# Chapter 2

```ICS: Introduction to
Computing Science
Binary Number System
Yes
40%
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No
No
60%
Yes
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Contents
• 2.1 Binary Values and Number System
• 2.2 Data and Computers
• 2.3 Representing Numeric Data
• 2.4 Representing Text
• 2.5 Representing Audio Information
• 2.6 Representing Image and Graphics
• 2.7 Representing Video
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2.1 Binary Values and Number System
• Number Categories
– Number: A unit of an abstract mathematical system
subject to the laws of arithmetic
– Natural number: The number 0 and any number obtained
by repeatedly adding 1 to it.
– Negative number: A value less than 0, with a sign opposite
to its positive counterpart
– Integer: Any of the natural numbers or any of the
negatives of these numbers
– Rational number: An integer or the quotient of two
integers (division by zero excluded)
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https://en.wikipedia.org/wiki/Irrational_number
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2.1 Binary Values and Number System
• Natural Number
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 …
19
20 21
……
101 102 103 ……
……
• Decimal (base-10)
• The base of a number system
– specifies the number of digits used in the system
• Base-8, base-5, ……, base-2 (binary)
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2.1 Binary Values and Number System
• Positional Notation
– Base-10
– Base-R
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The rightmost digit represents its value times
the base to the zeroth power. The digit to the
left of that one represents its value times the
base to the first power. The next digit
represents its value times the base to the second
power. The next digit represents its value times
the base to the third power, and so on.
Position value
A more formal way of defining positional
notation is that the value is represented as a
polynomial in the base of the number system.
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2.1 Binary Values and Number System
• Positional Notation
Example:
– Base-10
943
9 ∗ 102 + 4 ∗ 101 + 3
– Base-13
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Keep in mind that these two numbers have an
equivalent value. That is, they both represent
the same number of things. If a bag contains
943 (base 13) beans, and another bag contains
1576 (base 10) beans, both bags contain the
exact same number of beans. Number systems
just allow us to represent values in various
ways.
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2.1 Binary Values and Number System
– Binary number system (Base-2)
• 2 digits: 0-1
– Octal number system (Base-8)
• 8 digits: 0-7
• 16 digits
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2.1 Binary Values and Number System
Examples: convert the octal number 754 to decimal
Examples: convert the hexadecimal number ABC to decimal:
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2.1 Binary Values and Number System
Examples: convert the octal number 754 to decimal
Examples: convert the hexadecimal number ABC to decimal:
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2.1 Binary Values and Number System
Examples: convert a binary (base-2) number 1010110 to
decimal.
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2.1 Binary Values and Number System
Examples: convert a binary (base-2) number 1010110 to
decimal.
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2.1 Binary Values and Number System
Exercise:
convert the hexadecimal number 11D to decimal
convert the binary number 110110 to decimal
convert the octal number 172 to decimal
convert the octal number 7801 to decimal
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2.1 Binary Values and Number System
• Arithmetic in Other Bases
The rightmost digit reverts to 0,
and there is a carry into the next
position to the left.
– Subtraction
To accomplish this, you have to
―borrow one from the next
left digit of the number from
which you are subtracting.
More precisely, you borrow one
power of the base.
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minuend
subtractor
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2.1 Binary Values and Number System
• Arithmetic in Other Bases
Exercise
110011 + 110111 =
11100 – 10111 =
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2.1 Binary Values and Number System
• Power of Two Number System
– Binary and Octal
– convert from binary to octal
• Combine three digits as a
group ,from right to left, then
convert each group to a octal
digit using the mapping
relationship.
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2.1 Binary Values and Number System
• Power of Two Number System
– convert from binary to hexadecimal
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Binary
0000
0
0001
1
0010
2
0011
3
0100
4
0101
5
0110
6
0111
7
1000
8
1001
9
1010
A
1011
B
1100
C
1101
D
1110
E
1111
F
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2.1 Binary Values and Number System
• Converting from Base 10 to other Bases
– Converting algorithm
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2.1 Binary Values and Number System
• Converting from Base 10 to other Bases
Examples: convert the decimal number 2748 to hexadecimal
C
B
A
– The final answer is ABC.
