MATLAB image transformation

Question

The following code descripes the affine transformation of an image, I,

T = [sx, 0, 0;...
      0, sy, 0;...
      0, 0, 1];
tform = maketform('affine', T);
[J,cdata,rdata] = imtransform(I,tform);

After obtaining the transformed image, J, I want to find the appropriate pixel value of I(5,5) on image J.

What should I do?

My code is

function test()
    sx = 0.5; sy = 1;
    theta = pi/4;
    Ts = [sx, 0, 0;...
    0, sy, 0;...
    0, 0, 1];
    Tr = [cos(theta) -sin(theta) 0;...
          sin(theta) cos(theta), 0; ...
          0, 0, 1];
    T = Ts*Tr;

    I = imread('image_0002.jpg');
    tform = maketform('affine', T);
    [J,xdata,ydata] = imtransform(I,tform);
    h = ydata(2) - ydata(1);
    w = xdata(2) - xdata(1);

    %%
    gridOx = meshgrid(1:50:size(I,2), 1:50:size(I,1));
    gridOy = meshgrid(1:50:size(I,1), 1:50:size(I,2))';
    go = [gridOy(:), gridOx(:), ones(length(gridOx(:)),1)];

    for i=1:size(go,1)
        I = makept(I, go(i,1), go(i,2));
    end
    imshow(I);

    %%
    gt = T*go';

    TL = T*[1,1,1]';
    BL = T*[size(I,1), 1, 1]';
    TR = T*[1, size(I,2), 1]';
    BR = T*[size(I,1), size(I,2), 1]';
    minr = min([TL(1), TR(1), BL(1), BR(1)]);
    minc = min([TL(2), TR(2), BL(2), BR(2)]);

    %%
    gt = int32(round(gt));
    r = gt(1,:) - minr+1;
    c = gt(2,:) - minc+1;

    figure,hold on;
    for i=1:length(r)
        J = makept(J, r(i), c(i));
        plot(gt(1, i), gt(2, i));
    end
    figure, imshow(J);

    close all;
end

function I = makept(I, r, c)
    mr = max(r-2,1);
    mc = max(c-2,1);
    maxr = min(r+2,size(I,1));
    maxc = min(c+2,size(I,2));

    I(mr:maxr, mc:maxc,:) = 0;
    I(mr:maxr, mc:maxc,1) = 255;
end

Answer 1

I found another way.. use 'tformfwd'.. we need to apply a scaling as well as explained in http://blogs.mathworks.com/steve/2006/02/14/spatial-transformations-maketform-tformfwd-and-tforminv/

         function [new_im, grn, gcn] = transformImage(I, gr, gc, TM)
              [new_im xdata ydata] = imtransform(I, TM);
              w = xdata(2)-xdata(1) +1;
              h = ydata(2)-ydata(1)+1;
              scalex = size(new_im,2)/w;
              scaley = size(new_im,1)/h;

              % cartesian coordinates
              go = [gc(:), gr(:)];

              %% cartesian coordinates
              NPt = tformfwd(TM, go);

              % convert to image coordinates
              NPtImage(:,1) = NPt(:,1) - xdata(1) + 1;
              NPtImage(:,2) = NPt(:,2) - ydata(1) + 1;
              NPtImage(:,1) = NPtImage(:,1)*scalex;
              NPtImage(:,2) = NPtImage(:,2)*scaley;
              NPtImage = round(NPtImage);

              grn = NPtImage(:,2);
              gcn = NPtImage(:,1);
              grn = reshape(grn, size(gr,1), size(gr,2));
              gcn = reshape(gcn, size(gc,1), size(gc,2));

              showimgs = false;
              if showimgs
                  I = uint8(I);
                  new_im = uint8(new_im);

                  figure, imshow(I), hold on;
                  for i=1:size(go,1)
                      plot(go(i,1), go(i,2), 'r*');
                  end

                  figure, imshow(new_im, 'XData', xdata, 'YData', ydata);
                  hold on
                  axis on
                  for i=1:size(go,1)
                      plot(NPt(i,1), NPt(i,2), 'r*')
                  end

                  tmpim = new_im;
                  for i=1:size(NPtImage,1)
                      tmpim = makept(tmpim, NPtImage(i,2), NPtImage(i,1), 'b');
                  end
                  figure, imshow(tmpim);
              end

              end


              function I = makept(I, r, c, color)
              mr = max(r-2,1);
              mc = max(c-2,1);
              maxr = min(r+2,size(I,1));
              maxc = min(c+2,size(I,2));

              I(mr:maxr, mc:maxc,:) = 0;

              if strcmp(color, 'r')
                  I(mr:maxr, mc:maxc,1) = 255;
              else
                  I(mr:maxr, mc:maxc,3) = 255;
              end
              end

Answer 2

You transform the pixel coordinates just like you transformed the image.

appropriatePixel = J(T*[5;5;1]);

EDIT

The code you posted was already quite close. Always remember that with images, the origin for images is in the top left and that the order of image dimensions is columns-rows, unlike with matrices. You took this into account in some places, but not everywhere.

The code below should fix the problem. gt = T*go was changed to use the transpose of T, and minc and minr were swapped.

function airplane()
       close all
       sx = 0.5; sy = 1;
       theta = pi/4;
       Ts = [sx, 0, 0;...
       0, sy, 0;...
       0, 0, 1];
       Tr = [cos(theta) -sin(theta) 0;...
           sin(theta) cos(theta), 0; ...
           0, 0, 1];
       T = Ts*Tr;

       I = imread('airplane2.png');
       tform = maketform('affine', T);
       [J,xdata,ydata] = imtransform(I,tform);

        %% 
       gridOx = meshgrid(1:50:size(I,2), 1:50:size(I,1));
       gridOy = meshgrid(1:50:size(I,1), 1:50:size(I,2))';
       go = [gridOx(:), gridOy(:), ones(length(gridOx(:)),1)];

       figure; imshow(I); hold on;
       scatter(go(:,1), go(:,2), 25, 'rs', 'filled');

       %%
       gt = T'*go'; 

       TL = T*[1,1,1]';
       BL = T*[size(I,1), 1, 1]';
       TR = T*[1, size(I,2), 1]';
       BR = T*[size(I,1), size(I,2), 1]';
       minc = min([TL(1), TR(1), BL(1), BR(1)]);
       minr = min([TL(2), TR(2), BL(2), BR(2)]);


       %%
       gt = int32(round(gt));
       r = gt(1,:) - minr+1;
       c = gt(2,:) - minc+1;

       figure; imshow(J);hold on;
       scatter(c, r, 25, 'rs', 'filled');

end

Result: