Matrix Representation of Geometric Transformations

You can represent a linear geometric transformation as a numeric matrix. Each type of transformation, such as translation, scaling, rotation, and reflection, is defined using a matrix whose elements follow a specific pattern. You can combine multiple transformations by taking a composite of the matrices representing the transformations. For more information, see Create Composite 2-D Affine Transformations.

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 row must be [0 0 1].

• Use combinations of 2-D translation matrices to create a transltform2d object representing a translation transformation.

• Use combinations of 2-D translation and rotation matrices to create a rigidtform2d object representing a nonreflective rigid transformation.

• Use combinations of 2-D translation, rotation, and scaling matrices to create a simtform2d object representing a nonreflective similarity transformation.

• Use any combination of 2-D transformation matrices to create an affinetform2d object representing a general affine transformation.

2-D Affine Transformation	Example (Original and Transformed Image)	Transformation Matrix
Translation		$$\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]$$	tx specifies the displacement along the x axis ty specifies the displacement along the y axis. For more information about pixel coordinates, see Image Coordinate Systems.
Scale		$$\left[\begin{array}{ccc}{s}_x& 0& 0\\ 0& s_y& 0\\ 0& 0& 1\end{array}\right]$$	sx specifies the scale factor along the x axis sy specifies the scale factor along the y axis.
Shear		$$\left[\begin{array}{ccc}1& s{h}_x& 0\\ s{h}_y& 1& 0\\ 0& 0& 1\end{array}\right]$$	shx specifies the shear factor along the x axis. shy specifies the shear factor along the y axis.
Reflection		$$\left[\begin{array}{ccc}\text{cosd(2}\phi \text{)}& \text{sind(2}\phi \text{)}& 0\\ \text{sind(2}\phi \text{)}& -\text{cosd(2}\phi \text{)}& 0\\ 0& 0& 1\end{array}\right]$$	φ specifies the angle of the axis of reflection, in degrees. Two common reflections are vertical and horizontal reflection. Vertical reflection is reflection about the x-axis, so φ is 0 and the reflection matrix simplifies to: [1 0 0; 0 -1 0; 0 0 1]. Horizontal reflection is reflection about the y-axis, so φ is 90 and the reflection matrix simplifies to: [-1 0 0; 0 1 0; 0 0 1]
Rotation		$$\left[\begin{array}{ccc}\mathrm{cosd}(\text{θ})& -\mathrm{sind}(\text{θ})& 0\\ \mathrm{sind}(\text{θ})& \mathrm{cosd}(\text{θ})& 0\\ 0& 0& 1\end{array}\right]$$	θ specifies the angle of rotation about the origin, in degrees.

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 row of the transformation matrix. Use any composition of 2-D affine and projective transformation matrices to create a projtform2d object representing a general projective transformation.

2-D Projective Transformation	Example	Transformation Matrix
Tilt		$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ E& F& 1\end{array}\right]$$	E and F influence the vanishing point. When E and F are large, the vanishing point comes closer to the origin and thus parallel lines appear to converge more quickly.

3-D Affine Transformations

The 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. For 3-D affine transformations, the last row must be [0 0 0 1].

• Use combinations of 3-D translation matrices to create a transltform3d object representing a translation transformation.

• Use combinations of 3-D translation and rotation matrices to create a rigidtform3d object representing a nonreflective rigid transformation.

• Use combinations of 3-D translation, rotation, and scaling matrices to create a simtform3d object representing a nonreflective similarity transformation.

• Use any combination of 3-D transformation matrices to create an affinetform3d object representing a general affine transformation.

3-D Affine Transformation	Transformation Matrix
Translation	Translation by amount tx, ty, and tz in the x, y, and z directions, respectively: $$\left[\begin{array}{cccc}1& 0& 0& t_x\\ 0& 1& 0& t_y\\ 0& 0& 1& t_z\\ 0& 0& 0& 1\end{array}\right]$$
Scale	Scale by scale factor sx, sy, and sz in the x, y, and z dimensions, respectively: $$\left[\begin{array}{cccc}{s}_x& 0& 0& 0\\ 0& s_y& 0& 0\\ 0& 0& s_z& 0\\ 0& 0& 0& 1\end{array}\right]$$
Shear	Shear within the y-z plane: $$\left[\begin{array}{cccc}1& 0& 0& 0\\ s{h}_{xy}& 1& 0& 0\\ s{h}_{xz}& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ such that $$\begin{array}{l}x\text{'}=x\\ y\text{'}=y+s{h}_{xy}x\\ z\text{'}=z+s{h}_{xz}x\end{array}$$	Shear within the x-z plane: $$\left[\begin{array}{cccc}1& s{h}_{yx}& 0& 0\\ 0& 1& 0& 0\\ 0& s{h}_{yz}& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ such that $$\begin{array}{l}x\text{'}=x+s{h}_{yx}y\\ y\text{'}=y\\ z\text{'}=z+s{h}_{yz}y\end{array}$$	Shear within the x-y plane: $$\left[\begin{array}{cccc}1& 0& s{h}_{zx}& 0\\ 0& 1& s{h}_{zy}& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ such that $$\begin{array}{l}x\text{'}=x+s{h}_{zx}z\\ y\text{'}=y+s{h}_{zy}z\\ z\text{'}=z\end{array}$$
Reflection	Reflection across the y-z plane, negating the x coordinate: $$\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$	Reflection across the x-z plane, negating the y coordinate: $$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$	Reflection across the x-y plane, negating the z coordinate: $$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$$
Rotation	Rotation within the y-z plane, by angle θx about the x axis, in degrees: $$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \text{cosd}({\theta }_x)& -\text{sind}({\theta }_x)& 0\\ 0& \text{sind}({\theta }_x)& \text{cosd}({\theta }_x)& 0\\ 0& 0& 0& 1\end{array}\right]$$	Rotation within the x-z plane, by angle θy about the y axis, in degrees: $$\left[\begin{array}{cccc}\text{cosd}({\theta }_y)& 0& \text{sind}({\theta }_y)& 0\\ 0& 1& 0& 0\\ -\mathrm{sind}({\theta }_y)& 0& \text{cosd}({\theta }_y)& 0\\ 0& 0& 0& 1\end{array}\right]$$	Rotation within the x-y plane, by angle θz about the z axis, in degrees: $$\left[\begin{array}{cccc}\text{cosd}({\theta }_z)& -\text{sind}({\theta }_z)& 0& 0\\ \text{sind}({\theta }_z)& \text{cosd}({\theta }_z)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

3-D Projective and N-D Transformations

The imwarp function does not support 3-D projective transformations or N-D affine and projective transformations. Instead, you can create a spatial transformation structure from a geometric transformation matrix using the maketform function. Then, apply the transformation to an image using the tformarray function. For more information, see N-Dimensional Spatial Transformations.

The dimensions of the transformation matrix must be (N+1)-by-(N+1). The maketform and tformarray functions use the postmultiply matrix convention. Geometric transformation matrices in the postmultiply convention are the transpose of matrices in the premultiply convention. Therefore, for N-D affine transformation matrices, the last column must contain [zeros(N,1); 1] and there are no restrictions on the values of the last row.