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One-Minute Survey Result

Thank you for your responses

Kristen, Anusha, Ian, Christofer, Bernard, Greg, Michael, Shalini, Brian and Justin

Valentine’s challenge

Min: 30-45 minutes, Max: 5 hours, Ave: 2-3 hours

Muddiest points

Regular tree grammar (CS410 compiler or CS422: Automata)

Fractal geometry (“The fractal geometry of nature” by Mandelbrot)

Seeing the Connection

Remember the first story in Steve Jobs’ speech “Staying Hungry, Staying Foolish”?

In addition to Jobs and Shannon, I have two more examples: Charles Darwin and Bruce Lee

EE465: Introduction to Digital Image Processing

Data Compression Basics

Discrete source

Information=uncertainty

Quantification of uncertainty

Source entropy

Variable length codes

Motivation

Prefix condition

Huffman coding algorithm

EE465: Introduction to Digital Image Processing

Information

- What do we mean by information?
- “A numerical measure of the uncertainty of an experimental outcome” – Webster Dictionary
- How to quantitatively measure and represent information?
- Shannon proposes a statistical-mechanics inspired approach
- Let us first look at how we assess the amount of information in our daily lives using common sense

EE465: Introduction to Digital Image Processing

Information = Uncertainty

Zero information

Pittsburgh Steelers won the Superbowl XL (past news, no uncertainty)

Yao Ming plays for Houston Rocket (celebrity fact, no uncertainty)

Little information

It will be very cold in Chicago tomorrow (not much uncertainty since this is winter time)

It is going to rain in Seattle next week (not much uncertainty since it rains nine months a year in NW)

Large information

An earthquake is going to hit CA in July 2006 (are you sure? an unlikely event)

Someone has shown P=NP (Wow! Really? Who did it?)

EE465: Introduction to Digital Image Processing

Shannon’s Picture on Communication (1948)

channel

encoder

channel

decoder

source

channel

destination

super-channel

source

encoder

source

decoder

The goal of communication is to move information

from here to there and from now to then

Examples of source:

Human speeches, photos, text messages, computer programs …

Examples of channel:

storage media, telephone lines, wireless transmission …

EE465: Introduction to Digital Image Processing

Source-Channel Separation Principle*

The role of channel coding:

Fight against channel errors for reliable transmission of information

(design of channel encoder/decoder is considered in EE461)

We simply assume the super-channel achieves error-free transmission

The role of source coding (data compression):

Facilitate storage and transmission by eliminating source redundancy

Our goal is to maximally remove the source redundancy by intelligent designing source encoder/decoder

EE465: Introduction to Digital Image Processing

Discrete Source

A discrete source is characterized by a discrete random variable X

Examples

Coin flipping: P(X=H)=P(X=T)=1/2

Dice tossing: P(X=k)=1/6, k=1-6

Playing-card drawing: P(X=S)=P(X=H)=P(X=D)=P(X=C)=1/4

What is the redundancy with a discrete source?

EE465: Introduction to Digital Image Processing

Two Extreme Cases

source

encoder

source

decoder

tossing

a fair coin

channel

P(X=H)=P(X=T)=1/2: (maximum uncertainty)

Minimum (zero) redundancy, compression impossible

HHHH…

Head

or

Tail?

tossing a coin with

two identical sides

channel

duplication

TTTT…

P(X=H)=1,P(X=T)=0: (minimum redundancy)

Maximum redundancy, compression trivial (1bit is enough)

Redundancy is the opposite of uncertainty

EE465: Introduction to Digital Image Processing

Quantifying Uncertainty of an Event

Self-information

- probability of the event x
- (e.g., x can be X=H or X=T)

notes

must happen

(no uncertainty)

0

1

unlikely to happen

(infinite amount of uncertainty)

0

Intuitively, I(p) measures the amount of uncertainty with event x

EE465: Introduction to Digital Image Processing

Weighted Self-information

0

0

1/2

1

1/2

0

1

0

As p evolves from 0 to 1, weighted self-information

first increases and then decreases

Question:

Which value of p maximizes Iw(p)?

