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Image Compression Using the Haar Wavelet Transform

Image Compression Using the Haar Wavelet Transform. Peggy Morton and Arne Petersen 2004. 10. 18 Yoon HeeJoo. Contents. Introduction Averaging and Differencing Image Representation Using Linear Algebra Image Compressions. 1. Introduction. Done by compressing the image

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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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