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