Homography Matrix

Can somebody please help me in understanding how to calculate an homography matrix in matlab.

[Lalit Patil]

I am finding homography matrix in another way and it is 3*3

So, i want to know that in image where to apply or what to do of this homography matrix.?

Why we are finding this matrix..?

Answer 1

[David Young]

You don't say what you are starting from. If you have a set of matched input and output points, one possible method is given here. A simple implementation is below.

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

If your initial data is in some other form, such as camera position parameters relative to the plane, please say.

EDIT - added:

To apply the resulting matrix to a set of points, you can use the following function.

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

[Sean Lawson]

Hi

I have a question regarding the final transform matrix v.

I want to apply the homography(projective transform) in Matlab, I use this approach to calculate the matrix v. However, v(3,3) ~= 1, whereas Matlab requires v(3,3) = 1, do you know how to fix this? Or just v/v(3,3) works?

[David Young]

Hi Sean, I'm adding a function to my answer with I hope deals with this question.

[Royi Avital]

What if you want to impose Affine Matrix (Last row to be [0, 0, 1])?

[Jason Rebello]

Hi, First of all thanks for the code it really helped. But can you explain to me why exactly are u reshaping only the last column. Im new to matlab so I dont really pick up on things fast. v = (reshape(U(:,9), 3, 3)).'; ?? Thanks, Jason

[Erez Farhan]

Hi, David- thanks for your post. Please note: I believe your solution doesn't account for normalization of the homography matrix- which can be done by dividing it by it's second singular value.

[pankhuri]

can this be applied for 2D to 3D projection??

[sushanth kalva]

i want to know the difference between transformation matrix and homography matrix

[Tianya Zhang]

svd() function returns U, S and V. h can be obtained from the last colum of V. Why you use U to get h in your code? That's confusing.... Is that an obvious mistake?

[Edwin Stanley]

Also curious about Tianya's comment above!

"svd() function returns U, S and V. h can be obtained from the last colum of V. Why you use U to get h in your code? That's confusing.... Is that an obvious mistake?"

[Albert Dayn]

In response to Tianya and Edwin's question, U is correct. It looks like in this function, h is constructed as the transpose of the usual matrix (each point is represented by 2 columns here, instead of 2 rows). If it were constructed as the link (for the SVD method) specifies, you would indeed use the last column of V.

[PENIEL EMANUEL]

how to get target physical coordinates in millimetres

[Wei Han]

Hi, I just want to thank for your code, it's really helped me!

[Sahana Srivathsa]

This is very helpful. I think equation for q needs to be slightly altered.

q = v * [x; y; ones(1, size(x,2))];

Answer 2

[Matt J]

Ue fitgeotrans().

tform = fitgeotrans(movingPoints,fixedPoints,'projective')

[Matt J]

You shouldn't be extracting the matrix. You should be using tform.transformPointsForward().