附：相关需要的工具函数源代码（投影函数、校正矩阵计算等）见最下面
图像畸变一般有两种，第一种是透镜本身的形状有问题，使得图像发生径向畸变；第二种是透镜安装时与成像平面之间不完全平行，导致图像发生切向畸变。畸变会导致图像中物体的形状与实际物体的形状不相同，比如直线变成曲线、矩形拉长等。故而想要得到实际真实图像，必须要根据之前对相机进行标定得到的参数对图像进行畸变的去除。
(a)径向畸变
(b)切向畸变
图像畸变示意
畸变去除的基本思路为:
function [img_correct,newOrigin] = ImgDistortionCorrection(img,camera,OutputView)
%undistortImage Correct image for lens distortion.
% [J, newOrigin] = undistortImage(I, intrinsics) removes lens distortion
% from image I, and returns the result as image J. I can be a grayscale
% or a truecolor image. intrinsics is either cameraParameters or
% cameraIntrinsics object.
%
% newOrigin is a 2-element vector containing the [x,y] location of the
% origin of the output image J in the intrinsic coordinates of the input
% image I. Before using extrinsics, pointsToWorld, or triangulate
% functions you must add newOrigin to the coordinates of points detected
% in undistorted image J in order to transform them into the intrinsic
% coordinates of the original image I.
% If 'OutputView' is set to 'same', then newOrigin is [0, 0].
if OutputView=='same'
% Para init
[X,Y] = meshgrid(0:size(img,2)-1,0:size(img,1)-1);
c_x = camera.center(1,1);
c_y = camera.center(1,2);
f_x = camera.focal(1,1);
f_y = camera.focal(1,2);
k1 = camera.dis(1,1);
k2 = camera.dis(1,2);
g1 = camera.dis(1,3);
g2 = camera.dis(1,4);
k3 = camera.dis(1,5);
% calculate point in camrera coordinate system
x1 = (X-c_x)/f_x;
y1 = (Y-c_y)/f_y;
% calculate dis_point in camrera coordinate system
r2 = x1.^2+y1.^2;
x2 = x1.*(1+k1*r2+k2*r2.^2+k3*r2.^3)+2*g1*x1.*y1+g2*(r2+2*x1.^2);
y2 = y1.*(1+k1*r2+k2*r2.^2+k3*r2.^3)+2*g2*x1.*y1+g1*(r2+2*y1.^2);
% (u, v) undis (u_d, v_d)
% calculate dis point in image coordinate system
u_d = f_x*x2 + c_x+1;
v_d = f_y*y2 + c_y+1;
% interp to get undis image
img_correct = interp2(img, u_d, v_d);
newOrigin = [0,0];
end
end
注意事项：
（1）与OPENCV结果进行过对比，matlab中的矩阵序号需要减去1以成为坐标，但是仍然存在一定误差，具体原因不明确，希望有大佬给出解释。
进行畸变校正获取到真实图像后，为了方便下一步的立体匹配，还需要进行极线校正，即将左右两个相机平面变成一个平面，然后每一行像素相互对应。（下文代码大部分搬运自matlab标定工具箱）
% R、T are right camera to left camera,R*C2+T=C1
% P2=RP1+T
% Bring the 2 cameras in the same orientation by rotating them "minimally":
r_r = rodrigues(-camera2.R/2);
r_l = r_r';
t = r_r * camera2.T';
% Rotate both cameras so as to bring the translation vector in alignment with the (1;0;0) axis:
if abs(t(1)) > abs(t(2))
type_stereo = 0;
uu = [1;0;0]; % Horizontal epipolar lines
else
type_stereo = 1;
uu = [0;1;0]; % Vertical epipolar lines
end
if dot(uu,t)<0
uu = -uu; % Swtich side of the vector
end
% rotate to make the epipolar lines of the two camera images horizontal,the x-axis coincides with t.
