1. Image Compression Using Space-Filling Curves Michal Krátký, Tomáš Skopal, Václav Snášel Department of Computer Science, VŠB-Technical University of Ostrava Czech Republic

2. ITAT 20032 Presentation Outline Motivation Properties of Space-Filling Curves (SFC) Experiments –lossless compression (RLE, LZW) –lossy compression (delta compression) Conclusions

3. ITAT 20033 Space-Filling Curves bijective mapping of an n-dimensional vector space into a single-dimensional interval Computer Science: discrete finite vector spaces clustering tool in Data Engineering, indexing, KDD

4. ITAT 20034 Space-Filling Curves (examples)

5. ITAT 20035 Motivation Traditional methods of image processing: scanning rows or columns, i.e. along the C-curve Our assumption: other „scanning paths“ could improve the compression and could decrease errors when using lossy compression

6. ITAT 20036 Images scanned along SFC „Random“ Lena„Hilbert“ Lena „Z-ordered“ Lena„C-ordered“ Lena„Snake“ Lena„Spiral“ Lena

7. ITAT 20037 Properties of SFC SFCs partially preserve topological properties of the vector space. The topological (metric) quality of SFC: Points „close“ in the vector space are also „close“ on the curve. Two anomalies in a SFC shape: –“distance enlargements” in every SFC –symmetry of SFC: correlation of anomalies in all dimensions –jumping factor: number of “distance shrinking” occurences ( jumps over neighbours) distance shrinking distance enlargement

8. ITAT 20038 SFC symmetry, jumping factor Symmetry:C-curve = Snake < Random < Z-curve < Spiral < Hilbert Jumping factor:Hilbert = Spiral = Snake < C-curve < Z-curve < Random

9. ITAT 20039 Experiments, lossless compression neighbour color redundancy, applicability to RLE

10. ITAT 200310 Experiments, lossless compression pattern redundancy, applicability to LZW

11. ITAT 200311 Experiments, lossy compression delta compression, 6-bit delta  delta histograms Max. deltas = error pixels Tall “bell” = low entropy

12. ITAT 200312 Experiments, lossy compression visualization of error pixels (all color components) C-curve errors Snake curve errorsZ-curve errors

13. ITAT 200313 Experiments, lossy compression visualization of error pixels (all color components) Random curve errors Spiral curve errors Hilbert curve errors

14. ITAT 200314 Experiments, lossy compression entropy evaluation  arithmetical coding

15. ITAT 200315 Conclusions Choice of a suitable SFC can positively affect the compression rate (or entropy) as well as the quality of lossy compression. Experiments: symmetric curves with low (zero) jumping factor are the most appropriate  Hilbert curve