Lossless Image Compression

1
Lossless Image Compression

• Recall run length coding of binary and graphic
  images
• Why does it not work for gray-scale images?
• Image modeling revisited
• Predictive coding of gray-scale images
• 1D Predictive coding DPCM
• 2D fixed and adaptive prediction
• Applications Lossless JPEG and JPEG-LS

2
Lossless Image Compression

• No information loss i.e., the decoded image is
  mathematically identical to the original image
• For some sensitive data such as document or
  medical images, information loss is simply
  unbearable
• For others such as photographic images, we only
  care about the subjective quality of decoded
  images (not the fidelity to the original)

3
Data Compression Paradigm
Y
entropy coding
binary bit stream
source modeling
discrete source X
P(Y)
probability estimation
Probabilities can be estimated by counting
relative frequencies either online or offline
The art of data compression is the art of source
modeling
4
Recall Run Length Coding
Y
Transformation by run-length counting
Huffman coding
binary bit stream
discrete source X
P(Y)
probability estimation
Y is the sequence of run-lengths from which X can
be recovered losslessly
5
Image Example
col.
156 159 158 155 158 156 159 158
160 154 157 158 157 159 158 158
156 159 158 155 158 156 159 158
160 154 157 158 157 159 158 158
156 153 155 159 159 155 156 155
155 155 155 157 156 159 152 158
156 153 157 156 153 155 154 155
159 159 156 158 156 159 157 161
row
Runmax4
6
Why Short Runs?
7
Why RLC bad for gray-scale images?

• Gray-scale (also called photographic) images
  have hundreds of different gray levels
• Since gray-scale images are acquired from the
  real world, noise contamination is inevitable

You simply can not freely RUN in a gray-scale
image
8
Source Modeling Techniques

• Prediction
• Predict the future based on the causal past
• Transformation
• transform the source into an equivalent yet more
  convenient representation
• Pattern matching
• Identify and represent repeated patterns

9
The Idea of Prediction

• Remarkably simple just follow the trend
• Example I X is a normal persons temperature
  variation through day
• Example II X is intensity values of the first
  row of cameraman image
• Markovian school (short memory)
• Prediction does not count on the data a long time
  ago but the most recent ones (e.g., your
  temperature in the evening is more correlated to
  that in the afternoon than that in the morning)

10
1D Predictive Coding
1st order Linear Prediction
x1 x2 xn-1 xn xn1 xN
original samples
y1 y2 yn-1 yn yn1 yN
prediction residues
x1 x2 xN
y1 y2 yN
- Encoding
y1x1
initialize
ynxn-xn-1, n2,,N
prediction
- Decoding
y1 y2 yN
x1 x2 xN
x1y1
initialize
prediction
xnynxn-1, n2,,N
11
Numerical Example

90
original samples
92
91
93
93
95
a
b
a-b
prediction residues
90
2
-1
2
0
2
a
b
decoded samples

90
92
91
93
93
95
ab
12
Image Example (take one row)
H(X)6.56bpp
original row signal x (left) and its histogram
(right)
13
Source Entropy Revisited

• How to calculate the entropy for a given
  sequence (or image)?
• Obtain the histogram by relative frequency
  counting
• Normalized the histogram to obtain probabilities
  PkProb(Xk),k0-255
• Plug the probabilities into entropy formula

You will be asked to implement it in the
assignment
14
Cautious Notes

• The entropy value calculated in previous slide
  need to be understood as the result if we choose
  to model the image by an independent identically
  distributed (i.i.d.) random variable.
• It does not take spatially correlated and varying
  characteristics into account
• The true entropy is smaller!

15
Image Example (cont)
H(Y)4.80bpp
prediction residue signal y (left) and its
histogram (right)
16
Interpretation

• H(Y)ltH(X) justifies the role of prediction
  (intuitively it decorrelates the signal).
• Similarly, H(Y) is result if we choose to model
  the residue image by an independent identically
  distributed (i.i.d.) random variable.
• It is an improved model when compared with X due
  to the prediction
• The true entropy is smaller!

17
High-order 1D Prediction Coding
k-th order Linear Prediction
x1 x2 xn-1 xn xn1 xN
original samples
x1 x2 xN
y1 y2 yN
- Encoding
y1x1,y2x2,,ykxk
initialize
prediction
- Decoding
y1 y2 yN
x1 x2 xN
x1y1,x2y2,,xkyk
initialize
prediction
18
Why High-order?

