Wavelet Transform

1
Wavelet Transform
2
Wavelet Transform Coding Multiresolution approach
Unlike DFT and DCT, Wavelet transform is a
multiresolution transform.
3
Multiresolution

• If the objects are small in size / low in
  contrast high resolutions
• If the objects are large in size / high in
  contrast low resolutions (a coarse view)
• If both small large objects / low or high
  contrast objects are present simultaneously, it
  can be advantageous to study them at several
  resolutions multiresolution processing

4
Wavelet History Image Pyramid
If we smooth and then down sample an image
repeatedly, we will get a pyramidal image
Coarser, decrease (low) resolution
Finer, increase (high) resolution
Pyramidal structured image
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
5
Introduction

• The wavelet transform breaks an image down into
  four subsampled, or decimated, images.
• They are subsampled by keeping every other pixel.
  
• The results consist of
• one image that has been highpass filtered in both
  the horizontal and vertical directions,
• one that has been highpass filtered in the
  vertical and lowpass filtered in the horizontal,
• one that has been lowpassed in the vertical and
  highpassed in the horizontal, and
• one that has been lowpass filtered in both
  directions.

6
Decomposition
Standard decomposition

• One-dimensional DWT to all the columns and then
  one-dimensional DWTs to all the rows
• Two-dimensional wavelet by columns, then by rows
  in one scale only

nonstandard decomposition
7
Filters

• Numerous filters can be used to implement the
  wavelet transform, and two of the commonly used
  ones, the Daubechies and the Haar, will be
  explored here.
• These are separable, so they can be used to
  implement a wavelet transform by first convolving
  them with the rows and then the columns.

8
2 common Filters

• The Haar basis vectors are

• An example of Daubechies basis vectors (there are
  many others) follows

9
Wavelet Transformation step

1. Convolve the lowpass filter with the rows
   (remember that this is done by sliding,
   multiplying coincident terms, and summing the
   results) and save the results. (Note For the
   basis vectors as given, they do not need to be
   reversed for convolution.)
2. Convolve the lowpass filter with the columns (of
   the results from step 1) and subsample this
   result by taking every other value this gives us
   the lowpass-Iowpass version of the image
   LOW/LOW.
3. Convolve the result from step 1, the lowpass
   filtered rows, with the highpass filter on the
   columns. Subsample by taking every other value to
   produce the lowpass-highpass image LOW/HIGH
4. Convolve the original image with the highpass
   filter on the rows and save the result.
5. Convolve the result from step 4 with the lowpass
   filter on the columns subsample to yield the
   highpass-lowpass version HIGH/LOW of the image.
   
6. To obtain the highpass-highpass version
   HIGH/HIGH, convolve the columns of the result
   from step 4 with the highpass filter.

10
Wavelet Transformation multiresolution
decomposition process
11
2D Discrete Wavelet Transformation
Original image NxN
d diagonal detail (LOW/LOW) h horizontal
detail (HIGH/LOW) v vertical detail
(LOW/HIGH) a approximation (HIGH/HIGH)
h1
d1
a1
v1
d2
h2
Level/Band/Scale 1
Level/Band/Scale 3
v2
a2
d3
h3
Level/Band/Scale 2
a3
v3
12
2D Discrete Wavelet Transformation (cont.)
h2
h1
a3
h3
Original image NxN
d3
v3
d2
v2
d1
v1
Wavelet coefficients NxN
13
Example of 2D Wavelet Transformation
Original image
14
Example of 2D Wavelet Transformation (cont.)
The first level wavelet decomposition
15
Example of 2D Wavelet Transformation (cont.)
The second level wavelet decomposition
16
Example of 2D Wavelet Transformation (cont.)
HL3
LL3
HL2
HL1
HH3
LH3
LH2
HH2
LH1
HH1
The third level wavelet decomposition
17
Example of 2D Wavelet Transformation
Level 1
Level 2
18
Example of 2D Wavelet Transformation
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.
19
Examples Types of Wavelet Transform
Daubechies wavelets
Haar wavelets
Biorthogonal wavelets
Symlets
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2nd Edition.