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2102427Multimedia Compression Technology
Lecture 7
Wavelet Methods (I)
Dr. Supavadee AramvithDr. Supavadee AramvithChulalongkorn UniversityChulalongkorn University
[email protected]@chula.ac.thMultimedia Compression Technology 2
Outline
Averaging and DifferencingAveraging and DifferencingThe The HaarHaar TransformTransformSubbandSubband TransformsTransforms
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Wavelet Transform
A family of transformations that filters the data into low A family of transformations that filters the data into low resolution data plus detail dataresolution data plus detail data
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Wavelet Transform – Example (Enhanced)
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Wavelet Transform – Example (Actual)
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Wavelet Transform Compression
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Bit planes of Coefficients
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Why Wavelet Compression works
Wavelet coefficients are transmitted in bitWavelet coefficients are transmitted in bit--plane orderplane orderIn the most significant bit planes, many coefficients are zero sIn the most significant bit planes, many coefficients are zero so o they can be coded efficiently.they can be coded efficiently.Only some of the bit planes are transmitted (this is where qualiOnly some of the bit planes are transmitted (this is where quality ty is lost when doing is lost when doing lossylossy compression)compression)
Natural progressive transmissionNatural progressive transmission
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Wavelet Coding Methods
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Wavelet Transform Compression
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One Dimensional Average Transform (I)
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One Dimensional Average Transform (II)
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One Dimensional Average Transform (III)
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One Dimensional Average Transform (IV)
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One Dimensional Average Inverse Transform
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Complexity of the transform
The number of arithmetic operations as a function of The number of arithmetic operations as a function of the size of the datathe size of the data
)1(2)12(222121
21122 11
01−=−=−=−
−−
=−⎟⎠
⎞⎜⎝
⎛= +
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==∑∑ Nnn
nn
i
in
i
i
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Example
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Two Dimensional Transform (I)
2 approaches2 approachesStandard decompositionStandard decompositionPyramid decompositionPyramid decomposition
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Standard Image Wavelet Transform and Decomposition (I)
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Standard Image Wavelet Transform and Decomposition (II)
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Pyramid Image Wavelet Transform
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Two Dimensional Transform
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Wavelet Transformed Image
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Wavelet Transforms
Technically wavelet transforms are special kinds of linear Technically wavelet transforms are special kinds of linear transformations. Easiest to think of them as filterstransformations. Easiest to think of them as filters
The filters depend only on a constant number of values (bounded The filters depend only on a constant number of values (bounded support)support)Preserve energy (norm of pixels = norm of the coefficients)Preserve energy (norm of pixels = norm of the coefficients)Inverse filters also have bounded supportInverse filters also have bounded support
WellWell--known wavelet transformsknown wavelet transformsHaarHaar transform transform –– like the average but orthogonal to preserve like the average but orthogonal to preserve energy. Not used in practice.energy. Not used in practice.DaubechiesDaubechies 9/7 9/7 –– biorthogonalbiorthogonal (inverse is not the transpose). (inverse is not the transpose). Most commonly used in practice.Most commonly used in practice.
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An 8x8 Image and Its SubbandDecompostion
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The Subband Decomposition of a Diagonal Line
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Subbands and Levels in Wavelet Decomposition
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An Example of the Pyramid Image Wavelet Transform (I)
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An Example of the Pyramid Image Wavelet Transform (II)
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Highly Correlated Image (I)
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Its Haar Transform (II)
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A 128x128 Image with Activity on the Right and Its Transform
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Three Lossy Reconstruction of an 8x8 Image (I)
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Three Lossy Reconstruction of an 8x8 Image (II)
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Three Lossy Reconstruction of an 8x8 Image (III)
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Reconstructing a 128x128 Simple Image from 4% of its Coefficients
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Matlab Code for the Haar Transform of an Image (I)
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Matlab Code for the Haar Transform of an Image (II)
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Three Lossy Reconstruction of the 128x128 Lena Image (I)
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Three Lossy Reconstruction of the 128x128 Lena Image (II)
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Three Lossy Reconstruction of the 128x128 Lena Image (III)
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Lossy Wavelet Image Compression
LossyLossy involves the discarding of coefficientsinvolves the discarding of coefficientsSparseness ratio Sparseness ratio
The measurement of number of coefficients discardedThe measurement of number of coefficients discardedThe number of nonzero wavelet coefficients divided by number of The number of nonzero wavelet coefficients divided by number of coefficients left after some are discardedcoefficients left after some are discardedHigher sparseness ratio Higher sparseness ratio --> fewer coefficients left> fewer coefficients left
Better compression Better compression --> poorly reconstructed image> poorly reconstructed image
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The Haar Transform
∑ ∑∑∞
−∞=
∞
=
∞
−∞=
−+−=k j
jkj
kk ktdktctf
0, )2()()( ψφ
⎩⎨⎧ <≤
=otherwise
tt
,010,1
)(φ⎩⎨⎧
<≤−<≤
=15.0,15.00,1
)(t
ttψ
Basic scale functionBasic scale function Basic Basic HaarHaar wavelet wavelet –– step functionstep function
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The Haar Basis Scale and Wavelet Functions
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Haar Matrix Representation (I)
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Haar Matrix Representation (II)
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Haar Matrix Representation (III)
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Haar Matrix Representation (IV)
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Example of Matrix Wavelet Transform
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Example of Matrix Wavelet Transform
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Subband Transforms
Orthogonal transform Orthogonal transform Inner product of the data with a set of basis functionsInner product of the data with a set of basis functionsOutput is set of transform coefficients Output is set of transform coefficients
Discrete inner product of the two vectors Discrete inner product of the two vectors
Wavelet transform = Wavelet transform = subbandsubband transformtransformCompute a convolution of the data items with a set of Compute a convolution of the data items with a set of bandpassbandpass filtersfiltersEach resulting Each resulting subbandsubband encodes a particular portion of the frequency encodes a particular portion of the frequency content of the datacontent of the data
ii
i gfgf ∑=,
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Subband Transforms
Discrete convolution of two vectorsDiscrete convolution of two vectors
Wavelet transform = Wavelet transform = subbandsubband transformtransformCompute a convolution of the data items with a set of Compute a convolution of the data items with a set of bandpassbandpass filtersfiltersEach resulting Each resulting subbandsubband encodes a particular portion of the frequency encodes a particular portion of the frequency content of the datacontent of the data
)()()()()( jigjfigifihi
−=∗= ∑
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Convolution of x(t) and g(t)
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Example of Convolution
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Applying Convolution to Denoising a Function
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