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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
