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
Lo-Hi
Hi-Lo
Hi-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
Qstep = 15
• 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
4.65 1.15 0.80 1.70
H(X ) 



 2.07 bits / pel
4
4
4
4
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
Level2 Entropies  Level1 Entropies

Entropy

16
Bands
Level 2
Level 1
4
Entropy/Coeff
Entropy/pixel
LoLo
5.58
0.34
LoHi
2.22
0.14
HiLo
2.99
0.19
HiHi
1.75
0.11
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
Level3 Entropies  Level2 Entropies  Level1 Entropies

Entropy


64
16
4
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 Bands
level 3 coefficient
represents
64 pixels
Entropy/Coeff
Entropy/pixel
Level3 Entropies  Level2 Entropies  Level1 Entropies

LoHi
3.55
0.06
Entropy


Level 3
160.07
4
HiLo64
4.52
Level 2
Level 1
LoLo
6.42
0.10
HiHi
3.05
0.05
LoHi
2.22
0.14
HiLo
2.99
0.19
HiHi
1.75
0.11
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
GOAL – estimate a theoretical value for the entropy of one
of the subbands
So we can estimate x0 for the band by finding the standard
deviation of the coefficient values.
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
lk   log 2( pk )
lk   log 2(0.8)  0.32 bits
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
RLC
RLC coding to create “events”
13
-5 1
0
0
0 0 0
0 -1 0
0
0 0 0 0 0
0 0 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 event
2 x Run-of-2 zeros event
1x Run-of-8-zeros event
1 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
Symbol
Code
A
0
B
10
C
11
• 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.
Summary