ww = cross(t,uu);
ww = ww/norm(ww);
ww = acos(abs(dot(t,uu))/(norm(t)*norm(uu)))*ww; % Rotation angle modulo times the rotation axis
R2 = rodrigues(ww);
% Computation of the *new* intrinsic parameters for both left and right cameras:
% Vertical focal length *MUST* be the same for both images (here, we are trying to find a focal length that retains as much information contained in the original distorted images):
fc_y_new = min(camera1.focal(1,2),camera2.focal(1,2));
% For simplicity, let's pick the same value for the horizontal focal length as the vertical focal length (resulting into square pixels):
fc_left_new = round([fc_y_new;fc_y_new]);
fc_right_new = round([fc_y_new;fc_y_new]);
（2）计算前两步旋转变换之后，如果想将相机坐标系得到的中心点变换到图像坐标系，偏移应该是多少，以确保得到更多图像。首先将原始图像四个角点变换到归一化相机坐标系，然后将其投影到新的图像坐标系（旋转变换后的图像坐标系）下，用理论图像中心点减去四个角点的平均值，即可得到偏移值 c x , c y c_x,c_y cx,cy,为了考虑左右都取较多图像，因此需要取平均值，根据是水平校正还是竖直校正选取平均 c x 或 c y c_x或c_y cx或cy。
% Select the new principal points to maximize the visible area in the rectified images
% normalize_pixel: Transform the four corners of the original image to the normalized camera plane
% project_points2:Project the four corners of the normalized plane to the transformed camera coordinate plane, and set new f_x, f_y and c_x, c_y.
cc_left_new = [(nx-1)/2;(ny-1)/2] - mean(project_points2([normalize_pixel([0 nx-1 nx-1 0; 0 0 ny-1 ny-1],fc_left,cc_left,kc_left,alpha_c_left);[1 1 1 1]],rodrigues(R_L),zeros(3,1),fc_left_new,[0;0],zeros(5,1),0),2);
cc_right_new = [(nx-1)/2;(ny-1)/2] - mean(project_points2([normalize_pixel([0 nx-1 nx-1 0; 0 0 ny-1 ny-1],fc_right,cc_right,kc_right,alpha_c_right);[1 1 1 1]],rodrigues(R_R),zeros(3,1),fc_right_new,[0;0],zeros(5,1),0),2);
% For simplivity, set the principal points for both cameras to be the average of the two principal points.
if ~type_stereo
%-- Horizontal stereo
cc_y_new = (cc_left_new(2) + cc_right_new(2))/2;
cc_left_new = [cc_left_new(1);cc_y_new];
cc_right_new = [cc_right_new(1);cc_y_new];
else
%-- Vertical stereo
cc_x_new = (cc_left_new(1) + cc_right_new(1))/2;
cc_left_new = [cc_x_new;cc_left_new(2)];
cc_right_new = [cc_x_new;cc_right_new(2)];
end
% Of course, we do not want any skew or distortion after rectification:
（3）根据最新参数，重新计算图像变换以及插值。
% Pre-compute the necessary indices and blending coefficients to enable quick rectification:
% The original image is changed in the function when Irec_junk_left is calculated, so it cannot be directly used as the corrected image
[Irec_junk_left,ind_new_left,ind_1_left,ind_2_left,ind_3_left,ind_4_left,a1_left,a2_left,a3_ left,a4_left] = rect_index(zeros(ny,nx),R_L,fc_left,cc_left,kc_left,alpha_c_left,KK_left_new);
[Irec_junk_right,ind_new_right,ind_1_right,ind_2_right,ind_3_right,ind_4_right,a1_right,a2_right,a3_right,a4_right] = rect_index(zeros(ny,nx),R_R,fc_right,cc_right,kc_right,alpha_c_right,KK_right_new);
clear Irec_junk_left Irec_junk_right
%图像校正
img_left_rectified = zeros(ny,nx);
img_left_rectified(ind_new_left) = a1_left .* img_left(ind_1_left) + a2_left .* img_left(ind_2_left) + a3_left .* img_left(ind_3_left) + a4_left .* img_left(ind_4_left);
完整函数代码如如下
function [img_left_rectified, img_right_rectified,Q,R1] = PolarlineCorrection(img_left, ...
img_right, camera1,camera2,flag,alpha,type_stereo)
% rectifyStereoImages Rectifies a pair of stereo images.
% [img_left_rectified, img_left_rectified] = rectifyStereoImages(img_left, img_right, camera1,camera2)
% rectifies img_left and img_right, a pair of truecolor or grayscale stereo images.
% camera1 and camera2 are stereoParameters object containing the parameters of
% the stereo camera system. img_left_rectified and img_right_rectified are the rectified images.
%
% param flags Operation flags that may be zero or CALIB_ZERO_DISPARITY . If the flag is set CALIB_ZERO_DISPARITY,
% the function makes the principal points of each camera have the same pixel coordinates in the
% rectified views. And if the flag is not set, the function may still shift the images in the
% horizontal or vertical direction (depending on the orientation of epipolar lines) to maximize the
% useful image area.
