# A New Bayesian Edge-Linking Algorithm Using Single-Target Tracking Techniques

## Abstract

**:**

## 1. Introduction

## 2. Motivation (Idea)

## 3. Mathematical Background

#### 3.1. Gaussian Markov Random Field

#### 3.2. Circular State Space Model

#### 3.3. Data Association for Single-Object Tracking

## 4. Proposed Approach

#### 4.1. Improved Posterior with Marginalization of the EPs

#### 4.2. Approach Based on the Metropolis–Hastings Algorithm

#### 4.3. Gibbs Sampler-Based Approach

Algorithm 1 Gibbs sampler-based approach. | |

1: | Let ${\mathbf{c}}_{1:T}=[{c}_{1},{c}_{2},\cdots ,{c}_{t},\cdots ,{c}_{T}]$ be a vector with non-negative integers s.t. ${c}_{t}\in \{0,1,\cdots ,{N}_{t}\}$ for all $t=1,2,\cdots ,T$. |

2: | for $iter=1$ to ${N}_{iter}$ do |

3: | for $t=1$ to T do |

4: | for $i=0$ to ${N}_{t}$ do |

5: | ${\mathbf{c}}_{1:T}^{(t,i)}\leftarrow \left(\right)open="["\; close="]">{\mathbf{c}}_{1:t-1}^{*},{c}_{t}^{*}=i,{\mathbf{c}}_{t+1:T}$ and make ${\mathbf{w}}^{(t,i)}$ from the ${\mathbf{c}}_{1:T}^{(t,i)}$. |

6: | end for |

7: | Calculate the posterior $p({c}_{t}^{*}=i|{\mathbf{c}}_{1:t-1}^{*},{\mathbf{c}}_{t+1:T},\mathbf{y})=\frac{p({\mathbf{w}}^{(t,i)}|\mathbf{y},\tau ,\kappa )}{{\sum}_{j=0}^{{N}_{t}}p({\mathbf{w}}^{(t,j)}|\mathbf{y},\tau ,\kappa )}.$ |

8: | Draw a sample from the conditional posterior by ${c}_{t}^{*}\sim p({c}_{t}^{*}|{\mathbf{c}}_{1:t-1}^{*},{\mathbf{c}}_{t+1:N},{\mathbf{y}}_{1:T})$ |

9: | end for |

10: | Reconstruct $\mathbf{w}$ using ${\mathbf{c}}_{1:T}^{*}$ and find the best configuration $\widehat{\mathbf{w}}$ with the MAP estimate. |

11: | end for |

#### 4.4. Variational Bayes Using Mean-Field Approximation with Pseudo-Integration

Algorithm 2 Variational Bayes approach. |

1: Let ${\mathbf{c}}_{1:T}=[{c}_{1},{c}_{2},\cdots ,{c}_{t},\cdots ,{c}_{T}]$ be a vector with non-negative integers s.t. ${c}_{t}\in \{0,1,2,\cdots ,{N}_{t}\}$ for all $t=1,2,\cdots ,T$. |

2: Set ${\mathbf{p}}_{t}=[1/{N}_{t},1/{N}_{t},\cdots ,1/{N}_{t}]$ for all $t\in \{1,2,\cdots ,T\}$. |

3: Reconstruct $\mathbf{w}$ using ${\mathbf{c}}_{1:T}$, and set $\widehat{\mathbf{w}}=\mathbf{w}$. |

4: while $||{\mathbf{c}}_{1:T}-{\mathbf{c}}_{1:T}^{*}||>\u03f5$ (convergence) do |

5: ${\mathbf{c}}_{1:T}\leftarrow {\mathbf{c}}_{1:T}^{*}$. |

6: for $t=1$ to T do |

7: for $i=0$ to ${N}_{t}$ do |

8: ${\mathbf{c}}_{1:T}^{(t,i)}\leftarrow \left(\right)open="["\; close="]">{\mathbf{c}}_{1:i-1}^{*},{c}_{i}^{*}=i,{\mathbf{c}}_{i+1:N}$, and make ${\mathbf{w}}^{\left(i\right)}$ from ${\mathbf{c}}_{1:T}^{(t,i)}$. |

9: end for |

10: Calculate ${\pi}_{t,a}=\frac{p(\mathbf{y}|{c}_{t}=a,{\overline{\mathbf{c}}}_{-t})}{{\sum}_{b\in \{0,1,\cdots ,{N}_{t}\}}p(\mathbf{y}|{c}_{t}=b,{\overline{\mathbf{c}}}_{-t})}$ for all $a\in \{0,1,\cdots ,{N}_{t}\}$. |

11: Update ${p}_{t,a}^{*}={p}_{t,a}+{\pi}_{t,a}$ for all $a\in \{0,1,\cdots ,{N}_{t}\}$. |

12: Normalize the updated distribution, ${\widehat{p}}_{t,a}^{*}=\frac{{p}_{t,a}^{*}}{{\sum}_{b}{p}_{t,b}^{*}}$. |

13: Obtain ${c}_{t}^{*}={arg}_{a}max\phantom{\rule{3.33333pt}{0ex}}{\left\{{\widehat{p}}_{t,a}^{*}\right\}}_{a=0}^{{N}_{t}}$, and update ${\mathbf{p}}_{t}\leftarrow {\widehat{\mathbf{p}}}_{t}^{*}$. |

