Image Compression Using Space-Filling Curves

Michal Krátký, Tomáš Skopal, Václav Snášel

Department of Computer Science, VŠB-Technical University of Ostrava

Czech Republic

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Presentation Outline

• Motivation

• Properties of Space-Filling Curves (SFC)

• Experiments– lossless compression (RLE, LZW)– lossy compression (delta compression)

• Conclusions

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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

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Space-Filling Curves (examples)

C -curve H ilbert cu rveZ-curve

R andom curveSnake curve Sp ira l cu rve

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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

C -curve

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Images scanned along SFC

„Random“ Lena„Hilbert“ Lena

„Z-ordered“ Lena

„C-ordered“ Lena

„Snake“ Lena„Spiral“ Lena

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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

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SFC symmetry, jumping factor

C -curve H ilbert cu rveZ-curve

R andom curveSnake cu rve Sp ira l cu rve

Symmetry: C-curve = Snake < Random < Z-curve < Spiral < Hilbert Jumping factor: Hilbert = Spiral = Snake < C-curve < Z-curve < Random

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Experiments, lossless compression

• neighbour color redundancy, applicability to RLE

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Experiments, lossless compression

• pattern redundancy, applicability to LZW

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Experiments, lossy compression• delta compression, 6-bit delta delta histograms

Max. deltas

= error pixels

Tall “bell”

= low entropy

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Experiments, lossy compression• visualization of error pixels (all color components)

C-curve errors Snake curve errors Z-curve errors

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Experiments, lossy compression• visualization of error pixels (all color components)

Random curve errors Spiral curve errors Hilbert curve errors

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Experiments, lossy compression

• entropy evaluation arithmetical coding

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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