The Structure of Digital Imaging: The Haar Wavelet Transformation and Image Compression Adam L. Bruce and Alice R. Pitt December 7, 2009 Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Abstract The Haar wavelet transform allows images to be compressed and sent over a network of computers so that it is visable to anyone on the network. The mathematical apparatus for the transform is that of linear algebra and other elements of matrix theory, the main part of this being manipulations applied to the initial matrix representing a certian resolution of an image. It shall be shown how these techniques are used in conjuction with each other to create a given digital image of a certain resolution. Prerequisites: A rudimentary knowledge of matrices and arithmetical operations. Also, an indtroductory course in linear algebra would be helpful for the third section. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Overview ◮ How is an image sent over a computer network? Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Overview ◮ How is an image sent over a computer network? ◮ It’s possible through a technique called the Haar wavelet transformation . Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Overview ◮ How is an image sent over a computer network? ◮ It’s possible through a technique called the Haar wavelet transformation . ◮ There are two main mathematical processes averaging and differencing . Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Overview ◮ How is an image sent over a computer network? ◮ It’s possible through a technique called the Haar wavelet transformation . ◮ There are two main mathematical processes averaging and differencing . ◮ They are simple, but to use them in a large matrix requires some linear algebra. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Overview ◮ How is an image sent over a computer network? ◮ It’s possible through a technique called the Haar wavelet transformation . ◮ There are two main mathematical processes averaging and differencing . ◮ They are simple, but to use them in a large matrix requires some linear algebra. ◮ Most of the operations can also be programmed into Matlab. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
An Image as a Matrix ◮ Every Image represents a matrix Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
An Image as a Matrix ◮ Every Image represents a matrix ◮ The numbers within the matrix represent different shades of black and white or color. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
An Image as a Matrix ◮ Every Image represents a matrix ◮ The numbers within the matrix represent different shades of black and white or color. ◮ The values of the numbers range from 0 to some positive whole number. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
An Image as a Matrix ◮ Every Image represents a matrix ◮ The numbers within the matrix represent different shades of black and white or color. ◮ The values of the numbers range from 0 to some positive whole number. ◮ For this presentation we will only consider gray-scale images. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
An Image as a Matrix ◮ So consider the matrix: 123 145 222 234 68 66 76 12 34 19 187 188 47 182 209 4 144 55 23 27 111 29 250 0 48 38 79 247 72 77 112 14 A = 63 14 149 150 37 44 121 11 233 155 122 69 58 47 136 18 34 21 13 59 209 146 100 10 211 151 98 60 32 31 88 17 Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
An Image as a Matrix ◮ So consider the matrix: 123 145 222 234 68 66 76 12 34 19 187 188 47 182 209 4 144 55 23 27 111 29 250 0 48 38 79 247 72 77 112 14 A = 63 14 149 150 37 44 121 11 233 155 122 69 58 47 136 18 34 21 13 59 209 146 100 10 211 151 98 60 32 31 88 17 ◮ This could represent some portion of an image Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
An Image as a Matrix ◮ The image Figure: Bird is a 256 by 256 matrix. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ So how do we transport the image? Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ So how do we transport the image? ◮ Using averaging and differencing we can actually compress an image, making it easier to transport. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ So how do we transport the image? ◮ Using averaging and differencing we can actually compress an image, making it easier to transport. ◮ How is this done? Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ To average first we isolate a row, called a data string , such as � � S 1 = 122 145 222 234 68 66 76 12 This is row one of Matrix A . Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ To average first we isolate a row, called a data string , such as � � S 1 = 122 145 222 234 68 66 76 12 This is row one of Matrix A . ◮ We now can find the basic average of these terms, given by y = x 1 + x 2 . 2 where x 1 and x 2 are two elements in S 1 which are next to each other. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ The difference is found by taking the average and subtracting them from x 1 like d = x 1 − y Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ The difference is found by taking the average and subtracting them from x 1 like d = x 1 − y ◮ These differenced numbers are the detail coefficients . Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ Both averaging and differencing must take place 3 times with a string with length 8. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ Both averaging and differencing must take place 3 times with a string with length 8. ◮ This is because 2 3 = 8. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Averaging and Differencing ◮ Both averaging and differencing must take place 3 times with a string with length 8. ◮ This is because 2 3 = 8. ◮ Here is a table which summarizes the results on S 1 : Table: Averaging and Differencing of S 1 Compression No. average/detail coefficient 1 133.5 228 67 44 -11.5 -6 1 32 2 180.7 55.5 -47.2 12 -11.5 -6 1 32 3 118 67.7 -47.2 12 -11.5 -6 1 32 Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Basic Compression Matrices ◮ Consider images as matrices. There must be an efficient way to apply the previous idea of averaging and differencing to a matrix the size of 256 by 256. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
Basic Compression Matrices ◮ Consider images as matrices. There must be an efficient way to apply the previous idea of averaging and differencing to a matrix the size of 256 by 256. ◮ This would bring in a compression matrix: 1 / 2 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 − 1 / 2 0 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 − 1 / 2 0 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 − 1 / 2 0 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 − 1 / 2 Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
◮ This simplifies what could be a rather long process otherwise. Adam L. Bruce and Alice R. Pitt The Structure of Digital Imaging: The Haar Wavelet Transformation
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