1. An image processing apparatus comprising: computation means for determining an interpolation process starting point and an interpolation process gradient by comparing, when an image signal has a change point at which a difference between pixels values of adjacent pixels of the image signal is greater than zero and is less than or equal to a predetermined threshold value, a first continuous width of identical grayscale value pixels scanned earlier than the change point with a second continuous width of identical grayscale value pixels scanned later than the change point to determine a comparison result, when the comparison result is the first continuous width is smaller than the second continuous width, a position of a pixel scanned earlier by substantially one-half the first continuous width than the change point as the interpolation process starting point, and determining the interpolation process gradient on the basis of a difference between pixel grayscale values before and after the change point and the first continuous width, and setting, when the comparison result is the first continuous width is greater than the second continuous width, a position of a pixel scanned earlier by substantially one-half the second continuous width than the change point as the interpolation process starting point, and determining the interpolation process gradient on the basis of a difference between pixel grayscale values before and after the change point and the second continuous width; and conversion means for converting pixel grayscale values of the image signal on the basis of the interpolation process starting point and interpolation process gradient determined by the computation means so that a grayscale change from the interpolation process starting point in the image signal has a value in accordance with the interpolation process gradient.