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Image Compression Using the Haar Wavelet TransformPowerPoint Presentation

Image Compression Using the Haar Wavelet Transform

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Image Compression Using the Haar Wavelet Transform

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Image Compression Using the Haar Wavelet Transform

Peggy Morton and Arne Petersen

2004. 10. 18

Yoon HeeJoo

- Introduction
- Averaging and Differencing
- Image Representation
- Using Linear Algebra
- Image Compressions

- 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

- 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

- 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

- 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

- Use the averaging and differencing process

[Figure 1]

- 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 0.531 -29 27 -25]
- Step3 [32.5 00.5 0.5 31 -29 27 -25]

- rows the results

detail coefficients

row average

columns the results

choose some number (δ) and set equal to zero

δ = 5

- apply the inverse of the averaging the differencing operations

Original Image

Decompressed Image

[Figure 2]

- Step1

Average

Difference

- Step3

- Step2

- W = A1A2A3A

- A : Original Matrix
- W : Transforming Matrix
- T : Compressed Matrix

- 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

- Program
- Questions
- Additional Explanations