Specify the packed (x,y,z) coordinates of five input points. The packed coordinates are stored as a 5-by-3 matrix, where the first, second, and third columns contain the x-, y-, and z- coordinates,respectively.

XYZ = [5 25 20;10 5 25;15 10 5;20 15 10;25 20 15];

Define an inverse mapping function that accepts and returns points in packed (x,y,z) format.

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

Create a 3-D geometric transformation object, tform, that stores this inverse mapping function.

tform = geometricTransform3d(inverseFcn)

tform = 
  geometricTransform3d with properties:

        InverseFcn: @(c)[c(:,1)+c(:,2),c(:,1)-c(:,2),c(:,3).^2]
        ForwardFcn: []
    Dimensionality: 3

Apply the inverse transformation of this 3-D geometric transformation to the input points.

UVW = transformPointsInverse(tform,XYZ)

UVW = 5×3

    30   -20   400
    15     5   625
    25     5    25
    35     5   100
    45     5   225

Specify the x-, y- and the z-coordinate vectors of five points to transform.

x = [3 5 7 9 11];
y = [2 4 6 8 10];
z = [5 9 13 17 21];

Define the inverse and forward mapping functions that accept and return points in packed (x,y,z) format.

inverseFcn = @(c)[c(:,1).^2,c(:,2).^2,c(:,3).^2];
forwardFcn = @(c)[sqrt(c(:,1)),sqrt(c(:,2)),sqrt(c(:,3))];

Create a 3-D geometric transformation object, tform, that stores these inverse and forward mapping functions.

tform = geometricTransform3d(inverseFcn,forwardFcn)

tform = 
  geometricTransform3d with properties:

        InverseFcn: @(c)[c(:,1).^2,c(:,2).^2,c(:,3).^2]
        ForwardFcn: @(c)[sqrt(c(:,1)),sqrt(c(:,2)),sqrt(c(:,3))]
    Dimensionality: 3

Apply the inverse transformation of this 3-D geometric transformation to the input points.

[u,v,w] = transformPointsInverse(tform,x,y,z)

u = 1×5

     9    25    49    81   121

v = 1×5

     4    16    36    64   100

w = 1×5

    25    81   169   289   441

Apply the forward geometric transform to the transformed points u, v, and w.

[x,y,z] = transformPointsForward(tform,u,v,w)

x = 1×5

     3     5     7     9    11

y = 1×5

     2     4     6     8    10

z = 1×5

     5     9    13    17    21

Define an inverse mapping function that performs reflection about horizontal axis. The function must accept and return packed (x,y,z) coordinates, where the first, second, and third columns contain the x-, y-, and z-coordinates, respectively.

inverseFcn = @(xyz)[xyz(:,1),-xyz(:,2),xyz(:,3)];

Create a 3-D geometric transformation object, tform, that stores this inverse mapping function.

tform = geometricTransform3d(inverseFcn)

tform = 
  geometricTransform3d with properties:

        InverseFcn: @(xyz)[xyz(:,1),-xyz(:,2),xyz(:,3)]
        ForwardFcn: []
    Dimensionality: 3

Load and display an MRI volume to be transformed.

s = load('mri');
mriVolume = squeeze(s.D);

Use imwarp to apply the inverse geometric transform to the input MRI volume.

[mriVolumeTransformed] = imwarp(mriVolume,tform,'nearest','SmoothEdges',true);

Display the image slices from the input MRI volume as montage.

montage(mriVolume,'Size',[4 8],'BackgroundColor','w')
title('Image Slices from 3-D MRI','FontSize',14)

Display the image slices from the transformed MRI volume as a montage. The transformed image slices are the reflection of the input image slices across the x-axis.

montage(mriVolumeTransformed,'Size',[4 8],'BackgroundColor','w')
title('Image Slices from Inverse Geometric Transformation of 3-D MRI','FontSize',14)