Math t成功地取消了用作示例的图像的失真（我没有尝试进行高质量插值）：

死链

（这是我在博客上写的，作为比较：）

---

我盲目地实现了中的公式，因此我无法保证它能满足您的需要

使用

auto_zoom

获取

zoom

参数的值

def dist(x,y):
    return sqrt(x*x+y*y)

def fisheye_to_rectilinear(src_size,dest_size,sx,sy,crop_factor,zoom):
    """ returns a tuple of dest coordinates (dx,dy)
        (note: values can be out of range)
 crop_factor is ratio of sphere diameter to diagonal of the source image"""  
    # convert sx,sy to relative coordinates
    rx, ry = sx-(src_size[0]/2), sy-(src_size[1]/2)
    r = dist(rx,ry)

    # focal distance = radius of the sphere
    pi = 3.1415926535
    f = dist(src_size[0],src_size[1])*factor/pi

    # calc theta 1) linear mapping (older Nikon) 
    theta = r / f

    # calc theta 2) nonlinear mapping 
    # theta = asin ( r / ( 2 * f ) ) * 2

    # calc new radius
    nr = tan(theta) * zoom

    # back to absolute coordinates
    dx, dy = (dest_size[0]/2)+rx/r*nr, (dest_size[1]/2)+ry/r*nr
    # done
    return (int(round(dx)),int(round(dy)))


def fisheye_auto_zoom(src_size,dest_size,crop_factor):
    """ calculate zoom such that left edge of source image matches left edge of dest image """
    # Try to see what happens with zoom=1
    dx, dy = fisheye_to_rectilinear(src_size, dest_size, 0, src_size[1]/2, crop_factor, 1)

    # Calculate zoom so the result is what we wanted
    obtained_r = dest_size[0]/2 - dx
    required_r = dest_size[0]/2
    zoom = required_r / obtained_r
    return zoom

---

我找到了这个pdf文件，我已经证明了数学是正确的（除了行

vd=*xd**fv+v0，应该是vd=**yd**+fv+v0

）

它并没有使用OpenCV提供的所有最新的协同效率，但我相信它可以相当容易地进行调整

double k1 = cameraIntrinsic.distortion[0];
double k2 = cameraIntrinsic.distortion[1];
double p1 = cameraIntrinsic.distortion[2];
double p2 = cameraIntrinsic.distortion[3];
double k3 = cameraIntrinsic.distortion[4];
double fu = cameraIntrinsic.focalLength[0];
double fv = cameraIntrinsic.focalLength[1];
double u0 = cameraIntrinsic.principalPoint[0];
double v0 = cameraIntrinsic.principalPoint[1];
double u, v;


u = thisPoint->x; // the undistorted point
v = thisPoint->y;
double x = ( u - u0 )/fu;
double y = ( v - v0 )/fv;

double r2 = (x*x) + (y*y);
double r4 = r2*r2;

double cDist = 1 + (k1*r2) + (k2*r4);
double xr = x*cDist;
double yr = y*cDist;

double a1 = 2*x*y;
double a2 = r2 + (2*(x*x));
double a3 = r2 + (2*(y*y));

double dx = (a1*p1) + (a2*p2);
double dy = (a3*p1) + (a1*p2);

double xd = xr + dx;
double yd = yr + dy;

double ud = (xd*fu) + u0;
double vd = (yd*fv) + v0;

thisPoint->x = ud; // the distorted point
thisPoint->y = vd;

---

我采取了JMBR的做法，并基本上逆转了它。他计算了畸变图像的半径（Rd，即图像中心到像素的距离），并找到了Ru的公式，即未畸变图像的半径

你想走另一条路。对于未失真（处理过的图像）中的每个像素，您需要知道失真图像中对应的像素是什么。 换句话说，给定（许，余）->（xd，yd）。然后将未失真图像中的每个像素替换为失真图像中相应的像素

