1. N R 1 Image Compression Using the Haar Wavelet Transform Peggy Morton and Arne Petersen 2004. 10. 18 Yoon HeeJoo

2. 2 Contents  Introduction  Averaging and Differencing  Image Representation  Using Linear Algebra  Image Compressions

3. 3 1. Introduction  Done by compressing the image  Use a process called averaging and differencing to develop a new matrix representing the same image in a more concise manner.  Eliminate some of unnecessary information, and arrive at an approximation of our original image

4. 4 2. Averaging and Differencing (1/3)  8X8 matrix, process involve three steps (2³=8)  Step 1 [3 5 4 8 13 7 5 3] [4 6 10 4 -1 -2 3 1] ex. (3 + 5)/2 = 4 3 - 4 = -1 Averaging Differencing … detail coefficient

5. 5 2. Averaging and Differencing (2/3)  Step 2 [4 6 10 4 -1 -2 3 1] [5 7 -1 3 -1 -2 3 1] ex. (4 + 6)/2 = 5 4 - 5 = -1 Averaging Differencing … detail coefficient

6. 6 2. Averaging and Differencing (3/3)  Step 3 [5 7 -1 3 -1 -2 3 1] [6 -1 -1 3 -1 -2 3 1] ex. (5 + 7)/2 = 6 5 - 6 = -1 Averaging Differencing … detail coefficient row average

7. 7 3. Image Representation (1/6)  Use the averaging and differencing process [Figure 1]

8. 8 3. Image Representation (2/6)  first row [64 2 3 61 60 6 7 57]  Step1 [33 32 33 32 31 -29 27 -25]  Step2 [32.5 32.5 0.5 0.5 31 -29 27 -25]  Step3 [32.5 0 0.5 0.5 31 -29 27 -25]

9. 9 3. Image Representation (3/6)  rows the results row average detail coefficients

10. 10 3. Image Representation (4/6)  columns the results  choose some number (δ) and set equal to zero δ = 5

11. 11 3. Image Representation (5/6)  apply the inverse of the averaging the differencing operations

12. 12 3. Image Representation (6/6) [Figure 2] Decompressed ImageOriginal Image

13. 13 4. Using Linear Algebra (1/3)  Step1 AverageDifference

14. 14 4. Using Linear Algebra (2/3)  Step2  Step3

15. 15 4. Using Linear Algebra (3/3)  W = A 1 A 2 A 3 A  A : Original Matrix W : Transforming Matrix T : Compressed Matrix

16. 16 5. Image Compressions  Figure 3a.  Original Image  Figure 3b.  65,536 entries  631 non-zero entries  Compression Ratio – 103:1  Figure 3c.  65,536 entries  1,411 non-zero entries  Compression Ratio – 46:1

17. 17 Thank you for Attending My Presentation  Program Program  Questions  Additional Explanations