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EE465: Introduction to Digital Image Processing 1 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

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EE465: Introduction to Digital Image Processing 2 Data Compression Basics Discrete source Information=uncertainty Quantification of uncertainty Source entropy Variable length codes Motivation Prefix condition Huffman coding algorithm

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EE465: Introduction to Digital Image Processing 3 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

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EE465: Introduction to Digital Image Processing 4 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?)

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EE465: Introduction to Digital Image Processing 5 Shannon’s Picture on Communication (1948) source encoder channel source decoder sourcedestination Examples of source: Human speeches, photos, text messages, computer programs … Examples of channel: storage media, telephone lines, wireless transmission … super-channel channel encoder channel decoder The goal of communication is to move information from here to there and from now to then

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EE465: Introduction to Digital Image Processing 6 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 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

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EE465: Introduction to Digital Image Processing 7 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?

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EE465: Introduction to Digital Image Processing 8 Two Extreme Cases source encoder channel source decoder tossing a fair coin Head or Tail? channel duplication tossing a coin with two identical sides P(X=H)=P(X=T)=1/2: (maximum uncertainty) Minimum (zero) redundancy, compression impossible P(X=H)=1,P(X=T)=0: (minimum redundancy) Maximum redundancy, compression trivial (1bit is enough) HHHH… TTTT… Redundancy is the opposite of uncertainty

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EE465: Introduction to Digital Image Processing 9 Quantifying Uncertainty of an Event - probability of the event x (e.g., x can be X=H or X=T) 1 0 0 notes must happen ( no uncertainty ) unlikely to happen ( infinite amount of uncertainty ) Self-information Intuitively, I(p) measures the amount of uncertainty with event x

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EE465: Introduction to Digital Image Processing 10 Weighted Self-information 0 1 0 1/21 0 0 Question:Which value of p maximizes I w (p)? As p evolves from 0 to 1, weighted self-information first increases and then decreases

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EE465: Introduction to Digital Image Processing 11 p=1/e Maximum of Weighted Self-information*

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EE465: Introduction to Digital Image Processing 12 To quantify the uncertainty of a discrete source, we simply take the summation of weighted self- information over the whole set X is a discrete random variable Quantification of Uncertainty of a Discrete Source A discrete source (random variable) is a collection (set) of individual events whose probabilities sum to 1

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EE465: Introduction to Digital Image Processing 13 Shannon’s Source Entropy Formula (bits/sample) or bps Weighting coefficients

14 EE465: Introduction to Digital Image Processing 14 Source Entropy Examples Example 1: (binary Bernoulli source) Flipping a coin with probability of head being p (0

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EE465: Introduction to Digital Image Processing 15 Entropy of Binary Bernoulli Source

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EE465: Introduction to Digital Image Processing 16 Source Entropy Examples Example 2: (4-way random walk) N E S W

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EE465: Introduction to Digital Image Processing 17 Source Entropy Examples (Con’t) Example 3: 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 (source with geometric distribution)

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EE465: Introduction to Digital Image Processing 18 Source Entropy Calculation If we consider all possible events, the sum of their probabilities will be one. Then we can define a discrete random variable X with Check: Entropy: Problem 1 in HW3 is slightly more complex than this example

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EE465: Introduction to Digital Image Processing 19 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(x k) 0)

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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 20

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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 N identical particles, of which N i are in the i-th microscopic condition (range) of position and momentum. Use Stirling’s formula: log N! ~ NlogN-N and note that p i = N i /N, you will get S ~ ∑ p i log p i EE465: Introduction to Digital Image Processing 21

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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 22

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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 23

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EE465: Introduction to Digital Image Processing 24 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

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EE465: Introduction to Digital Image Processing 25 Code Redundancy* Average code length: Theoretical bound Practical performance l i : the length of codeword assigned to the i-th symbol Note: if we represent each symbol by q bits (fixed length codes), Then redundancy is simply q-H(X) bps

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EE465: Introduction to Digital Image Processing 26 How to achieve source entropy? 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. entropy coding discrete source X P(X) binary bit stream

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EE465: Introduction to Digital Image Processing 27 Data Compression Basics Discrete source Information=uncertainty Quantification of uncertainty Source entropy Variable length codes Motivation Prefix condition Huffman coding algorithm

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EE465: Introduction to Digital Image Processing 28 Recall: Variable Length Codes (VLC) Assign a long codeword to an event with small probability Assign a short codeword to an event with large probability 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.

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EE465: Introduction to Digital Image Processing 29 symbol k pkpk S W N E 0.5 0.25 0.125 fixed-length codeword 0.125 00 01 10 11 variable-length codeword 0 10 110 111 4-way Random Walk Example symbol stream : S S N W S E N N N W S S S N E S S fixed length: variable length: 00 00 01 11 00 10 01 01 11 00 00 00 01 10 00 00 0 0 10 111 0 110 10 10 111 0 0 0 10 110 0 0 32bits 28bits 4 bits savings achieved by VLC (redundancy eliminated)

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EE465: Introduction to Digital Image Processing 30 =0.5×1+0.25×2+0.125×3+0.125×3 =1.75 bits/symbol average code length: Toy Example (Con’t) source entropy: Total number of bits Total number of symbols (bps) fixed-length variable-length

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EE465: Introduction to Digital Image Processing 31 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 symbol S W N E VLC 0 1 10 11 S S N W S E … 0 0 1 11 0 10… S S W N S E …S S N W S E … e dd

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EE465: Introduction to Digital Image Processing 32 Uniquely Decodable Codes 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 a b: a is the prefix of b

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EE465: Introduction to Digital Image Processing 33 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

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EE465: Introduction to Digital Image Processing 34 Binary Codeword Tree 10 … 10110100 root Level 1 Level 2 # of codewords 2 2 2k2k Level k

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EE465: Introduction to Digital Image Processing 35 Prefix Condition Examples symbol x W E S N 0 1 10 11 codeword 1codeword 2 0 10 110 111 10 … 10110100 10 … 1011 codeword 1 codeword 2 111110

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EE465: Introduction to Digital Image Processing 36 How to satisfy prefix condition? Basic rule: If a node is used as a codeword, then all its descendants cannot be used as codeword. 10 1011 111110 Example …

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EE465: Introduction to Digital Image Processing 37 Kraft’s inequality l i : length of the i-th codeword Property of Prefix Codes W E S N 0 1 10 11 0 10 110 111 symbol xVLC- 1VLC-2 Example (proof skipped)

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EE465: Introduction to Digital Image Processing 38 Two Goals of VLC design –log 2 p(x) 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 ) achieve optimal code length (i.e., minimal redundancy) satisfy prefix condition code redundancy: Unless probabilities of events are all power of 2, we often have r>0

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EE465: Introduction to Digital Image Processing 39 Solution: Huffman Coding (Huffman’1952) – we will cover it later while studying JPEG Arithmetic Coding (1980s) – not covered by EE465 but EE565 (F2008)

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EE465: Introduction to Digital Image Processing 40 Golomb Codes for Geometric Distribution k12345678…k12345678… codeword 0 10 110 1110 11110 111110 1111110 11111110 … Optimal VLC for geometric source: P(X=k)=(1/2) k, k=1,2,… 0 1 10 10 10 …

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EE465: Introduction to Digital Image Processing 41 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

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