The Structure of Digital Imaging: The Haar Wavelet Transform. Wavelet Transformation and Image Compression
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1 The Structure of Digital Imaging: The Haar Wavelet Transformation and Image Compression December 7, 2009
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3 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.
4 Overview How is an image sent over a computer network?
5 Overview How is an image sent over a computer network? It s possible through a technique called the Haar wavelet transformation.
6 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.
7 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.
8 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.
9 An Image as a Matrix Every Image represents a matrix
10 An Image as a Matrix Every Image represents a matrix The numbers within the matrix represent different shades of black and white or color.
11 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.
12 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.
13 An Image as a Matrix So consider the matrix: A =
14 An Image as a Matrix So consider the matrix: A = This could represent some portion of an image
15 An Image as a Matrix The image Figure: Bird is a 256 by 256 matrix.
16 Averaging and Differencing So how do we transport the image?
17 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.
18 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?
19 Averaging and Differencing To average first we isolate a row, called a data string, such as S 1 = ( ) This is row one of Matrix A.
20 Averaging and Differencing To average first we isolate a row, called a data string, such as S 1 = ( ) 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.
21 Averaging and Differencing The difference is found by taking the average and subtracting them from x 1 like d = x 1 y
22 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.
23 Averaging and Differencing Both averaging and differencing must take place 3 times with a string with length 8.
24 Averaging and Differencing Both averaging and differencing must take place 3 times with a string with length 8. This is because 2 3 = 8.
25 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
26 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.
27 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
28 This simplifies what could be a rather long process otherwise.
29 This simplifies what could be a rather long process otherwise. Here is our compression matrix multiplied with our initial matrix three times to create matrix T: T =
30 Compression The process of applying the compression matrix to the initial matrix multiple times is wavelet transform.
31 Compression The process of applying the compression matrix to the initial matrix multiple times is wavelet transform. This process is meant to transform data in the matrix to zero or near zero.
32 Compression The process of applying the compression matrix to the initial matrix multiple times is wavelet transform. This process is meant to transform data in the matrix to zero or near zero. A matrix is considered sparse when it is highly composed of zeros.
33 Compression The process of applying the compression matrix to the initial matrix multiple times is wavelet transform. This process is meant to transform data in the matrix to zero or near zero. A matrix is considered sparse when it is highly composed of zeros. Compression involves choosing a threshold value e, let s set e = 20.
34 Compression The process of applying the compression matrix to the initial matrix multiple times is wavelet transform. This process is meant to transform data in the matrix to zero or near zero. A matrix is considered sparse when it is highly composed of zeros. Compression involves choosing a threshold value e, let s set e = 20. Then all numbers that fall within the absolute value of e will be made into a zero.
35 Compression The process of applying the compression matrix to the initial matrix multiple times is wavelet transform. This process is meant to transform data in the matrix to zero or near zero. A matrix is considered sparse when it is highly composed of zeros. Compression involves choosing a threshold value e, let s set e = 20. Then all numbers that fall within the absolute value of e will be made into a zero. This helps to maitain the images beginning integrity while saving space or transmission time.
36 This is our matrix T after the threshold value e = 20 has been applied. We ll call it matrix D: D =
37 This is our matrix T after the threshold value e = 20 has been applied. We ll call it matrix D: D = This has created 27 zero components in our matrix creating a 2:1 compression ratio.
38 Progressive Image Transmission Progressive Image transmission utilizes the previous techniques.
39 Progressive Image Transmission Progressive Image transmission utilizes the previous techniques. The compression process allows the first image we attain to be comparable to our matrix T.
40 Progressive Image Transmission Progressive Image transmission utilizes the previous techniques. The compression process allows the first image we attain to be comparable to our matrix T. Matrix T is brought up starting with the overall average,larger detail coefficients and finally the smallest detail coefficients.
41 Progressive Image Transmission Progressive Image transmission utilizes the previous techniques. The compression process allows the first image we attain to be comparable to our matrix T. Matrix T is brought up starting with the overall average,larger detail coefficients and finally the smallest detail coefficients. T matrix initial image is crude but as more wavelet coefficients are used the image slowly becomes an exact copy of the initial image.
