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Hello,

I have 10 points with x and y coordinate (cell locations imaged with microscope). I have used two different methods to image these cells, so I have two sets of coordinates that could be rotated, sheared, shifted, mirrored to each other. I would like to assign each point from one coordinate system to the matching point in the other coordinate system. How can I do that? In theory, some kind of transformation optimizing minimum distance between the points would be desirable.

In the end I would like to assign properties I obtained from the cells in one method to properties obtained with the other method.

Thank you.

Matt J
on 8 Oct 2019

Edited: Matt J
on 8 Oct 2019

If the shear component of the deformation isn't too strong, the matchpoints function defined below might work. It will sort the rows of B to correspond with A and also return the corresponding permutation indices. It relies on absor, mentioned by Bruno, which you will have to download

function [permIndices,Bsorted]=matchpoints(A,B)

%

%IN:

%

% A: an Nx2 matrix of points

% B: an Nx2 matrix of points from A transformed and unordered

%

%OUT:

%

% permIndices: permutation indices of rows of B thought to match A

% Bsorted: the Nx2 permuted version of B

La=landmarks(A);

Lb=landmarks(B);

B3=B.'; B3(3,:)=0;

reg=absor( Lb,La,'doScale',1);

C3=(reg.s*reg.R)*B3+reg.t;

C=C3(1:2,:).';

[~,permIndices]=pdist2(C,A,'euclidean','Smallest',1);

if nargout>1

Bsorted=B(permIndices,:);

end

function L=landmarks(P)

G=pdist2(P,P); G(~G)=nan;

[i,j]=find( G==min(G(:)) ,1);

I=P(i,:);

J=P(j,:);

K=mean(P,1);

if norm(I-K)<norm(J-K)

[I,J]=deal(J,I);

end

L=[I;J;K].';

L(3,:)=0;

end

end

Applying it to your example data, I obtain,

A = [376 455;421 489;465 537;353 512;355 535;329 571;377 593;417 598;482 575;355 634];

B = [168 88;107 138;69 194;126 229;163 232;199 267;272 239;228 210;235 155;215 68];

[permIndices,Bsorted]=matchpoints(A,B)

plot(graph(1:10,1:10),'EdgeColor','none','XData',A(:,1),'YData',A(:,2));

hold on

plot(graph(1:10,1:10),'EdgeColor','none','XData',Bsorted(:,1),'YData',Bsorted(:,2));

hold off

Matt J
on 2 May 2020

Here is a modification of the code that works on your example. You have a very large outlier in your coordinates, however, so it can't always be guaranteed that such cases would be robustly handled. Note that the code uses the FEX file minL1intlin.m (Download).

function [permIndices,Bsorted]=matchpoints(A,B)

%

%IN:

%

% A: an Nx2 matrix of points

% B: an Nx2 matrix of points from A transformed and unordered

%

%OUT:

%

% permIndices: permutation indices of rows of B thought to match A

% Bsorted: the Nx2 permuted version of B

La=landmarks(A);

Lb=landmarks(B);

B3=B.'; B3(3,:)=0;

reg=absor( Lb,La,'doScale',1);

C3=(reg.s*reg.R)*B3+reg.t;

C=C3(1:2,:).';

N=size(A,1);

e=ones(1,N);

E=speye(N);

Aeq=[ kron(E,e) ; kron(e,E) ]; beq=[e,e].';

Q=kron(E,C.');

d=reshape(A.',[],1);

lb=zeros(1,N^2);

ub=ones(1,N^2);

intcon=1:N^2;

P=minL1intlin(Q,d,intcon,[],[],Aeq,beq,lb,ub);

P=round(reshape(P,N,[]));

permIndices=(1:N)*P;

if nargout>1

Bsorted=B(permIndices,:);

end

function L=landmarks(P)

G=pdist2(P,P); G(~G)=nan;

[i,j]=find( G==min(G(:)) ,1);

I=P(i,:);

J=P(j,:);

K=mean(P,1);

if norm(I-K)<norm(J-K)

[I,J]=deal(J,I);

end

L=[I;J;K].';

L(3,:)=0;

end

end

Bruno Luong
on 8 Oct 2019

Look for the literature of image registration.

For simple rotation/scaling/translation you can use Matt J's submission

For more complex deformation, you need to apply spline deformation

There are a bunch of intermediate method for camera which take into account for camera cushion distortion or higher order. Pick one that is suitable for your need.

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