4c8 image compression. lossy compression effective bit rate = 8 bits/pixel effective bit rate = 1...

4C8

Image Compression

Lossy Compression

Effective bit rate = 8 bits/pixel

Effective bit rate = 1 bit/pixel (approx)

Transform Coding

• In the last set of slides we showed that transforming the image into a difference image reduces the entropy of image.

G(x,y) = I(x,y) – I(x-1,y)

Transform Coding

• This is because entropy is greatest when uniform

Histogram of the original image Histogram of the difference image

Signal Energy

Lossy Transform Coding

Lossless Lossy Lossless

Lossless

Energy Compaction with Xforms

The Haar Xform

LoLo Hi-Lo

Hi-HiLo-Hi

Implementation Details

• When calculating the haar transform for the image the mid gray value is typical.

• Colour Images are processed by treating each colour channel as separate gray scale images.– If YUV colourspace is used subsampling of

the U and V channels is probable.

Quantisation

• After we create the image we quantise the transform coefficients.

• Step size is shown by perceptual evaluation

• We can assign different step sizes to the different bands.• We can use different step sizes for the different colour

channels.

• We will consider a uniform step size, Qstep, for each band for now.

Entropy

Qstep = 15

Entropy

• Calculating the overall entropy is trickier– Each coefficient in a band represents 4 pixel

locations in the original image.– So bits/pixel = (bits/coefficient)/4

• So the entropy of the transformed and quantised lenna is

pelbitsXH /07.24

70.1

4

80.0

4

15.1

4

65.4)(

Qstep = 15

Mistake in Fig. 5 of handout

Red Dashed Line is the Histogram. Blue bars represent the “entropies” (ie. - p * log2(p) ) and not vice versa

Multilevel Haar Xform

Calculating the Entropy for Level 2 of the transform

• One Level 1 coefficient represents 4 pixels• One level 2 coefficient represents 16 pixels

4

Entropies 1 Level

16

Entropies 2 LevelEntropy

Bands Entropy/Coeff Entropy/pixel

Level 2

LoLo 5.58 0.34

LoHi 2.22 0.14

HiLo 2.99 0.19

HiHi 1.75 0.11

Level 1

LoHi 1.15 0.29

HiLo 1.70 0.43

HiHi 0.80 0.29

Total Entropy = 1.70 bits/pixel

Qstep = 15

Multilevel Haar Xform

Calculating the Entropy for Level 3 of the transform

• One Level 1 coefficient represents 4 pixels• One level 2 coefficient represents 16 pixels• One level 3 coefficient represents 64 pixels

4

Entropies 1 Level

16

Entropies 2 Level

64

Entropies 3 LevelEntropy

Qstep = 15

Calculating the Entropy for Level 3 of the transform

• One Level 1 coefficient represents 4 pixels• One level 2 coefficient represents 16 pixels• One level 3 coefficient represents 64 pixels

4

Entropies 1 Level

16

Entropies 2 Level

64

Entropies 3 LevelEntropy

Bands Entropy/Coeff Entropy/pixel

Level 3

LoLo 6.42 0.10

LoHi 3.55 0.06

HiLo 4.52 0.07

HiHi 3.05 0.05

Level 2

LoHi 2.22 0.14

HiLo 2.99 0.19

HiHi 1.75 0.11

Level 1

LoHi 1.15 0.29

HiLo 1.70 0.43

HiHi 0.80 0.29

Total Entropy = 1.62 bits/pixel

Qstep = 15

Multilevel Haar Xform

Qstep = 15

Measuring Performance

• Compression Efficiency - Entropy• Reconstruction Quality – Subjective Analysis

Haar Transform

Quantisation

Quantisation

Reconstruction Qstep = 15

Reconstruction Qstep = 30

Reconstruction Qstep = 30

Original Quantised Haar Transform + Quantisation

Laplacian Pdfs

So we can estimate x0 for the band by finding the standard deviation of the coefficient values.

GOAL – estimate a theoretical value for the entropy of one of the subbands

x1 = 0, x2 = Q/2

x1 = (k-1/2)Q, x2 = (k-1/2)Q

See Handout for Missing Steps Here

Measured Entropy is less than what we would expect for a laplacian pdf. This is because the actual decay of the histogram is greater than an exponential decay.

Practical Entropy Coding

Huffman Coding

Practical Results

The code is inefficient because level 0 as a probability >>0.5 (0.8 approx)

Remember the ideal codelength

So if pk = 0.8, then

However, the minimum code length we can use for a symbol is 1 bit.

Therefore, we need to find a new way of coding level 0 – use run length coding

)(2log kk pl

bits32.0)8.0(2log kl

RLC

RLC coding to create “events”

13 -5 1 0 -1 00 00 0 00 00 0 0 0 00 0 0

Define max run of zeros as 8, and we are coding runs of 1, 2, 4 and 8 zeros

Here we have 4 non-zero “events”1 x Run-of-4-Zeros event2 x Run-of-2 zeros event1x Run-of-8-zeros event1 x Run-of-1-zero event

Practical Results

Synchronisation

Say we have a source with symbols A, B and C. Say we wish to encode the message ABBCCBCABAA using the following code table

Symbol Code

A 0

B 10

C 11

The Coded message is therefore 010101111101101000

Q. What is the decoded message if the 6th bit in the stream is corrupted?

Ie. We receive 010100111101101000

Synchronisation

• 010100111101101000

• The decoded stream is ABBACCACABA• The problem is that 1 bit error causes subsequent

symbols to be decoded incorrectly as well.• The stream is said to have lost synchronisation.• A solution is to periodically insert synchronisation

symbols into the stream (eg. One at the start of each row). This limits how far errors can propagate.

Symbol Code

A 0

B 10

C 11

Summary