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I see the official document that the Matlab R2019a version already supports estimating the camera projection matrix, The condition is that at least 6 sets of points in the same plane can be solved, but the problem is whether the camera matrix P can be inferred to obtain the camera intrinsics K, the rotation matrix R, and the translation. Vector T?

I saw that a third party has a function to decompose it. I don't know if it is reliable. I pose here:

function [K, Rc_w, Pc, pp, pv] = DecomposeCamera(P)

% DECOMPOSECAMERA Decomposition of a camera projection matrix

%

% Usage: [K, Rc_w, Pc, pp, pv] = decomposecamera(P);

%

% P is decomposed into the form P = K*[R -R*Pc]

%

% Argument: P - 3 x 4 camera projection matrix

% Returns:

% K - Calibration matrix of the form

% | ax s ppx |

% | 0 ay ppy |

% | 0 0 1 |

%

% Where:

% ax = f/pixel_width and ay = f/pixel_height,

% ppx and ppy define the principal point in pixels,

% s is the camera skew.

% Rc_w - 3 x 3 rotation matrix defining the world coordinate frame

% in terms of the camera frame. Columns of R transposed define

% the directions of the camera X, Y and Z axes in world

% coordinates.

% Pc - Camera centre position in world coordinates.

% pp - Image principal point.

% pv - Principal vector from the camera centre C through pp

% pointing out from the camera. This may not be the same as

% R'(:,3) if the principal point is not at the centre of the

% image, but it should be similar.

%

% See also: RQ3

% Reference: Hartley and Zisserman 2nd Ed. pp 155-164

% Copyright (c) 2010 Peter Kovesi

% Centre for Exploration Targeting

% School of Earth and Environment

% The University of Western Australia

% peter.kovesi at uwa edu au

%

% Permission is hereby granted, free of charge, to any person obtaining a copy

% of this software and associated documentation files (the "Software"), to deal

% in the Software without restriction, subject to the following conditions:

%

% The above copyright notice and this permission notice shall be included in

% all copies or substantial portions of the Software.

%

% October 2010 Original version

% November 2013 Description of rotation matrix R corrected (transposed)

% Projection matrix from Hartley and Zisserman p 163 used for testing

if ~exist('P','var')

P = [ 3.53553e+2 3.39645e+2 2.77744e+2 -1.44946e+6

-1.03528e+2 2.33212e+1 4.59607e+2 -6.32525e+5

7.07107e-1 -3.53553e-1 6.12372e-1 -9.18559e+2];

end

% Convenience variables for the columns of P

p1 = P(:,1);

p2 = P(:,2);

p3 = P(:,3);

p4 = P(:,4);

M = [p1 p2 p3];

m3 = M(3,:)';

% Camera centre, analytic solution

X = det([p2 p3 p4]);

Y = -det([p1 p3 p4]);

Z = det([p1 p2 p4]);

T = -det([p1 p2 p3]);

Pc = [X;Y;Z;T];

Pc = Pc/Pc(4);

Pc = Pc(1:3); % Make inhomogeneous

% Pc = null(P,'r'); % numerical way of computing C

% Principal point

pp = M*m3;

pp = pp/pp(3);

pp = pp(1:2); % Make inhomogeneous

% Principal ray pointing out of camera

pv = det(M)*m3;

pv = pv/norm(pv);

% Perform RQ decomposition of M matrix. Note that rq3 returns K with +ve

% diagonal elements, as required for the calibration matrix.

[K,Rc_w] = RQ3(M);

% Check that R is right handed, if not give warning

if dot(cross(Rc_w(:,1), Rc_w(:,2)), Rc_w(:,3)) < 0

warning('Note that rotation matrix is left handed');

end

subfunction is here:

function [R,Q] = RQ3(A)

% RQ3 RQ decomposition of 3x3 matrix

%

% Usage: [R,Q] = rq3(A)

%

% Argument: A - 3 x 3 matrix

% Returns: R - Upper triangular 3 x 3 matrix

% Q - 3 x 3 orthonormal rotation matrix

% Such that R*Q = A

%

% The signs of the rows and columns of R and Q are chosen so that the diagonal

% elements of R are +ve.

%

% See also: DECOMPOSECAMERA

% Follows algorithm given by Hartley and Zisserman 2nd Ed. A4.1 p 579

% Copyright (c) 2010 Peter Kovesi

% Centre for Exploration Targeting

% School of Earth and Environment

% The University of Western Australia

% peter.kovesi at uwa edu au

%

% Permission is hereby granted, free of charge, to any person obtaining a copy

% of this software and associated documentation files (the "Software"), to deal

% in the Software without restriction, subject to the following conditions:

%

% The above copyright notice and this permission notice shall be included in

% all copies or substantial portions of the Software.

%

% October 2010

% February 2014 Incorporated modifications suggested by Mathias Rothermel to

% avoid potential division by zero problems

if ~all(size(A)==[3 3])

error('A must be 3x3');

end

eps = 1e-10;

% Find rotation Qx to set A(3,2) to 0

A(3,3) = A(3,3) + eps;

c = -A(3,3)/sqrt(A(3,3)^2+A(3,2)^2);

s = A(3,2)/sqrt(A(3,3)^2+A(3,2)^2);

Qx = [1 0 0; 0 c -s; 0 s c];

R = A*Qx;

% Find rotation Qy to set A(3,1) to 0

R(3,3) = R(3,3) + eps;

c = R(3,3)/sqrt(R(3,3)^2+R(3,1)^2);

s = R(3,1)/sqrt(R(3,3)^2+R(3,1)^2);

Qy = [c 0 s; 0 1 0;-s 0 c];

R = R*Qy;

% Find rotation Qz to set A(2,1) to 0

R(2,2) = R(2,2) + eps;

c = -R(2,2)/sqrt(R(2,2)^2+R(2,1)^2);

s = R(2,1)/sqrt(R(2,2)^2+R(2,1)^2);

Qz = [c -s 0; s c 0; 0 0 1];

R = R*Qz;

Q = Qz'*Qy'*Qx';

% Adjust R and Q so that the diagonal elements of R are +ve

for n = 1:3

if R(n,n) < 0

R(:,n) = -R(:,n);

Q(n,:) = -Q(n,:);

end

end

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