Wavelet Analysis

1
Image Processing, Computer Vision, and Pattern
Recognition
Fac. of Comp., Eng. Tech. Staffordshire
University
Wavelet Analysis
Dr. Claude C. Chibelushi
2
Outline

• Introduction
• Image Pyramids
• Gaussian pyramid
• Wavelet pyramid
• Wavelet Transform
• Forward / inverse transform
• Filter coefficients
• Sample Applications
• Summary

3
Introduction

• Multi-resolution (hierarchical) image processing
• Involves the creation of images at various
  spatial resolutions (image pyramid)
• high resolution image may contain irrelevant
  details
• low resolution image may omit or distort
  significant details
• most image features are often related across
  resolution hierarchy
• hence, judicious combination of information
  contained in the pyramid levels may lead to
  higher performance

4
Image Pyramids

• Collection of images (same scene) of varying
  resolution

• Possible techniques for varying resolution
• resolution reduction may be based on averaging
  (low-pass filtering) and downsampling
• resolution increase may use interpolation and
  upsampling

5
Image Pyramids

• Recursive filtering generates pyramid
• Many types of image pyramids
• Gaussian pyramid
• based on Gaussian (low-pass) filter
• Laplacian pyramid
• based on differences across levels of Gaussian
  pyramid
• wavelet pyramid
• based on wavelet transform

6
Image Pyramids

• Gaussian pyramid
• (example 1-D three-level pyramid)

low res.
high res.
7
Image Pyramids

• Three-level Gaussian pyramid

8
Image Pyramids

• Two-level image pyramid from wavelet transform

B blurred H horizontal edge sensitivity V
vertical edge sensitivity HV horiz. vert. edge
sensitivity
9
Wavelet Transform

• Example three-level image pyramid

10
Wavelet Transform

• Based on filtering
• uses special pair of filters
• low-pass filter and high-pass filter
• decomposition
• image ? B1, H1, V1, HV1 (pyramid level 1)
• B1 ? B2, H2, V2, HV2 (pyramid level 2)
• B2 ? ...
• reconstruction right-hand-side ? left-hand-side

11
Wavelet Transform

• Recursive filtering of input image with special
  pair of filters
• Quadrature Mirror Filter (QMF) pair
• low-pass filter and high-pass filter
• pass-bands mirrored about cut-off of low-pass
  filter
• separates signal into its low-frequency and
  high-frequency components

12
Wavelet Transform

• Convolution is afflicted by image-border problem
• possible solution image is assumed symmetric
  about vertical and horizontal borders
• mirror image to allow convolution near borders
• Recursive filtering generates pyramid
• total number of pixels equal to number in
  original image
• filtering of image usually implemented as 1D
  convolution along rows and columns

13
Wavelet Transform

• Low-pass filter generates blurred lower
  resolution image (approximation image) from
  higher resolution image
• High-pass filter generates details (detail
  image) required for reconstruction of higher
  resolution image from lower resolution
  approximation image
• detail image refines approximation image

14
Wavelet Transform

• Forward transform
• Procedure recursively apply QMF pair to image
• low-pass filter applied to higher resolution
  image
• generates approximation image
• high-pass filter applied to higher resolution
  image
• generates detail image

15
Wavelet Transform

• Forward transform
• Procedure (ctd.)
• down-sample images output by QMF filters
  (decimation discard every other pixel)
• feed down-sampled approximation image into QMF
  pair / down-sampler combination
• generates approximation and detail images at next
  lower-resolution
• process repeated until desired lowest resolution

16
Wavelet Transform

• Forward transform

Lower resolution images
Higher resolution image
17
Wavelet Transform

• Forward transform pseudo code
• / decompose image into given number of pyramid
  levels /
• fwdTransform(image, nLevelsToDo)
• curLevel nAllLevels - nLevelsToDo
• for index1 from 0 to 1
• // row-wise filtering
• tmpFltIm convolve(fwdQMFindex1, image,
  row)
• tmpFltDsIm downsample(tmpFltIm, row)
• // column-wise filtering
• for index2 from 0 to 1
• tmpFltIm convolve(fwdQMFindex2,
  tmpFltDsIm, col)
• pyrcurLevelindex1 2 index2
  downsample(tmpFltIm, col)
• --nLevelsToDo
• if (nLevels gt 0)
• // assume approximation stored as 0th entry at
  each pyramid level

