1. An image processing apparatus comprising: a band-limited image production means for producing, based on an original image, a plurality of band-limited images which represent components in a plurality of frequency bands in the original image; a transformation means for producing a plurality of transformed images by processing each pixel value in each of the band-limited images so that the absolute value of an output value becomes less than or equal to that of an input value; a multiplication means for multiplying pixel values in the plurality of transformed images by a predetermined subtraction coefficient to obtain multiplied pixel values; a subtraction processing means for obtaining pixel values in a processed image by subtracting the multiplied pixel values in the plurality of transformed images from pixel values in the original image; and a compression means for producing a compressed image by performing entropy encoding on the processed image, wherein the processing by the transformation means is non-linear transformation processing which is performed so that if the absolute value of the input value is less than or equal to a predetermined threshold value, the absolute value of the output value increases as the absolute value of the input value increases, or if the absolute value of the input value is larger than the predetermined threshold value, the absolute value of the output value becomes less than or equal to the absolute value of an output value corresponding to the predetermined threshold value.