【图像处理】透视变换 Perspective Transformation

短命女 2022-07-16 12:59 278阅读 0赞

透视变换（Perspective Transformation)是指利用透视中心、[像点][Link 1]、目标点三点共线的条件，按[透视旋转定律][Link 2]使承影面（透视面）绕[迹线][Link 3]（透视轴）旋转某一角度，破坏原有的投影光线束，仍能保持承影面上投影几何图形不变的变换。

![20140521144153765][]

透视变换(Perspective Transformation)是将图片投影到一个新的视平面(Viewing Plane)，也称作投影映射(Projective Mapping)。通用的变换公式为：

![20140521142738671][]

u,v是原始图片坐标，对应得到变换后的图片坐标x,y,其中![20140521142754687][]。  
变换矩阵![20140521142820609][]可以拆成4部分，![20140521142832296][]表示线性变换，比如scaling（缩放），shearing（错切）和ratotion（翻转）。![20140521142911312][]用于平移，![20140521142842390][]产生透视变换。所以可以理解成仿射（线性变换+平移）等是透视变换的特殊形式。经过透视变换之后的图片通常不是平行四边形（除非映射视平面和原来平面平行的情况）。

重写之前的变换公式可以得到（默认w为1？）：

![20140521143006406][]

所以，**已知变换对应的几个点就可以求取变换公式，**反之，特定的变换公式也能生成新的变换后的图片。简单的看一个正方形到四边形的变换：  
变换的4组对应点可以表示成：![20140521143117828][]

根据变换公式得到（默认a33为1？）：

![20140521143219625][]

定义几个辅助变量：

![20140521143302781][]

![20140521143339968][]都为0时变换平面与原来是平行的，可以得到：

![20140521143422484][]

![20140521143339968][]不为0时，得到：

![20140521143514640][]

求解出的变换矩阵就可以将一个正方形变换到四边形。反之，四边形变换到正方形也是一样的。于是，我们通过两次变换：四边形变换到正方形+正方形变换到四边形就可以将任意一个四边形变换到另一个四边形。

![20140521143605687][]

看一段代码：

**\[cpp\]** [view plain][] [copy][view plain]

[![在CODE上查看代码片][CODE]][CODE_CODE] [![派生到我的代码片][ico_fork.svg]][ico_fork.svg 1]

