Transcript Wavelets in Image Compression

Wavelets in Image
Compression
Bhushan D Patil
PhD Research Scholar
Department of Electrical Engineering
Indian Institute of Technology, Bombay
Powai, Mumbai 40076
What are the principles behind
compression?
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Two fundamental components of compression
are redundancy and irrelevancy reduction.
Redundancy reduction aims at removing
duplication from the signal source
(image/video).
Irrelevancy reduction omits parts of the
signal that will not be noticed by the signal
receiver, namely the Human Visual System
(HVS).
Lossless vs. Lossy Compression
Reconstructed
image
Compression rate
Lossless
Lossy
numerically
identical to
the original
image
2:1 (at most
3:1)
contains
degradation
relative to the
original
high
compression
(visually
lossless)
Image compression steps:
Original image
(reconstructed)
Source encoder
(inverse T)
linear transform to
decorrelate the
image data
(lossless)
Compressed
image
Entropy Coding
(decoding)
of the resulting
quantized values
(lossless)
Quantization
(dequantization)
of basis
functions
coefficients
(lossy)
Basic ideas of linear
transformation
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We change the coordinate system in which we
represent a signal in order to make it much
better suited for processing (compression).
We should be able to represent all the useful
signal features and important phenomena in as
compact manner as possible.
Important to compact the bulk of the signal
energy into the fewest number of transform
coefficients.
Which options do we have for
linear transformation?
A possible choice for the linear
transformation are:
 DFT
 or, avoiding complex coefficients, the DCT
 JPEG (decomposition into smaller
subimages of size 8x8 or 16x16, followed
by DCT as the compression algorithm)
Why Wavelet-based
Compression?
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No need to block the input image and its basis functions
have variable length to avoid blocking artifacts.
More robust under transmission and decoding errors.
Better matched to the HVS characteristics
Good frequency resolution at lower frequencies, good time
resolution at higher frequencies – good for natural images.
Wavelet Decomposition
Iterated 2-D filter bank
Energy Compactness
No compression yet
EZW: ZeroTree Coding
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Uses “parent-child” dependencies between
subband coefficients at same spatial location
‘Bit-plane’ coding: enables an embedded
bitstream wrt distortion
Significance Pass
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Significant Coefficient y w.r.t. Threshold T: |y|≥T
􀁺 In a significance pass, all as yet insiginfant
coefficients are examined and declared either:
􀁺 Significant positive
􀁺 Significant negative
􀁺 Root of a zerotree (All children and root insigificant)
􀁺 Isolated insisignificant
 􀁺 At each pass, T ←T/2
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Refinement Pass
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All coefficient previously declared
significant are refined by one bit:
􀁺 y-estimate quantized to + or – T/4
􀁺 Coefficients are visited by decreasing
magnitude, then in raster order
EZW Example
EZW Example
Set Partitioning in Hierarchical Trees
(SPIHT)
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Same sort of idea as EZW
More efficient
Based on two types of zerotrees (not including
root):
􀁺 All descendants of a node are insignificant
(Type A)
All descendants of a node starting with the
grandchildren are insignificant (Type B)
SPIHT
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Coefficients and trees are stored in lists
processed in sequence
􀁺 List of Significant Coefficients (LSC)
􀁺 List of Insignificant Coefficients (LIC)
􀁺 List of Insignificant Sets (LIS)
􀁺 The lists enable a more efficient scan
order of the different trees and
coefficients
Coding passes
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All nodes from low-res LL in LIC, all those with
descendants in LIS
Examine nodes in LIC. If become significant, “1” and
their sign, move to LSC; otherwise “0”
􀁺 Examine sets in LIS.
If remains insignificant, “0”.
Else “1” and:
If Type A:
Encode all children’s current bit (and sign), move them to end of LIC
or LSC
Change Type to B, move to end of LIS
If Type B: delete tree from LIS. Add each child at end of LIS
as Type A.
􀁺 Refinement: refine all coefficients in LSC
SPIHT
-Result
MATLAB Implementation
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Analysis at various Compression rate.
PSNR