Embedded Zero Tree Wavelet Coding

Compare the two matrices576 704 1152 1280 1344 1472 1536 1536

704 640 1156 1088 1344 1408 1536 1600

768 832 1216 1472 1472 1536 1600 1600

832 832 960 1344 1536 1536 1600 1536

832 832 960 1216 1536 1600 1536 1536

960 896 896 1088 1600 1600 1600 1536

768 768 832 832 1280 1472 1600 1600

448 768 704 640 1280 1408 1600 1600

1212 -306 -146 -54 -24 -68 -40 4

30 36 90 2 8 -20 8 -4

-50 -10 -20 -24 0 72 -16 -16

82 38 -24 68 48 -64 32 8

8 8 -32 16 -48 -48 -16 16

20 20 -56 -16 -16 32 -16 -16

-8 8 -48 0 -16 -16 -16 -16

44 36 0 8 80 -16 -16 0

A Multi-resolution Analysis Example

Wavelet Transform

Discrete Wavelet Transform

– Sub bands arise from separable application of filters

LL1

LH1

HL1

HH1

First stage

LH1

HL1

HH1

Second stage

LL2

LH2

HL2

HH2

Embedded Zero tree Wavelet algorithm (EZW)

• A simple, yet remarkable effective, image compression algorithm, having the property that the bits in the bit stream are generated in order of importance, giving a fully embedded (progressive) code.

• The compressed data stream can have any bit rate desired. Any bit rate is only possible if there is information loss somewhere so that the compressor is lossy. However, lossless compression is also possible with less spectacular results.

6

EZW - observations

1. Natural images in general have a low pass spectrum, so the wavelet coefficients will, on average, be smaller in the higher subbands than in the lower subbands. This shows that progressive encoding is a very natural choice for compressing wavelet transformed images, since the higher subbands only add detail.

2. Large wavelet coefficients are more important than smaller wavelet coefficients.

631 544 86 10 -7 29 55 -54 730 655 -13 30 -12 44 41 32 19 23 37 17 -4 –13 -13 39 25 -49 32 -4 9 -23 -17 -35 32 -10 56 -22 -7 -25 40 -10 6 34 -44 4 13 -12 21 24 -12 -2 -8 -24 -42 9 -21 45 13 -3 -16 -15 31 -11 -10 -17

typical wavelet coefficients for a 8*8 block in a real image

Zero Tree Coding

Parent – Child relationship

coefficients that are in the same spatial location consist of a quad-tree.

EZW Algorithm

EZW Algorithm contd..

Scanning order of sub bands

EZW - example

EZW – Example contd..

Contd..

Contd..

Contd..

18

EZW - example

Set Partitioning in Hierarchical Trees (SPIHT) Algorithm

SPIHT

References

• Shapiro, J.M.; “Embedded Image Coding Using Zerotrees of Wavelet Coefficients”, IEEE Transactions on Signal Processing, Volume: 41 , No: 12 , Dec. 1993 Pages: 3445 – 3462

• Khalid Sayood, “Introduction to Data Compression”, 2/E, 2000

• A. Said and W. Pearlman, “A new, fast and efficient image codec based on set partitioning”, IEEE Trans. Circuits Syst. VideoTechnol., vol. 6, pp. 243-250, June 1996.