlargement, by determining a pixel value of a target pixel of an enlarged image for the filtering processing based on weighted addition of pixel values of respective reference pixels of the enlarged image; and wherein the edge-preserving enlarging unit sets weighting coefficients for the pixel values of the respective reference pixels of the enlarged image: (i) based on differences between a pixel value of the input image corresponding to the target pixel of the enlarged image and the pixel values of the respective reference pixels of the enlarged image, and (ii) based on distances between the target pixel of the enlarged image and the respective reference pixels of the enlarged image.