1. Wavelet Transform

2. Wavelet Transform Coding: Multiresolution approach Wavelet transform Quantizer Symbol encoder Input image (NxN) Compressed image Inverse wavelet transform Symbol decoder Decompressed image Decoder Encoder 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 Pyramidal structured image Coarser, decrease (low) resolution Finer, increase (high) resolution If we smooth and then down sample an image repeatedly, we will get a pyramidal image: (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd 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 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

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 An example of Daubechies basis vectors (there are many others) follows: The Haar basis vectors are

9. 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. Wavelet Transformation step

10. Wavelet Transformation – multiresolution decomposition process

11. 2D Discrete Wavelet Transformation d2d2h2h2 v2v2a2a2 a1a1 h1h1d1d1 v1v1 Original image NxN a3a3 d3d3h3h3 v3v3 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)

12. 2D Discrete Wavelet Transformation (cont.) d2d2 h2h2 v2v2 h1h1 d1d1v1v1 a3a3 d3d3 h3h3 v3v3 Original image NxN Wavelet coefficients NxN

13. Original Image Example of 2D Wavelet Transformation Original image

14. LH 1 HL 1 HH 1 LL 1 Example of 2D Wavelet Transformation (cont.) The first level wavelet decomposition

15. LL 2 LH 1 HL 1 HH 1 LH 2 HH 2 HL 2 LL 2 Example of 2D Wavelet Transformation (cont.) The second level wavelet decomposition

16. LH 1 HH 1 LH 2 HH 2 HL 2 HL 3 HH 3 LH 3 LL 3 HL 1 The third level wavelet decomposition Example of 2D Wavelet Transformation (cont.)

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, 2 nd Edition.

19. Examples: Types of Wavelet Transform (Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2 nd Edition. Haar wavelets Symlets Daubechies wavelets Biorthogonal wavelets