Today. Data Representation. Break: Binary Numbers. Representing Text. Representing Images. 18:30(ish) 15 Minute Break

Transcription

1

2 Introduction

3 Today Data Representation Binary Numbers Representing Text Representing Images Break: 18:30(ish) 15 Minute Break

4 Why Data Representation? GCSE syllabus has an entire Module on data representation.

5 Binary Numbers

6 Lets Talk About Bits Computers use bits to represent data: bit: 0 or 1 Larger collections of bits are given special names: A byte is a sequence of 8 bits. A kilobyte (KB) is 1024 (210) bytes. A megabyte (MB) is 1024 kilobytes. A gigabyte (GB) is 1024 megabytes. These sequences of bits are used to store all data.

7 Representing Decimal Values How can we represent decimal numbers (base 10) using bits?

8 Representing Decimal Values How can we represent decimal numbers (base 10) using bits? Answer: Binary Examples: 010 = = = =

9 Converting Decimal to Binary To convert a decimal number D to binary follow these steps: While D >= 2 1. Divide D by 2, keeping track of the Quotient and Remainder. 2. Write the remainder down. 3. D = Quotient. Print: The last quotient followed by the list of remainders in reverse order.

10 Converting to Binary Number (D) Quotient (D Div 2) Remainder (D Mod 2) Last quotient and remainder column backwards =

11 Binary Tasks Convert the following numbers into binary: ,719 78,425

12 Binary Solutions Convert the following numbers into binary: 11 = = = ,719 = ,425 =

13 Converting Binary to Decimal To convert a binary number B to decimal follow these steps: 1. Write down the binary number. 2. Above each digit, right the sequence 0 n-1 from right to left. 3. Now take each binary digit D and the number above it N. Then multiply D by 2N. 4. Add together all the results.

14 Converting to Decimal Step 1: Write down the binary number Step 2: Write down 0...n-1 from left to right x 22 = 4, 1 x 20 = 1. Step 4: = 13 Result: 13 Step 3: 1 x 23 = 8, 3 0 x 21 = 0,

15 Binary Tasks (2) Convert the following binary numbers into decimal:

16 Binary Solutions(2) Convert the following binary numbers into decimal: 10 = = = = = 255

17 Converting Decimal to Base K To convert a number D to base K follow these steps: While D >= K 1. Divide D by K, keeping track of the Quotient and Remainder. 2. Write the remainder down. 3. D = Quotient. Print: The last quotient followed by the list of remainders in reverse order.

18 Converting Base K to Decimal To convert a base K number H to decimal follow these steps: 1. Write down the binary number. 2. Above each digit, right the sequence 0 n-1 from right to left. 3. Now take each binary digit D and the number above it N. Then multiply D by KN. 4. Add together all the results.

19 Hexadecimal: Base 16 Hexadecimal is also commonly used within computer science. Notice that the letters A to F represent the numbers 10 to 15. When using the algorithms for conversion take this into account.

20 Harder Tasks What would a unary number system look like (Base 1)? Convert the following to hexadecimal: Convert the following to Decimal: 10 AB FF1 How can we convert from hex to binary and back?

21 Harder Task Solutions What would a unary number system look like (Base 1)? Just one value (think of stick counting). Convert the following to hexadecimal: 10 = A 100 = = 84 Convert the following to Decimal: 10 = 16 AB = 171 FF1 = 4081 How can we convert from hex to binary and back? Convert hex digit to block of 4 bits and vice versa.

22 Representing Text

23 Encoding Text In a similar manner to Numbers, we can also represent text. Can anyone think of one possible representation? -- Remember we can only use bits!

24 ASCII Assign each letter a number, convert the number to Binary. Examples: A represented by the number 65.

25 ASCII to Bits As each letter is represented by a number, we can then convert that number to binary using the techniques we have already seen. Some Notes: In total there are 256 ASCII characters. Hence each letter is represented by 1 byte (8 bits = 1 byte).