从JMBR做的地方开始，我做相反的事情，发现Rd是Ru的函数。我得到：

Rd = f * sqrt(2) * sqrt( 1 - 1/sqrt(r^2 +1))

其中f是以像素为单位的焦距（我将在后面解释），并且

r=Ru/f

我相机的焦距是2.5毫米。我的CCD上每个像素的大小是6平方毫米。因此f为2500/6=417像素。这可以通过反复试验找到

查找Rd允许您使用极坐标在扭曲的图像中查找相应的像素

每个像素与中心点的角度相同：

theta=arctan（（yu-yc）/（xu-xc））

其中xc，yc是中心点

那么

确保你知道你在哪个象限

这是我使用的C代码

 public class Analyzer
 {
      private ArrayList mFisheyeCorrect;
      private int mFELimit = 1500;
      private double mScaleFESize = 0.9;

      public Analyzer()
      {
            //A lookup table so we don't have to calculate Rdistorted over and over
            //The values will be multiplied by focal length in pixels to 
            //get the Rdistorted
          mFisheyeCorrect = new ArrayList(mFELimit);
            //i corresponds to Rundist/focalLengthInPixels * 1000 (to get integers)
          for (int i = 0; i < mFELimit; i++)
          {
              double result = Math.Sqrt(1 - 1 / Math.Sqrt(1.0 + (double)i * i / 1000000.0)) * 1.4142136;
              mFisheyeCorrect.Add(result);
          }
      }

      public Bitmap RemoveFisheye(ref Bitmap aImage, double aFocalLinPixels)
      {
          Bitmap correctedImage = new Bitmap(aImage.Width, aImage.Height);
             //The center points of the image
          double xc = aImage.Width / 2.0;
          double yc = aImage.Height / 2.0;
          Boolean xpos, ypos;
            //Move through the pixels in the corrected image; 
            //set to corresponding pixels in distorted image
          for (int i = 0; i < correctedImage.Width; i++)
          {
              for (int j = 0; j < correctedImage.Height; j++)
              {
                     //which quadrant are we in?
                  xpos = i > xc;
                  ypos = j > yc;
                     //Find the distance from the center
                  double xdif = i-xc;
                  double ydif = j-yc;
                     //The distance squared
                  double Rusquare = xdif * xdif + ydif * ydif;
                     //the angle from the center
                  double theta = Math.Atan2(ydif, xdif);
                     //find index for lookup table
                  int index = (int)(Math.Sqrt(Rusquare) / aFocalLinPixels * 1000);
                  if (index >= mFELimit) index = mFELimit - 1;
                     //calculated Rdistorted
                  double Rd = aFocalLinPixels * (double)mFisheyeCorrect[index]
                                        /mScaleFESize;
                     //calculate x and y distances
                  double xdelta = Math.Abs(Rd*Math.Cos(theta));
                  double ydelta = Math.Abs(Rd * Math.Sin(theta));
                     //convert to pixel coordinates
                  int xd = (int)(xc + (xpos ? xdelta : -xdelta));
                  int yd = (int)(yc + (ypos ? ydelta : -ydelta));
                  xd = Math.Max(0, Math.Min(xd, aImage.Width-1));
                  yd = Math.Max(0, Math.Min(yd, aImage.Height-1));
                     //set the corrected pixel value from the distorted image
                  correctedImage.SetPixel(i, j, aImage.GetPixel(xd, yd));
              }
          }
          return correctedImage;
      }
}

