In this paper, we propose a highly accurate method for estimating the quality of images compressed using fractal image compression. Using an iterated function system, fractal image compression compresses images by exploiting their self-similarity, thereby achieving high levels of performance; however, we cannot always use fractal image compression as a standard compression technique because some compressed images are of low quality. Generally, sufficient time is required for encoding and decoding an image before it can be determined whether the compressed image is of low quality or not. Therefore, in our previous study, we proposed a method to estimate the quality of images compressed using fractal image compression. Our previous method estimated the quality using image features of a given image without actually encoding and decoding the image, thereby providing an estimate rather quickly; however, estimation accuracy was not entirely sufficient. Therefore, in this paper, we extend our previously proposed method for improving estimation accuracy. Our improved method adopts a new image feature, namely lacunarity. Results of simulation showed that the proposed method achieves higher levels of accuracy than those of our previous method.