# Spatiotemporal Background Modeling

## Why do we reasearch on this subject

• Many machine vision applications suffer from the effect of unstable light conditions. Selecting the invariance features of the targets and compensating the lighting changes for better segmentation, tracking, recognition or searching performance are difficult tasks.

• The skin detector of 4th floor Cameraman is affected by day-light shifting. The last attempt to increase the accuracy of skin detector is successful although limited by it's structure. More researches on time-related lighting changes and invariance color features selection would be very interesting and necessary.

## Frequency Domain Analysis

• For each pixel, the H, S, V variance in time can be modeled as multiple distributions. Once the modeling process has been completed, the probability for each color occurs in a pixel is known. Taking the advantage of this information, it is possible to label the pixel as background or foreground.

## Space Domain Analysis

• Hypothesis 1: Assuming the camera is not moving.

## References

• Nir Friedman[1] proposed a method to model the pixels as road, shadow and vehicle, then created background gaussian mix model(then the K of the GMM is 3)for moving object extraction.
• Yuanhong Li; Ming Dong; Jing Hua; [2]Simultaneous Localized Feature Selection and Model Detection for Gaussian Mixtures

## Gaussian Mixture Models

• definition

$X_{i} = (H_{i}, S_{i}, V_{i}), i=1, 2,..., L\,$

$P(X_{i}) = \sum_{i=1}^{L}K_{i,t}^{j} \cdot \Phi(X_{i}, \mu_{i,t}^{j}, \Sigma_{i,t}^{j}) \,$

In which, $K_{i,t}^{j}\,$ means the weight of pixel i at time t belongs to j gaussian distribution. The $\mu_{i,t}^{j}\,$ means the mean value of jth gaussian distribution. The $\Sigma_{i,t}^{j}=(\sigma_{i,t}^{j})^2I$means the covariance of jth gaussian distribution.

$\Phi(X_{i}, \mu_{i,t}^{j}, \Sigma_{i,t}^{j})= \frac{1}{(2\pi)^\frac{d}{2}|\Sigma_{i,t}^{j}|^{\frac{1}{2}}}\times exp[-\frac{1}{2}(X_{j}-\mu_{i,t}^{j})^T](\Sigma_{i,t}^{j})^{-1}(X_{j}-\mu_{i,t}^j)]$

In which, the d stands for dimension of Xi$\Sigma_{i,t}^{j}$ is a d*d matrix.

• updata

$if(distance((X_{i}, \mu_{i,t}^j)>2.5(\sigma_{i,t}^{j}))$

then

$K_{i,t+1}^{j}={(1-\alpha)}K_{i,t}^{j}+\alpha(M_{i,t}^{j})$

$\mu_{i,t}^{j}=(1-p)\mu_{i,t}^{j}+p \cdot X_{i}$

$(\sigma_{i,t+1}^{j})^{2}=(1-p)(\sigma_{i,t}^{j})^2+p(X_{i}-\mu_{i,t}^{j})^{T}(X_{i}-\mu_{i,t}^{j})$

$p=\frac{\alpha}{K_{i,t}^{j}}$

Note the $\alpha\,$ is the learning rate.

## Problems

• In GMM framework, do we have to set the value of K? In other word, do we have to know how many distributions are there at first?