image compression using space-filling curves michal krátký, tomáš skopal, václav snášel...
TRANSCRIPT
![Page 1: Image Compression Using Space-Filling Curves Michal Krátký, Tomáš Skopal, Václav Snášel Department of Computer Science, VŠB-Technical University of Ostrava](https://reader036.vdocument.in/reader036/viewer/2022062314/56649ee65503460f94bf62e1/html5/thumbnails/1.jpg)
Image Compression Using Space-Filling Curves
Michal Krátký, Tomáš Skopal, Václav Snášel
Department of Computer Science, VŠB-Technical University of Ostrava
Czech Republic
![Page 2: Image Compression Using Space-Filling Curves Michal Krátký, Tomáš Skopal, Václav Snášel Department of Computer Science, VŠB-Technical University of Ostrava](https://reader036.vdocument.in/reader036/viewer/2022062314/56649ee65503460f94bf62e1/html5/thumbnails/2.jpg)
ITAT 2003 2
Presentation Outline
• Motivation
• Properties of Space-Filling Curves (SFC)
• Experiments– lossless compression (RLE, LZW)– lossy compression (delta compression)
• Conclusions
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ITAT 2003 3
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
![Page 4: Image Compression Using Space-Filling Curves Michal Krátký, Tomáš Skopal, Václav Snášel Department of Computer Science, VŠB-Technical University of Ostrava](https://reader036.vdocument.in/reader036/viewer/2022062314/56649ee65503460f94bf62e1/html5/thumbnails/4.jpg)
ITAT 2003 4
Space-Filling Curves (examples)
C -curve H ilbert cu rveZ-curve
R andom curveSnake curve Sp ira l cu rve
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ITAT 2003 5
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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ITAT 2003 6
Images scanned along SFC
„Random“ Lena„Hilbert“ Lena
„Z-ordered“ Lena
„C-ordered“ Lena
„Snake“ Lena„Spiral“ Lena
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ITAT 2003 7
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
![Page 8: Image Compression Using Space-Filling Curves Michal Krátký, Tomáš Skopal, Václav Snášel Department of Computer Science, VŠB-Technical University of Ostrava](https://reader036.vdocument.in/reader036/viewer/2022062314/56649ee65503460f94bf62e1/html5/thumbnails/8.jpg)
ITAT 2003 8
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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ITAT 2003 9
Experiments, lossless compression
• neighbour color redundancy, applicability to RLE
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ITAT 2003 10
Experiments, lossless compression
• pattern redundancy, applicability to LZW
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ITAT 2003 11
Experiments, lossy compression• delta compression, 6-bit delta delta histograms
Max. deltas
= error pixels
Tall “bell”
= low entropy
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ITAT 2003 12
Experiments, lossy compression• visualization of error pixels (all color components)
C-curve errors Snake curve errors Z-curve errors
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ITAT 2003 13
Experiments, lossy compression• visualization of error pixels (all color components)
Random curve errors Spiral curve errors Hilbert curve errors
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ITAT 2003 14
Experiments, lossy compression
• entropy evaluation arithmetical coding
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ITAT 2003 15
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