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