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Efficient Image Partitioning and Compression Techniques Using Four-Color Theorem

This research explores advanced image partition encoding methods focusing on a three-layered image structure designed for documents with a distinct text layer. By leveraging layer-specific compression algorithms, we enhance compression efficiency, inspired by existing systems like DjVu from AT&T. The paper discusses methods to segment natural images, compress each segment separately, and the critical role of a partition map for reassembly. We delve into the challenges of shape coding using four-coloring theorems and various strategies including greedy algorithms and spectral methods to achieve effective encoding.

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Efficient Image Partitioning and Compression Techniques Using Four-Color Theorem

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  1. Explorations in Image Partition Encoding Sameer Agarwal Department of Computer Science and Engineering University of California, San Diego

  2. Background • Multiple Raster Content Specification • Three layered image structure • Aimed at documents with a distinct text layer. • Layer specific compression algorithms for better compression • DjVu from At&T is a working system which implements a scheme like this.

  3. Generalization to natural images • Implement the MRC idea to natural scenes/ video. • Use a robust image segmentation method (e.g. N-Cut) to break the image into pieces. • Compress each segment separately. • Many different ways of compressing the segments possible including, texture modelling and wavelet based compression.

  4. How do you reassemble an image from partitions ? • Besides the individuals segments a image partition map needs to be stored for re-assembly of the image. How do you efficiently store the image partition map ?

  5. The shape coding problem • Shape descriptors like a fourier transform of the curve. • Chain-coding of the boundary. • Bitmap compression • Label each region with a small integer and compress the resulting low bitrate image.

  6. The Naïve solution Label each segment by its segment number. Problem: The number of bits per pixel depends on the number of segments. (Press Enter for the smart solution)

  7. The four color theorem

  8. The smart solution • The four color theorem allows you to encode an image partition with just 4 colors. • A strict upper bound of 2 bpp. Problem: How to four color ? How to find the most compressible 4-coloring ?

  9. Explorations in Four-coloring • Polynomial time algorithms exist but are quite useless • Alternate solutions • Integer -> Linear programming based solutions • Heuristic based coloring • Backtracking • First two methods do not guarantee a 4 coloring. • Backtracking is exponential. • None of them solve the most-compressible 4-coloring problem.

  10. Four-coloring Our attempts : Greedy backtracking • Greedy on color use, hence tries to minimize color-entropy • Inefficient and does not get the best coloring, random does better sometimes.

  11. Attempts (contd.) Spectral coloring • Approximate coloring method, based on using the top two eigenvectors of the laplacian. • Gives approximate coloring, but makes mistakes. Probably these mistakes can be repaired. • work still in progress..

  12. Attempts (contd.) Balkanization • Coloring becomes difficult with increasing connectivity of the graph. • Break segments with very high degree into pieces with lower degree. • The size of the graph increases too fast for the backtracking based methods. • The spectral methods do not show any difference.

  13. Acknowledgements • Serge for all the hours. • Pam Cosman and Yan Ye for their JBIG2 encoder. and the music of Nickelback for keeping me company.

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