1. An image quality enhancing apparatus comprising: a section which stores a plurality of first feature quantities which can express a feature quantity in any image with a linear sum thereof; a section which stores a plurality of second feature quantities which indicate a feature quantity of a high image quality corresponding to the first feature quantities; a coefficient calculating section which calculates, from a feature quantity of an image targeted for image quality enhancement, a coefficient for expressing the feature quantity with the linear sum of the plurality of first feature quantities; and a section which generates an image which is the image with the image quality enhanced by replacing each of the first feature quantities included in the linear sum using the coefficient calculated by the coefficient calculating section with corresponding second feature quantities; wherein the coefficient calculating section comprises: a selecting section which selects fewer first feature quantities than the total number of the plurality of first feature quantities; and a calculating section which calculates a coefficient to be multiplied by the first feature quantities selected by the selecting section as a sparse solution by calculating an analytic solution without using the iteration method, based on the first feature quantities selected by the selecting section and the feature quantity of the image.