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N R 1 Image Compression Using the Haar Wavelet Transform Peggy Morton and Arne Petersen 2004. 10. 18 Yoon HeeJoo
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2 Contents Introduction Averaging and Differencing Image Representation Using Linear Algebra Image Compressions
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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
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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
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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
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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
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7 3. Image Representation (1/6) Use the averaging and differencing process [Figure 1]
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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]
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9 3. Image Representation (3/6) rows the results row average detail coefficients
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10 3. Image Representation (4/6) columns the results choose some number (δ) and set equal to zero δ = 5
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11 3. Image Representation (5/6) apply the inverse of the averaging the differencing operations
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12 3. Image Representation (6/6) [Figure 2] Decompressed ImageOriginal Image
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13 4. Using Linear Algebra (1/3) Step1 AverageDifference
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14 4. Using Linear Algebra (2/3) Step2 Step3
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15 4. Using Linear Algebra (3/3) W = A 1 A 2 A 3 A A : Original Matrix W : Transforming Matrix T : Compressed Matrix
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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
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17 Thank you for Attending My Presentation Program Program Questions Additional Explanations
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