18
Wavelet Transform

• Example two-level image pyramid

C horizontal-edge sensitivity (high frequency
along column, low frequency along row) R
vertical-edge sensitivity (high frequency along
row, low frequency along column)
RC horizontal- and vertical-edge sensitivity
(high frequency along row, high frequency along
column)
19
Wavelet Transform

• Inverse transform
• Procedure
• introduce pixel with value 0 between pixel-pairs
  in both approximation and detail image
• generate higher-resolution images through
• low-pass filtering
• high-pass filtering
• add filtered up-sampled images
• reconstructs approximation image at next level
• repeat above steps until original resolution
  obtained

20
Wavelet Transform

• Inverse transform

Rows
Detail RC
Detail R
Lower resolution images
Higher resolution image
Detail C
Approx.
21
Wavelet Transform

• Inverse transform pseudo code
• / reconstruct image from given number of pyramid
  levels /
• invTransform(pyr, nLevelsToDo)
• curLevel nLevelsToDo - 1
• for index1 from 0 to 1
• // column-wise filtering
• for index2 from 0 to 1
• tmpFltUsIm upsample(pyrcurLevelindex1 2
  index2, col)
• tmpFltImindex2 convolve(invQMFindex2,
  tmpFltUsIm, col)
• tmpFltImSum add(tmpFltIm0, tmpFltIm1)
• // row-wise filtering
• tmpFltUsIm upsample(tmpFltImSum, row)
• imageindex1 convolve(invQMFindex1,
  tmpFltUsIm, row)
• tmpFltImSum add(image0, image1)
• // approximation stored as 0th entry at each
  pyramid level
• pyrcurLevel0 scale(tmpFltImSum, 4)

22
Wavelet Transform

• Filter coefficients
• Forward transform (decomposition)
• coefficients in reverse order of corresponding
  filter in inverse transform

23
Wavelet Transform

• Filter coefficients
• Relation between reconstruction and decomposition
  filter coefficients

reverse order h(-n)
24
Wavelet Transform

• Filter coefficients
• Inverse transform (reconstruction)
• high-pass filter coefficients of low-pass filter
  in reverse order and shifted rightwards by 1,
  with alternating sign
• let h(n) be coefficients of low-pass filter, and
  g(n) be coefficients of high-pass filter
• g(n) (-1)1-n h(1-n)
• i.e. if (n is even) g(n) - h(1-n) else g(n)
  h(1-n)

25
Wavelet Transform

• Filter coefficients
• Relation between coefficients of low-pass and
  high-pass reconstruction filters

1. reverse order h(-n)
2. shift right h(1 - n)
3. negate even-numbered coefficients g(n)
(-1)1-n h(1-n)
0
26
Wavelet Transform

• Filter coefficients
• Many wavelet filters are available e.g.
• a Haar low-pass filter
• a set of Daubechies QMF filters

27
Sample Applications

• Multi-resolution image processing has been used
  in a wide variety of applications
• image segmentation
• image matching
• image compression
• texture analysis
• noise removal
• ...

28
Sample Applications

• Image compression
• High-pass filtered image (edges)
• has most values concentrated near zero
• higher redundancy (than original image) offers
  possibility of higher compression ratio
• hence, Laplacian or wavelet pyramid provide basis
  for compression
• JPEG 2000 is based on wavelet transform

29
Sample Applications

• Image compression
• Forward transform used in compression
• transform, followed by quantisation, Huffman
  coding, ...
• Inverse transform used in decompression
• Huffman decoding, ... followed by transform

30
Summary

• Multi-resolution image processing
• processing at several image scales
• generation of image pyramid recursive filtering
• some pyramid types
• Gaussian, Laplacian, wavelet pyramids
• Sample applications
• image compression, segmentation, matching, ...