1. An image processing device comprising: a storing unit storing the position of a focus detecting pixel of an image-capturing sensor which contains a plurality of pixels having spectroscopic characterizations corresponding to respective a plurality of color components and the focus detecting pixel having spectroscopic characterization different from the spectroscopic characterizations of the plurality of the pixels; a pixel interpolating unit generating an interpolation pixel value of the focus detecting pixel by using a pixel value of one of pixels in the neighborhood of the focus detecting pixel in an image generated by the image-capturing sensor; a neighborhood-pixel estimating unit calculating an estimation pixel value corresponding to a pixel value when one of the pixels in the neighborhood of the focus detecting pixel has the same spectroscopic characterization as the focus detecting pixel; a high-frequency component calculating unit calculating a high frequency component of the image by using a pixel value of the focus detecting pixel and the estimation pixel value; and a high frequency component adding unit adding the interpolation pixel value with the high frequency component to calculate a pixel value of the focus detecting pixel for imaging.