Download - Transform Coding II
-
8/9/2019 Transform Coding II
1/19
TRANSFORM CODING II
-
8/9/2019 Transform Coding II
2/19
Introduction The coding techniques which operate directly on the
pixels of an image are the spatial domain methods.
The coding is known as waveform coding. The coding techniques which are based on modifying
the transform of an image is known as transform
coding.
In transform coding a reversible linear transform is
used to map the image into a set of transform
coefficients which are then quantized and coded.
-
8/9/2019 Transform Coding II
3/19
A Transform coding System
-
8/9/2019 Transform Coding II
4/19
Important Steps in Transform Coding Transform selection
The transform which packs most of the information inthe smallest number of transform coefficients is
selected to achieve the best compression. Sub section selection
The sub section size which will reduce thereconstruction error and the computational complexity
is selected. For image data sub image size of ! or "# is common.
$arger sub image sizes increases the blocking artifacts. For speech data a block of "% to &%msec is selected for
processing
-
8/9/2019 Transform Coding II
5/19
Important Steps in Transform Coding 'it allocation(
The reconstruction error depends on the number and
relative importance of the transform coefficients that arediscarded and the precision used to represent the
retained coefficients.
The overall process of truncating quantizing and codingthe coefficients of the transformed sub image is called
bit allocation.
-
8/9/2019 Transform Coding II
6/19
Quantization and Coding of
Transform coefficients
If the amount of information conveyed by each
coefficient is different it makes sense to assign
differing numbers of bits to the different coefficients. There are two approaches to assign bits
)ne approach relies on the average properties of the
transform coefficients while the other approach assigns
bits as needed by individual transform coefficients
-
8/9/2019 Transform Coding II
7/19
Quantization and Coding of
Transform coefficients In the first approach we first obtain an
estimate of the variances of the transform
coefficients )n the basis of maximum variance * +onal coding
)n the basis of maximum magnitude * Threshold coding
These estimates can be used by one of twoalgorithms to assign the number of bits used
to quantize each of the coefficients
-
8/9/2019 Transform Coding II
8/19
ona! Coding ,e assume that the relative variance of the coefficients
corresponds to the amount of information contained in each
coefficient.
Thus coefficients with higher variance are assigned more bitsthan coefficients with smaller variance.
$et us find an expression for the distortion then find the bitallocation that minimizes the distortion.
To perform the minimization we will use the method of
$agrange.
-
8/9/2019 Transform Coding II
9/19
ona! Coding To find the number of bits to be allotted(
If the average number of bits per sample to be used
by the transform coding system is R, and the averagenumber of bits per sample used by the k th coefficientis Rk then
- * o( of transform coefficients.
/"0
-
8/9/2019 Transform Coding II
10/19
ona! Coding The reconstruction error variance for the kth quantizer
1rk & is related to the kth quantizer input variance 12k &
by
3k 4 factor that depends on the input distribution and the
quantizer.
The total reconstruction error is given by
-
8/9/2019 Transform Coding II
11/19
ona! Coding So we need to find 5 k to minimize the error at
the same time keeping the average number of
bits to 5. 6ssuming that 3k is a constant 3 for all
k we can set up the minimization problem in
terms of $agrange multiplier as
-
8/9/2019 Transform Coding II
12/19
ona! Coding Taking the derivative of 7 with respect to 5 k and
setting it equal to zero we can obtain the
expression for 5 k as
Substituting for 5 k in equation /"0 we get the
value of 8 as
-
8/9/2019 Transform Coding II
13/19
ona! Coding ow substituting this expression for 8 in the equation
for 5 k we obtain
The values obtained for 5 k will not be positiveintegers. So the standard approach is to set thenegative 5 k s to zero. This will increase the average bitrate above 5. Therefore the non zero 5 k s are
uniformly reduced until the average rate is equal to 5.
-
8/9/2019 Transform Coding II
14/19
T"res"o!d Coding The underlying principle is that for any sub image the
transform coefficients of largest magnitude make the
most significant contribution to reconstructed sub imagequality.
6fter applying the threshold masks the resulting nxn
array is reordered in a zigzag fashion and is run length
encoded. 6nd variable length coding is done to the
resulting sequence.
-
8/9/2019 Transform Coding II
15/19
T"res"o!d Coding
There are three ways to threshold the
transformed image 6 single global thresholding can be applied to all
sub images.
6 different threshold can be applied for each sub
image. The threshold can be varied as a function of
location of each coefficient within the sub image.
-
8/9/2019 Transform Coding II
16/19
T"res"o!d Coding 9sually the third method is used where the
thresholding and quantization is implemented using a
single equation
.0/
0./0/:
0/
0/0/:
arrayionnormalizat transformtheof element vu Z
vuT of ionapproximat quantized and d thresholdevuT where
vu Z
vuT round vuT
−
−
=
-
8/9/2019 Transform Coding II
17/19
T"res"o!d Coding
-
8/9/2019 Transform Coding II
18/19
-
8/9/2019 Transform Coding II
19/19