在分析基于雾度线估计大气光的文章时,雾度线是根据霍夫变换中的图像处理基础原理所计算出来的,但是一直是使用opencv的函数库,不知道霍夫变换的工作原理。在冈萨雷斯的图像处理的书中看了半天,也没怎么看明白,所以就在网上找到了一篇关于霍夫变换的博客,作者对霍夫变换的解释,清晰明了,所以就直接转载了原文。
前言
今天群里有人问到一个图像的问题,但本质上是一个基本最小二乘问题,涉及到霍夫变换(Hough Transform),用到了就顺便总结一下。内容为自己的学习记录,其中多有参考他人,最后一并给出链接。
一、霍夫变换(Hough)
A-基本原理
一条直线可由两个点A=(X1,Y1)和B=(X2,Y2)确定(笛卡尔坐标)
另一方面,![]()
也可以写成关于(k,q)的函数表达式(霍夫空间):
![](https://image.shishitao.com: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%3D.jpg?w=700&webp=1)
![](https://image.shishitao.com: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%3D.jpg?w=700&webp=1)
对应的变换可以通过图形直观表示:
变换后的空间成为霍夫空间。即:笛卡尔坐标系中一条直线,对应霍夫空间的一个点。
反过来同样成立(霍夫空间的一条直线,对应笛卡尔坐标系的一个点):
再来看看A、B两个点,对应霍夫空间的情形:
一步步来,再看一下三个点共线的情况:
可以看出如果笛卡尔坐标系的点共线,这些点在霍夫空间对应的直线交于一点:这也是必然,共线只有一种取值可能。
如果不止一条直线呢?再看看多个点的情况(有两条直线):
其实(3,2)与(4,1)也可以组成直线,只不过它有两个点确定,而图中A、B两点是由三条直线汇成,这也是霍夫变换的后处理的基本方式:选择由尽可能多直线汇成的点。
看看,霍夫空间:选择由三条交汇直线确定的点(中间图),对应的笛卡尔坐标系的直线(右图)。
到这里问题似乎解决了,已经完成了霍夫变换的求解,但是如果像下图这种情况呢?
k=∞是不方便表示的,而且q怎么取值呢,这样不是办法。因此考虑将笛卡尔坐标系换为:极坐标表示。
在极坐标系下,其实是一样的:极坐标的点→霍夫空间的直线,只不过霍夫空间不再是[k,q]的参数,而是![]()
的参数,给出对比图:
![](https://image.shishitao.com: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%3D.jpg?w=700&webp=1)
是不是就一目了然了?
给出霍夫变换的算法步骤:
对应code:
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function [ Hough, theta_range, rho_range ] = naiveHough(I)
%NAIVEHOUGH Peforms the Hough transform in a straightforward way.
%
[rows, cols] = size(I);
theta_maximum = 90;
rho_maximum = floor(sqrt(rows^2 + cols^2)) - 1;
theta_range = -theta_maximum:theta_maximum - 1;
rho_range = -rho_maximum:rho_maximum;
Hough = zeros(length(rho_range), length(theta_range));
for row = 1:rows
for col = 1:cols
if I(row, col) > 0 %only find: pixel > 0
x = col - 1;
y = row - 1;
for theta = theta_range
rho = round((x * cosd(theta)) + (y * sind(theta))); %approximate
rho_index = rho + rho_maximum + 1;
theta_index = theta + theta_maximum + 1;
Hough(rho_index, theta_index) = Hough(rho_index, theta_index) + 1;
end
end
end
end
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其实本质上就是:
交点怎么求解呢?细化成坐标形式,取整后将交点对应的坐标进行累加,最后找到数值最大的点就是求解的![]()
,也就求解出了直线。
![](https://image.shishitao.com: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%3D.jpg?w=700&webp=1)
B-理论应用
这里给出MATLAB自带的一个应用,主要是对一幅图像进行直线检验,原图像为:
首先是对其进行边缘检测:
边缘检测后并二值化,就可以通过找非零点的坐标确定数据点。从而对数据点进行霍夫变换。对应映射到霍夫空间的结果为:
找出其中数值较大的一些点,通常可以给定一个阈值。
这就完成了霍夫变换的整个过程。这个时候求解出来了其实就是多条直线的斜率k以及截距q,通常会根据直线的特性进一步判断,从而将直线变为线段:
不过这一步更类似后处理,其实已经不是霍夫变换本身的特性了。
给出对应的代码:
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clc;clear all;close all;
I = imread(\'circuit.tif\');
rotI = imrotate(I,40,\'crop\');
subplot 221
fig1 = imshow(rotI);
BW = edge(rotI,\'canny\');
title(\'原图像\');
subplot 222
imshow(BW);
[H,theta,rho] = hough(BW);
title(\'图像边缘检测\');
subplot 223
imshow(imadjust(mat2gray(H)),[],\'XData\',theta,\'YData\',rho,...
