CSE 490 G Introduction to Data Compression Wavelet Transform ...
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CSE 490 G Introduction to Data Compression Wavelet Transform ...
most of the details are small so they are very dark.
Detail subbands
CSE 490g - Lecture 12 - Winter 2006
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CSE 490g - Lecture 12 - Winter 2006
Wavelet Transform Compression wavelet transform
transformed image (coefficients)
wavelet coding
sign plane
bit stream
+-+++
1 Decoder wavelet decoding
2 transformed image (approx coefficients)
inverse wavelet transform
distorted image
Wavelet coder transmits wavelet transformed image in bit plane order with the most significant bits first. CSE 490g - Lecture 12 - Winter 2006
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Bit Planes of Coefficients
Encoder image (pixels)
2
5
3 4 . . .
00011 00001 01000 10100
. . . Coefficients are normalized between –1 and 1 CSE 490g - Lecture 12 - Winter 2006
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1
Rate-Fidelity Curve
Why Wavelet Compression Works
PSNR
– In most significant bit planes most coefficients are 0 so they can be coded efficiently. – Only some of the bit planes are transmitted. This is where fidelity is lost when compression is gained.
• Natural progressive transmission compressed bit planes
• Wavelet coefficients are transmitted in bit-plane order.
bits per pixel
truncated compressed bit planes More bit planes of the wavelet transformed image that is sent the higher the fidelity. CSE 490g - Lecture 12 - Winter 2006
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CSE 490g - Lecture 12 - Winter 2006
Wavelet Coding Methods
Wavelet Transform
• EZW - Shapiro, 1993
Encoder
– Embedded Zerotree coding.
image (pixels)
• SPIHT - Said and Pearlman, 1996 – Set Partitioning in Hierarchical Trees coding. Also uses “zerotrees”.
• JPEG 2000 – new standard based largely on EBCOT • GTW – Hong, Ladner 2000 – Uses group testing which is closely related to Golomb codes – Like GTW but uses arithmetic coding CSE 490g - Lecture 12 - Winter 2006
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One-Dimensional Average Transform (1)
(x+y)/2
detail y x
inverse wavelet transform
distorted image
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One-Dimensional Average Transform (2)
x
y
(x+y)/2 = L
L
low pass filter
x
(y-x)/2 (y-x)/2 = H
x=L-H y=L+H
high pass filter
How do we represent two data points at lower resolution?
H Transform
CSE 490g - Lecture 12 - Winter 2006
bit stream
CSE 490g - Lecture 12 - Winter 2006
y y
wavelet coding
A wavelet transform decomposes the image into a low resolution version and details. The details are typically very small so they can be coded in very few bits.
• PACW - Ladner, Askew, Barney 2003
average
transformed image (coefficients)
Decoder
– Uses arithmetic coding with context.
• EBCOT – Taubman, 2000
x
8
11
detail
CSE 490g - Lecture 12 - Winter 2006
Inverse Transform
12
2
One-Dimensional Average Transform (3)
One-Dimensional Average Transform (4)
Low Resolution Version
L
A
B
L
H
H Note that the low resolution version and the detail together have the same number of values as the original.
Detail
CSE 490g - Lecture 12 - Winter 2006
1 1 n B[i] = A[2i] + A[2i + 1], 0 ≤ i < 2 2 2 1 1 n B[n/2 + i] = − A[2i] + A[2i + 1], 0 ≤ i < 2 2 2
13
One-Dimensional Average Inverse Transform
L = B[0..n/2-1] H = B[n/2..n-1]
CSE 490g - Lecture 12 - Winter 2006
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Two Dimensional Transform (1) low resolution subband horizontal transform
