# 哈里斯邊角偵測

## 哈里斯邊角偵測的發展 [1]

${\displaystyle M={\underset {(x,y)\in W}{\sum }}{\begin{bmatrix}I_{x}^{2}&I_{x}I_{y}\\I_{x}I_{y}&I_{y}^{2}\end{bmatrix}}={\begin{bmatrix}{\underset {(x,y)\in W}{\sum }}I_{x}^{2}&{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}\\{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}&{\underset {(x,y)\in W}{\sum }}I_{y}^{2}\end{bmatrix}}}$

${\displaystyle f(x,y)={\underset {(x_{k},y_{k})\in W}{\sum }}\left(I(x_{k},y_{k})-I(x_{k}+\Delta x,y_{k}+\Delta y)\right)^{2}}$

${\displaystyle I(x+\Delta x,y+\Delta y)}$ 可以用泰勒展開去近似，以${\displaystyle I_{x}}$${\displaystyle I_{y}}$分別代表${\displaystyle I}$在x及y方向的偏微分，於是我們可以近似成

${\displaystyle I(x+\Delta x,y+\Delta y)\approx I(x,y)+I_{x}(x,y)\Delta x+I_{y}(x,y)\Delta y}$ 所以${\displaystyle f(x,y)}$可以寫成下列
${\displaystyle f(x,y)\approx {\underset {(x,y)\in W}{\sum }}\left(I_{x}(x,y)\Delta x+I_{y}(x,y)\Delta y\right)^{2},}$ 如果以矩陣的形式來表達,
${\displaystyle f(x,y)\approx {\begin{pmatrix}\Delta x&\Delta y\end{pmatrix}}M{\begin{pmatrix}\Delta x\\\Delta y\end{pmatrix}},}$ 在這裡M代表結構張量,
${\displaystyle M={\underset {(x,y)\in W}{\sum }}{\begin{bmatrix}I_{x}^{2}&I_{x}I_{y}\\I_{x}I_{y}&I_{y}^{2}\end{bmatrix}}={\begin{bmatrix}{\underset {(x,y)\in W}{\sum }}I_{x}^{2}&{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}\\{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}&{\underset {(x,y)\in W}{\sum }}I_{y}^{2}\end{bmatrix}}}$

## 哈里斯邊角偵測的流程[3][4]

1. 彩色影像轉換成灰階影像
2. 空間微分的計算
3. 建構結構張量
4. 計算哈里斯響應
5. 非極大值抑制

### 計算哈里斯響應

${\displaystyle \lambda _{min}\approx {\frac {\lambda _{1}\lambda _{2}}{(\lambda _{1}+\lambda _{2})}}={\frac {det(M)}{trace(M)}}}$

${\displaystyle trace(M)=m_{11}+m_{22}}$

${\displaystyle R=det(M)-k(trace(M))^{2}=\lambda _{1}\lambda _{2}-k(\lambda _{1}+\lambda _{2})^{2}}$

## 改進的偵測演算法[5][6]

1. 哈里斯-拉普拉斯邊角偵測[7]

2. Differential Morphological Decomposition Based Corner Detector[8]

3. Multi-scale Bilatera Structure Tensor Based Corner Detector[9]

## 應用

1. 影像對齊，影像縫合，影像配準[10]

2. 建立二維馬賽克[11]

3. 三維場景建模及重建[12]

4. 動作偵測[13]

5. 物體識別[14]

6. 基于内容的影像檢索[15]

7. 影片追蹤[16]

## 參考資料

1. Chris Harris and Mike Stephens (1988). "A Combined Corner and Edge Detector". Alvey Vision Conference. 15.
2. ^ Konstantinos G. Derpanis (2004). The harris corner detector. York University.
3. ^ Harris Operator Corner Detection using Sliding Window Method - Google Scholar. scholar.google.com. [2015-11-29].
4. ^ The Comparison and Application of Corner Detection Algorithms - Google Scholar. scholar.google.com. [2015-11-29].
5. ^ Bellavia, F.; Tegolo, D.; Valenti, C. Improving Harris corner selection strategy. IET Computer Vision. 2011-03-01, 5 (2). doi:10.1049/iet-cvi.2009.0127.
6. ^ Rosten, Edward; Drummond, Tom. Leonardis, Aleš; Bischof, Horst; Pinz, Axel, 编. Machine Learning for High-Speed Corner Detection. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 2006-05-07: 430–443. ISBN 978-3-540-33832-1. doi:10.1007/11744023_34#page-1 （英语）.
7. ^ A Comparison of Affine Region Detectors - Google Scholar. scholar.google.com. [2015-11-29].
8. ^ Gueguen, L.; Pesaresi, M. Multi scale Harris corner detector based on Differential Morphological Decomposition. Pattern Recognition Letters: 1714–1719. doi:10.1016/j.patrec.2011.07.021.
9. ^ A Multi-scale Bilateral Structure Tensor Based Corner Detector - Google Scholar. scholar.google.com. [2015-11-29].
10. ^ Kang, Juan; Xiao, Chuangbai; Deng, M.; Yu, Jing; Liu, Haifeng. Image registration based on harris corner and mutual information. 2011 International Conference on Electronic and Mechanical Engineering and Information Technology (EMEIT). 2011-08-01, 7: 3434–3437. doi:10.1109/EMEIT.2011.6023066.
11. ^ Underwater Mosaic Creation using Video sequences from Different Altitudes - Google Scholar. scholar.google.com. [2015-12-02].
12. ^ Automated reconstruction of 3D scenes from sequences of images - Google Scholar. scholar.google.com. [2015-12-02].
13. ^ Liu, Meng; Wu, Chengdong; Zhang, Yunzhou. Multi-resolution optical flow tracking algorithm based on multi-scale Harris corner points feature. Control and Decision Conference, 2008. CCDC 2008. Chinese. 2008-07-01: 5287–5291. doi:10.1109/CCDC.2008.4598340.
14. ^ Object Recognition from Local Scale-Invariant Features - Google Scholar. scholar.google.com. [2015-11-29].
15. ^ Salient Points for Content Based Retrieval - Google Scholar. scholar.google.com. [2015-12-02].
16. ^ Tracking and Recognition of Objects using SURF Descriptor and Harris Corner Detection - Google Scholar. scholar.google.com. [2015-12-02].