公共类分析器
{
私人ArrayList mFisheyeCorrect；
私有int mFELimit=1500；
私人双mScaleFESize=0.9；
公共分析器（）
{
//一个查找表，这样我们就不必反复计算Rdistored
//这些值将乘以以像素为单位的焦距，以
//把这个故事讲清楚
mFisheyeCorrect=新阵列列表（mFELimit）；
//i对应于Rundist/focalLength像素*1000（用于获取整数）
for（int i=0；ixc；
ypos=j>yc；
//找出离中心的距离
双xdif=i-xc；
双ydif=j-yc；
//距离的平方
双Rusquare=xdif*xdif+ydif*ydif；
//从中心的角度
双θ=Math.Atan2（ydif，xdif）；
//查找查找表的索引
int索引=（int）（数学Sqrt（Rusquare）/aFocalLinPixels*1000）；
如果（索引>=mFELimit）索引=mFELimit-1；
//计算失真
double Rd=aFocalLinPixels*（double）mFisheyeCorrect[索引]
/mScaleFESize；
//计算x和y距离
double-xdelta=Math.Abs（Rd*Math.Cos（θ））；
double ydelta=Math.Abs（Rd*Math.Sin（θ））；
//转换为像素坐标
int xd=（int）（xc+（xpos？xdelta:-xdelta））；
int yd=（int）（yc+（ypos？ydelta:-ydelta））；
xd=Math.Max（0，Math.Min（xd，aImage.Width-1））；
yd=数学最大值（0，数学最小值（yd，目标高度-1））；
//设置扭曲图像的校正像素值
校正了mage.SetPixel（i，j，aImage.GetPixel（xd，yd））；
}
}
返回正确的图像；
}
}

我不太明白你在找什么。鱼眼从球体映射到图片平面。反向映射将从图片返回到球体，对吗？你在寻找什么样的直线坐标？@mtrw我的源图像是鱼眼扭曲的，我想让它不失真。你要找的图片是什么？是的，经过校正的图片，例如通过OpenCV，或者一个公式来校正图片中的任何点。威尔，你有没有得到一个结论性的答案？我很想看看你最终得到的任何代码。这取决于你是否可以使用有问题的摄像机。我不知道，我只是在看录像。另外，我突然想到相机是批量生产的，而且不会有太多的变化？原始文章中链接的工具不需要有人拿着棋盘站在相机前面！？仅通过调整缩放，同一摄影机的摄影机参数会有所不同。此外，您可以依赖自动校准技术，而不是使用棋盘。无论如何，我已经编辑了我的答案，以解决你问题的第一部分，为你提供了你想要的公式。谢谢

#!/usr/bin/python

# ./undistort 0_0000.jpg 1367.451167 1367.451167 0 0 -0.246065 0.193617 -0.002004 -0.002056

import sys
import cv

def main(argv):
    if len(argv) < 10:
    print 'Usage: %s input-file fx fy cx cy k1 k2 p1 p2 output-file' % argv[0]
    sys.exit(-1)

    src = argv[1]
    fx, fy, cx, cy, k1, k2, p1, p2, output = argv[2:]

    intrinsics = cv.CreateMat(3, 3, cv.CV_64FC1)
    cv.Zero(intrinsics)
    intrinsics[0, 0] = float(fx)
    intrinsics[1, 1] = float(fy)
    intrinsics[2, 2] = 1.0
    intrinsics[0, 2] = float(cx)
    intrinsics[1, 2] = float(cy)

    dist_coeffs = cv.CreateMat(1, 4, cv.CV_64FC1)
    cv.Zero(dist_coeffs)
    dist_coeffs[0, 0] = float(k1)
    dist_coeffs[0, 1] = float(k2)
    dist_coeffs[0, 2] = float(p1)
    dist_coeffs[0, 3] = float(p2)

    src = cv.LoadImage(src)
    dst = cv.CreateImage(cv.GetSize(src), src.depth, src.nChannels)
    mapx = cv.CreateImage(cv.GetSize(src), cv.IPL_DEPTH_32F, 1)
    mapy = cv.CreateImage(cv.GetSize(src), cv.IPL_DEPTH_32F, 1)
    cv.InitUndistortMap(intrinsics, dist_coeffs, mapx, mapy)
    cv.Remap(src, dst, mapx, mapy, cv.CV_INTER_LINEAR + cv.CV_WARP_FILL_OUTLIERS,  cv.ScalarAll(0))
    # cv.Undistort2(src, dst, intrinsics, dist_coeffs)

    cv.SaveImage(output, dst)


if __name__ == '__main__':
    main(sys.argv)

