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Using Wavelets for Recognition of Cognitive Pattern Primitives

Using Wavelets for Recognition of Cognitive Pattern Primitives. Dasu Aravind Feature Group PRISM/ASU. 3DK – September 21, 2000. WAVELETS. Robust and Objective Identification of all CPs. Wavelets. 3DK – September 21, 2000. WHAT IS A CORNER POINT ?. 3DK – September 21, 2000.

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Using Wavelets for Recognition of Cognitive Pattern Primitives

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  1. Using Wavelets for Recognition of Cognitive Pattern Primitives Dasu Aravind Feature Group PRISM/ASU 3DK – September 21, 2000

  2. WAVELETS Robust and Objective Identification of all CPs Wavelets 3DK – September 21, 2000

  3. WHAT IS A CORNER POINT ? 3DK – September 21, 2000

  4. It is an abrupt change in the orientation of a vessel wall or a distinct angle in the joining of vessel parts such as neck or body Anthropologist : Subjective Engineer : Objective 3DK – September 21, 2000

  5. ABRUPT CHANGE What are the tools to capture high frequency information ?? HIGH FREQUENCY INFORMATION 3DK – September 21, 2000

  6. Fourier Transform • Short Time Fourier Transform • Continuous Wavelet Transform • Discrete Wavelet Transform 3DK – September 21, 2000

  7. Fourier Transform FT decomposes a signal into complex exponential functions of different frequencies. The way it does this, is defined by the following two equations 3DK – September 21, 2000

  8. Fourier Transform A signal S1 and its FT, FT(S1) 3DK – September 21, 2000

  9. Fourier Transform Another signal S2 and its FT, FT(S2) 3DK – September 21, 2000

  10. Fourier Transform • S1 and S2 have the same frequency components, but these components • occur at different times. Therefore their FTs look alike • Fourier transform tells whether a certain frequency component • exists or not. This information is independent of where in time this • component appears 3DK – September 21, 2000

  11. Solution is STFT (Short Term Fourier Transform) There is only a minor difference between STFT and FT. In STFT, the signal is divided into small enough segments, where these segments (portions) of the signal can be assumed to be stationary(all frequency components exist at all times). For this purpose, a window function "w" is chosen. 3DK – September 21, 2000

  12. STFT Consider the signal S3 and its STFT if the window is very wide 3DK – September 21, 2000

  13. STFT Consider the signal S3 and its STFT if the window is very narrow 3DK – September 21, 2000

  14. STFT The problem, is choosing a window function, once and for all, and use that window in the entire analysis. The answer, of course, is application dependent 3DK – September 21, 2000

  15. Solution is CWT Wavelet analysis is a measure of similarity between the basis functions (wavelets) and the signal itself. Here the similarity is in the sense of similar frequency content. The calculated CWT coefficients refer to the closeness of the signal to the wavelet at the current scale 3DK – September 21, 2000

  16. CWT This equation shows how a function f(t) is decomposed into a set of basis functions , called the wavelets. The variables s and (tau), scale and translation, are the new dimensions after the wavelet transform. Scale is the inverse of frequency. 3DK – September 21, 2000

  17. CWT The wavelets are generated from a single basic wavelet psi(t), the mother wavelet, by scaling and translation 3DK – September 21, 2000

  18. CWT Once the mother wavelet is chosen the computation starts with s=1 and the continuous wavelet transform is computed for all values of scale from s=1 and will continue for the increasing values of s , i.e., the analysis will start from high frequencies and proceed towards low frequencies. This first value of s will correspond to the most compressed wavelet. As the value of s is increased, the wavelet will dilate. 3DK – September 21, 2000

  19. CWT The obtained CWT looks like this 3DK – September 21, 2000

  20. CWT Lower scales (higher frequencies) have better scale resolution (narrower in scale, which means that it is less ambiguous what the exact value of the scale) which correspond to poorer frequency resolution . Similarly, higher scales have scale frequency resolution (wider support in scale, which means it is more ambiguous what the exact value of the scale is) , which correspond to better frequency resolution of lower frequencies 3DK – September 21, 2000

  21. CWT The most important properties of wavelets are the admissibility and the regularity conditions and these are the properties which gave wavelets their name. The admissibility condition implies that the Fourier transform of psi(t) vanishes at the zero frequency, i.e. This means that wavelets must have a band-pass like spectrum 3DK – September 21, 2000

  22. CWT A zero at the zero frequency also means that the average value of the wavelet in the time domain must be zero, and therefore it must be oscillatory. In other words, psi(t) must be a wave 3DK – September 21, 2000

