Reconstruct 3D-Coordinates in Camera Coordinate System from 2D - Pixels with side condition

Question

I am trying to reconstruct 3D-Coordinates from the 2D-Pixel-Coordinates in a Camera Picture using a side condition (in MatLab). I do have extrinsic and intrinsic camera parameters.

Using homogenous transformation I can transform 3D-Coordinates from a initial World Coordinate System to my Camera Corrdinate System. So here I have my extrinsic parameters in my Transform Matrix R_world_to_Camera:

R_world_to_Camera = [ r_11, r_12, r_13, t1;
r_21, r_22, r_23, t2;
r_31, r_32, r_33, t3;
0, 0, 0, 1];

For intrinsic parameters I used Caltech's "Camera Calibration Toolbox for MatLab" and got these parameters:

Calibration results (with uncertainties): 

 Focal Length:          fc = [ 1017.21523   1012.54901 ] ± [ NaN   NaN ] 
 Principal point:       cc = [ 319.50000   239.50000 ] ± [ NaN   NaN ] 
 Skew:             alpha_c = [ 0.00000 ] ± [ NaN  ]   => angle of pixel axes = 90.00000 ± NaN degrees 
 Distortion:            kc = [ 0.00000   0.00000   0.00000   0.00000  0.00000 ] ± [ NaN   NaN   NaN   NaN    NaN ] 
 Pixel error:          err = [ 0.11596   0.14469 ] 

 Note: The numerical errors are approximately three times the standard deviations (for reference).

So I then get the Camera-Calibration-Matrix K (3x3)

K = [1.017215234570303e+03, 0, 3.195000000000000e+02; 
0, 1.012549014668498e+03,2.395000000000000e+02; 
0, 0, 1.0000];

and using this I can calculate the 3D -> 2D - Projection-Matrix P (3x4) with:

P = K * [eye(3), zeros(3,1)];

When converting a Point in World-Coordinates [X, Y, Z]_World I transform it first to Camera-Coordinates and then project it to 2D:

% Transformation
P_world = [X; Y; Z; 1]; % homogenous coordinates in World coordinate System
P_camera = R_world_to_Camera * [X; Y; Z; 1];

% Projection
P_pixels = P * camera;
P_pixels = P_pixels / P_pixels(3); % normalize coordinates

So my question now is how to reverse these steps? As side condition I want to set the Z-Coordinate to be known (zero in world-coordinates). I tried the solution proposed here on Stackoverflow, but somehow I get wrong coordinates. Any idea? Every help is appreciated a lot!!

Answer 1

You cannot reverse the step in general: depth and scale information is lost when 3D points are projected onto a 2D image. However if, as you indicate, all your 3D points are on the Z=0 plane, then getting them back from their projections is trivial: compute the inverse Ki = K^-1 of the camera matrix, and apply it to the image points in homogeneous coordinates.

P_camera = Ki * [u, v, 1]'

where [u, v] are the image coordinates, and the apostrophe denotes transposition. The 3D points you want lie on the rays from the camera centre to the P_camera's. Express both in world coordinates:

P_world = [R|t]_camera_to_world * [P_camera, 1]'

C_world = [R|t]_camera_to_world * [0, 0, 0, 1]'

where [R|t] is the 4x4 coordinate transform. Now, the set of points on each ray is expressed as

P = C_world + lambda * P_world;

where lambda is a scalar (the coordinate along the ray). You can now impose the condition that P(3) = 0 to find the value of lambda that places your points on the Z = 0 plane.

Answer 2

thanks to some red wine and thorough reading I found the answer in a (german) Master Thesis :) My "testcode" is as follows:

% Clean-Up First 
clear all; 
close all; 
clc; 

% Caamera-Calibration-Matrix
 K = [1.017215234570303e+03, 0, 3.195000000000000e+02, 0; 
0, 1.012549014668498e+03,2.395000000000000e+02, 0; 
0, 0, 1.0000, 0; 
0, 0, 0, 0]; 

% Transforma Matrix from 3D-World-Coordinate System to 3D-Camera-Coordinate System (Origin on CCD-Chip) 
% 
% The Camera is oriented "looking" into positive X-Direction of the World-Coordinate-System. On the picture,
% positive Y-Direction will be to the left, positive Z-Direction to the top. (right hand coordinate system!)
 R_World_to_Cam = [-0.0113242625465167   -0.999822053685344   0.0151163536128891   141.173585444427; 
0.00842007509644635   -0.0152123858102325   -0.999848810645587   1611.96528372161; 
0.999900032304804   -0.0111955728474261   0.00859117128537919   847.090629282911; 
0   0   0   1]; 

% Projection- and Transforma Matrix P 
 P = K * R_World_to_Cam; 

% arbitrary Points X_World in World-Coordinate-System [mm] (homogenous Coordinates) 
% forming a square of size 10m x 4m 
 X_World_1 = [20000; 2000; 0; 1]; 
 X_World_2 = [20000; -2000; 0; 1]; 
 X_World_3 = [10000; 2000; 0; 1]; 
 X_World_4 = [10000; -2000; 0; 1]; 

% Transform and Project from 3D-World -> 2D-Picture 
 X_Pic_1 = P * X_World_1; 
 X_Pic_2 = P * X_World_2; 
 X_Pic_3 = P * X_World_3; 
 X_Pic_4 = P * X_World_4; 