Rin = 2 f sin(w/2) -> sin(w/2)= Rin/2f
Rout= f tan(w)     -> tan(w)= Rout/f

(Rin/2f)^2 = [sin(w/2)]^2 = (1 - cos(w))/2  ->  cos(w) = 1 - 2(Rin/2f)^2
(Rout/f)^2 = [tan(w)]^2 = 1/[cos(w)]^2 - 1

-> (Rout/f)^2 = 1/(1-2[Rin/2f]^2)^2 - 1

Rout = Rin*(1 + A*Rin^2 + B*Rin^4 + ...)

(Rout/f)^2 = 1/(1-2[Rin/2f]^2)^2 - 1
-> Rout/f = [Rin/f] * sqrt(1-[Rin/f]^2/4)/(1-[Rin/f]^2/2)

Rin/f = [Rout/f] / sqrt( sqrt(([Rout/f]^2+1) * (sqrt([Rout/f]^2+1) + 1) / 2 )

def dist(x,y):
    return sqrt(x*x+y*y)

def correct_fisheye(src_size,dest_size,dx,dy,factor):
    """ returns a tuple of source coordinates (sx,sy)
        (note: values can be out of range)"""
    # convert dx,dy to relative coordinates
    rx, ry = dx-(dest_size[0]/2), dy-(dest_size[1]/2)
    # calc theta
    r = dist(rx,ry)/(dist(src_size[0],src_size[1])/factor)
    if 0==r:
        theta = 1.0
    else:
        theta = atan(r)/r
    # back to absolute coordinates
    sx, sy = (src_size[0]/2)+theta*rx, (src_size[1]/2)+theta*ry
    # done
    return (int(round(sx)),int(round(sy)))

def dist(x,y):
    return sqrt(x*x+y*y)

def fisheye_to_rectilinear(src_size,dest_size,sx,sy,crop_factor,zoom):
    """ returns a tuple of dest coordinates (dx,dy)
        (note: values can be out of range)
 crop_factor is ratio of sphere diameter to diagonal of the source image"""  
    # convert sx,sy to relative coordinates
    rx, ry = sx-(src_size[0]/2), sy-(src_size[1]/2)
    r = dist(rx,ry)

    # focal distance = radius of the sphere
    pi = 3.1415926535
    f = dist(src_size[0],src_size[1])*factor/pi

    # calc theta 1) linear mapping (older Nikon) 
    theta = r / f

    # calc theta 2) nonlinear mapping 
    # theta = asin ( r / ( 2 * f ) ) * 2

    # calc new radius
    nr = tan(theta) * zoom

    # back to absolute coordinates
    dx, dy = (dest_size[0]/2)+rx/r*nr, (dest_size[1]/2)+ry/r*nr
    # done
    return (int(round(dx)),int(round(dy)))


def fisheye_auto_zoom(src_size,dest_size,crop_factor):
    """ calculate zoom such that left edge of source image matches left edge of dest image """
    # Try to see what happens with zoom=1
    dx, dy = fisheye_to_rectilinear(src_size, dest_size, 0, src_size[1]/2, crop_factor, 1)

