wavelet transform. wavelet transform coding: multiresolution approach wavelet transform quantizer...
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Wavelet Transform
Wavelet Transform Coding: Multiresolution approachWavelet Transform Coding: Multiresolution approach
Wavelet transform Quantizer Symbol
encoder
Input image(NxN)
Compressedimage
Inverse wavelet
transform
Symboldecoder
Decompressedimage
Decoder
Encoder
Unlike DFT and DCT, Wavelet transform is a multiresolution transform.
Multiresolution 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
Wavelet History: Image PyramidWavelet History: Image Pyramid
Pyramidal structured image
Coarser, decrease (low) resolution
Finer, increase (high) resolution
If we smooth and then down sample an image repeatedly, we willget a pyramidal image:
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
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.
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
Standard decomposition
nonstandard decomposition
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.
2 common Filters
• An example of Daubechies basis vectors (there are many others) follows:
• The Haar basis vectors are
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 sub sample 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.
Wavelet Transformation stepWavelet Transformation step
Wavelet Transformation – Wavelet Transformation – multiresolution decomposition process multiresolution decomposition process
2D Discrete Wavelet Transformation 2D Discrete Wavelet Transformation
d2 h2
v2 a2
a1
h1d1
v1
Original imageNxN
a3
d3 h3
v3
Level/Band/Scale 1
Level/Band/Scale 2
Level/Band/Scale 3
d = diagonal detail (LOW/LOW)h = horizontal detail (HIGH/LOW)v = vertical detail (LOW/HIGH)a = approximation (HIGH/HIGH)
2D Discrete Wavelet Transformation (cont.) 2D Discrete Wavelet Transformation (cont.)
d2
h2
v2h1
d1v1
a3d3h3
v3
Original imageNxN
Wavelet coefficientsNxN
OriginalImage
Example of 2D Wavelet Transformation Example of 2D Wavelet Transformation
Original image
LH1
HL1
HH1
LL1LL1
Example of 2D Wavelet Transformation (cont.) Example of 2D Wavelet Transformation (cont.)
The first level wavelet decomposition
LL2
LH1
HL1
HH1
LH2 HH2
HL2LL2
Example of 2D Wavelet Transformation (cont.) Example of 2D Wavelet Transformation (cont.)
The second level wavelet decomposition
LH1 HH1
LH2 HH2
HL2
HL3
HH3LH3
LL3
HL1
The third level wavelet decomposition
Example of 2D Wavelet Transformation (cont.) Example of 2D Wavelet Transformation (cont.)
Example of 2D Wavelet TransformationExample of 2D Wavelet Transformation
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
Examples: Types of Wavelet TransformExamples: Types of Wavelet Transform
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
Haarwavelets
Symlets
Daubechieswavelets
Biorthogonalwavelets