## 2-D and 3-D Geometric Transformation Process Overview

To perform a 2-D or 3-D geometric transformation, first create a geometric transformation object that stores information about the transformation. Then, pass the image to be transformed and the geometric transformation object to the imwarp function. You optionally can provide spatial referencing information about the input image to imwarp.

imwarp uses the geometric transformation to map coordinates in the output image to the corresponding coordinates in the input image (inverse mapping). Then, imwarp uses the coordinate mapping to interpolate pixel values within the input image and calculate the output pixel value.

### Create Geometric Transformation Object

Different types of geometric transformation objects store different information about the transformation.

There are several ways to create a geometric transformation object.

Approach to Create Geometric Transformation

transltform2d

transltform3d

rigidtform2d

rigidtform3d

simtform2d

simtform3d

affinetform2d

affinetform3d

projective2d

geometricTransform2d

geometricTransform3d

Other Geometric Transformations
Specify Translation, Rotation, or Scale ParametersXX
Specify Transformation MatrixXXXX
Specify Custom Point-Wise Mapping Function    X
Estimate Transformation from Control Point Pairs X (2-D)X (2-D)X (2-D) X
Estimate Transformation Using Similarity OptimizationX (2-D)X (2-D)X (2-D)
Estimate Transformation Using Phase CorrelationX (2-D)X (2-D)
Generate Random Affine Transformations  X

#### Specify Translation, Rotation, or Scale Parameters

If you know the amount of translation, the rotation angle, and the scale factor, then you can create a transformation by specifying these parameters.

• Specify translation to create transltform2d and transltform3d objects that represent translation transformations.

• Specify translation, rotation angles, or both to create rigidtform2d and rigidtform3d objects that represent rigid transformations.

• Specify any combination of translation, rotation, and an isotropic scale factor to create simtform2d and simtform3d objects that represent nonreflective similarity transformations.

The following example defines a translation and rotation angle, then creates a rigidtform2d geometric transformation object from the specified parameters.

theta = 30;
translation = [10 20.5];
tform = rigidtform2d(theta,translation)
tform =

rigidtform2d with properties:

Dimensionality: 2
RotationAngle: 30
Translation: [10 20.5000]
R: [2×2 double]
A: [3×3 double]

#### Specify Transformation Matrix

For more complex linear geometric transformations, you can represent the transformation as a matrix. For example, use a matrix representation for projective transformations or for affine transformations involving reflection, anisotropic scaling, shear, or compositions of linear transformations. Specify the transformation matrix to create an affinetform2d, affinetform3d, or projtform2d object. For more information about creating a transformation matrix, see Matrix Representation of Geometric Transformations.

The following example defines the transformation matrix for anisotropic scaling and reflection about the y axis, then creates an affinetform2d geometric transformation object from the transformation matrix.

scaleX = 0.8;
scaleY = 1.5;
A = [scaleX 0 0; 0 -scaleY 0; 0 0 1];
tform = affinetform2d(A)
tform =

affinetform2d with properties:

Dimensionality: 2
A: [3×3 double]

#### Specify Custom Point-Wise Mapping Function

If you have an inverse point-wise mapping function, then you can create a custom 2-D and 3-D geometric transformation using the geometricTransform2d and the geometricTransform3d objects respectively.

The following example specifies an inverse mapping function that accepts and returns 2-D points in packed (x,y) format. Then, the example creates a geometricTransform2d geometric transformation object from the inverse mapping function.

inversefn = @(c) [c(:,1)+c(:,2),c(:,1).^2]
inversefn =

function_handle with value:

@(c)[c(:,1)+c(:,2),c(:,1).^2]
tform = geometricTransform2d(inversefn)
tform =

geometricTransform2d with properties:

InverseFcn: [function_handle]
ForwardFcn: []
Dimensionality: 2

Similarly, the following example creates a geometricTransform3d geometric transformation object using the inverse mapping function. The example specifies an inverse mapping function that accepts and returns 3-D points in packed (x,y,z) format.

inversefn = @(c)[c(:,1)+c(:,2),c(:,1)-c(:,2),c(:,3).^2]
inversefn =

function_handle with value:

@(c)[c(:,1)+c(:,2),c(:,1)-c(:,2),c(:,3).^2]
tform = geometricTransform3d(inversefn)
tform =

geometricTransform3d with properties:

InverseFcn: [function_handle]
ForwardFcn: []
Dimensionality: 3

#### Estimate Transformation from Control Point Pairs

You can create a geometric transformation object by passing pairs of control points to the fitgeotform2d function. The fitgeotform2d function automatically estimates the transformation from these points and returns one of the geometric transformation objects.

Different transformations require a varying number of points. For example, affine transformations require three non-collinear points in each image (a triangle) and projective transformations require four points (a quadrilateral).

This example defines two pairs of control points, then uses the fitgeotform2d to create an affinetform2d geometric transformation object.

movingPoints = [11 11;21 11; 21 21];
fixedPoints = [51 51;61 51;61 61];
tform = fitgeotform2d(movingPoints,fixedPoints,"affine")
tform =

affinetform2d with properties:

Dimensionality: 2
A: [3×3 double]

#### Estimate Transformation Using Similarity Optimization

If you have a fixed image and a moving image that are slightly misaligned, then you can use the imregtform function to estimate an affine geometric transformation that aligns the images. imregtform optimizes the mean squares or Mattes mutual information similarity metric of the two images, using a regular step gradient descent or one-plus-one evolutionary optimizer. For more information, see Create an Optimizer and Metric for Intensity-Based Image Registration.

#### Estimate Transformation Using Phase Correlation

If you have a fixed image and a moving image that are severely misaligned, then you can use the imregcorr function to estimate an affine geometric transformation that improves the image alignment. You can refine the resulting transformation by using similarity optimization.

#### Generate Random Affine Transformations

You can create an affine geometric transformation with randomized transformation parameters using the randomAffine2d and randomAffine3d functions. These functions support all affine parameters including reflection about each axis, rotation, shearing, and anisotropic scale factors. Randomized affine transformations are commonly used as a data augmentation technique for deep learning.