    # Calculate zoom so the result is what we wanted
    obtained_r = dest_size[0]/2 - dx
    required_r = dest_size[0]/2
    zoom = required_r / obtained_r
    return zoom

double k1 = cameraIntrinsic.distortion[0];
double k2 = cameraIntrinsic.distortion[1];
double p1 = cameraIntrinsic.distortion[2];
double p2 = cameraIntrinsic.distortion[3];
double k3 = cameraIntrinsic.distortion[4];
double fu = cameraIntrinsic.focalLength[0];
double fv = cameraIntrinsic.focalLength[1];
double u0 = cameraIntrinsic.principalPoint[0];
double v0 = cameraIntrinsic.principalPoint[1];
double u, v;


u = thisPoint->x; // the undistorted point
v = thisPoint->y;
double x = ( u - u0 )/fu;
double y = ( v - v0 )/fv;

double r2 = (x*x) + (y*y);
double r4 = r2*r2;

double cDist = 1 + (k1*r2) + (k2*r4);
double xr = x*cDist;
double yr = y*cDist;

double a1 = 2*x*y;
double a2 = r2 + (2*(x*x));
double a3 = r2 + (2*(y*y));

double dx = (a1*p1) + (a2*p2);
double dy = (a3*p1) + (a1*p2);

double xd = xr + dx;
double yd = yr + dy;

double ud = (xd*fu) + u0;
double vd = (yd*fv) + v0;

thisPoint->x = ud; // the distorted point
thisPoint->y = vd;

Rd = f * sqrt(2) * sqrt( 1 - 1/sqrt(r^2 +1))

xd = Rd * cos(theta) + xc
yd = Rd * sin(theta) + yc

 public class Analyzer
 {
      private ArrayList mFisheyeCorrect;
      private int mFELimit = 1500;
      private double mScaleFESize = 0.9;

      public Analyzer()
      {
            //A lookup table so we don't have to calculate Rdistorted over and over
            //The values will be multiplied by focal length in pixels to 
            //get the Rdistorted
          mFisheyeCorrect = new ArrayList(mFELimit);
            //i corresponds to Rundist/focalLengthInPixels * 1000 (to get integers)
          for (int i = 0; i < mFELimit; i++)
          {
              double result = Math.Sqrt(1 - 1 / Math.Sqrt(1.0 + (double)i * i / 1000000.0)) * 1.4142136;
              mFisheyeCorrect.Add(result);
          }
      }

      public Bitmap RemoveFisheye(ref Bitmap aImage, double aFocalLinPixels)
      {
          Bitmap correctedImage = new Bitmap(aImage.Width, aImage.Height);
             //The center points of the image
          double xc = aImage.Width / 2.0;
          double yc = aImage.Height / 2.0;
          Boolean xpos, ypos;
            //Move through the pixels in the corrected image; 
            //set to corresponding pixels in distorted image
          for (int i = 0; i < correctedImage.Width; i++)
          {
              for (int j = 0; j < correctedImage.Height; j++)
              {
                     //which quadrant are we in?
                  xpos = i > xc;
                  ypos = j > yc;
                     //Find the distance from the center
                  double xdif = i-xc;
                  double ydif = j-yc;
                     //The distance squared
                  double Rusquare = xdif * xdif + ydif * ydif;
                     //the angle from the center
                  double theta = Math.Atan2(ydif, xdif);
                     //find index for lookup table
                  int index = (int)(Math.Sqrt(Rusquare) / aFocalLinPixels * 1000);
                  if (index >= mFELimit) index = mFELimit - 1;
                     //calculated Rdistorted
                  double Rd = aFocalLinPixels * (double)mFisheyeCorrect[index]
                                        /mScaleFESize;
                     //calculate x and y distances
                  double xdelta = Math.Abs(Rd*Math.Cos(theta));
                  double ydelta = Math.Abs(Rd * Math.Sin(theta));
                     //convert to pixel coordinates
                  int xd = (int)(xc + (xpos ? xdelta : -xdelta));
                  int yd = (int)(yc + (ypos ? ydelta : -ydelta));
                  xd = Math.Max(0, Math.Min(xd, aImage.Width-1));
                  yd = Math.Max(0, Math.Min(yd, aImage.Height-1));
                     //set the corrected pixel value from the distorted image
                  correctedImage.SetPixel(i, j, aImage.GetPixel(xd, yd));
              }
          }
          return correctedImage;
      }
}