Create Composite 2-D Affine Transformations

This example shows how to create a composite of 2-D translation and rotation transformations. The order of the transformation matters, so there are two approaches to creating a composition:

1) Create a matrix that represents the individual transformations, then create the composite transformation by multiplying the matrices together, and finally store the transformation matrix as an affinetform2d object. This approach has the most flexibility because you can use all types of affine transformations including shear, reflection, anisotropic scaling, and other composite affine transformations. You can also control the order of transformations through the order of the matrix multiplication.

2) Create a simtform2d, rigidtform2d, or transltform2d geometric object. This approach makes it easy to create a composite transformation when you know the scale factor, rotation angle, and translation distance of a nonreflective similarity transformation. However, the geometric transformation objects always perform the relevant transformations in a fixed order: scaling first, then rotation, then translation. If you need to perform the transformations in a different order, then you must use matrices and create an affine transformation object.

This example demonstrates both approaches for a rigid transformation consisting of translation and rotation.

Create a checkerboard image that will undergo transformation. Also create a spatial reference object for the image.

cb = checkerboard(4,2);
cb_ref = imref2d(size(cb));

To illustrate the spatial position of the image, create a flat background image. Overlay the checkerboard over the background, highlighting the position of the checkerboard in green.

background = zeros(150);
imshowpair(cb,cb_ref,background,imref2d(size(background)))

Create a translation matrix that shifts an image horizontally and vertically.

tx =100;
ty =0;
T = [1 0 tx; 0 1 ty; 0 0 1]

T = 3×3

     1     0   100
     0     1     0
     0     0     1

Create a rotation matrix that rotates an image clockwise about the origin.

theta =30;
R = [cosd(theta) -sind(theta) 0; sind(theta) cosd(theta) 0; 0 0 1]

R = 3×3

    0.8660   -0.5000         0
    0.5000    0.8660         0
         0         0    1.0000

TR = T*R

TR = 3×3

    0.8660   -0.5000  100.0000
    0.5000    0.8660         0
         0         0    1.0000

tform_tr = affinetform2d(TR);
[out,out_ref] = imwarp(cb,cb_ref,tform_tr);
imshowpair(out,out_ref,background,imref2d(size(background)))

RT = R*T

RT = 3×3

    0.8660   -0.5000   86.6025
    0.5000    0.8660   50.0000
         0         0    1.0000

tform_rt = affinetform2d(RT);
[out,out_ref] = imwarp(cb,cb_ref,tform_rt);
imshowpair(out,out_ref,background,imref2d(size(background)))

tform_rigid = rigidtform2d(theta,[tx ty]);

tform_rigid.A

ans = 3×3

    0.8660   -0.5000  100.0000
    0.5000    0.8660         0
         0         0    1.0000

[out,out_ref] = imwarp(cb,cb_ref,tform_rigid);
imshowpair(out,out_ref,background,imref2d(size(background)))

See Also

imwarp

Related Topics

• 2-D and 3-D Geometric Transformation Process Overview
• Matrix Representation of Geometric Transformations