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2.1 Binary Values and Number System
• Converting from Base 10 to other Bases
Examples: convert the decimal number 148 to binary
Examples: convert the decimal number 548 to octal
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Contents
• 2.1 Binary Values and Number System
• 2.2 Data and Computers
• 2.3 Representing Numeric Data
• 2.4 Representing Text
• 2.5 Representing Audio Information
• 2.6 Representing Image and Graphics
• 2.7 Representing Video
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2.2 Data and Computers
• Modern computers are binary machines
– Modern computers are designed to use and manage
binary values
– each storage location within a computer either contains a
low-voltage signal or a high-voltage signal
– These devices (storage and management) only have to
represent one of two possible values.
– Therefore they
reliable
are far less expensive and far more
– Each storage location can have one of two states, it is
logical to equate those states to 0 and 1.
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2.2 Data and Computers
• Modern computers are binary machines
– There are many ways to implement two states,
for example, low voltage/ high voltage, low
current/high current.
– In fact, modern computer adopts voltage signal.
– In general,
• A signal in the range of 0 to 2 volts is considered “low”
and is interpreted as a binary 0.
• A signal in the range of 2 to 5 volts is considered
“high” and is interpreted as a binary 1.
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2.2 Data and Computers
• Analog and Digital Information
– Analog data is a continuous representation,
analogous to the actual information it represents.
mercury
– Digital data is a discrete representation, breaking
the information up into separate elements.
So instead, we digitize information by breaking it into
pieces and representing those pieces separately.
Thermometer
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2.2 Data and Computers
• Analog and Digital Information
– All electronic signals degrade (退化) as they move down a
line. That is, the voltage of the signal fluctuates due to
environmental effects.
– The trouble is as soon as an analog signal degrades,
information is lost.
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2.2 Data and Computers
• Analog and Digital Information
– Digital signals jump sharply between two extremes.
– A digital signal can degrade quite a bit before any information is
lost. Because any voltage value above a certain threshold is
considered a high value, and any value below that threshold is
considered a low value.
– Periodically, a digital signal is reclocked to regain its original
shape. As long as it is reclocked before too much degradation
occurs, no information is lost.
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2.2 Data and Computers
• Modern computers are binary machines
– Each storage unit is called a binary unit or bit for short.
– Bits are grouped together into bytes (8bits)
– Eight bits is one byte
– Kilobyte: 1024 bytes = 210 bytes
KB
– Megabyte :1024 Kilobytes=220 bytes
MB
– Gigabyte: 1024 Megabytes = 230 bytes
GB
– Terabyte: 1024 Gigabyte = 240 bytes
TB
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2.2 Data and Computers
• Binary Representations
– One bit can be either 0 or 1. There are no other
possibilities. Therefore, one bit can represent only two
things
– To represent more than two things, we need multiple bits
– In general, n bits can represent 2n things
– How many bits do you need to represent, say, 27 unique
things?
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Contents
• 2.1 Binary Values and Number System
• 2.2 Data and Computers
• 2.3 Representing Numeric Data
• 2.4 Representing Text
• 2.5 Representing Audio Information
• 2.6 Representing Image and Graphics
• 2.7 Representing Video
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2.3 Representing Numeric Data
• Representing Negative Number
– Signed-Magnitude Representation
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2.3 Representing Numeric Data
• Representing Negative Number
– Fixed-Sized Numbers
Subtraction
(Note that the carries are discarded)
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2.3 Representing Numeric Data
• Representing Negative Number
– Fixed-Sized Numbers
• In last example, we have assumed a fixed size of 100
values, and kept our numbers small enough to use the
number line to calculate the negative representation of a
number.
• This representation of negative numbers is called the
ten’s complement.
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2.3 Representing Numeric Data
• Representing Negative Number
– Two’s Complement
• The complement strategy is actually easier in some
ways when it comes to electronic calculations.
• Everything in a modern computer is stored in binary,
we use the binary equivalent of the ten‘s complement,
called the two’s complement.
