## 3 Vibration Source Tracking Based on Various Types of Image Features
### Particle Filter Tracking Algorithm  
In particle filtering, the particle states are described by affine transformation parameters, each of which is a six-dimensional vector. For each particle, the variables are distributed randomly and obey a probability distribution in the state space. The next most probable state is estimated by probability calculation according to the previous state:  
\[M=[\begin{array}{cccccc}x&y&s_{c}&r_{o}&r_{a}&s_{a}\end{array}] \tag{8}\]  
where \(x\) and \(y\) are the abscissa and ordinate of the center of the particle sample; \(s_{c}\) is the length-width ratio of the sample; \(r_{o}\) is the rotation angle of the particle sample; \(r_{a}\) is the height-width ratio of the particle sample; and, \(s_{a}\) is the gradient of the tracking window.  
Particle filtering is an important resampling process that places a number of particles, by certain rules, in the current frame. According to the placement rules, the particles are either placed evenly or denser near the target. The similarity between the particles and the target template is measured by particle weights. For simplicity, the weights should be normalized so that the sum of weights of all particles equals 1.  
In this paper, the particle filter tracking algorithm is implemented as follows. First, \(n\) particle samples were obtained by random sampling in the initial frame. The weights of the particles were set to \(1/n\). Let \(\left\{s_{t-1}^{i}\right\}_{i\in[1,n]}\) be the state of the \(n\) particles at time \(t-1\) and \(\left\{w_{t-1}^{i}\right\}_{i\in[1,n]}\) be the weights of these particles. Then, \(n\) particle samples were selected from the particle set based on the weights. The normalized weight probability set \(\left\{C_{t-1}^{i}\right\}_{i\in[1,n]}\) can be expressed as:  
\[C_{t-1}^{i}=\sum_{k=1}^{i}w_{t-1}^{k}/\sum_{k=1}^{n}w_{t-1}^{k} \tag{9}\]  
The \(n\) sets of variables evenly distributed between 0 and 1 were randomly generated and denoted as \(\left\{r^{i}\right\}_{i\in[1,n]}\). Then, \(\left\{C_{t-1}^{i}\right\}_{i\in[1,n]}\), the set of \(n\) minimum indices \(\left\{Idx_{t-1}^{i}\right\}_{i\in[1,n]}\), was established \(\left(\left\{C_{t-1}^{ldxi}\right\}_{i\in[1,n]}\geq 1\right)\). Then, \(s_{t-1}^{i}\) was instated to the \(s_{t-1}^{ldxi}\), marking the end of the resampling process.  
The updated particle set was transmitted via the system state-change equation. When a new frame arrives, the state of the particle state can be obtained as:  
\[s_{t}=A\cdot s_{t-1}+v_{t-1} \tag{10}\]  
where \(A\) is the state transition matrix and \(v_{t-1}\) are the multivariate Gaussian variables randomly generated by affine transformation parameters. The confidence of each particle can be obtained as:  
\[E(y)=\rho(y)w(y) \tag{11}\]  
where \(\rho(y)\) is the similarity between the histogram of gradient directions of the candidate sample and that of the target template and \(w(y)\) is the particle weight obtained by GoogLeNet. The maximum confidence particle was considered as the final estimation of the output frame. In this way, the texture features were combined with the shape features, and the sample with maximum similarity was identified to enhance the tracking accuracy.