  23. CWT The wavelet transform as described so far still has properties that make it difficult to use directly in the form shown so far. The is redundancy in the CWT. The wavelet transform is calculated by continuously shifting a continuously scalable function over a signal and calculating the correlation between the two. These scaled functions are not orthogonal basis and the obtained wavelet coefficients will therefore be highly redundant. 3DK – September 21, 2000

  24. DWT To overcome this problem discrete wavelets have been introduced. Discrete wavelets are not continuously scalable and translatable but can only be scaled and translated in discrete steps. This is achieved by modifying the wavelet representation to j and k are integers and s0 > 1 is a fixed dilation step. The translation factor tau0 depends on the dilation step. We usually choose s0 = 2 so that the sampling of the frequency axis corresponds to dyadic sampling. For the translation factor we usually choose tau 0 = 1 so that we also have dyadic sampling of the time axis. Yielding s=2j and t =k*2j. 3DK – September 21, 2000

  25. DWT Even with discrete wavelets we still need an infinite number of scalings and translations to calculate the wavelet transform. The easiest way to tackle this problem is simply not to use an infinite number of discrete wavelets We know that the wavelet has a band-pass like spectrum. From Fourier theory we know that compression in time is equivalent to stretching the spectrum and shifting it upwards: 3DK – September 21, 2000

  26. DWT This means that a time compression of the wavelet by a factor of 2 will stretch the frequency spectrum of the wavelet by a factor of 2 and also shift all frequency components up by a factor of 2. Using this insight we can cover the finite spectrum of our signal with the spectra of dilated wavelets 3DK – September 21, 2000

  27. DWT Every time you stretch the wavelet in the time domain with a factor of 2, its bandwidth is halved. In other words, with every wavelet stretch you cover only half of the remaining spectrum, which means that you will need an infinite number of wavelets to get the job done. The solution to this problem is simply not to try to cover the spectrum all the way down to zero with wavelet spectra, but to use a cork to plug the hole when it is small enough. This cork then is a low-pass spectrum and it belongs to the so-called scaling function. The scaling function was introduced by Mallat. Because of the low-pass nature of the scaling function spectrum it is sometimes referred to as the averaging filter. 3DK – September 21, 2000

  28. DWT If we regard the wavelet transform as a filter bank, then we can consider wavelet transforming a signal as passing the signal through this filter bank. The outputs of the different filter stages are the wavelet- and scaling function transform coefficients and the technique is called subband coding. 3DK – September 21, 2000

  29. DWT Filtering a signal corresponds to the mathematical operation of convolution of the signal with the impulse response of the filter. The convolution operation in discrete time is defined as follows: 3DK – September 21, 2000

  30. DWT After passing the signal through a half band lowpass filter, half of the samples can be eliminated according to the Nyquist’s rule. Hence simply discarding every other sample will subsample the signal by two, and the signal will then have half the number of points. The scale of the signal is now doubled. The subsampling process changes the scale. Resolution, on the other hand, is related to the amount of information in the signal, and therefore, it is affected by the filtering operations. Half band lowpass filtering removes half of the frequencies, which can be interpreted as losing half of the information. Therefore, the resolution is halved after the filtering operation. 3DK – September 21, 2000

  31. DWT The original signal x[n] is first passed through a halfband highpass filter g[n] and a lowpass filter h[n]. 3DK – September 21, 2000

  32. ABRUPT CHANGE Wavelets is the tool to capture high frequency information HIGH FREQUENCY INFORMATION 3DK – September 21, 2000

  33. 2D-DWT Horizontal edges LL LH HL HH Vertical edges Diagonal edges Convolution Kernels Sub-band Mallat’s structure 3DK – September 21, 2000

  34. ANGLE MEASUREMENT ALONG THE PERIPHERY Angle #1 Angle #k This is a 1D signal Angles #1, #2, ………#n 3DK – September 21, 2000

  35. Low pass coefficients Wavelet Transformer 1 D signal + High pass coefficients Edge information • Basic Haar filter • Odd Symmetric Daubechies filter in FP • Odd Symmetric Daubechies filter FxP • Even Symmetric Daubechies filter FP • Even Symmetric Daubechies filter in FxP Filter options 3DK – September 21, 2000

  36. Flat regions A,B CP CP Flat regions A,B Flat region C Flat region C High pass coefficients The x axis corresponds to half the total number of points. That is, every couple of points in the original plot is approximated by one in the wavelet transformed plot. The direction from left to right along the X axis corresponds to scanning the pot from left to right as in the curvature plot. The dips in the plot correspond to what could be corner points. 3DK – September 21, 2000

  37. Method tested for 57 pots Anthropologist : Subjective Engineer : Objective Exact match for all test cases 3DK – September 21, 2000

  38. Future Work: • Cognitive Pattern Primitives (CPP’s) 3DK – September 21, 2000

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