Problem
Suppose that we have a 1D signal where we need to detect spikes, i.e., significant changes in signal values. Of course, what makes a given change significant is domain- or problem-dependent. 1D HWT can be used to detect spikes. In this post, I will write down a few thoughts on how one can define spikes conceptually. In the next post, I will detail a JAVA implementation of detecting spikes with 1D HWT.
Sample Images & Signals
Let us take a look at a concrete image where spikes enable us to detect useful features. Fig. 1 gives an image of a road with two lanes. The original image has been grayscaled and blurred with Gaussian blur using a 7x7 window for denoising. We can take a 1D horizontal sub-signal from this image (or several such sub-signals) and use 1D HWT of that subsignal to detect spikes.
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| Figure 1. Sample road image |
Fig. 2 gives a graphical representation of a subsignal. Specifically, the horizontal white line is a row from which a signal of 128 samples is taken at row 87 on the left of the image, i.e., from columns 0 to 127. The two vertical lines represent the beginning and the end of a spike detected over the left lane. Fig. 3 depicts another signal of 128 samples taken at the same row on the right side of the image and a spike detected over the right lane.
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| Figure 2. Signal of 128 samples from row 87 |
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| Figure 3. Signal of 128 samples from row 87 |
1D HWT Analysis of Signals for Spike Detection
Fig. 4 is an graph of grayscale values in the signal depicted as the horizontal line in Fig. 2.
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| Figure 4. Graph of grayscale values of 128 sample signals in Fig. 2 |
If we apply 1 scale of 1D Ordered HWT to this signal, we get the graph shown in Fig. 5. The first 64 values represent a downsampled signal whereas the remaining 64 values are the corresponding Haar wavelet coefficients.
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| Figure 5. 1 scale of 1D HWT of the signal in Fig. 4 |
Figures 6 and 7 gives graphs of the two parts of Fig. 5, i.e., the downsampled signal and the corresponding wavelets.
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| Figure 6. Downsampled signal after 1 scale of 1D HWT of the signal in Fig. 4 |
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| Figure 8. Haar wavelets after 1 scale of 1D HWT of the signal in Fig. 4 |
Definition of Spikes
1D HWT wavelets can be used to detect spikes. Spikes can be broken into two broad categories: up-down spikes and down-up spikes. An up-down spike consists of three structural segments: an upward segment (signal samples increasing), a flat segment (signal samples remain more or less on the same level, i..e, neither increase or decrease), and a downward segment (signal samples decreasing). Similarly, a down-up spike also consists of three structural segments: a downward segment (signal samples decreasing), a flat segment (signal samples neither increasing nor decreasing), and an upward segment (signal samples increasing). In up-down and down-up spikes the flat segments are optional. A specific segment can be detected by analyzing values of 1D HWT wavelets similar to those given in Fig. 8. After a spike is detected, the exact coordinates of the segments can be used to locate a spike to the original signal, e.g., a 2D grayscale image. Detected spikes can subsequently be used to identify various objects in the original signal, i.e., road lanes.