EE465: Introduction to Digital Image Processing

Quantification of Uncertainty of a Discrete Source

- A discrete source (random variable) is a collection (set) of individual events whose probabilities sum to 1

X is a discrete random variable

- To quantify the uncertainty of a discrete source, we simply take the summation of weighted self-information over the whole set

EE465: Introduction to Digital Image Processing

Shannon’s Source Entropy Formula

(bits/sample)

or bps

Weighting

coefficients

EE465: Introduction to Digital Image Processing

- Example 1: (binary Bernoulli source)

Flipping a coin with probability of head being p (0<p<1)

Check the two extreme cases:

As p goes to zero, H(X) goes to 0 bps compression gains the most

As p goes to a half, H(X) goes to 1 bps no compression can help

EE465: Introduction to Digital Image Processing

Entropy of Binary Bernoulli Source

EE465: Introduction to Digital Image Processing

N

- Example 2: (4-way random walk)

W

E

S

EE465: Introduction to Digital Image Processing

Source Entropy Examples (Con’t)

- Example 3:

(source with geometric distribution)

A jar contains the same number of balls with two different colors: blue and red.

Each time a ball is randomly picked out from the jar and then put back. Consider

the event that at the k-th picking, it is the first time to see a red ball – what is the

probability of such event?

Prob(event)=Prob(blue in the first k-1 picks)Prob(red in the k-th pick )

=(1/2)k-1(1/2)=(1/2)k

EE465: Introduction to Digital Image Processing

Source Entropy Calculation

If we consider all possible events, the sum of their probabilities will be one.

Check:

Then we can define a discrete random variable X with

Entropy:

Problem 1 in HW3 is slightly more complex than this example

EE465: Introduction to Digital Image Processing

Properties of Source Entropy

- Nonnegative and concave
- Achieves the maximum when the source observes uniform distribution (i.e., P(x=k)=1/N, k=1-N)
- Goes to zero (minimum) as the source becomes more and more skewed (i.e., P(x=k)1, P(xk) 0)

EE465: Introduction to Digital Image Processing

History of Entropy

- Origin: Greek root for “transformation content”
- First created by Rudolf Clausius to study thermodynamical systems in 1862
- Developed by Ludwig Eduard Boltzmann in 1870s-1880s (the first serious attempt to understand nature in a statistical language)
- Borrowed by Shannon in his landmark work “A Mathematical Theory of Communication” in 1948

EE465: Introduction to Digital Image Processing

A Little Bit of Mathematics*

- Entropy S is proportional to log P (P is the relative probability of a state)
- Consider an ideal gas of Nidentical particles, of which Ni are in the i-th microscopic condition (range) of position and momentum.
- Use Stirling’s formula: log N! ~ NlogN-N and note that pi = Ni /N, you will get S ~ ∑ pi log pi

EE465: Introduction to Digital Image Processing

Entropy-related Quotes

“My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage. ”

--Conversation between Claude Shannon and John von Neumann regarding what name to give to the “measure of uncertainty” or attenuation in phone-line signals (1949)

EE465: Introduction to Digital Image Processing

Other Use of Entropy

- In biology
- “the order produced within cells as they grow and divide is more than compensated for by the disorder they create in their surroundings in the course of growth and division.” – A. Lehninger
- Ecological entropy is a measure of biodiversity in the study of biological ecology.
- In cosmology
- “black holes have the maximum possible entropy of any object of equal size” – Stephen Hawking

EE465: Introduction to Digital Image Processing

What is the use of H(X)?

Shannon’s first theorem (noiseless coding theorem)

For a memoryless discrete source X, its entropy H(X)

defines the minimum average code length required to

noiselessly code the source.

Notes:

1. Memoryless means that the events are independently

generated (e.g., the outcomes of flipping a coin N times

are independent events)

2. Source redundancy can be then understood as the

difference between raw data rate and source entropy

EE465: Introduction to Digital Image Processing

Theoretical bound

Practical performance

li: the length of

codeword assigned

to the i-th symbol

Average code length:

Note: if we represent each symbol by q bits (fixed length codes),

Then redundancy is simply q-H(X) bps

EE465: Introduction to Digital Image Processing

How to achieve source entropy?

entropy

coding

binary

bit stream

discrete

source X

P(X)

Note: The above entropy coding problem is based on simplified

assumptions are that discrete source X is memoryless and P(X)

is completely known. Those assumptions often do not hold for

real-world data such as images and we will recheck them later.

