Losslessy Compression of Multimedia Data

Losslessy Compression of Multimedia Data

Hao JiangComputer Science Department

Sept. 25, 2007

Lossy Compression

Apart from lossless compression, we can further reduce the bits to represent media data by discarding “unnecessary” information.

Media such as image, audio and video can be “modified” without seriously affecting the perceived quality.

Lossy multimedia data compression standards include JPEG, MPEG, etc.

Methods of Discarding Information

Reducing resolution

Original image 1/2 resolution and zoom in

Reduce pixel color levels

½ color levelsOriginal image

For audios and videos we can similarly reduce the sampling rate, the sample levels, etc.

These methods usually introduce large distortion. Smarter schemes are necessary!

2.3bits/pixel (JPEG)

Distortion Distortion: the amount of difference between

the encoded media data and the original one.

Distortion measurement– Mean Square Error (MSE)

mean( ||xorg – xdecoded||2)

– Signal to Noise Ratio (SNR) SNR = 10log10 (Signal_Power)/(MSE) (dB)

– Peak Signal to Noise Ratio PSNR = 10log10(255^2/MSE) (dB)

The Relation of Rate and Distortion

The lowest possible rate (average codeword length per symbol) is correlated with the distortion.

D

Bit Rate

0 D_max

H

Quantization

Maps a continuous or discrete set of values into a smaller set of values.

The basic method to “throw away” information.

Quantization can be used for both scalars (single numbers) or vectors (several numbers together).

After quantization, we can generate a fixed length code directly.

Uniform Scalar Quantization

xmin xmax

Quantization step =(xmax-xmin)/N

Decision boundaries

Quantization value

Assume x is in [xmin, xmax]. We partition the intervaluniformly into N nonoverlapping regions.

A quantizer Q(x) maps x to the quantization value in the regionwhere x falls in.

Quantization ExampleQ(x) = [floor(x/) + 0.5]

Q(x)/

x/0 1 2 3-3 -2 -1

0.5

1.5

-0.5

-1.5

-2.5

2.5

Midrise quantization

Quantization ExampleQ(x) = [round(x/)]

Q(x)/

x/0 1 2 3-3 -2 -1

1

2

-1

-2

-3

3

Midrise quantization

Quantization Error

To minimize the possible maximum error, the quantization value should be at the center of each decision interval.

If x randomly occurs, Q(x) is uniformly distributed in [-/2, /2]

xnxn+1

Quantization error

x Quantization value

Quantization and Codewords

xmin xmax

Each quantization value can be associated with a binarycodeword.

In the above example, the codeword corresponds to theindex of each quantization value.

000 001 010 011 100 101

Another Coding Scheme

Gray code

xmin xmax

000 001 011 010 110 111

• The above codeword is different in only 1bit for each neighbors.

• Gray code is more resistant to bit errors than the natural binary code.

Bit Assignment

If the # of quantization interval is N, we can use log2(N) bits to represent each quantized value.

For uniform distributed x, The SNR of Q(x) is proportional to 20log(N) = 6.02n, where N=2n

bits

dB

1 more bit

About 6db gain

Non-uniform Quantizer For audio and visual, the tolerance of a

distortion is proportional to the signal size.

So, we can make quantization step proportional to the signal level.

If signal is not uniformly distributed, we also prefer non-uniform quantization.

0

Perceived distortion ~ / s

Vector Quantization

Decision Region

Quantization Value

Predictive Coding

Lossless difference coding revisited

1 3 4 5 3 2 1 0 3 4 5 6 70

1 2 1 1 -2 -1 -1 -1 3 1 1 1 10

1 3 4 5 3 2 1 0 3 4 5 6 7

+ + … +

+-

+-

+-…+

-

encoder

decoder

Local decoder

Predictive Coding in Lossy Compression

1 3 4 5 3 2 1 0 3 4 5 6 70

1 1 1 1 -1 -1 -1 -1 1 1 1 1 1

0 1 2 3 4 3 2 1 0 1 2 3 4 5

Q

+-

Q

+ +

+ +

+

- …

Encoder

…

-+-

Q

Q(x) = 1 if x > 0, 0 if x == 0 and –1 if x < 0

A Different Notation

Buffer

+Audio samples or image pixels

Entropycoding

0101…

Lossless Predictive Encoder Diagram

-

A Different Notation

Buffer

+Reconstructedaudio samples or image pixels+

Entropydecoding

Lossless Predictive Decoder Diagram

Codestream

Local Decompression

A different Notation

Buffer

+Audio samples or image pixels

-

Coding

0101…

Q

+

Lossy Predictive Coding

General Prediction Method For image:

For Audio:

Issues with Predictive Coding– Not resistant to bit errors.– Random access problem.

C B

A X

ABCD X

Transform Coding

1 3 4 5 3 2 1 0 3 4 5 6

2 4.5 2.5 0.5 3.5 5.5

+ + + + + +

1 3 4 5 3 2 1 0 3 4 5 6

+ + + + + +- - - - - -

½ ½ ½ ½ ½ ½

-1 -0.5 0.5 0.5 -0.5 0.5

½ ½ ½ ½ ½ ½

Transform and Inverse Transform

y1y2

= ½ ½½ -½

x1x2

We did a transform for a block of input data using

The inverse transform is:

x1x2

= 1 11 -1

y1y2

Transform Coding

A proper transform focuses the energy into small number of numbers.

We can then quantize these values differently and achieve high compression ratio.

Useful transforms in compressing multimedia data:– Fourier Transform– Discrete Cosine Transform