42 Figure: The progressive transmission of figure 1 (3rd compression to 1st compression)
43 Programming Compressions with Matlab Creating compressions of large matrices, such as 256 by 256 is difficult and time consuming by hand
44 Programming Compressions with Matlab Creating compressions of large matrices, such as 256 by 256 is difficult and time consuming by hand Therefore we program the compression matrices with Matlab
45 Programming Compressions with Matlab Creating compressions of large matrices, such as 256 by 256 is difficult and time consuming by hand Therefore we program the compression matrices with Matlab This is by defining the matrix as a function and writting code for a matrix with variable dimentions
46 Programming Compressions with Matlab Frist we define our function and variable. Using the name indmat
47 Programming Compressions with Matlab Frist we define our function and variable. Using the name indmat we code this like function a = indmat(n) b=[1;1]/2 c=[1;-1]/2
48 Programming Compressions with Matlab Frist we define our function and variable. Using the name indmat we code this like function a = indmat(n) b=[1;1]/2 c=[1;-1]/2 This creates two vectors, b, and c which consists of < 1 2, 1 2 > and < 1 2, 1 2 > respectively.
49 Programming Compressions with Matlab next we specify the first half of the matrix which will average the image.
50 Programming Compressions with Matlab next we specify the first half of the matrix which will average the image. this is while min(size(b))< n/2 b=[b, zeros(max(size(b)),min(size(b)));... zeros(max(size(b)),min(size(b))), b]; end
51 Programming Compressions with Matlab next we specify the first half of the matrix which will average the image. this is while min(size(b))< n/2 b=[b, zeros(max(size(b)),min(size(b)));... zeros(max(size(b)),min(size(b))), b]; end Then, to specify the differencing half, we use the same code, replacing c s for the b s.
52 Programming Compressions with Matlab next we specify the first half of the matrix which will average the image. this is while min(size(b))< n/2 b=[b, zeros(max(size(b)),min(size(b)));... zeros(max(size(b)),min(size(b))), b]; end Then, to specify the differencing half, we use the same code, replacing c s for the b s. We finally create a larger matrix out of the two of these with a=[b,c]
53 Programming Compressions with Matlab Now that we have the compression matrix we apply indmat(256) to our original image. This creates the 1st compression
54 Programming Compressions with Matlab Now that we have the compression matrix we apply indmat(256) to our original image. This creates the 1st compression To compress again we apply the matrix 2ndcompression=[indmat(128),zeroes(128);zeroes(128),eye(128)]
55 Programming Compressions with Matlab Now that we have the compression matrix we apply indmat(256) to our original image. This creates the 1st compression To compress again we apply the matrix 2ndcompression=[indmat(128),zeroes(128);zeroes(128),eye(128)] For the next compression the dimention is reduced by 2 again, but there must be more blocks. 3rdcompression=[indmat(64),zeroes(64),zeroes(64),zeroes(64); zeroes(64),eye(64),zeroes(64),zeroes(64);zeroes(64), zeroes(64),eye(64),zeroes(64);zeroes(64),zeroes(64),zeroes(64),eye(64)
56 Programming Compressions with Matlab Now that we have the compression matrix we apply indmat(256) to our original image. This creates the 1st compression To compress again we apply the matrix 2ndcompression=[indmat(128),zeroes(128);zeroes(128),eye(128)] For the next compression the dimention is reduced by 2 again, but there must be more blocks. 3rdcompression=[indmat(64),zeroes(64),zeroes(64),zeroes(64); zeroes(64),eye(64),zeroes(64),zeroes(64);zeroes(64), zeroes(64),eye(64),zeroes(64);zeroes(64),zeroes(64),zeroes(64),eye(64) Thus, eventually the image becomes compressed to a significant enough degree to be sent
57 The Linear Algebra of Image Compression We can generalize the averaging and differencing constituting the compression
58 The Linear Algebra of Image Compression We can generalize the averaging and differencing constituting the compression For any string the equation c 1 = sa 1 represents the first compression, where A is the general compression matrix.
59 The Linear Algebra of Image Compression We can generalize the averaging and differencing constituting the compression For any string the equation c 1 = sa 1 represents the first compression, where A is the general compression matrix. for the second compression, the equation is c 2 = c 1 A 2 the A matrix is the block matrix ( ) A 0 A 2 = 0 I
60 The Linear Algebra of Image Compression We can generalize the averaging and differencing constituting the compression For any string the equation c 1 = sa 1 represents the first compression, where A is the general compression matrix. for the second compression, the equation is c 2 = c 1 A 2 the A matrix is the block matrix ( ) A 0 A 2 = 0 I for the 3rd compression the equation is c 3 = c 2 A 3 with the A matrix being A A 3 = 0 I I I
61 The Linear Algebra of Image Compression in general the equation for the nth compression will be c n = c n 1 A n
62 The Linear Algebra of Image Compression in general the equation for the nth compression will be c n = c n 1 A n The matrix A n is A I c n = c n I I 2 n 1 1
63 The Linear Algebra of Image Compression We can multiply all of the A matrices together to do the entire series of compressions in one step
64 The Linear Algebra of Image Compression We can multiply all of the A matrices together to do the entire series of compressions in one step We call this W and say that W = A 1 A 2 A 3... A n where n is the total amount of compressions.