EE465: Introduction to Digital Image Processing

- Discrete source
- Information=uncertainty
- Quantification of uncertainty
- Source entropy
- Variable length codes
- Motivation
- Prefix condition
- Huffman coding algorithm

EE465: Introduction to Digital Image Processing

Recall:

Self-information

It follows from the above formula that a small-probability event contains

much information and therefore worth many bits to represent it. Conversely,

if some event frequently occurs, it is probably a good idea to use as few bits

as possible to represent it. Such observation leads to the idea of varying the

code lengths based on the events’ probabilities.

Assign a long codeword to an event with small probability

Assign a short codeword to an event with large probability

EE465: Introduction to Digital Image Processing

4-way Random Walk Example

variable-length

codeword

fixed-length

codeword

pk

symbol k

0.5

S

00

0

N

0.25

01

10

E

0.125

10

110

W

0.125

11

111

symbol stream :

S S N W S E N N N W S S S N E S S

32bits

fixed length:

00000111001001011100000001100000

variable length:

28bits

0010111011010101110001011000

4 bits savings achieved by VLC (redundancy eliminated)

EE465: Introduction to Digital Image Processing

• source entropy:

=0.5×1+0.25×2+0.125×3+0.125×3

=1.75 bits/symbol

• average code length:

Total number of bits

(bps)

Total number of symbols

fixed-length

variable-length

EE465: Introduction to Digital Image Processing

Problems with VLC

- When codewords have fixed lengths, the boundary of codewords is always identifiable.
- For codewords with variable lengths, their boundary could become ambiguous

S S N W S E …

symbol

VLC

e

S

0

00111010…

N

1

00111010…

00111010…

10

E

d

d

11

S S W N S E …

S S N W S E …

W

EE465: Introduction to Digital Image Processing

- To avoid the ambiguity in decoding, we need to enforce certain conditions with VLC to make them uniquely decodable
- Since ambiguity arises when some codeword becomes the prefix of the other, it is natural to consider prefix condition

Example:

p pr pre pref prefi prefix

ab: a is the prefix of b

EE465: Introduction to Digital Image Processing

Prefix condition

No codeword is allowed to be the prefix of any other codeword.

We will graphically illustrate this condition

with the aid of binary codeword tree

EE465: Introduction to Digital Image Processing

Binary Codeword Tree

root

# of codewords

Level 1

1

0

2

Level 2

11

10

01

00

22

… …

Level k

2k

EE465: Introduction to Digital Image Processing

symbol x

codeword 1

codeword 2

S

0

0

N

1

10

10

110

E

W

11

111

1

0

1

0

11

10

01

00

11

10

111

110

… …

… …

codeword 1

codeword 2

EE465: Introduction to Digital Image Processing

How to satisfy prefix condition?

- Basic rule: If a node is used as a codeword, then all its descendants cannot be used as codeword.

1

0

Example

11

10

111

110

…

EE465: Introduction to Digital Image Processing

Property of Prefix Codes

Kraft’s inequality

li: length of the i-th codeword

(proof skipped)

Example

symbol x

VLC- 1

VLC-2

S

0

0

1

10

N

10

110

E

W

11

111

EE465: Introduction to Digital Image Processing

Two Goals of VLC design

• achieve optimal code length (i.e., minimal redundancy)

For an event x with probability of p(x), the optimal

code-length is , where x denotes the

smallest integer larger than x (e.g., 3.4=4 )

–log2p(x)

code redundancy:

Unless probabilities of events are all power of 2,

we often have r>0

• satisfy prefix condition

EE465: Introduction to Digital Image Processing

Huffman Coding (Huffman’1952) –

we will cover it later while studying JPEG

Arithmetic Coding (1980s) –

not covered by EE465 but EE565 (F2008)

EE465: Introduction to Digital Image Processing

Golomb Codes for Geometric Distribution

Optimal VLC for geometric source: P(X=k)=(1/2)k, k=1,2,…

codeword

0

10

110

1110

11110

111110

1111110

11111110

… …

k

1

2

3

4

5

6

7

8

…

0

1

1

0

1

0

1

0

…

EE465: Introduction to Digital Image Processing

Summary of Data Compression Basics

Shannon’s Source entropy formula (theory)

Entropy (uncertainty) is quantified by weighted self-information

VLC thumb rule (practice)

Long codeword small-probability event

Short codeword large-probability event

bps

EE465: Introduction to Digital Image Processing

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