%
% param alpha Free scaling parameter. If it is -1 or absent, the function performs the default
% scaling. Otherwise, the parameter should be between 0 and 1. alpha=0 means that the rectified
% images are zoomed and shifted so that only valid pixels are visible (no black areas after
% rectification). alpha=1 means that the rectified image is decimated and shifted so that all the
% pixels from the original images from the cameras are retained in the rectified images (no source
% image pixels are lost). Obviously, any intermediate value yields an intermediate result between
% those two extreme cases.
if flag==0 && alpha == -1
%Test: init para
alpha_c_left = 0;
kc_left =zeros(1,5);
alpha_c_right = 0;
kc_right =zeros(1,5);
fc_left =camera1.focal';
fc_right =camera2.focal';
cc_left = camera1.center';
cc_right = camera2.center';
% R、T are right camera to left camera,R*C2+T=C1
% P2=RP1+T
% Bring the 2 cameras in the same orientation by rotating them "minimally":
r_r = rodrigues(-camera2.R/2);
r_l = r_r';
t = r_r * camera2.T';
% Rotate both cameras so as to bring the translation vector in alignment with the (1;0;0) axis:
if abs(t(1)) > abs(t(2))
type_stereo = 0;
uu = [1;0;0]; % Horizontal epipolar lines
else
type_stereo = 1;
uu = [0;1;0]; % Vertical epipolar lines
end
if dot(uu,t)<0
uu = -uu; % Swtich side of the vector
end
% rotate to make the epipolar lines of the two camera images horizontal,the x-axis coincides with t.
ww = cross(t,uu);
ww = ww/norm(ww);
ww = acos(abs(dot(t,uu))/(norm(t)*norm(uu)))*ww; % Rotation angle modulo times the rotation axis
R2 = rodrigues(ww);
% Global rotations to be applied to both views:
R_R = R2 * r_r;
R_L = R2 * r_l;
% The resulting rigid motion between the two cameras after image rotations (substitutes of om, R and T):
R_new = eye(3);
om_new = zeros(3,1);
T_new = R_R*camera2.T';
nx = size(img_left,2);
ny = size(img_left,1);
% Computation of the *new* intrinsic parameters for both left and right cameras:
% Vertical focal length *MUST* be the same for both images (here, we are trying to find a focal length that retains as much information contained in the original distorted images):
fc_y_new = min(camera1.focal(1,2),camera2.focal(1,2));
% For simplicity, let's pick the same value for the horizontal focal length as the vertical focal length (resulting into square pixels):
fc_left_new = round([fc_y_new;fc_y_new]);
fc_right_new = round([fc_y_new;fc_y_new]);
% Select the new principal points to maximize the visible area in the rectified images
% normalize_pixel: Transform the four corners of the original image to the normalized camera plane
% project_points2:Project the four corners of the normalized plane to the transformed camera coordinate plane, and set new f_x, f_y and c_x, c_y.
cc_left_new = [(nx-1)/2;(ny-1)/2] - mean(project_points2([normalize_pixel([0 nx-1 nx-1 0; 0 0 ny-1 ny-1],fc_left,cc_left,kc_left,alpha_c_left);[1 1 1 1]],rodrigues(R_L),zeros(3,1),fc_left_new,[0;0],zeros(5,1),0),2);
cc_right_new = [(nx-1)/2;(ny-1)/2] - mean(project_points2([normalize_pixel([0 nx-1 nx-1 0; 0 0 ny-1 ny-1],fc_right,cc_right,kc_right,alpha_c_right);[1 1 1 1]],rodrigues(R_R),zeros(3,1),fc_right_new,[0;0],zeros(5,1),0),2);
% For simplivity, set the principal points for both cameras to be the average of the two principal points.