\'InitialMagnification\',\'fit\');
xlabel(\'\theta (degrees)\'), ylabel(\'\rho\');
axis on, axis normal, hold on;
colormap(hot)
P = houghpeaks(H,5,\'threshold\',ceil(0.7*max(H(:))));
x = theta(P(:,2));
y = rho(P(:,1));
plot(x,y,\'s\',\'color\',\'black\');
lines = houghlines(BW,theta,rho,P,\'FillGap\',5,\'MinLength\',7);
title(\'Hough空间\');
subplot 224, imshow(rotI), hold on
max_len = 0;
for k = 1:length(lines)
xy = [lines(k).point1; lines(k).point2];
plot(xy(:,1),xy(:,2),\'LineWidth\',2,\'Color\',\'green\');
% Plot beginnings and ends of lines
plot(xy(1,1),xy(1,2),\'x\',\'LineWidth\',2,\'Color\',\'yellow\');
plot(xy(2,1),xy(2,2),\'x\',\'LineWidth\',2,\'Color\',\'red\');
% Determine the endpoints of the longest line segment
len = norm(lines(k).point1 - lines(k).point2);
if ( len > max_len)
max_len = len;
xy_long = xy;
end
end
% highlight the longest line segment
plot(xy_long(:,1),xy_long(:,2),\'LineWidth\',2,\'Color\',\'red\');
title(\'直线检测\');
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对比自带的Hough与编写的Hough:
效果还是比较接近的。
看到*上的一个答案,觉得很好,收藏一下:
这篇文章中,对于霍夫变换的原理解释相当简单明了,所以不再赘述。因为霍夫变换是为了检测图像中的像素集是否位于指定形状的曲线或直线上的一种技术方法。一旦检测到,这些像素就是形成指定形状的像素,如直线,圆等更复杂的图像函数。
在冈萨雷斯的图像处理的书中,例举了一个关于基于霍夫变换的边缘连接问题:在一张航拍的机场图像中,使用霍夫变换提取竖直跑道的两条边。其标准操作是:1)第一步是使用canny算法得到一幅边缘图像,2)通过逐级递增θ和ρ得到霍夫参数空间,3)根据机场图像特点,竖直的跑道对应θ=+-90°,平且是这个方向最长的直线,所以选择对应该θ值的并包含最高数量的单元为为跑到线的霍夫空间的点。 连接这些点就可得到跑道两边的线。
这和博主所举的直线检测的例子有很大的相似之处。
---------------------
作者:lanmengyiyu
下面的补充也是在网上博主的内容:
另外,在做霍夫变换时,我们不可能将所有的点都做映射,这样计算量太大,因此我们一般会
1)先用canny算子对边缘轮廓做提取,主要目的是降维,这样可以尽可能的去掉平滑区域的点减少计算量,
2)之后利用二值的方法(包括开闭操作等)提取轮廓,去掉噪声等扰动,主要目的是在拟合线性方程时尽可能准确,
3)霍夫变换完成空间映射
4)计数器统计出现次数最高的未知系数组合,如(ρ,θ)