Temporal Wavelet Analysis

The third step of our fire detection algorithm is to keep track of the frequency history of pixels in the fire colored region and analyze the history. In order to detect flicker or oscillations in pixels due to fire in a reliable manner, the video capture rate should be high enough to capture high-frequency flicker in flames. To capture 10 Hz flicker, the video should capture at least 20 frames per second (fps). However, in some surveillance systems, the video capture rate is below 20 Hz. If the video is available at a lower capture rate, aliasing occurs but flicker due to flames can still be observed in the video. For example, 8 Hz sinusoid appears as 2 Hz sinusoid in a 10 fps video.

Each pixel $x_n[k,l]$ of the binary mask $Fire$ is fed to a two stage-filter bank as shown in Fig. . The signal $x_n[k,l]$ is a one-dimensional signal representing the temporal variations in color values at location $[k,l]$ in the $n^{th}$ frame. Temporal wavelet analysis can be carried out using either the luminance (Y component) in YUV color representation or the red component in RGB color representation. In our implementation the red channel values of the pixels are used. The two-channel subband decomposition filter bank is composed of half-band high-pass and low-pass filters with filter coefficients {-0.25, 0.5, -0.25} and {0.25, 0.5, 0.25}, respectively, as shown in Fig. . The filter bank produces wavelet subsignals $d_n[k,l]$ and $e_n[k,l]$. If there is high frequency activity at pixel location $[k,l]$, high-band subsignals $d$ and $e$ get non-zero values. However, in a stationary pixel, the values of these two subsignals should be equal to zero or very close to zero because of high-pass filters used in subband analysis. If the pixel is part of a flame boundary at some time (see Fig. ), then there will be several spikes in one second due to transitions from background colors to flame colors and vice versa. If there is an ordinary fire-colored moving object going through pixel $[k,l]$, then there will be a single spike in one of these wavelet subsignals because of the transition from the background pixel to the object pixel as shown in Fig. . The number of zero crossings of the subband signals $d_n$ and $e_n$ in a few seconds is used to discriminate between a flame pixel and an ordinary fire colored object pixel. If this number is above some threshold, then an alarm can be issued for this pixel.

The temporal history of the red channel of a pixel $x_n[111, 34]$ which is part of a flame, and the corresponding wavelet signals are shown in Fig. . A flicker in the red channel values of this flame pixel is obvious from the figure. The pixel is part of a flame for image frames $I_n$, n=1, 2, 3, 19, 23, 24, 41 and 50. It becomes part of the background for n=12,...,17, 20, 21, 26, 27, 31,...,39, 45, 52,..., and 60. Wavelet domain subsignals $d_n$ and $e_n$ reveal the fluctuations of the pixel at [111, 34] with several zero crossings. Due to a down-sampling operation during wavelet computation, the length of wavelet signals are halved after each stage of subband filtering. As a result, the value of a sample in a subband signal corresponds to several samples in the original signal, e.g., the value of $d_5[111,34]$ corresponds to the values of $x_{10}[111,34]$ and $x_{11}[111,34]$, and the value of $e_4[111,34]$ corresponds to the values of $x_{12}[111,34]$, $x_{13}[111,34]$, $x_{14}[111,34]$ and $x_{15}[111,34]$, in the original signal.

The temporal history of the red channel of a pixel $x_n[18, 34]$, which is part of a fire colored object, and the corresponding wavelet signals are shown in Fig. . As shown in this figure, neither the original nor the wavelet signals exhibit oscillatory behavior. The pixel is part of a white-colored background for n=1, 2, and 3, becomes part of a fire colored object for n=4, 5, 6, 7, and 8, then it becomes part of the background again for $n>8$. Corresponding wavelet signals $d_n$ and $e_n$ do not exhibit oscillatory behavior as shown in Fig. . Small variations due to noise around zero after the $10^{th}$ frame are ignored by setting up a threshold.

The number of wavelet stages needed for used in flame flicker analysis is determined by the video capture rate. In the first stage of dyadic wavelet decomposition, the low-band subsignal and the high-band wavelet subsignal $d_n[k,l]$ of the signal $x_n[k,l]$ are obtained. The subsignal $d_n[k,l]$ contains [2.5 Hz, 5 Hz] frequency band information of the original signal $x_n[k,l]$ in 10 Hz video frame rate. In the second stage the low-band subsignal is processed once again using a dyadic filter bank, and the wavelet subsignal $e_n[k,l]$ covering the frequency band [1.25 Hz, 2.5 Hz] is obtained. Thus by monitoring wavelet subsignals $e_n[k,l]$ and $d_n[k,l]$, one can detect fluctuations between 1.25 to 5 Hz in the pixel $[k,l]$.