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2.3 Representing Numeric Data
• Representing Negative Number
– Two’s Complement
• Assume that a number must be represented in eight
bits.
• Addition and subtraction are accomplished the same
way as in two‘s complement arithmetic:
• Note that : with this representation, the leftmost bit in
a negative number is always a 1. Therefore you can say
that the leftmost position is a sign position.
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2.3 Representing Numeric Data
• Representing Negative Number
– Two’s Complement
– The reasons why we use two’s complement
1)It has only one representation of zero.
2)Addition and subtraction are accomplished the same
way. It is not easy for computer to do borrow.
3)The sign bit participates in the calculation.
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2.3 Representing Numeric Data
• Representing Negative Number
– Two’s Complement
– Exercise:
• 126 - 127
• -126 + 8
• -126 - 3
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Assume that we use eight bits to represent
numbers
1. Convert the decimal number to binary
2. Use two’s complement representation
to represent negative number
4. Write the result down and check
whether it is right
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2.3 Representing Numeric Data
• Representing Negative Number
– Number Overflow
• Overflow occurs when the value that we compute
cannot fit into the number of bits we have allocated
for the result.
• For example, if each value is stored using eight bits,
adding 127 to 3 would overflow:
• 100000010 in our scheme represents -126, not +130.
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2.3 Representing Numeric Data
• Representing Real Number
Decimal point
– In decimal
104.32
100s
position
10s
position
1s
position
10-1s
position
tenth
– In binary
100-1s
position
hundredth
101.01
22s
position
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21 s
position
20s
position
2-1s
position
2-2s
position
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2.3 Representing Numeric Data
• Representing Real Number
– How to convert a real number from decimal to
binary?
– eg. 4.125
– eg. 0.3
– Fixed point and floating point
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2.3 Representing Numeric Data
• Representing Real Number
– Floating point
• In decimal
-148.69
14869 *10
2
• In binary
1 bit
52 bits
11 bits
0
0000000000000000000000000000000000011100000000000000
00000000011
sign
mantissa
exp
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Contents
• 2.1 Binary Values and Number System
• 2.2 Data and Computers
• 2.3 Representing Numeric Data
• 2.4 Representing Text
• 2.5 Representing Audio Information
• 2.6 Representing Image and Graphics
• 2.7 Representing Video
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2.4 Representing Text
• To represent a text document in digital form, we
simply need to be able to represent every possible
character that appear.
• There are a finite number of characters.
• So the general approach for representing characters
is to list them all and assign each character a binary
string.
• To store a particular letter, we store the appropriate
bit string.
• A character set is simply a list of characters and the
codes used to represent each one.
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2.4 Representing Text
• The ASCII Character Set (American Standard Code for
Information Interchange)
– The ASCII character set originally used seven bits to
represent each character, allowing for 128 unique characters.
– Later ASCII evolved so that all eight bits were used to
represent a character.
– This eight-bit version is formally called the Latin–1 Extended
ASCII character set.
– Then the extended ASCII set allows for 256 characters and
includes accented letters as well as several additional special
symbols.
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2.4 Representing Text
• The Unicode Character Set
– The Unicode character set uses 16 bits per character.
– Therefore, the Unicode character set can represent 216, or
over 65 thousand characters.
The goal of the
people who
created Unicode is
to represent every
character in every
language used in
the entire world.
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2.4 Representing Text
• The Unicode Character Set
– The Unicode character set uses 16 bits per character.
– For consistency, Unicode was designed to a superset of
ASCII. The first 256 characters in the Unicode character set
correspond exactly to the extended ASCII character set.
– There are also other character set, such as
• GBK (simple Chinese)
• UTF-8
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2.4 Representing Text
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2.4 Representing Text
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2.4 Representing Text
• Text Compression
– Data compression—reducing the amount of space needed to
store a piece of data.
– The compression ratio gives an indication of how much
compression occurs.
– The compression ratio is the size of the compressed data
divided by the size of the original data.
– Lossless: the data can be retrieved without losing any of the
original information.
– Lossy: some information is lost in the process of compaction.
