This example shows how to align or register two images that differ by a rotation and a scale change. You can use `fitgeotrans`

to find the rotation angle and scale factor after manually picking corresponding points. You can then transform the distorted image to recover the original image.

Read an image into the workspace.

original = imread('cameraman.tif'); imshow(original) text(size(original,2),size(original,1)+15, ... 'Image courtesy of Massachusetts Institute of Technology', ... 'FontSize',7,'HorizontalAlignment','right')

scale = 0.7; distorted = imresize(original,scale); % Try varying the scale factor. theta = 30; distorted = imrotate(distorted,theta); % Try varying the angle, theta. imshow(distorted)

Use the Control Point Selection Tool to pick at least two pairs of control points.

movingPoints = [151.52 164.79; 131.40 79.04]; fixedPoints = [135.26 200.15; 170.30 79.30];

You can run the rest of the example with these pre-picked points, but try picking your own points to see how the results vary.

`cpselect(distorted,original,movingPoints,fixedPoints);`

Save control points by choosing the **File** menu, then the **Save Points to Workspace** option. Save the points, overwriting variables `movingPoints`

and `fixedPoints`

.

Fit a nonreflective similarity transformation to your control points.

`tform = fitgeotrans(movingPoints,fixedPoints,'nonreflectivesimilarity');`

After you have done Steps 5 and 6, repeat Steps 4 through 6 but try using 'affine' instead of 'NonreflectiveSimilarity'. What happens? Are the results as good as they were with 'NonreflectiveSimilarity'?

The geometric transformation, `tform`

, contains a transformation matrix in `tform.T`

. Since you know that the transformation includes only rotation and scaling, the math is relatively simple to recover the scale and angle.

Let $\mathrm{sc}=\mathrm{scale}*\mathrm{cos}\left(\mathrm{theta}\right)$

Let $\mathrm{ss}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{scale}*\mathrm{sin}\left(\mathrm{theta}\right)$

Then, `Tinv = invert(tform)`

, and `Tinv.T = `

$\left[\begin{array}{ccc}\mathrm{sc}& -\mathrm{ss}& 0\\ \mathrm{ss}& \mathrm{sc}& 0\\ \mathrm{tx}& \mathrm{ty}& 1\end{array}\right]$

where `tx`

and `ty`

are x and y translations, respectively.

tformInv = invert(tform); Tinv = tformInv.T; ss = Tinv(2,1); sc = Tinv(1,1); scale_recovered = sqrt(ss*ss + sc*sc)

scale_recovered = 0.7000

theta_recovered = atan2(ss,sc)*180/pi

theta_recovered = 29.3741

The recovered values of `scale_recovered`

and `theta_recovered`

should match the values you set in **Step 2: Resize and Rotate the Image**.

Recover the original image by transforming `distorted`

, the rotated-and-scaled image, using the geometric transformation `tform`

and what you know about the spatial referencing of `original`

. The 'OutputView' name-value pair is used to specify the resolution and grid size of the resampled output image.

```
Roriginal = imref2d(size(original));
recovered = imwarp(distorted,tform,'OutputView',Roriginal);
```

Compare `recovered`

to `original`

by looking at them side-by-side in a montage.

montage({original,recovered})

The `recovered`

(right) image quality does not match the `original`

(left) image because of the distortion and recovery process. In particular, the image shrinking causes loss of information. The artifacts around the edges are due to the limited accuracy of the transformation. If you were to pick more points in **Step 3: Select Control Points**, the transformation would be more accurate.

`cpselect`

| `fitgeotrans`

| `imref2d`

| `imresize`

| `imrotate`

| `imwarp`