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 2D Translation Transformation.
2D Affine Transformations
The table lists 2D affine transformations with the transformation matrix used to define them. For 2D affine transformations, the last column must contain [0 0 1] homogeneous coordinates.
Use any combination of 2D transformation matrices to create an affine2d
geometric transformation object. Use combinations of 2D translation
and rotation matrices to create a rigid2d
geometric
transformation object.
2D Affine Transformation  Example (Original and Transformed Image)  Transformation Matrix  

Translation 
For more information about pixel coordinates, see Image Coordinate Systems.  
Scale 

 
Shear 

 
Rotation 


2D 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 3by3 matrix. Unlike affine transformations, there are no restrictions on the last column of the transformation matrix.
2D Projective Transformation  Example  Transformation Matrix  

Tilt 
$$\left[\begin{array}{ccc}1& 0& E\\ 0& 1& F\\ 0& 0& 1\end{array}\right]$$

When 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
pairs using cpselect
. Then, fit a projective
transformation matrix to control point pairs using fitgeotrans
and setting the transformationType
to
'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 imwarp
.
Create Composite 2D 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 2D 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.
3D Affine Transformations
The following table lists the 3D affine transformations with the transformation matrix used to define them. Note that in the 3D 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 3D transformation matrices to create an affine3d
geometric transformation object. Use combinations of 3D translation
and rotation matrices to create a rigid3d
geometric
transformation object.
3D Affine Transformation  Transformation Matrix  

Translation 
$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ {t}_{x}& {t}_{y}& {t}_{z}& 1\end{array}\right]$$
 
Scale 
$$\left[\begin{array}{cccc}{s}_{x}& 0& 0& 0\\ 0& {s}_{y}& 0& 0\\ 0& 0& {s}_{z}& 0\\ 0& 0& 0& 1\end{array}\right]$$
 
Shear  x,y shear:
$$\begin{array}{l}x\text{'}=x+az\\ y\text{'}=y+bz\\ z\text{'}=z\end{array}$$
$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ a& b& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$
 x,z shear:
$$\begin{array}{l}x\text{'}=x+ay\\ y\text{'}=y\\ z\text{'}=z+cy\end{array}$$
$$\left[\begin{array}{cccc}1& 0& 0& 0\\ a& 1& c& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$
 y, z shear:
$$\begin{array}{l}x\text{'}=x\\ y\text{'}=y+bx\\ z\text{'}=z+cx\end{array}$$
$$\left[\begin{array}{cccc}1& b& c& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

Rotation  About x axis:
$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \mathrm{cos}(a)& \mathrm{sin}(a)& 0\\ 0& \mathrm{sin}(a)& \mathrm{cos}(a)& 0\\ 0& 0& 0& 1\end{array}\right]$$
 About y axis:
$$\left[\begin{array}{cccc}\mathrm{cos}(a)& 0& \mathrm{sin}(a)& 0\\ 0& 1& 0& 0\\ \mathrm{sin}(a)& 0& \mathrm{cos}(a)& 0\\ 0& 0& 0& 1\end{array}\right]$$
 About z axis:
$$\left[\begin{array}{cccc}\mathrm{cos}(a)& \mathrm{sin}(a)& 0& 0\\ sin(a)& \mathrm{cos}(a)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

For ND affine transformations, the last column must contain [zeros(N,1);
1]
. imwarp
does not support transformations of more than
three dimensions.
See Also
imwarp
 fitgeotrans
 affine2d
 affine3d
 rigid2d
 rigid3d
 projective2d