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Some results of the fractal program are shown below in Figure 13.6. We have used the same five grey scale images as in Chapter 11. All images have been compressed with a domain density value of 2.

File | Quality | Starting Size | Compressed Size | Ratio | RMS Error |
---|---|---|---|---|---|

CHEETAH.GS | 1 | 64000 | 12157 | 82% | 9.9 |

CHEETAH.GS | 2 | 64000 | 12157 | 82% | 9.9 |

CHEETAH.GS | 3 | 64000 | 12140 | 82% | 9.9 |

CHEETAH.GS | 4 | 64000 | 12094 | 82% | 9.9 |

CHEETAH.GS | 5 | 64000 | 11993 | 82% | 9.9 |

CHEETAH.GS | 10 | 64000 | 10017 | 85% | 10.4 |

CHEETAH.GS | 15 | 64000 | 6613 | 90% | 12.4 |

CHEETAH.GS | 20 | 64000 | 3888 | 94% | 15.1 |

CLOWN.GS | 1 | 64000 | 11961 | 82% | 7.9 |

CLOWN.GS | 2 | 64000 | 11206 | 83% | 8.0 |

CLOWN.GS | 3 | 64000 | 10137 | 85% | 8.0 |

CLOWN.GS | 4 | 64000 | 9571 | 86% | 8.0 |

CLOWN.GS | 5 | 64000 | 9203 | 86% | 8.1 |

CLOWN.GS | 10 | 64000 | 6144 | 91% | 9.2 |

CLOWN.GS | 15 | 64000 | 3937 | 94% | 10.9 |

CLOWN.GS | 20 | 64000 | 2643 | 96% | 12.7 |

LISAW.GS | 1 | 64000 | 12136 | 82% | 2.7 |

LISAW.GS | 2 | 64000 | 11246 | 83% | 2.7 |

LISAW.GS | 3 | 64000 | 7945 | 88% | 3.0 |

LISAW.GS | 4 | 64000 | 5208 | 92% | 3.4 |

LISAW.GS | 5 | 64000 | 3704 | 95% | 3.8 |

LISAW.GS | 10 | 64000 | 1296 | 98% | 5.7 |

LISAW.GS | 15 | 64000 | 951 | 99% | 6.9 |

LISAW.GS | 20 | 64000 | 796 | 99% | 7.4 |

ROSE.GS | 1 | 64000 | 8707 | 87% | 8.0 |

ROSE.GS | 2 | 64000 | 8703 | 87% | 8.0 |

ROSE.GS | 3 | 64000 | 8597 | 87% | 8.0 |

ROSE.GS | 4 | 64000 | 8349 | 87% | 8.1 |

ROSE.GS | 5 | 64000 | 7885 | 88% | 8.1 |

ROSE.GS | 10 | 64000 | 5065 | 93% | 9.0 |

ROSE.GS | 15 | 64000 | 3369 | 95% | 10.3 |

ROSE.GS | 20 | 64000 | 2280 | 97% | 12.5 |

MOUSE.GS | 1 | 64000 | 9580 | 86% | 2.8 |

MOUSE.GS | 2 | 64000 | 7546 | 89% | 2.8 |

MOUSE.GS | 3 | 64000 | 5323 | 92% | 2.9 |

MOUSE.GS | 4 | 64000 | 4411 | 94% | 3.1 |

MOUSE.GS | 5 | 64000 | 3571 | 95% | 3.3 |

MOUSE.GS | 10 | 64000 | 1379 | 98% | 5.1 |

MOUSE.GS | 15 | 64000 | 989 | 99% | 6.3 |

MOUSE.GS | 20 | 64000 | 798 | 99% | 7.1 |

** Figure 13.6** Compression Results

Figure 13.6 can be compared with Figure 11.13 giving the results of DCT compression with some caution: the RMS error value is not an accurate measure of image quality, and the two programs are simple implementations which can both be improved. Yet from the comparison it is apparent that DCT compression performs better than fractal compression at low compression ratios, and that fractal compression is much better at high compression ratios. This is confirmed by visual inspection of the images.

The fractal compression program can achieve compression ratios that the DCT program cannot reach without extreme degradation of the image. For example, the LISAW image can be compressed to 1752 bytes (97% compression ratio) with fractal compression and still retain a natural aspect. Of course some degradation of the image is unavoidable at such a compression ratio, but the image quality is still acceptable at least for some applications where high-fidelity is not an absolute requirement. With the DCT compression program, LISAW compressed to 4170 bytes (more than twice as large as the fractal version) has an extremely “blocky” aspect which is immediately apparent.

Even a production-quality JPEG compressor cannot reach the quality of our simple fractal compression program at such high compression ratios. This is clearly visible in Figure 13.7, where both compressed files have approximately the same size (1752 bytes for the fractal version, 1741 bytes for the JPEG version). The fractal version was produced with quality factor 8 and domain density 2; the JPEG quality factor was chosen to get an equivalent size for the compressed file.

Fractal compression, 1752 bytes

**Figure 13-7** JPEG compression, 1741 bytes

In general, there is little improvement in the image quality between quality factors 2 and 1. This is due to the simple nature of our algorithm, which reaches its limits at high quality factors and low compression ratios. However even with a more complex algorithm, a fractal-based method may not be able to beat DCT-based methods at low compression ratios. Fractal compression really shines for high compression ratios, or when zooming on a portion of an image or enlarging a complete image.

The compression ratios obtained with our simple fractal compression program could be further improved by taking into account possible symmetries and rotations between ranges and domains, by using non-square ranges and domains, by using more bits to encode contrast and brightness in. the affine transformations, and by using Huffman or arithmetic encoding on the parameters of affine transformations.

The fractal compression program is slower than the DCT program, but the decompression is faster even in this simple implementation. The decompression speed could be further improved significantly by taking advantage of the resolution-independence of fractal compression: the image could be decoded at a lower resolution in the first iterations, and decoded at the full image size only for the last two iterations. While working at a lower resolution, the decoder need only work with a fraction of all the pixels of the original image, thus the total number of instructions required for decoding is much reduced.

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