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Image Compression Using the Haar Wavelet Transform

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

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

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

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

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

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

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

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

  13. 4. Using Linear Algebra (1/3) • Step1 Average Difference

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

  15. 4. Using Linear Algebra (3/3) • W = A1A2A3A • A : Original Matrix • W : Transforming Matrix • T : Compressed Matrix

  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. Thank you for Attending My Presentation • Program • Questions • Additional Explanations

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