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Zerotree Coding Toolbox

Image compression based on wavelets, using zerotrees of wavelet coefficients. Uses Wavelet and Quality Assessment toolboxes. The SPIHT binaries required for function spspiht.m can be downloaded from the SPIHT image compression homepage, see below.

Functions

  • dead_zone_q - Quantisation with a central dead-zone around zero
  • ezw - EZW (Embedded Zerotree Wavelet) image compression
  • pdf_opt - Optimisation (pdf) of reconstructed values of quantised wavelet coefficients
  • spiht_stream_dec - Performs decoding of the bitstream produced by spiht_wpackets.m
  • spiht_wpackets - SPIHT image compression using Wavelet Packets (WP) decomposition
  • spspiht - Script for batch execution of DOS SPIHT binaries

DOS SPIHT binaries required for function spspiht.m can be found at SPIHT image compression homepage. Here's direct link to the package.

Examples

SPIHT algorithm flow-chart:

SPIHT algorithm flow-chart

spiht_wpackets with dyadic DWT, as in the original SPIHT:

>> [Arec,bitstream,PSNR]=spiht_wpackets('Lena512.png',[0.1 1],'CDF_9x7',6);
6 decompositions, 0 packet depth, CDF_9x7 wavelet, highest bitrate 1.000000 bpp
Image transformed in 0.85 seconds
Maximum magnitudes computed in 0.24 seconds

SPIHT algorithm...

1. bitrate - 0.099995 bpp
No. output bits - 26216(3277 bytes; 3 excess bits)-> 0.100006 bpp
MSE (Mean Squared Error) = 68.231052
PSNR (Peak Signal / Noise Ratio) = 29.790983 dB

2. bitrate - 0.999996 bpp
No. output bits - 262144(32768 bytes; 1 excess bits)-> 1.000000 bpp
MSE (Mean Squared Error) = 6.535866
PSNR (Peak Signal / Noise Ratio) = 39.977772 dB

...LIS (checking children) loop out - completed in 8.91 seconds

Statistics collected during encoding in SPIHTlog.txt, where 'bpc' stands for 'bits per significant coefficient':

                    LIP loop                LIS loop              Total significant
 pass(start bit)   bits/coeffs.            bits/coeffs.              coeffs.(LSP start bit)
 1.(      52)       65/1     =65.00 bpc     64/0     =  Inf bpc           1(     181)
 2.(     181)       81/18    = 4.50 bpc     89/5     =17.80 bpc          24(     351)
 3.(     352)       67/12    = 5.58 bpc    246/32    = 7.69 bpc          68(     665)
 4.(     689)      150/39    = 3.85 bpc    749/111   = 6.75 bpc         218(    1588)
 5.(    1656)      362/80    = 4.53 bpc   1650/235   = 7.02 bpc         533(    3668)
 6.(    3886)      927/237   = 3.91 bpc   4549/624   = 7.29 bpc        1394(    9362)
 7.(    9895)     2512/621   = 4.05 bpc   9202/1334  = 6.90 bpc        3349(   21609)
 8.(   23003)     5424/1489  = 3.64 bpc  17931/2677  = 6.70 bpc        7515(   46358)
 9.(   49707)    10719/2998  = 3.58 bpc  33035/5101  = 6.48 bpc       15614(   93461)
10.(  100976)    20709/5739  = 3.61 bpc  68089/10551 = 6.45 bpc       31904(  189774)
11.(  205388)    45154/12562 = 3.59 bpc  11602/2519  = 4.61 bpc       46985(       0)
Total:
                147206/23189 = 3.62 bpc  86170/23189 = 6.35 bpc       46985

The following example uses functions from Wavelet and Quality Assessment Toolboxes. It demonstrates that using the pdf-optimised reconstruction of quantised wavelet coefficients a meagre improvement of ~0.02 dB can be expected on high bit-rates. Functions used are dead_zone_q and pdf_opt:

>> D=dwt_dyadic_decomp('Lena256.png','CDF_9x7',6);
>> qstep = 4; %quantisation step is 4
>> Dq = dead_zone_q(D,qstep); 
>> Dq_opt = pdf_opt(Dq,qstep);
>> iq_measures(idwt_dyadic_recon(Dq,'CDF_9x7',6),'Lena256.png',1,1); 
MSE (Mean Squared Error)= 2.556086
PSNR (Peak SNR)= 44.055048 dB
>> iq_measures(idwt_dyadic_recon(Dq_opt,'CDF_9x7',6),'Lena256.png','disp');
MSE (Mean Squared Error)= 2.544909
PSNR (Peak SNR)= 44.074081 dB

DOS SPIHT binaries using function spspiht:

>> [Arec,bitstream,PSNR]=spspiht('Lena512.png',[0.1 1],'fast');
Number of bits = 262160(32770 bytes, 1.0001 bpp)

1. bitrate - 0.1000 bpp
MSE (Mean Squared Error)= 67.644821
PSNR (Peak SNR)= 29.828458 dB

2. bitrate - 1.0000 bpp
MSE (Mean Squared Error)= 6.532291
PSNR (Peak SNR)= 39.980148 dB

Total execution time - 0.84 seconds

EZW (Embedded Zerotree Wavelet) algorithm with function ezw:

>> ezw('ezw_testdata1.mat',2,2,'CDF_9x7');
Image ezw_testdata1.mat, 2 decompositions, CDF9_7 wavelet, bitrate 2.000000 bpp

Example matrix:
    55    42    24   -28     6    16     0     3     1     8     1     8
    31    48    22    29     2     6     0     2    -8     2    -8     2
    25   -21    26    20     6    18     2     7    -9     4    -9     4
     1     0    29    18    16    10     5     3     1     2     1     2
    17     2     7   -16     6     3    -4     6     1     0     7     0
    13    -9    16    15     1     7     7     9     7     0     7     0
   -18     0     1     2     6    -3     5     5    -7     8    -7     8
    10     2     0     8    -1     9     5     9     6     3     6     3

1. pass
p z z z z z t t t t t t t t t t t t t t t t t t 
1
2. pass
z p z z z z t t t t t t p t t t t t t t t t t t t t t t 
100
3. pass
z z z z p z z t t z t t z n t z t t p t t t t t n p p t t p t t t n t t t t t t t t
EZW algorithm executed in 0.047000 seconds

Number of output bits: 192
MSE (Mean Squared Error)= 60.529236
PSNR (Peak SNR)= 30.311152 dB