homography

capture_screenshot.png

fig.png

homography_example.m

function homography_example

    close all
    hold on
    axis([-100 800 -100 800]);
    axis square;
    set(gca, 'ydir', 'reverse');

    in = [ ...
         92 711 601 188; ...
        398 398  93  90];

    out = [ ...
         10 710 710  10; ...
        710 710  10  10];

    plot( ...
        [in(1,:); ...
         out(1,:)], ...
        [in(2,:); ...
         out(2,:)], ...
        'y--');
    plot(in(1,:), in(2,:), 'bo');
    plot(out(1,:), out(2,:), 'bx');

    v = homography_solve(in, out)

    in2 = [ ...
          114 199 297 395 495 592 213 305 396 487 587 226 310 397 482 565 238 317 396 476 555 249 324 396 471 547  91 131 149 163 177 189 258 328 396 467 538 602 617 632 645 664 684 707 606 503 396 289 183; ...
          325 329 332 333 334 334 267 269 269 270 270 214 213 214 217 218 166 167 169 170 171 125 126 126 129 130 394 265 211 166 125  93  92  93  92  93  94  95 131 173 219 272 333 398 400 401 401 400 398];

    out2 = homography_transform(in2, v)

    plot( ...
        [in2(1,:); ...
         out2(1,:)], ...
        [in2(2,:); ...
         out2(2,:)], ...
        'y--');
    plot(in2(1,:), in2(2,:), 'go');
    plot(out2(1,:), out2(2,:), 'rx');

end

% ------------------------------------------------------

% following two functions copied from
% http://www.mathworks.com/matlabcentral/answers/26141-homography-matrix

function v = homography_solve(pin, pout)
    % HOMOGRAPHY_SOLVE finds a homography from point pairs
    %   V = HOMOGRAPHY_SOLVE(PIN, POUT) takes a 2xN matrix of input vectors and
    %   a 2xN matrix of output vectors, and returns the homogeneous
    %   transformation matrix that maps the inputs to the outputs, to some
    %   approximation if there is noise.
    %
    %   This uses the SVD method of
    %   http://www.robots.ox.ac.uk/%7Evgg/presentations/bmvc97/criminispaper/node3.html
    % David Young, University of Sussex, February 2008
    if ~isequal(size(pin), size(pout))
        error('Points matrices different sizes');
    end
    if size(pin, 1) ~= 2
        error('Points matrices must have two rows');
    end
    n = size(pin, 2);
    if n < 4
        error('Need at least 4 matching points');
    end
    % Solve equations using SVD
    x = pout(1, :); y = pout(2,:); X = pin(1,:); Y = pin(2,:);
    rows0 = zeros(3, n);
    rowsXY = -[X; Y; ones(1,n)];
    hx = [rowsXY; rows0; x.*X; x.*Y; x];
    hy = [rows0; rowsXY; y.*X; y.*Y; y];
    h = [hx hy];
    if n == 4
        [U, ~, ~] = svd(h);
    else
        [U, ~, ~] = svd(h, 'econ');
    end
    v = (reshape(U(:,9), 3, 3)).';
end

function y = homography_transform(x, v)
    % HOMOGRAPHY_TRANSFORM applies homographic transform to vectors
    %   Y = HOMOGRAPHY_TRANSFORM(X, V) takes a 2xN matrix, each column of which
    %   gives the position of a point in a plane. It returns a 2xN matrix whose
    %   columns are the input vectors transformed according to the homography
    %   V, represented as a 3x3 homogeneous matrix.
    q = v * [x; ones(1, size(x,2))];
    p = q(3,:);
    y = [q(1,:)./p; q(2,:)./p];
end