DCC ‘99 - Adaptive Prediction Lossless Image Coding
Adaptive Linear Prediction
Lossless Image Coding
Giovanni Motta, James A. Storer
Bruno Carpentieri
Brandeis University
Volen Center for Complex Systems
Computer Science Department
Waltham MA-02454, US
{gim, storer}@cs.brandeis.edu
Universita' di Salerno
Dip. di Informatica ed Applicazioni
"R.M. Capocelli”
I-84081 Baronissi (SA), Italy
[email protected]
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Problem
Graylevel lossless image compression
addressed from the point of view of
the achievable compression ratio
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Outline
Motivations
Main Idea
Algorithm
Predictor Assessment
Entropy Coding
Final Experimental Results
Conclusion
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Past Results / Related Works
Until TMW, the best existing lossless digital image
compressors (CALIC, LOCO-I, etc..) seemed unable to
improve
compression
by
using
image-by-image
optimization techniques or more sophisticate and
complex algorithms
A year ago, B. Meyer and P. Tischer were able, with
their TMW, to improve some current best results by
using global optimization techniques and multiple
blended linear predictors.
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Past Results / Related Works
In spite of the its high computational complexity, TMW’s
results are in any case surprising because:
 Linear predictors are not effective in capturing
image edginess;
 Global optimization seemed to be ineffective;
 CALIC was thought to achieve a data rate close
to the entropy of the image.
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Motivations
Investigation on an algorithm that uses:
Multiple Adaptive Linear Predictors
Pixel-by-pixel optimization
Local image statistics
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Main Idea
Explicit use of local
statistics to:
 Classify the
context of the
current pixel;
 Select a Linear
Predictor;
 Refine it.
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Prediction Window
Rp+1
2Rp+1
Encoded Pixels
Window Wx,y(Rp)
Current Context
Current Pixel I(x,y)
Statistics are collected inside the window Wx,y(Rp)
Not all the samples in Wx,y(Rp) are used to refine the predictor
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Prediction Context
w0
w4
w1
w2
w5
-1
w3
 6 pixels
 fixed shape
 weights w0,…,w5 change to
minimize error energy inside
Wx,y(Rp)
Prediction:
I’(x,y) = int(w0 * I(x,y-2) + w1 * I(x-1,y-1) +
w2 * I(x,y-1) + w3 * I(x+1,y-1) + w4 * I(x-2,y) + w5 * I(x-1,y))
Error:
Err(x,y) = I’(x,y) - I(x,y)
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Predictor Refinement
Rp+1
2Rp+1
Gradient descent is used to
refine the predictor
Encoded Pixels
Window Wx,y(Rp)
Current Context
Current Pixel I(x,y)
min E(x, y)  min
w 0 ,..., w 5
w 0 ,..., w5
 (ERR(x' ,y' ))2
I (x' , y' )W ' x , y (R p )
E
wi (t 1)  wi (t)  
wi
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Algorithm
for every pixel I(x,y) do begin
/* Classification */
Collect samples Wx,y(Rp)
Classify the samples in n clusters (LBG on the contexts)
Classify the context of the current pixel I(x,y)
Let Pi={w0, .., w5} be the predictor that achieves
the smallest error on the current cluster Ck
/* Prediction */
Refine the predictor Pi on the cluster Ck
Encode and send the prediction error ERR(x,y)
end
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Results Summary
 Compression is better when structures and
textures are present
 Compression is worse on high contrast zones

Local Adaptive LP seems to capture
features not exploited by existing systems
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Test Images
9 “pgm” images,720x576 pixels, 256 greylevels (8 bits)
Balloon
Barb
Girl
Barb2
Gold
Board
Hotel
Boats
Zelda
downloaded from the ftp site of X. Wu:
”ftp:\\ftp.csd.uwo.ca/pub/from_wu/images”
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Outline
Motivations
Main Idea
Algorithm
Predictor Assessment
Entropy Coding
Final Experimental Results
Conclusion
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
File Size vs. Number of Predictors. (Rp=6)
Using an adaptive AC
# of predictors
Balloon
Barb
Barb2
