Global vision object detection using an improved Gaussian Mixture model based on contour

Single Gaussian model

The single Gaussian model is based on this assumption: The grayscale values of each pixel in a certain image follow Gaussian distribution on the timeline (Dong et al., 2018). The Gaussian distributions of each point do not affect each other. Assuming X_t represents the pixel value of the image at time t, u_tandΣ_t are the mean and variance at time t. So the following Eq. (1) can be used to represent a single Gaussian model: (1)${\displaystyle \eta \left(X_t,u_t,{\Sigma }_t\right)=\frac{1}{{\left(2\pi \right)}^{\frac{n}{2}}|{\Sigma }_t{|}^{\frac{1}{2}}}{e}^{-\frac{1}{2}{\left(X_t-u_t\right)}^T{\Sigma }_t^{-1}\left(X_t-u_t\right)}}$

Then update the mean and variance with Eqs. (2) and (3), where ρ is the learning rate:

(2)${\displaystyle u_t=\left(1-\rho \right)\times u_{t-1}+\rho \times X_t}$ (3)${\displaystyle {\sigma }_t^2=\left(1-\rho \right)\times {\sigma }_{t-1}^2+\rho \times {\left(X_t-u_{t-1}\right)}^T\times \left(X_t-u_{t-1}\right)}$

Gaussian mixture model

In the single Gaussian model, the probability density function obeys normal distribution. This Gaussian model can achieve ideal results in relatively simple environments. If the background image is complex and there is a lot of interference, there are many deviations in the detection results. It is necessary to use multiple Gaussian distributions for object detection. The Gaussian mixture model (GMM) is an extension of the single Gaussian model (Xue & Jiang, 2018). It is the precise quantification of variable distribution using multiple Gaussian probability density functions which decomposes the variable distribution into several statistical models based on the Gaussian probability density function. The GMM resolves the need to update the background model parameters by continuous iteration because it can smoothly approximate the distribution of any shape. The mixed Gaussian model consists of the following steps (Yan et al., 2020):

Establish a background model:

Using K Gaussian distributions as the background, the probability function is shown in Eq. (4): (4)${\displaystyle P\left(X_{i,t+1}\right)=\sum _{i=1}^k{\omega }_{i,t+1}\eta \left(X_{i,t+1};{\mu }_{i,t+1},{\Sigma }_{i,t+1}\right)}$

In the above equation, K is the number of Gaussian models, and its value typically ranges from 3 to 5. X_i,t+1 is the value of the pixel at time t+1. The weight, mean, and covariance of this pixel are represented by, ω_i,t+1, μ_i,t+1,  and Σ_i,t+1, respectively. The Gaussian distribution probability density function η is represented as Eq. (5): (5)${\displaystyle \eta \left(X_{i,t+1};{\mu }_{i,t+1},{\Sigma }_{i,t+1}\right)=\frac{1}{{\left(2\pi \right)}^{n/2}|{\Sigma }_{i,t+1}{|}^{1/2}}{e}^{-\frac{1}{2}{\left(X_{i,t+1}-{\mu }_{i,t+1}\right)}^T{\Sigma }_{i,t+1}^{-1}\left(X_{i,t+1}-{\mu }_{i,t+1}\right)}}$

Model matching: In the generated K single Gaussian models, if the absolute value of the difference between the current pixel value and the mean of a Gaussian distribution is less than the standard deviation σ D times, it can be determined that the two match. D is the deviation threshold, which is one of the basic parameters of the model. It is represented as Eq. (6). (6)${\displaystyle \left|x_t-u_{i,t}\right|<\mathrm{D}{\sigma }_{i,t}}$

Model updating: If the difference between the pixel values at time t and the mean of a certain Gaussian model satisfies Eq. (6), the two values match and the model parameters ω_i,t, μ_i,t, and Σ_i,t are updated using Eqs. (7), (8), (9) and (10). If the two do not match, a new Gaussian model with a smaller weight, larger variance, and a mean of that pixel value is used to update the model with the lowest priority in the old mixed Gaussian model. The following are the updated formulas:

(7)${\displaystyle {\omega }_{i,t+1}=\left(1-\alpha \right){\omega }_{i,t}+\alpha M_{i,t}}$ (8)${\displaystyle \rho =\alpha /{\omega }_{i,t}}$ (9)${\displaystyle {\mu }_{i,t+1}=\left(1-\rho \right){\mu }_{i,t}+\rho X_{i,t}}$ (10)${\displaystyle {\sigma }_{i,t+1}^2=\left(1-\rho \right){\sigma }_{i,t}^2+\rho {\left(X_{i,t+1}-{\mu }_{i,t+1}\right)}^T\left(X_{i,t+1}-{\mu }_{i,t+1}\right)}$

In the above equations, α and ρ are learning rates for weights and means, respectively. If the Gaussian distribution matches, then M_i,t = 1, otherwise M_i,t = 0.

Background model: This step mainly extracts the pixels that best describe the background effect. Sort $\frac{{\omega }_{i,t}}{{\sigma }_{i,t}}$ in descending order and select the first B Gaussian models as the background model. (11)${\displaystyle \mathrm{B}=arg\underset{b}{min}\sum _{i=1}^b{\omega }_{i,t}>T}$

In Eq. (11), the threshold value T represents the proportion of each Gaussian component to the background model, usually taken as 0.75.