四边形透视变换.docx

登录|注册小魏的修行路Talk is cheap. Show me the code.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：u,v是原始图片左边，对应得到变换后的图片坐标x,y,其中。变换矩阵可以拆成4部分，表示线性变换，比如scaling，shearing和ratotion。用于平移，产生透视变换。所以可以理解成仿射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：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：根据变换公式得到：定义几个辅助变量：都为0时变换平面与原来是平行的，可以得到：不为0时，得到：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。看一段代码：cppview plaincopy1. PerspectiveTransform:PerspectiveTransform(floatinA11,floatinA21,2. floatinA31,floatinA12,3. floatinA22,floatinA32,4. floatinA13,floatinA23,5. floatinA33):6. a11(inA11),a12(inA12),a13(inA13),a21(inA21),a22(inA22),a23(inA23),7. a31(inA31),a32(inA32),a33(inA33)8. 9. PerspectiveTransformPerspectiveTransform:quadrilateralToQuadrilateral(floatx0,floaty0,floatx1,floaty1,10. floatx2,floaty2,floatx3,floaty3,floatx0p,floaty0p,floatx1p,floaty1p,floatx2p,floaty2p,11. floatx3p,floaty3p)12. PerspectiveTransformqToS=PerspectiveTransform:quadrilateralToSquare(x0,y0,x1,y1,x2,y2,x3,y3);13. PerspectiveTransformsToQ=14. PerspectiveTransform:squareToQuadrilateral(x0p,y0p,x1p,y1p,x2p,y2p,x3p,y3p);15. returnsToQ.times(qToS);16. 17. 18. PerspectiveTransformPerspectiveTransform:squareToQuadrilateral(floatx0,floaty0,floatx1,floaty1,floatx2,19. floaty2,floatx3,floaty3)20. floatdx3=x0-x1+x2-x3;21. floatdy3=y0-y1+y2-y3;22. if(dx3=0.0f&dy3=0.0f)23. PerspectiveTransformresult(PerspectiveTransform(x1-x0,x2-x1,x0,y1-y0,y2-y1,y0,0.0f,24. 0.0f,1.0f);25. returnresult;26. else27. floatdx1=x1-x2;28. floatdx2=x3-x2;29. floatdy1=y1-y2;30. floatdy2=y3-y2;31. floatdenominator=dx1*dy2-dx2*dy1;32. floata13=(dx3*dy2-dx2*dy3)/denominator;33. floata23=(dx1*dy3-dx3*dy1)/denominator;34. PerspectiveTransformresult(PerspectiveTransform(x1-x0+a13*x1,x3-x0+a23*x3,x0,y1-y035. +a13*y1,y3-y0+a23*y3,y0,a13,a23,1.0f);36. returnresult;37. 38. 39. 40. PerspectiveTransformPerspectiveTransform:quadrilateralToSquare(floatx0,floaty0,floatx1,floaty1,floatx2,41. floaty2,floatx3,floaty3)42. /Here,theadjointservesastheinverse:43. returnsquareToQuadrilateral(x0,y0,x1,y1,x2,y2,x3,y3).buildAdjoint();44. 45. 46. PerspectiveTransformPerspectiveTransform:buildAdjoint()47. /Adjointisthetransposeofthecofactormatrix:48. PerspectiveTransformresult(PerspectiveTransform(a22*a33-a23*a32,a23*a31-a21*a33,a21*a3249. -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. returnresult;52. 53. 54. PerspectiveTransformPerspectiveTransform:times(PerspectiveTransformother)55. PerspectiveTransformresult(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+a3157. *other.a33,a12*other.a11+a22*other.a12+a32*other.a13,a12*other.a21+a2258. *other.a22+a32*other.a23,a12*other.a31+a22*other.a32+a32*other.a33,a1359. *other.a11+a23*other.a12+a33*other.a13,a13*other.a21+a23*other.a22+a3360. *other.a23,a13*other.a31+a23*other.a32+a33*other.a33);61. returnresult;62. 63. 64. voidPerspectiveTransform:transformPoints(vector&points)65. intmax=points.size();66. for(inti=0;imax;i+=2)67. floatx=pointsi;68. floaty=pointsi+1;69. floatdenominator=a13*x+a23*y+a33;70. pointsi=(a11*x+a21*y+a31)/denominator;71. pointsi+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：1. intmain()2. Matimg=imread(boy.png);3. intimg_height=img.rows;4. intimg_width=img.cols;5. Matimg_trans=Mat:zeros(img_height,img_width,CV_8UC3);6. PerspectiveTransformtansform=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,/topleft12. 771,0,/topright13. 0,1023,/bottomleft14. 650,102315. );16. vectorponits;17. for(inti=0;iimg_height;i+)18. for(intj=0;jimg_width;j+)19. ponits.push_back(j);20. ponits.push_back(i);21. 22. 23. tansform.transformPoints(ponits);24. for(inti=0;iimg_height;i+)25. uchar*t=img_trans.ptr(i);26. for(intj=0;jimg_width;j+)27. inttmp=i*img_width+j;28. intx=ponitstmp*2;29. inty=ponitstmp*2+1;30. if(x(img_width-1)|y(img_height-1)31. continue;32. uchar*p=img.ptr(y);33. tj*3=px*3;34. tj*3+1=px*3+1;35. tj*3+2=px*3+2;36. 37. 38. imwrite(trans.png,img_trans);39. return0;40.