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2.4 Representing Text
• Text Compression
– Keyword Encoding
• The thought of this method is that there are many
words which frequently occur in our text document or
language. For example, in English language, the word
“the”, “and”, “that” occur very often. If we use a
single character to replace this word, the file size will
shrink.
• Keyword Encoding is a text compression method, in
which frequently used words are replaced with a single
character.
• To decompress the document, you reverse the process:
replace the single characters with the appropriate full
word.
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2.4 Representing Text
Example:
The original paragraph is:
The human body is composed of many independent
systems, such as the circulatory system, the
respiratory system, and the reproductive system. Not
only must all systems work independently, they must
interact and cooperate as well. Overall health is a
function of the well-being of separate systems, as
well as how those separate systems work in concert.
351
characters
The encoded paragraph is:
316
characters
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The human body is composed of many independent systems, such ^ ~
circulatory system, ~ respiratory system, + ~ reproductive system. Not
only & all systems work independently, they & interact + cooperate ^ %.
Overall health is a function of ~ %-being of separate systems, ^ % ^
how # separate systems work in concert.
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2.4 Representing Text
• Text Compression
– Keyword Encoding
Example
• There are a total of 351 characters in the original paragraph
including spaces and punctuation.
• The encoded paragraph contains 316 characters, resulting
in a savings of 35 characters.
• The compression ratio for this example is 316/351 or
approximately 0.9.
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2.4 Representing Text
• Text Compression
– Keyword Encoding
Characteristics
• This method is not very efficient. The compression ratio is
too high.
• There is a limitation to use this compression method. We
must make sure all single symbols we use to replace the full
words can not appear in the document.
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2.4 Representing Text
• Text Compression
– Run-length Encoding
• In some situations, a single character may be repeated
over and over again in a long sequence.
• A text compression technique called run-length encoding
capitalizes on these situations. It is sometimes called
recurrence coding.
• In run-length encoding, a sequence of repeated
characters is replaced by a flag character, followed by
the repeated character, followed by a single digit that
indicates how many times the character is repeated.
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2.4 Representing Text
• Text Compression
– Run-length Encoding
• Example: consider the following string : AAAAAAA
• If we use the ‘*’ character as our flag, this string would
be encoded as: *A7
• The encoded string: *n5*x9ccc*h6 some other text
*k8eee
• would be decoded into the following original text:
nnnnnxxxxxxxxxccchhhhhh some other text
kkkkkkkkeee
• The compression ratio in this example of 35/51 or
approximately 0.68.
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2.4 Representing Text
• Text Compression
– Huffman Encoding
• This approach is contrary to the idea of a character set, in
which each character is represented by a fixed-length bit
string (such as 8 or 16).
• The idea behind this approach is that if we use only a
few bits to represent characters that appear often and
reserve longer bit strings for characters that don‘t
appear often, the overall size of the document being
represented is small.
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2.4 Representing Text
• Text Compression
– Huffman Encoding
Encoding Example:
– The word DOORBELL would be encoded in binary as:
1011110110111101001100100
– The compression ratio：
– If we used a fixed-size bit string to represent each character
(say, 8 bits), then the binary form of the original string would
be 8 characters times 8 bits or 64 bits.
– The Huffman encoding for that string is 25 bits long, giving a
compression ratio of 25/64, or approximately 0.39.
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2.4 Representing Text
• Text Compression
– Huffman Encoding
Encoding Example:
– The word DOORBELL would be encoded in binary as:
1011110110111101001100100
– The compression ratio：
– If we used a fixed-size bit string to represent each character
(say, 8 bits), then the binary form of the original string would
be 8 characters times 8 bits or 64 bits.
– The Huffman encoding for that string is 25 bits long, giving a
compression ratio of 25/64, or approximately 0.39.
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2.4 Representing Text
• Text Compression
– Huffman Encoding
• Decoding Example:
– 1010110001111011
– These characters are represented by different bit strings.
When we do decoding, we don’t know how many bits we
should include for each character.
– It seems that we might get confused. Let us scan this bit
string from left to right.
– it must be decoded into the word BOARD.
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2.4 Representing Text
• Text Compression
– Huffman Encoding
• Decoding Example:
– 1010110001111011
– it must be decoded into the word BOARD.