Board
Boats
Girl
Gold
Hotel
Zelda
Total (bytes)
1
2
4
154275
227631
250222
193059
210229
204001
235682
236037
195052
150407
223936
250674
190022
208018
202004
237375
236916
193828
150625
224767
254582
190504
209408
202326
238728
239224
194535
6
150221
225219
256896
190244
209536
202390
239413
240000
195172
8
150298
225912
258557
190597
210549
202605
240352
240733
195503
1906188 1893180 1904699 1909091 1915106
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
File Size vs. window radius RP (# pred.=2)
Using an adaptive AC
Rp
Balloon
Barb
Barb2
Board
Boats
Girl
Gold
Hotel
Zelda
Total (bytes)
6
8
10
12
14
150407
223936
250674
190022
208018
202004
237375
236916
193828
149923
223507
249361
190319
206630
201189
235329
235562
193041
149858
224552
246147
190911
206147
201085
234229
235856
192840
150019
225373
247031
191709
206214
201410
234048
236182
192911
150277
226136
246265
192509
206481
201728
234034
236559
193111
1893180 1884861 1881625 1884897 1887100
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Prediction Error
5.50
5.00
4.50
4.00
3.50
LOCO-I (Error Entropy after Context Modeling)
LOCO-I (Entropy of the Prediction Error)
3.00
2 Predictors, Rp=10, Single Adaptive AC
2.50
baloon barb
barb2
board
boats
Image
girl
gold
hotel
zelda
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Prediction Error
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Prediction Error (histogram)
Test image “Hotel”
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Prediction Error (magnitude and sign)
Test image “Hotel”
Magnitude
Sign
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Prediction Error (magnitude and sign)
Test image “Board”
Magnitude
Sign
DCC ‘99
DCC
- Adaptive
‘99 - Adaptive
LinearPrediction
PredictionLossless
Losless Image Coding
Outline
Motivations
Main Idea
Algorithm
Predictor Assessment
Entropy Coding
Final Experimental Results
Conclusion
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Entropy Coding
 AC model determined in a window Wx,y(Re)
 Two different ACs for typical and non typical symbols
(for practical reasons)
 Global determination of the cutting point
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Compressed File Size vs. error window radius Re
(# of predictors = 2 and Rp=10)
Re
8
10
12
14
16
18
147227
216678
234714
186351
202168
197243
230619
229259
189246
147235
216082
233303
186171
201585
197013
229706
228623
188798
147341
215906
232696
186187
201446
197040
229284
228441
188576
147479
215961
232455
186303
201504
197143
229111
228491
188489
147620
216135
232399
186467
201623
197245
229026
228627
188461
147780
216370
232473
186646
201775
197356
229012
228785
188469
Total (bytes) 1833505 1828516
1826917
1826936 1827603
1828666
balloon
barb
barb2
board
boats
girl
gold
hotel
zelda
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Outline
Motivations
Main Idea
Algorithm
Predictor Assessment
Entropy Coding
Final Experimental Results
Conclusion
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Comparisons
balloon barb barb2 board boats girl
SUNSET
LOCO-I
UCM
Our
CALIC
TMW
2.89
2.90
2.81
2.84
2.78
2.65
4.64
4.65
4.44
4.16
4.31
4.08
4.71
4.66
4.57
4.48
4.46
4.38
3.72
3.64
3.57
3.59
3.51
3.99
3.92
3.85
3.89
3.78
3.61
3.90
3.90
3.81
3.80
3.72
gold hotel zelda Avg.
4.60
4.47
4.45
4.42
4.35
4.28
4.48
4.35
4.28
4.41
4.18
3.79
3.87
3.80
3.64
3.69
4.08
4.04
3.95
3.91
3.86
3.80
Compression rate in bit per pixel.
(# of predictors = 2, Rp=10)
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Comparisons
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Conclusion
 Compression is better when structures and
textures are present
 Compression is worse on high contrast zones

Local Adaptive LP seems to capture
features not exploited by existing systems
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
Future Research
Compression
 Better context classification (to improve on
high contrast zones)
 Adaptive windows
 MAE minimization (instead of MSE min.)
Complexity
 Gradient Descent
 More efficient entropy coding
Additional experiments
 On different test sets
DCC ‘99 - Adaptive Linear Prediction Lossless Image Coding
DCC ‘99 - Adaptive Prediction Lossless Image Coding