1.  PerspectiveTransform::PerspectiveTransform(**float** inA11, **float** inA21, 
2.  **float** inA31, **float** inA12, 
3.  **float** inA22, **float** inA32, 
4.  **float** inA13, **float** inA23, 
5.  **float** inA33) : 
6.   a11(inA11), a12(inA12), a13(inA13), a21(inA21), a22(inA22), a23(inA23), 
7.   a31(inA31), a32(inA32), a33(inA33) \{\} 
8.  
9.  PerspectiveTransform PerspectiveTransform::quadrilateralToQuadrilateral(**float** x0, **float** y0, **float** x1, **float** y1, 
10. **float** x2, **float** y2, **float** x3, **float** y3, **float** x0p, **float** y0p, **float** x1p, **float** y1p, **float** x2p, **float** y2p, 
11. **float** x3p, **float** y3p) \{ 
12.  PerspectiveTransform qToS = PerspectiveTransform::quadrilateralToSquare(x0, y0, x1, y1, x2, y2, x3, y3); 
13.  PerspectiveTransform sToQ = 
14.  PerspectiveTransform::squareToQuadrilateral(x0p, y0p, x1p, y1p, x2p, y2p, x3p, y3p); 
15. **return** sToQ.times(qToS); 
16. \} 
17. 
18. PerspectiveTransform PerspectiveTransform::squareToQuadrilateral(**float** x0, **float** y0, **float** x1, **float** y1, **float** x2, 
19. **float** y2, **float** x3, **float** y3) \{ 
20. **float** dx3 = x0 - x1 + x2 - x3; 
21. **float** dy3 = y0 - y1 + y2 - y3; 
22. **if** (dx3 == 0.0f && dy3 == 0.0f) \{ 
23.  PerspectiveTransform result(PerspectiveTransform(x1 - x0, x2 - x1, x0, y1 - y0, y2 - y1, y0, 0.0f, 
24.  0.0f, 1.0f)); 
25. **return** result; 
26.  \} **else** \{ 
27. **float** dx1 = x1 - x2; 
28. **float** dx2 = x3 - x2; 
29. **float** dy1 = y1 - y2; 
30. **float** dy2 = y3 - y2; 
31. **float** denominator = dx1 \* dy2 - dx2 \* dy1; 
32. **float** a13 = (dx3 \* dy2 - dx2 \* dy3) / denominator; 
33. **float** a23 = (dx1 \* dy3 - dx3 \* dy1) / denominator; 
34.  PerspectiveTransform result(PerspectiveTransform(x1 - x0 + a13 \* x1, x3 - x0 + a23 \* x3, x0, y1 - y0 
35.  \+ a13 \* y1, y3 - y0 + a23 \* y3, y0, a13, a23, 1.0f)); 
36. **return** result; 
37.  \} 
38. \} 
39. 
40. PerspectiveTransform PerspectiveTransform::quadrilateralToSquare(**float** x0, **float** y0, **float** x1, **float** y1, **float** x2, 
41. **float** y2, **float** x3, **float** y3) \{ 
42. // Here, the adjoint serves as the inverse:
43. **return** squareToQuadrilateral(x0, y0, x1, y1, x2, y2, x3, y3).buildAdjoint(); 
44. \} 
45. 
46. PerspectiveTransform PerspectiveTransform::buildAdjoint() \{ 
47. // Adjoint is the transpose of the cofactor matrix:
48.  PerspectiveTransform result(PerspectiveTransform(a22 \* a33 - a23 \* a32, a23 \* a31 - a21 \* a33, a21 \* a32 
49.  \- a22 \* a31, a13 \* a32 - a12 \* a33, a11 \* a33 - a13 \* a31, a12 \* a31 - a11 \* a32, a12 \* a23 - a13 \* a22, 
50.  a13 \* a21 - a11 \* a23, a11 \* a22 - a12 \* a21)); 
51. **return** result; 
52. \} 
53. 
54. PerspectiveTransform PerspectiveTransform::times(PerspectiveTransform other) \{ 
55.  PerspectiveTransform result(PerspectiveTransform(a11 \* other.a11 + a21 \* other.a12 + a31 \* other.a13, 
56.  a11 \* other.a21 + a21 \* other.a22 + a31 \* other.a23, a11 \* other.a31 + a21 \* other.a32 + a31 
57.  \* other.a33, a12 \* other.a11 + a22 \* other.a12 + a32 \* other.a13, a12 \* other.a21 + a22 
58.  \* other.a22 + a32 \* other.a23, a12 \* other.a31 + a22 \* other.a32 + a32 \* other.a33, a13 
59.  \* other.a11 + a23 \* other.a12 + a33 \* other.a13, a13 \* other.a21 + a23 \* other.a22 + a33 
60.  \* other.a23, a13 \* other.a31 + a23 \* other.a32 + a33 \* other.a33)); 
61. **return** result; 
62. \} 
63. 
64. **void** PerspectiveTransform::transformPoints(vector<**float**> &points) \{ 
65. **int** max = points.size(); 
66. **for** (**int** i = 0; i < max; i += 2) \{ 
67. **float** x = points\[i\]; 
68. **float** y = points\[i + 1\]; 
69. **float** denominator = a13 \* x + a23 \* y + a33; 
70.  points\[i\] = (a11 \* x + a21 \* y + a31) / denominator; 
71.  points\[i + 1\] = (a12 \* x + a22 \* y + a32) / denominator; 
72.  \} 
73. \}

对一张透视图片变换回正面图的效果：

**\[cpp\]** [view plain][] [copy][view plain]

[![在CODE上查看代码片][CODE]][CODE_CODE] [![派生到我的代码片][ico_fork.svg]][ico_fork.svg 1]

