transform coding ii

8/9/2019 Transform Coding II

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TRANSFORM CODING II


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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.


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A Transform coding System


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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


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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.


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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


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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


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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.


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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


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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


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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


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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


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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.


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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.


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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.


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T"res"o!d Coding 9sually the third method is used where the

thresholding and quantization is implemented using a

single equation

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0./0/:

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0/0/:

arrayionnormalizat transformtheof element vu Z

vuT of ionapproximat quantized and d thresholdevuT where

vu Z

vuT round vuT

−

−

=


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T"res"o!d Coding


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transform coding ii

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