– An important characteristic of any Huffman encoding is that
no bit string used to represent a character is the prefix of
any other bit string used to represent a character.
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Contents
• 2.1 Binary Values and Number System
• 2.2 Data and Computers
• 2.3 Representing Numeric Data
• 2.4 Representing Text
• 2.5 Representing Audio Information
• 2.6 Representing Image and Graphics
• 2.7 Representing Video
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2.5 Representing Audio Information
• Sound
– A sound is defined in nature by the wave of air that interacts
with our eardrum.
– The sound wave is an analog signal .
– To represent audio information on a computer, we must
digitize the sound wave, somehow breaking it into discrete,
manageable pieces.
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2.5 Representing Audio Information
• Digitization
– To digitize the signal we periodically measure the voltage
of the signal and record the appropriate numeric value.
This process is called sampling.
– To reproduce the sound, the stored voltage values are
used to create a new continuous electronic signal.
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2.5 Representing Audio Information
• Sampling rate
– A sampling rate of around 40,000 times per second is
enough to create a reasonable sound
– If the sampling rate is much lower than that, the human
ear begins to hear distortions (失真/畸变).
– A higher sampling rate produces better quality sound, but
after a certain point the extra data is irrelevant because
the human ear can‘t hear the difference.
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2.5 Representing Audio Information
• A vinyl record album VS. Compact Disc (CD)
– A vinyl record album is an analog representation
of the sound wave.
• The needle of a record player (turntable) rides up and
down in the spiral groove of the album.
• The rise and fall of the needle is analogous to the
voltage changes of the signal that represents the
sound.
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2.5 Representing Audio Information
• A vinyl record album VS. Compact Disc (CD)
– A compact disc (CD), stores audio information digitally.
• On the surface of the CD are microscopic pits that represent
binary digits.
• For a CD player, a low intensity laser is pointed at the disc. The
laser light reflects strongly if the surface is smooth and reflects
poorly if the surface is pitted.
• A receptor analyzes the reflection and produces the appropriate
string of binary data. This data represents the numeric voltage
values. Then the signal is reproduced and sent to the speaker.
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2.5 Representing Audio Information
• MP3 (MPEG–2, audio layer 3 file)
– MPEG is an acronym for the Moving Picture Experts Group,
which is an international committee that develops
standards for digital audio and video compression.
– First it analyzes the frequency spread and compares it to
mathematical models of human psychoacoustics (心理声

humans.
– Then the bit stream is compressed using a form of Huffman
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Contents
• 2.1 Binary Values and Number System
• 2.2 Data and Computers
• 2.3 Representing Numeric Data
• 2.4 Representing Text
• 2.5 Representing Audio Information
• 2.6 Representing Image and Graphics
• 2.7 Representing Video
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2.6 Representing Images and Graphics
• Representing Color
– Color is our perception of the various frequencies of light that
reach the retinas (虹膜) of our eyes.
– Our retinas have three types of color photoreceptor （感光器）
cone cells that respond to different sets of frequencies.
– Three photoreceptor categories correspond to the colors of red,
green, and blue.
– All colors perceptible by the human eye can be made by
combining various amounts of these three colors.
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2.6 Representing Images and Graphics
• Representing Color
– Color is often expressed in a computer as an RGB (red-greenblue) value.
– RGB value is actually three numbers that indicate the relative
contribution of each of these three primary colors.
– If each number in the triple is given on a scale of 0 to 255, then
0 means no contribution of that color, and 255 means full
contribution of that color.
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2.6 Representing Images and Graphics
• Representing Color
– Color Depth : the amount of data that is used to represent a
color, it is usually expressed in terms of the number of bits
that are used to represent its color.
– TrueColor indicates a 24-bit color depth. Therefore, each
number in an RGB value gets eight bits, which gives the range
of 0 to 255 for each. This results in the ability to represent
over 16.7 million unique colors.
– HiColor is a term that indicates a 16-bit color depth. Five bits
are used for each number in an RGB value and the extra bit is
sometimes used to represent transparency.