26 ASCII Tasks 1. Convert the following Text to ASCII: CABAEACABCDEAEEFCEFF 2. Convert the first three letters to their representation in bits. 3. How many bits are needed to convert the entire piece of text. 4. Can we use less bits? Maybe an alternative to ASCII?

27 ASCII Solutions 1. Convert the following Text to ASCII: CABAEACABCDEAEEFCEFF = Convert the first three letters to their representation in bits = How many bits are needed to convert the entire piece of text = Can we use less bits? Maybe an alternative to ASCII? = Yes! -- Huffman Encoding.

28 Huffman Encoding ASCII uses a fixed encoding for all Strings. The Huffman Encoding changes depending on the text to be encoded. Idea: Analyse the number of occurrences of each letter. Encode letters with more occurrences with fewer bits. Result: A shorter encoding? always?

29 Huffman: Frequencies Consider the text: CABAEACABCDEAEEFCEFF Now analyse the frequencies of each character: Character A B C D E F Frequency Clearly A and E should have the shortest encoding

30 Huffman: The Encoding Idea: Build a binary tree with characters as leaves. Then label branches with bits. Frequent near top giving shorter encoding

31 Constructing The Tree (1) Start with the two values with lowest frequency (B and D). Build the tree: Also: add new combined value to table instead of individual values.

32 Constructing The Tree (2) Repeat this step with the next two smallest (F and B+D): Also: add new combined value to table instead of individual values.

33 Constructing The Tree (3) Repeat again! (A and C): Notice: We had a choice of A or E, but either works!

34 Constructing The Tree (4) (E and F+B+D):

35 Constructing The Tree (5) Finally (A+C and E+F+B+D):

36 Last Step: Label The Branches Label left branches with 0 and right branches with 1: Path from root to leave gives character encoding.

37 Huffman Tasks 1. Convert the following text to bits using the given tree: CABAEACABCDEAEEFCEFF 2. How many bits do we now need? 3. Computer the Huffman Encoding for the following text: BACADAEAFABBAAAGAH Hint: Start with the frequency table

38 Solution to Tasks (the beginning) Bits! (much better than the 160 for ASCII) 3.

39 Representing Images

40 Images in Bits We can even represent images through a basic bit encoding. Images are made up of pixels. A pixel is a small block of colour, like the picture of a tv screen opposite.

41 Black and White If an image is nothing but a grid of black and white pixels, then we can use 1 to represent black, 0 to represent white. What would be the encoding of this image?

42 Black and White If an image is nothing but a grid of black and white pixels, then we can use 1 to represent black, 0 to represent white

43 Adding Grey How could we represent grayscale images? (Hint: bytes?)

44 Adding Grey A byte can be used to represent values from So if we have a byte representing a pixel, we can encode 256 shades of grey = Black = White All other values are greys.

45 Image Tasks 1. Using a similar idea to grayscale images, discuss how we could represent colour images. (Hint: 3 colours per pixel red, green and Blue.) 2. Can anyone think of a compression method for images. (Hint: Consider the black and white image to start with.)

46 Image Solutions (1) Using a similar idea to grayscale images, discuss how we could represent colour images. Use 3 bytes per pixel, one byte for each colour: Red Green Blue Outcome Black White Mid Grey Bright Red Purple(ish)

47 Image Solutions (2) Can anyone think of a compression method for images. (Hint: Consider the black and white image to start with.) Bit pattern for our b&w image: Lots of runs of 0 s -- Count them and encode numbers?

48 Example Compression Bit pattern for our b&w image: If we count occurrences of 0 s between 1 s we get: 3,0,5,2,3,4,1,6,0,6,1,4,3,2,5,0,3 Note: 0-length runs where we hit two black pixels. Now encode these numbers (need 3 bits per number):

49 Example Compression: The Saving Encoding runs of 0 s requires: 54 bits (17 numbers, 3 bits each). rather than: 64 bits for the original picture (8 rows of 8 bits). On larger pictures, the effect is likely to be bigger!

50 Conclusion Data Representation. Binary Numbers (Conversions between bases). Representing Text (ASCII and Huffman). Representing Images (Pixels, Greys and Colours, Compression). Next Week: Networks and Cryptography