Multiresolution analysis is useful for identifying peaks and valleys of noisy signals. This method makes wavelet-based peak detection more accurate and robust than threshold or curve-fitting-based peak detection methods.
Signals usually contain both low-frequency components and high-frequency components. Low-frequency components vary slowly with time and require fine frequency resolution but coarse time resolution. High-frequency components vary quickly with time and require fine time resolution but coarse frequency resolution. Therefore, a multiresolution analysis method is useful for analyzing a signal that contains both low- and high-frequency components.
The multiresolution analysis method can help you recognize both the long-term trend and short-term variations of a signal. Information on the coarser resolution of a signal can help you locate the features, such as peaks, in which you are interested. Observation of the finer resolution levels can refine the gross features and provide more details. The following figure shows the multiresolution refinement process of the wavelet-based peak detection method.
Figure 3: The multiresolution process of wavelet-based peak detection
In this example, you perform five levels of undecimated wavelet transforms (UWT) on the input signal. The signal can be represented using the following equation:
Signal = A1 + D1
= A2 + D2 + D1
= A5 + D5 + D4 + D3 + D2 + D1
where A denotes the approximation coefficients and D denotes the detail coefficients.
After applying the UWT to the signal, the wavelet-based peak detection method first checks D5 for zero crossings, which correspond to the peaks in A4. Then, this method checks D4 for zero crossings. More zero crossings might exist in D4 than in D5 due to higher-frequency components or noise, so this method selects the zero-crossing points that are nearest to those in D5, which correspond to the peaks in A3. This method repeats the process until it reaches the finest scale in D1. The following figure shows the refinement process of the first detected peak.
Figure 4: The refinement process of the first detected peak
In D5 of the previous figure, the first zero crossing is at index 83. In D4, the zero crossing nearest to 83 is 84. The refinement process repeats until the peak is refined to index 86 in D1. You can see from the Signal array that index 86 is indeed a local maximum.