65 The Linear Algebra of Image Compression We can multiply all of the A matrices together to do the entire series of compressions in one step We call this W and say that W = A 1 A 2 A 3... A n where n is the total amount of compressions. So the equation c = sw, where c is the complete compression of the image, will do all of the avereging and differencing in one step.
66 The Linear Algebra of Image Compression The beauty of this process is that it is invertable; once compressed, the original image can be reconstructed from the compression.
67 The Linear Algebra of Image Compression The beauty of this process is that it is invertable; once compressed, the original image can be reconstructed from the compression. Not suprisingly, the equation s = cw 1 does this.
68 The Linear Algebra of Image Compression The beauty of this process is that it is invertable; once compressed, the original image can be reconstructed from the compression. Not suprisingly, the equation s = cw 1 does this. This is called the Inverse Haar Wavelet Transform.
69 The Linear Algebra of Image Compression The beauty of this process is that it is invertable; once compressed, the original image can be reconstructed from the compression. Not suprisingly, the equation s = cw 1 does this. This is called the Inverse Haar Wavelet Transform. For the general case, the equation W 1 = A 1 n A 1 n 1 A 1 n 2... A 1 1 gives the matrix W
70 The Linear Algebra of Image Compression For a String of length 2 n, W must have a dimension of n by n, and therefore so must the A matrices. Also note that n matrices are neededto completely compress the image.
71 The Linear Algebra of Image Compression For a String of length 2 n, W must have a dimension of n by n, and therefore so must the A matrices. Also note that n matrices are neededto completely compress the image. We can derive a product series which aids in the calculation of W and W 1 for strings of various lengths.
72 The Linear Algebra of Image Compression This is and W = W 1 = A n 0 I I I 2 i 1 1 i=1 A I I I 2 i 1 1 i=n 1
73 The Linear Algebra of Image Compression This is and W = W 1 = A n 0 I I I 2 i 1 1 i=1 A I I I 2 i 1 1 i=n 1 The general A matrix represented here is in blocks, the dimension of which are D/2 i 1 where D is the original dimension of the image.
74 The Linear Algebra of Image Compression Up to this point we have only been averaging the rows of the entire matrix (which we have called strings). Now we will talk about how to average the columns.
75 The Linear Algebra of Image Compression Up to this point we have only been averaging the rows of the entire matrix (which we have called strings). Now we will talk about how to average the columns. What we have computed so far is called the row-reduced form of a matrix. The complete compression will be a row-and-column-reduced form.
76 The Linear Algebra of Image Compression Up to this point we have only been averaging the rows of the entire matrix (which we have called strings). Now we will talk about how to average the columns. What we have computed so far is called the row-reduced form of a matrix. The complete compression will be a row-and-column-reduced form. the simplest way to do this is by transposing our equations and T = ((PW ) T W ) T = W T PW P = ((T ) T W 1 ) T = (W 1 ) T TW 1 Where T is the compressed image, and P is the original.
77 The Linear Algebra of Image Compression Up to this point we have only been averaging the rows of the entire matrix (which we have called strings). Now we will talk about how to average the columns. What we have computed so far is called the row-reduced form of a matrix. The complete compression will be a row-and-column-reduced form. the simplest way to do this is by transposing our equations and T = ((PW ) T W ) T = W T PW P = ((T ) T W 1 ) T = (W 1 ) T TW 1 Where T is the compressed image, and P is the original. If we normalized the columns of W we could simplify this even more, because the columns would then be orthonormal, making W an orthogonal matrix, thus W 1 = W T.
78 The Linear Algebra of Image Compression Then the equation for P would be P = ((T ) T W 1 o ) T = (Wo 1 ) T TWo 1 = (Wo T ) T TWo T = W o TWo T
79 The Linear Algebra of Image Compression Then the equation for P would be P = ((T ) T W 1 o ) T = (Wo 1 ) T TWo 1 = (Wo T ) T TWo T = W o TWo T This is now much easier to calculate than before, which would optimize the use of a computer s capacity if it was to be included in an algorithm.
80 Questions Any questions?
81 Bibliography Colm Mulcahy. Image Compression Using the Haar Wavelet Transform
Image Compression Using the Haar Wavelet Transform
College of the Redwoods http://online.redwoods.cc.ca.us/instruct/darnold/laproj/fall2002/ames/ 1/33 Image Compression Using the Haar Wavelet Transform Greg Ames College of the Redwoods Math 45 Linear Algebra
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