if ~type_stereo
%-- Horizontal stereo
cc_y_new = (cc_left_new(2) + cc_right_new(2))/2;
cc_left_new = [cc_left_new(1);cc_y_new];
cc_right_new = [cc_right_new(1);cc_y_new];
else
%-- Vertical stereo
cc_x_new = (cc_left_new(1) + cc_right_new(1))/2;
cc_left_new = [cc_x_new;cc_left_new(2)];
cc_right_new = [cc_x_new;cc_right_new(2)];
end
% Of course, we do not want any skew or distortion after rectification:
alpha_c_left_new = 0;
alpha_c_right_new = 0;
kc_left_new = zeros(5,1);
kc_right_new = zeros(5,1);
% The resulting left and right camera matrices:
KK_left_new = [fc_left_new(1) fc_left_new(1)*alpha_c_left_new cc_left_new(1);0 fc_left_new(2) cc_left_new(2); 0 0 1];
KK_right_new = [fc_right_new(1) fc_right_new(1)*alpha_c_right cc_right_new(1);0 fc_right_new(2) cc_right_new(2); 0 0 1];
% The sizes of the images are the same:
nx_right_new = nx;
ny_right_new = ny;
nx_left_new = nx;
ny_left_new = ny;
% Save the resulting extrinsic and intrinsic paramters into a file:
fprintf(1,'Saving the *NEW* set of intrinsic and extrinsic parameters corresponding to the images *AFTER* rectification under Calib_Results_stereo_rectified.mat...\n\n');
save Calib_Results_stereo_rectified om_new R_new T_new fc_left_new cc_left_new kc_left_new alpha_c_left_new KK_left_new fc_right_new cc_right_new kc_right_new alpha_c_right_new KK_right_new nx_right_new ny_right_new nx_left_new ny_left_new
% Let's rectify the entire set of calibration images:
fprintf(1,'Pre-computing the necessary data to quickly rectify the images (may take a while depending on the image resolution, but needs to be done only once - even for color images)...\n\n');
% Pre-compute the necessary indices and blending coefficients to enable quick rectification:
% The original image is changed in the function when Irec_junk_left is calculated, so it cannot be directly used as the corrected image
[Irec_junk_left,ind_new_left,ind_1_left,ind_2_left,ind_3_left,ind_4_left,a1_left,a2_left,a3_ left,a4_left] = rect_index(zeros(ny,nx),R_L,fc_left,cc_left,kc_left,alpha_c_left,KK_left_new);
[Irec_junk_right,ind_new_right,ind_1_right,ind_2_right,ind_3_right,ind_4_right,a1_right,a2_right,a3_right,a4_right] = rect_index(zeros(ny,nx),R_R,fc_right,cc_right,kc_right,alpha_c_right,KK_right_new);
clear Irec_junk_left Irec_junk_right
%图像校正
img_left_rectified = zeros(ny,nx);
% rect_Img_left(ind_new_left) = uint8(a1_left .* src_Img_left(ind_1_left) + a2_left .* src_Img_left(ind_2_left) + a3_left .* src_Img_left(ind_3_left) + a4_left .* src_Img_left(ind_4_left));
img_left_rectified(ind_new_left) = a1_left .* img_left(ind_1_left) + a2_left .* img_left(ind_2_left) + a3_left .* img_left(ind_3_left) + a4_left .* img_left(ind_4_left);
img_right_rectified = zeros(ny,nx);
% rect_Img_right(ind_new_right) = uint8(a1_right .*
% src_Img_right(ind_1_right) + a2_right .* src_Img_right(ind_2_right) + a3_right .* src_Img_right(ind_3_right) + a4_right .* src_Img_right(ind_4_right));
img_right_rectified(ind_new_right) = a1_right .* img_right(ind_1_right) + a2_right .* img_right(ind_2_right) + a3_right .* img_right(ind_3_right) + a4_right .* img_right(ind_4_right);
Q = zeros(4,4);
Q(1,1)=1;
Q(1,4)=-cc_left_new(1);
Q(2,2)=1;
Q(2,4)=-cc_left_new(2);
Q(3,4)=fc_left_new(1);
Q(4,3)=-1/T_new(1);
Q(4,4) = (cc_left_new(1)-cc_right_new(1))/(T_new(1));
R1= R_L;
end
end
1.畸变校正与极线校正本质上是两件事，因此可以单独先进行畸变校正，再进行极线校正（畸变参数设置为0），也可以直接极线校正时顺带将畸变校正完成（输入畸变参数）。
2.旋转矩阵变换时，当坐标系2到坐标系1的旋转矩阵为R，平移为T，设同一点在坐标系1为
P
1
P_1
P1,坐标系2为
P
2
P_2
P2,则可得
P
2
=
R
P
1
+
T
P_2=RP_1+T
P2=RP1+T，而非
P
1
=
R
P
2
+
T
P_1=RP_2+T
P1=RP2+T。
3.此函数使用时需要调用部分maltba标定工具箱的计算函数，具体源代码可见https://download.csdn.net/download/zhangpan333/85445628
参考文章：
[1]畸变校正原理 https://blog.csdn.net/shyjhyp11/article/details/109506149
[2] 极线校正原理 https://blog.csdn.net/weixin_44083110/article/details/117635824