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1
he arithmetic coders used in JPEG, JPEG 2000 and JBIG are called QM
coder
1
. It handles only binary strings or input and it is designed for
simplicity and speed. It uses approximation for multiplication operation,
fixedprecision integer arithmetic with renormalization of the probability interval
from time to time.
The main idea of the QMcoder is to classify the input bit as
M
ore
P
robable
S
ymbol
(
MPS)
and
L
ess
P
robable
S
ymbol
(
LPS)
. Before the next bit is input, the
QMcoder uses a statistical model (using a context, typically a twodimensional
context of black and white pixel in an image) to predict which one of the bits (0 or
1) will be the
MPS
. If the predicted
MPS
bit does not match with the actual bit,
then the QMcoder will classify this as
LPS
; otherwise, it will continue to be
classified as
MPS
. The output of the coder is simply a compressed version of a
stream of
MPS
or
LPS
, which are assigned probability values dynamically. The
decoder has only the knowledge of whether the next predicted bit is
MPS
or
LPS
.
It uses the same statistical model as that of the encoder to obtain the actual values
of the bit. Recall the range update equations we used for arithmetic coding: Let
L
and
H
denote the current ‘low’ and ‘high’, respectively,
and current ‘range’
A=H
L
. Let the current incoming symbol be
a
i,
,
its probability be
p (a
i
)
and its
cumulative low and high probability be
P
(
a
i1
) and
P
(
a
i
), respectively, then the
new low and new high
become
L: = L + (H – L) * P (a
i1
)
H: = L + (H – L) * P (a
i
)
Then the new range becomes ( by subtracting new
L
from
new
H
)
A: =A [P (a
i
)  P (a
i1
)] =A* p (a
i
)
Let us assign a probability
Q
to
LPS
and assign the lower interval to
MPS
with
1
There is an error in the book [Data Compression: The Complete Reference, 2nd Edition]:
Page 121, Table 2.64: The renormalized values of C are wrong. The correct table is:
Symbol
C
A
Renor. A
Renor. C

Initially
0
1
S1 (LPS)
0+10.1=0.9
0.1
0.8
0.9*2^3=
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This note was uploaded on 06/09/2011 for the course CAP 5015 taught by Professor Mukherjee during the Spring '11 term at University of Central Florida.
 Spring '11
 Mukherjee

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