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Wavelet Transform

Wavelet Transform . What Are Wavelets?. In general, a family of representations using: hierarchical (nested) basis functions finite (“compact”) support basis functions often orthogonal fast transforms, often linear-time. MULTIRESOLUTION ANALYSIS (MRA). Wavelet Transform

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Wavelet Transform

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  1. Wavelet Transform

  2. What Are Wavelets? In general, a family of representations using: • hierarchical (nested) basis functions • finite (“compact”) support • basis functions often orthogonal • fast transforms, often linear-time

  3. MULTIRESOLUTION ANALYSIS (MRA) • Wavelet Transform • An alternative approach to the short time Fourier transform to overcome the resolution problem • Similar to STFT: signal is multiplied with a function • Multiresolution Analysis • Analyze the signal at different frequencies with different resolutions • Good time resolution and poor frequency resolution at high frequencies • Good frequency resolution and poor time resolution at low frequencies • More suitable for short duration of higher frequency; and longer duration of lower frequency components

  4. PRINCIPLES OF WAVELET TRANSFORM • Split Up the Signal into a Bunch of Signals • Representing the Same Signal, but all Corresponding to Different Frequency Bands • Only Providing What Frequency Bands Exists at What Time Intervals

  5. Wavelet Transform (WT) • Wavelet transform decomposes a signal into a set of basis functions. • These basis functions are called wavelets • Wavelets are obtained from a single prototype wavelet y(t) called mother wavelet by dilations and shifting: • (1) where a is the scaling parameter and b is the shifting parameter

  6. The continuous wavelet transform (CWT) of a function f is defined as • If y is such that f can be reconstructed by an inverse wavelet transform:

  7. SCALE • Scale • a>1: dilate the signal • a<1: compress the signal • Low Frequency -> High Scale -> Non-detailed Global View of Signal -> Span Entire Signal • High Frequency -> Low Scale -> Detailed View Last in Short Time • Only Limited Interval of Scales is Necessary

  8. Wavelet transform vs. Fourier Transform • The standard Fourier Transform (FT) decomposes the signal into individual frequency components. • The Fourier basis functions are infinite in extent. • FT can never tell when or where a frequency occurs. • Any abrupt changes in time in the input signal f(t) are spread out over the whole frequency axis in the transform output F() and vice versa. • WT uses short window at high frequencies and long window at low frequencies (recall a and b in (1)). It can localize abrupt changes in both time and frequency domains.

  9. Better time resolution; Poor frequency resolution Frequency Better frequency resolution; Poor time resolution Time • Each box represents a equal portion • Resolution in STFT is selected once for entire analysis RESOLUTION OF TIME & FREQUENCY

  10. Discrete Wavelet Transform • Discrete wavelets • In reality, we often choose • In the discrete case, the wavelets can be generated from dilation equations, for example, f(t) = [h(0)f(2t) + h(1)f(2t-1) + h(2)f(2t-2) + h(3)f(2t-3)]. (2) • Solving equation (2), one may get the so called scaling function f(t). • Use different sets of parameters h(i)one may get different scaling functions.

  11. Discrete WT Continued • The corresponding wavelet can be generated by the following equation y (t)= [h(3)f(2t) - h(2)f(2t-1) + h(1)f(2t-2) - h(0)f(2t-3)]. (3) • When and equation (3) generates the D4 (Daubechies) wavelets.

  12. Discrete WT continued • In general, consider h(n) as a low pass filter and g(n) as a high-pass filter where • g is called the mirror filter of h. g and h are called quadrature mirror filters (QMF). • Redefine • Scaling function

  13. Discrete Formula • Wavelet function • Decomposition and reconstruction of a signal by the QMF. where and is down-sampling and is up-sampling

  14. Generalized Definition • Let be matrices, where are positive integers is the low-pass filter and is the high-pass filter. • If there are matrices and which satisfy: where is an identity matrix. Gi and Hi are called a discrete wavelet pair. • If and The wavelet pair is said to be orthonormal.

  15. For signal let and • One may have • The above is called the generalized Discrete Wavelet Transform (DWT) up to the scale is called the smooth part of the DWT and is called the DWT at scale • In terms of equation

  16. Multilevel Decomposition • A block diagram 2 2

  17. Haar Wavelets Example: Haar Wavelet

  18. x 1 Summary on Haar Transform • Two major sub-operations • Scaling captures info. at different frequencies • Translation captures info. at different locations • Can be represented by filtering and downsampling • Relatively poor energy compaction

  19. 2D Wavelet Transform • We perform the 2-D wavelet transform by applying 1-D wavelet transform first on rows and then on columns. Rows Columns LL f(m, n) LH HL HH H 2 2 H G 2 H 2 G 2 G 2

  20. Applications • Signal processing • Target identification. • Seismic and geophysical signal processing. • Medical and biomedical signal and image processing. • Image compression (very good result for high compression ratio). • Video compression (very good result for high compression ratio). • Audio compression (a challenge for high-quality audio). • Signal de-noising.

  21. 3-D Wavelet Transform for Video Compression Original Video Sequence Reconstructed Video Sequence

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