# 霍夫变换

## 歷史

Duda, R. O. and P. E. Hart, "Use of the Hough Transformation to Detect Lines and Curves in Pictures," Comm. ACM, Vol. 15, pp. 11–15 (January, 1972),

O'Gorman and Clowes' variation出自

O'Gorman, Frank; Clowes, MB. Finding Picture Edges Through Collinearity of Feature Points. IEEE Trans. Comput. 1976, 25 (4): 449–456.,

Hart, P. E., "How the Hough Transform was Invented" (PDF, 268 kB), IEEE Signal Processing Magazine, Vol 26, Issue 6, pp 18 – 22 (November, 2009).

## 理論

${\displaystyle y=m_{0}x+b_{0}}$

${\displaystyle r=x\cos \theta +y\sin \theta }$

${\displaystyle r}$是從原點到直線的距離，${\displaystyle \theta }$${\displaystyle {\vec {r}}}$${\displaystyle x}$軸的夾角。利用參數空間${\displaystyle (r,\theta )}$解決了原本參數空間${\displaystyle (m,b)}$發散的問題， 進而能夠比較每一個線段的參數，有人將${\displaystyle (r,\theta )}$平面稱為二維直線的霍夫空間(Hough space)。這個表示方法讓霍夫變換跟二維的拉東變換非常相似，可以說是一體兩面 [6]

${\displaystyle r=x_{0}\cos \theta +y_{0}\sin \theta \Rightarrow r={\sqrt {x_{0}^{2}+y_{0}^{2}}}\left({\frac {x_{0}}{\sqrt {x_{0}^{2}+y_{0}^{2}}}}\cos \theta +{\frac {y_{0}}{\sqrt {x_{0}^{2}+y_{0}^{2}}}}\sin \theta \right)\Rightarrow r={\sqrt {x_{0}^{2}+y_{0}^{2}}}\left(\cos \phi \cos \theta +\sin \phi \sin \theta \right)}$
${\displaystyle \Rightarrow r={\sqrt {x_{0}^{2}+y_{0}^{2}}}\cos(\theta -\phi )}$

## 霍夫變換的變形與延伸

Kernel-based Hough transform (KHT)

## 注釋

1. ^ Shapiro, Linda and Stockman, George. "Computer Vision", Prentice-Hall, Inc. 2001
2. ^ Duda, R. O. and P. E. Hart, "Use of the Hough Transformation to Detect Lines and Curves in Pictures," Comm. ACM, Vol. 15, pp. 11–15 (January, 1972)
3. ^ Hough, P.V.C. Method and means for recognizing complex patterns, U.S. Patent 3,069,654, Dec. 18, 1962
4. ^ P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, Proc. Int. Conf. High Energy Accelerators and Instrumentation, 1959
5. ^ Richard O. Duda and Peter E. Hart. Use of the Hough Transformation to Detect Lines and Curves in Pictures (PDF). Artificial Intelligence Center (SRI International). April 1971 [2017-06-30]. （原始内容存档 (PDF)于2012-03-13）.
6. ^ CiteSeerX — A short introduction to the Radon and Hough transforms and how they relate to each other. [2017-06-30]. （原始内容存档于2012-10-16）.
7. ^ Image Transforms - Hough Transform. Homepages.inf.ed.ac.uk. [2009-08-17]. （原始内容存档于2021-02-11）.