• By looking at more past samples, we can have a
  better prediction of the current one
• Compare c_, ic_ , dic_ and predic_
• It is a tradeoff between performance and
  complexity
• The performance quickly diminishes as the order
  increases
• Optimal order is often signal-dependent

19
1D Predictive Coding Summary
Y
entropy coding
binary bit stream
Linear Prediction
discrete source X
P(Y)
probability estimation
Prediction residue sequence Y usually contains
less uncertainty (entropy) than the original
sequence X
20
From 1D to 2D
1D
X(n)
n
future
causal past
2D



raster-scanning
Zigzag-scanning
21
2D Predictive Coding
raster scanning order
causal half-plane
Xm,n
22
Ordering Causal Neighbors
6
4
2
3
1
5
Xm,n
where
Xk the k nearest causal neighbors of Xm,n in
terms of Euclidean distance
prediction residue
23
Lossless JPEG
Notes
1 2 3 4 5 6 7
Predictor w n nw nw-nw w-(n-nw)/2 n-(w-nw)/2 (nw
)/2
horizontal
vertical
n
nw
diagonal
x
w
3rd-order
2nd-order
24
Numerical Examples
1D
a
b
X156 159 158 155
Y156 3 -1 -3
a-b
2D
Initialization no prediction applied
156 159 158 155 160 154 157
158 156 159 158 155 160 154 157
158
156 3 -1 -3 160 -6 3 1
156 3 -1 -3 160 -6 3 1
horizontal predictor
X
Y
Note
2D horizontal prediction can be viewed as the
vector case of 1D prediction of each row
25
Numerical Examples (Cont)
156 159 158 155 160 154 157
158 156 159 158 155 160 154 157
158
156 159 158 155 4 -5 -1
3 -4 5 1 -3 4
-5 -1 3
vertical predictor
X
Y
Note
2D vertical prediction can be viewed as
the vector case of 1D prediction of each column
Q Given a function of horizontal prediction, can
you Use this function to implement vertical
prediction? A Apply horizontal prediction to the
transpose of the image and then transpose the
prediction residue again
26
Image Examples
Comparison of residue images generated by
different predictors
vertical predictor
horizontal predictor
H(Y)5.05bpp
H(Y)4.67bpp
Q why vertical predictor outperforms horizontal
predictor?
27
Analysis with a Simplified Edge Model
100 100 50 50
100 50 100 50
n
nw
50 50 100 100
50 100 50 100
x
w
H_edge
V_edge
H_predictor
Y?50
Y0
V_predictor
Y?50
Y0
Conclusion when the direction of predictor
matches the direction of edges, prediction
residues are small
28
Horizontal vs. Vertical
Do we see more vertical edges than horizontal
edges in natural images? Maybe yes, but why?
29
Importance of Adaptation

• Wouldnt it be nice if we can switch the
  direction of predictor to locally match the edge
  direction?
• The concept of adaptation was conceived several
  thousands ago in an ancient Chinese story of how
  to win a horse racing

emperor
general
good
90
80
gt
How to win?
gt
fair
70
60
poor
50
40
gt
30
Median Edge Detection (MED) Prediction
n
nw
x
w
Key
MED use the median operator to adaptively select
one from three candidates (Predictors 1,2,4 in
slide 44) as the predicted value.
31
Another Way of Implementation
n
nw
x
If
w
Q which one is faster? You need to find it out
using MATLAB yourself
else if
32
Proof by Enumeration
Case 1 nwgtmax(n,w)
If nwgtngtw, then n-nwlt0 and w-nwlt0, so nw-nwltwltn
and median(n,w,nw-nw)min(n,w)w
If nwgtwgtn, then n-nwlt0 and w-nwlt0, so nw-nwltnltw
and median(n,w,nw-nw)min(n,w)n
Case 2 nwltmin(n,w)
If nwltnltw, then n-nwgt and w-nwgt0, so nw-nwgtwgtn
and median(n,w,nw-nw)max(n,w)w
If nwltwltn, then n-nwgt0 and w-nwgt0, so nw-nwgtngtw
and median(n,w,nw-nw)max(n,w)n
Case 3 nltnwltw or wltnwltn nw-nw also lies between
n and w, so median(n,w,nw-nw)nw-nw
33
Numerical Examples
H_edge
V_edge
n100,w50, nw100
n50,w100, nw100
100 50 100 50
100 100 50 50
nw-nw50
nw-nw50
Note how we can get zero prediction residues
regardless of the edge direction
34
Image Example
Fixed vertical predictor H4.67bpp
Adaptive (MED) predictor H4.55bpp
35
JPEG-LS (the new standard for lossless image
compression)
36
Summary of Lossless Image Compression

• Importance of modeling image source
• Different classes of images need to be handled by
  different modeling techniques, e.g., RLC for
  binary/graphic and prediction for photographic
• Importance of geometry
• Images are two-dimensional signals
• In 2D world, issues such as scanning order and
  orientation are critical to modeling