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2.6 Representing Images and Graphics
• Digitized Image
– A photograph is an analog representation of an image. It is
continuous across its surface. In order to represent and store
image on a computer, we should digitize picture.
– Digitizing a picture is the act of representing it as a collection of
individual dots called pixels, a term that stands for picture
elements.
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2.6 Representing Images and Graphics
• Digitized Image
– The number of pixels used to represent a picture is called the
resolution.
– If enough pixels are used (high resolution), and are then
presented in the proper order side by side, the human eye
can be fooled into thinking it‘s viewing a continuous picture.
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2.6 Representing Images and Graphics
• Raster-graphics format
• The storage of image information on a pixel-by-pixel basis
• Several Popular Raster File Formats
– BMP (bitmap)
• A bitmap file contains the pixel color values of the image from left to
right and top to bottom.
• A bitmap file supports 24-bit TrueColor, though usually the color depth
can be specified to reduce the file size.
• A bitmap file may be compressed using run-length encoding.
– GIF(Graphics Interchange Format)
• It uses indexed color to reduce file size, which limits the number of
available colors to 256.
• If even fewer colors are required, the color depth can usually be
specified to fewer bits.
• GIF files are best used for graphics and images with few colors, and are
therefore considered optimal for line art.
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2.6 Representing Images and Graphics
• Several Popular Raster File Formats
– JPEG
• JPEG is a commonly used method of lossy compression for digital
images
• It supports different compression ratio. The degree of compression
and image quality.
• The JPEG format is designed to exploit the nature of our eyes.
Humans are more sensitive to gradual changes of brightness and
color over distance than we are to rapid changes.
• Therefore, the data that the JPEG format stores averages out the
color hues over short distances.
• A fairly complicated compression scheme is adopted to significantly
reduce the file sizes.
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2.6 Representing Images and Graphics
• Vector Representation of Graphics
– A vector-graphics format describes an image in terms of lines
and geometric shapes.
– A vector graphic is a series of commands that describe a line‘s
direction, thickness, and color.
– The file sizes for vector-graphics format tend to be small .
– A raster graphic must be encoded multiple times for different
sizes and proportions.
– Vector graphics can be resized mathematically, and these changes
can be calculated dynamically as needed.
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2.6 Representing Images and Graphics
• Vector Representation of Graphics
– Vector graphics is not good for representing real-world images.
JPEG images are far superior in that regard
– but vector graphics is good for line art and cartoon-style
drawings.
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Contents
• 2.1 Binary Values and Number System
• 2.2 Data and Computers
• 2.3 Representing Numeric Data
• 2.4 Representing Text
• 2.5 Representing Audio Information
• 2.6 Representing Image and Graphics
• 2.7 Representing Video
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2.7 Representing Video
• Video
– Video clips contain the equivalent of many still images (静

– if the change between two consecutive images is small,
the movement between two images seems continuous.
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2.7 Representing Video
• Codec (COmpressor/DECompressor)
– A video codec refers to the methods used to shrink the
size of a movie to allow it to be played on a computer or
over a network.
– Almost all video codecs use lossy compression to minimize
the huge amounts of data associated with video.
– Video codecs employ two types of compression: temporal
and spatial.
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2.7 Representing Video
• Codec (COmpressor/DECompressor)
– Temporal compression
• Temporal compression looks for differences between
consecutive frames.
• A keyframe is chosen as the basis to compare the
differences, and its entire image is stored.
• For consecutive images, only the changes (called delta
frames) are stored.
• Temporal compression is effective in video that changes
little from frame to frame, such as a scene that contains
little movement.
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2.7 Representing Video
• Codec (COmpressor/DECompressor)
– Temporal compression
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2.7 Representing Video
• Codec (COmpressor/DECompressor)
– Spatial compression
• Spatial compression removes redundant information within
a frame.
• Spatial video compression often groups pixels into blocks
that have the same color, such as a portion of a clear blue
sky.
• Instead of storing each pixel, the color and the coordinates
(坐标) of the area are stored instead.
• This idea is similar to run-length.
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2.7 Representing Video
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The End
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