Wavelets, for the Layman. Mike Acton 25 Jan 08

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1 Wavelets, for the Layman Mike Acton 25 Jan 08

2 Introduction This won t be math-heavy. De-mystify the jargon. Enough understanding to use it yourself. Just an introduction From here you ll know what to look for. It s really easy.

3 What s are these wavelets? It does something with a sequence (vector) of sample values, but is it Some kind of fancy pants compression? Some kind of tricky math? Some kind of scam to give people to write something a thesis about?

4 It s a transform When you hear wavelets, you should really think wavelet transform Take data from one format, change it to another format that is better organized. There is no compression!

5 It s a transform Take data from one format, change it to another format that is better organized. In a raw signal, everything has the same weight (equally important). We want to reduce the energy needed to produce the exact same signal. (What this means depends on context. But, for example: number of bits needed for compression) i.e. Lower the entropy. NOTE: Implies that there is order to be found in the system. (Not random!) We also want to sort information by importance.

6 It s a transform Think something like the Burrows-Wheeler transform. Find the order in the system. BWT (as most text transforms) organizes by entry in the palette. ^BANANA@ BNN^AA@A Now there s some exploitable order. e.g. Now usually takes less energy to describe with just RLE

7 It s a transform The transform from 20,000 ft: Given signal vector [x] of length n, F(x) transforms [X] into vector [F] of length n/2 G(x) transforms [X] info vector [G] of length n/2 H(F(x),G(x)) transforms [F],[G] back into [X]

8 It s a transform F(x) can be considered Low-pass filter, or Prediction G(x) can be considered High-pass filter, or Update H(f,g) can be considered Band-pass combine, or Inverse

9 It s a transform Wavelet refers to the shape of F(x) and G(x)

10 It s a transform Why Wavelet and not Wave? A wave cycles, a wavelet doesn t. Only acts on local area of the input vector. Then with all the local wavelet results, we can recombine to the larger, full signal.

11 It s a transform In practice, any functions for F(x), G(x) and H(x) that satisfy the conditions you want can be considered. Choosing the right functions is a bit of a challenge. But there are lots of good ones.

12 Discrete Wavelet Transform Recursively transform LOW, F(x) That s it.

13 Wavelet Packet Transform Recursively transform both, F(x) and G(x)

14 Best Basis Sometimes, less processed vectors end up cheaper (use less energy) e.g. [F0] might be cheaper than [f1], [gf1] So we keep F[0] instead. (Requires an extra list of what was kept)

15 Start with an example. The classic favorite: Average and difference F(x) = 0.5 * (x[n] + x[n+1]) G(x) = x[n] F(x) (H) Reconstruct with simple line segment.

16 Average and Difference, con t (H) Reconstruct with simple line segment. X[n] = G[n/2] + F[n/2] X[n+1] = G[n/2] F[n/2]

19 Example 1 Notice: [F] and [G] are each half length of [X] i.e {[F],[G]} is same length as [X] The range of [G] is much smaller than [X] Less energy in this case = less bits needed. [G] has order: Larger values are more important than smaller values. Particularly for images.

20 It s a transform Why else a Wavelet and not Wave? i.e. Given [X] of length n, Ax = b Matrix [A] is n by n and contains F(x) and G(x) Vector [B] is length n and contains the vectors [F] and [G] Matrix [A] contains F(x) and G(x) Matrix [A] is sparse and only non-zero in the diagonal (more-or-less).

21 2D Example Let s look at a 2D DWT image example using average and difference (Difference might be hard to see. Look closely!)

28 2D Example How do you do DWT in 2D? Easiest answer: You don t. Combine two 1D signals. Row-transform (or column) first, then treat the full result as the input for column-transform (or row). Note: G(x) is filtered twice.

30 2D Example What are the tricky bits with images? Filter outside range of signal. (i.e. before or after the row/column) For most images, best result is to treat it as a cyclic signal around each row/column. What if it doesn t fit in memory (i.e. on SPU) Use a sliding window of source data with borders equal to ½ length of the filter. Then stitch them back together.

31 2D Example What are the tricky bits with images? Filter outside range of signal. (i.e. before or after the row/column) For most images, best result is to treat it as a cyclic signal around each row/column. What if it doesn t fit in memory (i.e. on SPU) Use a sliding window of source data with borders equal to ½ length of the filter. Then stitch them back together.

32 Final Example Use Daubuchies-4 Filter

33 Final Example Use Daubuchies-4 Filter const float d4_low[4] = { f, f, f, f }; const float d4_high[4] = { f, f, f, f };

34 Final Example Recurse until 1 pixel Organize into Zero-Tree Zeros in parent mean likely zeros in child Encode with arithmetic coder Q-Coder, in this case. Decide how big you want the data in bpp. Throw out any bits after (bpp*width*height)

42 Then Decompress Stop at 0.75 bpp Source data = 24bpp 1024x768x24bpp = 2,359,296 bytes + header Compressed version, 1024x768x0.75bpp = 73,728 bytes + header e.g. 32:1 compression Still room to improve quality AND compression!

44 Then Decompress Let s flip between them. Next image: original The one after that: compressed then decompressed.

47 Conclusion It s simple. Of course, there s a lot of devil in the details, but you should now know what to look for. Recommended reading: Ripples in Mathematics: The Discrete Wavelet Transform by A.Jensen, A.la Cour-Harbo This is the best book on wavelets that I ve found.

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