Single Gaussian Background Modeling and GBH

In GBH, a cumulative frequency is generated from the frequency of each individual intensity level as well as those of its neighboring levels in the histogram. In other words, the frequency of each intensity level and the frequency of its neighboring levels are summed to form the group-based histogram. First note that the frequency or probability of a conventional histogram is updated by using a single intensity, while the probability of GMM is constructed from a group of intensities. Thus, the GMM is more suited than a simple histogram for representing intensity distribution of the background image. GBH possesses similar merits to GMM because it takes the variation of pixel intensity into account. This operation effectively handles the problem of sensor noises. Further, the GBH algorithm gives a reliable Gaussian

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background model by using several image frames.

The GBH can be generated by using an average filter of width of 2w+1 where w is number of intensity levels. In GBH, the maximum probability density

p of a pixel at

* is smaller than a certain value, the maximum peak of the GBH will be located at a position closer to the center of a Gaussian model than the original histogram. This is because the filter smooths the histogram curve. Thus, the intensity that has the maximum frequency in the GBH can be treated as the mean intensity µ_u,v of the background model:

A smaller window width can save computation time for building up the GBH, while a larger width can produce a smoother GBH. For further discussion on the determination of window width, an example given below employs 13 Gaussians that were generated by using a Gaussian random number generator. The means of all Gaussians were chosen to be 205 and the standard deviations varied from 3 to 15. The histogram and the GBH generated with different widths were used to estimate the mean of each Gaussian; the error rates are shown in

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Table 3-1. The error rates that fall within ±2% are highlighted in the table. The results show that the estimation of the proposed method is superior to that of a conventional histogram.

One can conclude from the simulation results that a larger window width of an average filter will be needed for high-accuracy performance as the standard deviation increases.

Keeping the error rate of mean estimation within ±2%, and using the simulation result, the width can be determined as follows:

 

In the following derivation of the Gaussian model, C(i) is used for recording the GBH frequency of intensity i. The mean intensity _µu,v can be obtained from the maximum-value counter. The system does not process all counters when a new intensity l is captured, because the new intensity only increases the adjacent counters of counter

l . The proposed

Table 3-1 Estimation Error Rate of Gaussian Mean using Histogram and GBH.

Standard

deviation 3 4 5 6 7 8 9 10 11 12 13 14 15

-1.5% -1.5% -2.0% -2.4% -2.4% -2.9% -3.4% -2.9% -2.9% -4.4% -2.9% -4.9% -4.4%

width

w

1 0.5% 1.0% 0.0% -0.5% -0.5% 2.0% -3.4% -2.4% -2.9% -1.5% -3.4% 0.5% 1.0%

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algorithm for obtaining the mean of Gaussian model is summarized below.

step 1. Record the current intensity l of a pixel.

After the center of the background model is found, the variance can be computed as follows:

where σ′ is the maximum standard deviation of the Gaussian background models (the value can be experimentally obtained by analyzing the background models from image sequences offline).

Fig. 3-1(a) shows an example of intensity histogram of a pixel in a traffic image sequence. It is clear from the figure that the background intensity level is distributed in the range from 195 to 215. Further analysis of the sampled data in the background-intensity region was performed by using MINITAB Statistical Software (release 13.32 for WINDOWS;

Minitab, Inc, State College, PA). The result shows that the data can be modeled as a Gaussian [67] with a mean and standard deviation of 203.65 and 3.88, respectively. However, one

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(a) (b) Fig. 3-1. Statistical analysis of pixel intensity. (a) Histogram. (b) Group-based histogram of Fig.

3-1(a).

cannot determine the center of the background model from the histogram because three intensities have the same maximum frequency. On the contrary, the background intensity can be easily found in the group-based histogram (with width w = 3), as shown in Fig. 3-1(b). The mean and the standard deviation of the estimated Gaussian model of Fig. 3-1(b) are 205 and 4, respectively. The error rate of the mean and the standard deviation is 0.67% and 3.17%. The results confirm that the derived probability density function satisfactorily fits the background intensities.

Note that the proposed GBH only uses addition and comparison to estimate the mean of background pixels; on the contrary, other methods such as GMM involve much complex computation, such as multiplication, sorting, and division operations. Because estimation of the mean of background model is less time-consuming than the estimation of standard deviation, the mean intensity, the GBH, and the histogram are updated at every sample interval. The standard deviation is updated every 30 (or more) frames. To reduce the computation burden, the estimation of standard deviation is completed step-by-step in each sample interval. In traffic monitoring applications, the GBH, the histogram, and the background model are renewed every 15 minute to cope with illumination variations. The

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computational load of the proposed method is significantly lower than the methods presented in [24] and [29]. Thus, it is more suitable for the real-time requirement of visual tracking of vehicles.