Matrix Representation of Geometric Transformations
You can use a geometric transformation matrix to perform a global transformation of an
image. First, define a transformation matrix and use it to create a geometric transformation
object. Then, apply a global transformation to an image by calling
imwarp with the geometric transformation object. For an example, see Perform Simple 2-D Translation Transformation.
2-D Affine Transformations
The table lists 2-D affine transformations with the transformation matrix used to define them. For 2-D affine transformations, the last column must contain [0 0 1] homogeneous coordinates.
Use any combination of 2-D transformation matrices to create an
affine2d geometric transformation object. Use combinations of 2-D translation
and rotation matrices to create a
|2-D Affine Transformation||Example (Original and Transformed Image)||Transformation Matrix|
For more information about pixel coordinates, see Image Coordinate Systems.
2-D Projective Transformations
Projective transformation enables the plane of the image to tilt. Parallel lines can converge towards a vanishing point, creating the appearance of depth.
The transformation is a 3-by-3 matrix. Unlike affine transformations, there are no restrictions on the last column of the transformation matrix.
|2-D Projective Transformation||Example||Transformation Matrix|
Note that when
Projective transformations are frequently used to register images that are out of
alignment. If you have two images that you would like to align, first select control point
cpselect. Then, fit a projective
transformation matrix to control point pairs using
fitgeotrans and setting the
'projective'. This automatically creates a
projective2d geometric transformation object. The transformation matrix is
stored as a property in the
projective2d object. The transformation can
then be applied to other images using
Create Composite 2-D Affine Transformations
You can combine multiple transformations into a single matrix using matrix multiplication. The order of the matrix multiplication matters.
This example shows how to create a composite of 2-D translation and rotation transformations.
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, and store it as an
affine2d geometric transformation object. This translation will shift the image horizontally by 100 pixels.
T = [1 0 0;0 1 0;100 0 1]; tform_t = affine2d(T);
Create a rotation matrix, and store it as an
affine2d geometric transformation object. This translation will rotate the image 30 degrees clockwise about the origin.
R = [cosd(30) sind(30) 0;-sind(30) cosd(30) 0;0 0 1]; tform_r = affine2d(R);
Translation Followed by Rotation
Perform translation first and rotation second. In the multiplication of the transformation matrices, the translation matrix
T is on the left, and the rotation matrix
R is on the right.
TR = T*R; tform_tr = affine2d(TR); [out,out_ref] = imwarp(cb,cb_ref,tform_tr); imshowpair(out,out_ref,background,imref2d(size(background)))
Rotation Followed by Translation
Reverse the order of the transformations: perform rotation first and translation second. In the multiplication of the transformation matrices, the rotation matrix
R is on the left, and the translation matrix
T is on the right.
RT = R*T; tform_rt = affine2d(RT); [out,out_ref] = imwarp(cb,cb_ref,tform_rt); imshowpair(out,out_ref,background,imref2d(size(background)))
Notice how the spatial position of the transformed image is different than when translation was followed by rotation.
3-D Affine Transformations
The following table lists the 3-D affine transformations with the transformation matrix used to define them. Note that in the 3-D case, there are multiple matrices, depending on how you want to rotate or shear the image. The last column must contain [0 0 0 1].
Use any combination of 3-D transformation matrices to create an
affine3d geometric transformation object. Use combinations of 3-D translation
and rotation matrices to create a
|3-D Affine Transformation||Transformation Matrix|
y, z shear:
About x axis:
About y axis:
About z axis:
For N-D affine transformations, the last column must contain
imwarp does not support transformations of more than