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Digital Image Processing Lecture 21: Lossy Compression

Digital Image Processing Lecture 21: Lossy Compression. Prof. Charlene Tsai. Reminder …. The Thursday class (5/29) is moved to Monday (5/28) for next week. JPEG Algorithm. Lossy compression trades some acceptable data loss for greater rate of compression.

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Digital Image Processing Lecture 21: Lossy Compression

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  1. Digital Image Processing Lecture 21: Lossy Compression Prof. Charlene Tsai

  2. Reminder … • The Thursday class (5/29) is moved to Monday (5/28) for next week.

  3. JPEG Algorithm • Lossy compression trades some acceptable data loss for greater rate of compression. • There are many available, but the most popular is the one developed by JPEG – transform coding. • Coding not done on pixel values, but on a transform.

  4. Transform • Discrete Cosine Transform (DCT) • Applying to 8x8 blocks, the forward and inverse DCT are • DCT are real-valued, high information-packing capability, and separable.

  5. Comparison with FFT • Given the following sequence 10 25 40 55 70 85 100 115 • If using FFT, the inverse of the first 4 FFT coefficients gives 49 41 56 57 71 70 85 90 • If using DCT, the result is 11 23 41 56 69 84 102 114

  6. JPEG Compression • For each 8x8 image block, perform the compression • Subtracting 128 from each value • Apply DCT • Normalization by dividing by matrix Q (the lossy part) • Turning most of the elements zero • Change the matrix into a vector by reading the nonzero element in a zigzag fashion.

  7. (con’d) • The first element (DC coefficient) of the vector is encoded as the difference between itself and the DC of previous block. • Keeps all values (except the very first one) small • Compressed using RLE • Other values (AC coefficients) are compressed using Huffman coding • Rate of compression is controlled by scaling Q => more scaling, more compression

  8. Decompression • Inverse the operations: • Decode Huffman encoding and RLE • Put the vector back to 8x8 matrix • Multiply by Q • Inverse DCT • Shift back by 128

  9. The Lossless Parts • Huffman Coding and RLE for AC and DC terms, respectively. • We’ll focus the discussion on the AC terms • Each nonzero value x is assigned a category k • The number of preceding 0s (run). • Values: 7 -1 10 0 2 1 0 0 2 4 0 1 0 -1 • Category: 1 4 2 1 2 3 1 1 • Run: 0 0 1 0 2 0 1 1

  10. Huffman Code Table • Part of JPEG baseline standard • See the handout for the code table and application.

  11. Wavelet Transform • For both FT and DCT, we assume so kind of periodicity in the image. • Wavelet Transform keeps the wave concept, but drop the eriodicity. • Wavelet is a little part of a wave wave wavelet

  12. What to do with a wavelet? • Given a wavelet function , we can • Dilate it by applying a scaling factor to x. e.g. would squash and would expand. • Translate by adding/subtracting a value from x. e.g. shift the wavelet 2 to the right, and shift 3 to the left. • Change it height by multiplying the function by a constant. • All together, we get

  13. Applications • Noise reduction • Edge detection • Compression • Adopted by JPEG2000

  14. A Simple Wavelet Transform • All wavelet transform work by taking weighted averages of input values and providing extra information for inversion. • Here the example is: averaging of two values and differencing. • If a and b are two numbers, we get average s and difference d by:

  15. (cont) • Given a simple vector v of 8 elements, we create two new vectors v1 and v2 of 4 elements each: • v1 containing the averages • v2 containing the differences • DWT at 1 scale is: • We can keep going for another 2 levels. • Let’s go through the example in pg425 of the handout. (available online)

  16. Haar Wavelet • Simplest wavelet, defined as • Haar wavelet can be written in terms of a simpler pulse function

  17. Mother wavelet Father wavelet (scaling function)

  18. Discrete Wavelet transform • The forward transformation • The inverse is

  19. are called the filter coefficients (or taps) • A wavelet is completely specified by its taps.

  20. Back to Haar Wavelet Low pass • where • values are what we have in a DWT matrix • Let’s look at the example in pg 429. High pass

  21. Two-Dimensional Wavelet (STD)

  22. Two-Dimensional Wavelet (non-STD)

  23. Example: one scale

  24. Image Compression • For a given value d, set all values x in the DWT for which to 0.

  25. High-Pass Filtering • Except from the top left image, the rest is high-frequency information. • If setting the top-left corner to 0, the result after inversion would be a high-pass filtered image.

  26. Denoising • Very similar to compression by using thresholding. • An example is given in pg446.

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