CPS Grade 5 Math Pacing Guide 2011-12
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Outline
2102427
Averaging and Differencing
The Haar Transform
Multimedia Compression Technology
Subband Transforms
Lecture 7
Wavelet Methods (I)
Dr. Supavadee Aramvith
Chulalongkorn University
Supavadee.A@chula.ac.th
Multimedia Compression Technology 2
Wavelet Transform Wavelet Transform – Example (Enhanced)
A family of transformations that filters the data into low
resolution data plus detail data
Multimedia Compression Technology 3 Multimedia Compression Technology 4‰
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Wavelet Transform – Example (Actual) Wavelet Transform Compression
Multimedia Compression Technology 5 Multimedia Compression Technology 6
Bit planes of Coefficients Why Wavelet Compression works
Wavelet coefficients are transmitted in bit-plane order
In the most significant bit planes, many coefficients are zero so
they can be coded efficiently.
Only some of the bit planes are transmitted (this is where quality
is lost when doing lossy compression)lossy
Natural progressive transmission
Multimedia Compression Technology 7 Multimedia Compression Technology 8Wavelet Coding Methods
Wavelet Transform Compression
Multimedia Compression Technology 9 Multimedia Compression Technology 10
One Dimensional Average Transform (I) One Dimensional Average Transform (II)
Multimedia Compression Technology 11 Multimedia Compression Technology 12„
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One Dimensional Average Transform (III) One Dimensional Average Transform (IV)
Multimedia Compression Technology 13 Multimedia Compression Technology 14
One Dimensional Average Inverse Transform
Complexity of the transform
The number of arithmetic operations as a function of
the size of the data
n +1
n n
⎛ ⎞ 1 −2
i i n +1 n
2 = 2 −1 = −1 = 2 −2 = 2(2 −1) = 2(N −1)
⎜ ⎟
∑ ∑
1 −2
i =1 ⎝ i =0 ⎠
Multimedia Compression Technology 15 Multimedia Compression Technology 16‰
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Example
Two Dimensional Transform (I)
2 approaches
Standard decomposition
Pyramid decomposition
Multimedia Compression Technology 17 Multimedia Compression Technology 18
Standard Image Wavelet Transform and Standard Image Wavelet Transform and
Decomposition (I) Decomposition (II)
Multimedia Compression Technology 19 Multimedia Compression Technology 20„
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Pyramid Image Wavelet Transform
Two Dimensional Transform
Multimedia Compression Technology 21 Multimedia Compression Technology 22
Wavelet Transforms
Wavelet Transformed Image
Technically wavelet transforms are special kinds of linear
transformations. Easiest to think of them as filters
The filters depend only on a constant number of values (bounded
support)support)
Preserve energy (norm of pixels = norm of the coefficients)
Inverse filters also have bounded support
Well-known wavelet transforms-
HaarHaar transform – liklike the average but orthogonal to preserve
energy. Not used in practice.
energy
Daubechies 9/7 – biorthogonal (inverse is not the transpose).
Daubechies biorthogonal
Most commonly used in practice.
Multimedia Compression Technology 23 Multimedia Compression Technology 24An 8x8 Image and Its Subband The Subband Decomposition of a
Decompostion Diagonal Line
Multimedia Compression Technology 25 Multimedia Compression Technology 26
Subbands and Levels in Wavelet An Example of the Pyramid Image
Decomposition Wavelet Transform (I)
Multimedia Compression Technology 27 Multimedia Compression Technology 28An Example of the Pyramid Image
Highly Correlated Image (I)
Wavelet Transform (II)
Multimedia Compression Technology 29 Multimedia Compression Technology 30
Its Haar Transform (II) A 128x128 Image with Activity on the
Right and Its Transform
Multimedia Compression Technology 31 Multimedia Compression Technology 32Three Lossy Reconstruction of an 8x8 Three Lossy Reconstruction of an 8x8
Image (I) Image (II)
Multimedia Compression Technology 33 Multimedia Compression Technology 34
Three Lossy Reconstruction of an 8x8 Reconstructing a 128x128 Simple Image
Image (III) from 4% of its Coefficients
Multimedia Compression Technology 35 Multimedia Compression Technology 36Matlab Code for the Haar Transform of an Matlab Code for the Haar Transform of an
Image (I) Image (II)
Multimedia Compression Technology 37 Multimedia Compression Technology 38
Three Lossy Reconstruction of the Three Lossy Reconstruction of the
128x128 Lena Image (I) 128x128 Lena Image (II)
Multimedia Compression Technology 39 Multimedia Compression Technology 40
