Explorations in Image Partition Encoding

Sameer AgarwalDepartment of Computer Science and EngineeringUniversity of California, San Diego

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.

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.

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 ?

The shape coding problem

1. Shape descriptors like a fourier transform of the curve.

2. Chain-coding of the boundary.3. Bitmap compression

1. Label each region with a small integer and compress the resulting low bitrate image.

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)

The four color theorem

The smart solution

1. The four color theorem allows you to encode an image partition with just 4 colors.

2. A strict upper bound of 2 bpp.

Problem:How to four color ?How to find the most compressible 4-coloring ?

Explorations in Four-coloring

1. Polynomial time algorithms exist but are quite useless

2. Alternate solutions Integer -> Linear programming based

solutions Heuristic based coloring Backtracking

3. First two methods do not guarantee a 4 coloring.4. Backtracking is exponential.5. None of them solve the most-compressible 4-

coloring problem.

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.

Attempts (contd.)

Spectral coloring1. Approximate coloring method,

based on using the top two eigenvectors of the laplacian.

2. Gives approximate coloring, but makes mistakes. Probably these mistakes can be repaired.

3. work still in progress..

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.

Acknowledgements

1. Serge for all the hours.2. Pam Cosman and Yan Ye for their

JBIG2 encoder.

and the music of Nickelback for keeping me company.