% normalize homogenous Coordinates (3rd Element has to be 1!) 
 X_Pic_1 = X_Pic_1 / X_Pic_1(3); 
 X_Pic_2 = X_Pic_2 / X_Pic_2(3); 
 X_Pic_3 = X_Pic_3 / X_Pic_3(3); 
 X_Pic_4 = X_Pic_4 / X_Pic_4(3); 

% Now for reverse procedure take arbitrary points in Camera-Picture... 
% (for simplicity, take points from above and "go" 30px to the right and 40px down) 
 X_Pic_backtransform_1 = X_Pic_1(1:3) + [30; 40; 0]; 
 X_Pic_backtransform_2 = X_Pic_2(1:3) + [30; 40; 0]; 
 X_Pic_backtransform_3 = X_Pic_3(1:3) + [30; 40; 0]; 
 X_Pic_backtransform_4 = X_Pic_4(1:3) + [30; 40; 0]; 

% ... and transform back following the formula from the Master Thesis (in German):
% Ilker Savas, "Entwicklung eines Systems zur visuellen Positionsbestimmung von Interaktionspartnern" 
 M_Mat = P(1:3,1:3);                 % Matrix M is the "top-front" 3x3 part 
 p_4 = P(1:3,4);                     % Vector p_4 is the "top-rear" 1x3 part 
 C_tilde = - inv( M_Mat ) * p_4;     % calculate C_tilde 

% Invert Projection with Side-Condition ( Z = 0 ) and Transform back to 
% World-Coordinate-System 
 X_Tilde_1 = inv( M_Mat ) * X_Pic_backtransform_1; 
 X_Tilde_2 = inv( M_Mat ) * X_Pic_backtransform_2; 
 X_Tilde_3 = inv( M_Mat ) * X_Pic_backtransform_3; 
 X_Tilde_4 = inv( M_Mat ) * X_Pic_backtransform_4; 

 mue_N_1 = -C_tilde(3) / X_Tilde_1(3); 
 mue_N_2 = -C_tilde(3) / X_Tilde_2(3); 
 mue_N_3 = -C_tilde(3) / X_Tilde_3(3); 
 mue_N_4 = -C_tilde(3) / X_Tilde_4(3); 

% Do the inversion of above steps... 
 X_World_backtransform_1 = mue_N_1 * inv( M_Mat ) * X_Pic_backtransform_1 + C_tilde; 
 X_World_backtransform_2 = mue_N_2 * inv( M_Mat ) * X_Pic_backtransform_2 + C_tilde; 
 X_World_backtransform_3 = mue_N_3 * inv( M_Mat ) * X_Pic_backtransform_3 + C_tilde; 
 X_World_backtransform_4 = mue_N_4 * inv( M_Mat ) * X_Pic_backtransform_4 + C_tilde; 


% Plot everything
figure(1); 
% First Bird Perspective of World-Coordinate System... 
subplot(1,2,1); 
xlabel('Y-World'); 
ylabel('X-World'); 
grid on; 
axis([-3000 3000 0 22000]); 
hold on; 


plot( -X_World_1(2), X_World_1(1), 'bo' ); 
plot( -X_World_2(2), X_World_2(1), 'bo' ); 
plot( -X_World_3(2), X_World_3(1), 'bo' ); 
plot( -X_World_4(2), X_World_4(1), 'bo' ); 
line([-X_World_1(2) -X_World_2(2) -X_World_4(2) -X_World_3(2) -X_World_1(2)], [X_World_1(1) X_World_2(1) X_World_4(1) X_World_3(1) X_World_1(1)], 'Color', 'blue' ); 

plot( -X_World_backtransform_1(2), X_World_backtransform_1(1), 'ro' ); 
plot( -X_World_backtransform_2(2), X_World_backtransform_2(1), 'ro' ); 
plot( -X_World_backtransform_3(2), X_World_backtransform_3(1), 'ro' ); 
plot( -X_World_backtransform_4(2), X_World_backtransform_4(1), 'ro' ); 
line([-X_World_backtransform_1(2) -X_World_backtransform_2(2) -X_World_backtransform_4(2) -X_World_backtransform_3(2) -X_World_backtransform_1(2)], [X_World_backtransform_1(1) X_World_backtransform_2(1) X_World_backtransform_4(1) X_World_backtransform_3(1) X_World_backtransform_1(1)], 'Color', 'red' ); 


hold off; 

% ...then the camera picture (perspective!)
subplot(1,2,2); 
hold on; 
image(ones(480,640).*255); 
colormap(gray(256)); 
axis([0 640 -480 0]); 
line([X_Pic_1(1) X_Pic_2(1) X_Pic_4(1) X_Pic_3(1) X_Pic_1(1)], -1*[X_Pic_1(2) X_Pic_2(2) X_Pic_4(2) X_Pic_3(2) X_Pic_1(2)], 'Color', 'blue' ); 
line([X_Pic_backtransform_1(1) X_Pic_backtransform_2(1) X_Pic_backtransform_4(1) X_Pic_backtransform_3(1) X_Pic_backtransform_1(1)], -1*[X_Pic_backtransform_1(2) X_Pic_backtransform_2(2) X_Pic_backtransform_4(2) X_Pic_backtransform_3(2) X_Pic_backtransform_1(2)], 'Color', 'red' ); 
hold off;