14: end for |

15: Reconstruct $\mathbf{w}$ using ${\mathbf{c}}_{1:T}^{*}$, and find the best configuration $\widehat{\mathbf{w}}$ using the MAP estimate. |

16: end while |

#### 4.5. Variance Estimation Using Markov Chain Monte Carlo

## 5. Results

#### 5.1. Noise-Free Image

#### 5.2. Images with Occlusion

#### 5.3. Image Segmentation with Varying Noise Levels

#### 5.4. Application to More Complicated Raw Images

## 6. Discussion

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Marginalized Posterior

#### Appendix A.2. MATLAB Source Code

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**Figure 1.**Motivation and steps of the proposed idea: $\left(\mathbf{a}\right)\to \left(\mathbf{b}\right)\to \left(\mathbf{c}\right)\to \left(\mathbf{d}\right)$. (

**a**) Raw image; (

**b**) detected points; (

**c**) T intersections in a radial way; (

**d**) transformed tracking space.

**Figure 2.**Used graphical models: (

**a**) the second order Gaussian Markov random field; and (

**b**) circular state space model.

**Figure 3.**Comparison of segmentation by three different proposed algorithms for a noise-free image. (

**a**) Detected Points; (

**b**) Detected trajectory via variational Bayes; (

**c**) Segmentation.

**Figure 4.**Comparison of the segmentation by the three different proposed algorithms for an image with occlusion. (

**a**) Detected points; (

**b**) Detected trajectory via variational Bayes; (

**c**) Segmentation.

**Figure 5.**Raw image, hidden true segmentation for ground truth determination, points detected by the “Sobel” detector and the linked boundaries. (

**a**) Raw image; (

**b**) True segmentation; (

**c**) Detected points; (

**d**) Linked edges.

**Figure 6.**Comparison of edge linking with varying noisy rate, ${\sigma}_{\u03f5}$. (

**a**) ${\sigma}_{\u03f5}=0$; (

**b**) ${\sigma}_{\u03f5}=0.001$; (

**c**) ${\sigma}_{\u03f5}=0.005$; (

**d**) ${\sigma}_{\u03f5}=0.01$.

**Table 1.**Comparison of the overlapping rate, $S(\mathcal{K},{\mathcal{K}}_{r})$ and execution time. VB, variational Bayes.

Overlapping Rate (Average ± Std) | ||||

${\mathit{\sigma}}_{\mathit{\u03f5}}$ | MATLAB | MCMC | Gibbs | VB |

0 | $0.5\times {10}^{-2}\pm \phantom{\rule{3.33333pt}{0ex}}0.004$ | 0.77 $\pm $ 0.10 | $0.71\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.13$ | $0.71\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.14$ |

$0.001$ | $0.2\times {10}^{-2}\pm \phantom{\rule{3.33333pt}{0ex}}0.003$ | 0.76 $\pm $ 0.08 | $0.71\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.09$ | $0.74\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.08$ |

$0.005$ | $0.8\times {10}^{-3}\pm \phantom{\rule{3.33333pt}{0ex}}0.003$ | 0.72 $\pm $ 0.15 | $0.67\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.10$ | $0.67\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.12$ |

$0.01$ | $1.2\times {10}^{-3}\pm \phantom{\rule{3.33333pt}{0ex}}0.003$ | 0.64 $\pm $ 0.14 | $0.56\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.13$ | $0.48\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.14$ |

Execution Time (Average ± Std) | ||||

${\mathit{\sigma}}_{\mathit{\u03f5}}$ | MATLAB | MCMC | Gibbs | VB |

0 | $0.92\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.18$ | $196.97\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}18.15$ | $215.01\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}14.46$ | $13.72\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}5.77$ |

$0.001$ | $0.95\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.19$ | $236.11\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}33.91$ | $238.97\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}39.36$ | $13.53\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3.30$ |

$0.005$ | $1.49\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.47$ | $272.71\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}33.26$ | $294.75\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}35.81$ | $14.06\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3.48$ |

$0.01$ | $1.97\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.82$ | $332.64\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}73.67$ | $368.88\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}72.96$ | $15.20\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4.99$ |

Image | Execution Time | |||
---|---|---|---|---|

MATLAB | MCMC | Gibbs | VB | |

(a) | 1.7519 | 405.45 | 371.02 | 13.751 |

(b) | 2.1913 | 447.51 | 525.08 | 19.322 |

(c) | 1.5243 | 355.21 | 434.16 | 9.8595 |

(d) | 1.7146 | 359.09 | 450.6 | 10.566 |

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**MDPI and ACS Style**

Yoon, J.W.
A New Bayesian Edge-Linking Algorithm Using Single-Target Tracking Techniques. *Symmetry* **2016**, *8*, 143.
https://doi.org/10.3390/sym8120143

**AMA Style**

Yoon JW.
A New Bayesian Edge-Linking Algorithm Using Single-Target Tracking Techniques. *Symmetry*. 2016; 8(12):143.
https://doi.org/10.3390/sym8120143

**Chicago/Turabian Style**

Yoon, Ji Won.
2016. "A New Bayesian Edge-Linking Algorithm Using Single-Target Tracking Techniques" *Symmetry* 8, no. 12: 143.
https://doi.org/10.3390/sym8120143