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

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

ITAT 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

ITAT Space-Filling Curves (examples)

ITAT 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

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

ITAT 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

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

ITAT Experiments, lossless compression neighbour color redundancy, applicability to RLE

ITAT Experiments, lossless compression pattern redundancy, applicability to LZW

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

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

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

ITAT Experiments, lossy compression entropy evaluation  arithmetical coding

ITAT 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