1.  **int** main()\{ 
2.   Mat img=imread("boy.png"); 
3.  **int** img\_height = img.rows; 
4.  **int** img\_width = img.cols; 
5.   Mat img\_trans = Mat::zeros(img\_height,img\_width,CV\_8UC3); 
6.   PerspectiveTransform tansform = PerspectiveTransform::quadrilateralToQuadrilateral( 
7.   0,0, 
8.   img\_width-1,0, 
9.   0,img\_height-1, 
10.  img\_width-1,img\_height-1, 
11.  150,250, // top left
12.  771,0, // top right
13.  0,1023,// bottom left
14.  650,1023 
15.  ); 
16.  vector<**float**> ponits; 
17. **for**(**int** i=0;i<img\_height;i++)\{ 
18. **for**(**int** j=0;j<img\_width;j++)\{ 
19.  ponits.push\_back(j); 
20.  ponits.push\_back(i); 
21.  \} 
22.  \} 
23.  tansform.transformPoints(ponits); 
24. **for**(**int** i=0;i<img\_height;i++)\{ 
25.  uchar\* t= img\_trans.ptr<uchar>(i); 
26. **for** (**int** j=0;j<img\_width;j++)\{ 
27. **int** tmp = i\*img\_width+j; 
28. **int** x = ponits\[tmp\*2\]; 
29. **int** y = ponits\[tmp\*2+1\]; 
30. **if**(x<0||x>(img\_width-1)||y<0||y>(img\_height-1)) 
31. **continue**; 
32.  uchar\* p = img.ptr<uchar>(y); 
33.  t\[j\*3\] = p\[x\*3\]; 
34.  t\[j\*3+1\] = p\[x\*3+1\]; 
35.  t\[j\*3+2\] = p\[x\*3+2\]; 
36.  \} 
37.  \} 
38.  imwrite("trans.png",img\_trans); 
39. **return** 0; 
40. \}

![20140521144153765][]

另外在OpenCV中也实现了基础的透视变换操作，有关函数使用请见下一篇： [【OpenCV】透视变换 Perspective Transformation（续）][OpenCV_ Perspective Transformation]

转载自：http://blog.csdn.net/xiaowei\_cqu/article/details/26471527

[Link 1]: http://baike.baidu.com/view/3438358.htm
[Link 2]: http://baike.baidu.com/view/2077152.htm
[Link 3]: http://baike.baidu.com/view/984087.htm
[20140521144153765]: /images/20220715/353e36c8f91d4d038946952b800d86ca.png
[20140521142738671]: /images/20220715/ddf06949934142ab8c3a96bf3b29bf06.png
[20140521142754687]: /images/20220715/f5c211f296a6481685b961ee7260d9dd.png
[20140521142820609]: https://img-blog.csdn.net/20140521142820609
[20140521142832296]: https://img-blog.csdn.net/20140521142832296
[20140521142911312]: https://img-blog.csdn.net/20140521142911312
[20140521142842390]: /images/20220715/22f3b6c5f38947f0b949f59fb0887f80.png
[20140521143006406]: /images/20220715/537707fdf91e45e4815de85fd4e1b2e5.png
[20140521143117828]: /images/20220715/f934d4f8945743629e7b13c1412725cf.png
[20140521143219625]: /images/20220715/b93665a4733a4551b59dff6a09066f26.png
[20140521143302781]: /images/20220715/a29db8cc92f9432da6db0840b1f635de.png
[20140521143339968]: /images/20220715/c7fcf89d8ae94143ab14b738a82a0f02.png
[20140521143422484]: /images/20220715/31ef43a82c2048af97b5598b2a343a65.png
[20140521143514640]: /images/20220715/0a1e27bb413a4473afa63ec50ae4477a.png
[20140521143605687]: /images/20220715/3b3072d68f7e42f58fe01ac9ead9d065.png
[view plain]: http://blog.csdn.net/xiaowei_cqu/article/details/26471527#
[CODE]: https://code.csdn.net/assets/CODE_ico.png
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