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This example shows many properties of geometric transformations by applying different transformations to a checkerboard image.

A two-dimensional geometric transformation is a mapping that associates each point in a Euclidean plane with another point in a Euclidean plane. In these examples, the geometric transformation is defined by a rule that tells how to map the point with Cartesian coordinates (x,y) to another point with Cartesian coordinates (u,v). A checkerboard pattern is helpful in visualizing a coordinate grid in the plane of the input image and the type of distortion introduced by each transformation.

`checkerboard`

produces an image that has rectangular tiles and four unique corners, which makes it easy to see how the checkerboard image gets distorted by geometric transformations.

After you have run this example once, try changing the image `I`

to a larger checkerboard, or to your favorite image.

```
I = checkerboard(10,2);
imshow(I)
title('original')
```

Nonreflective similarity transformations may include a rotation, a scaling, and a translation. Shapes and angles are preserved. Parallel lines remain parallel. Straight lines remain straight.

For a nonreflective similarity,

`T`

is a 3-by-3 matrix that depends on 4 parameters.

% Try varying these 4 parameters. scale = 1.2; % scale factor angle = 40*pi/180; % rotation angle tx = 0; % x translation ty = 0; % y translation sc = scale*cos(angle); ss = scale*sin(angle); T = [ sc -ss 0; ss sc 0; tx ty 1];

Since nonreflective similarities are a subset of affine transformations, create an `affine2d`

object using:

t_nonsim = affine2d(T); I_nonreflective_similarity = imwarp(I,t_nonsim,'FillValues',.3); figure, imshow(I_nonreflective_similarity); title('nonreflective similarity')

About Translation: If you change either `tx`

or `ty`

to a non-zero value, you will notice that it has no effect on the output image. If you want to see the coordinates that correspond to your transformation, including the translation, try this:

[I_nonreflective_similarity,RI] = imwarp(I,t_nonsim,'FillValues',.3); figure, imshow(I_nonreflective_similarity,RI) axis on title('nonreflective similarity')

Notice that passing the output spatial referencing object `RI`

from imwarp reveals the translation. To specify what part of the output image you want to see, use the 'OutputView' name-value pair in the `imwarp`

function.

In a similarity transformation, similar triangles map to similar triangles. Nonreflective similarity transformations are a subset of similarity transformations.

For a similarity, the equation is the same as for a nonreflective similarity:

`T`

is a 3-by-3 matrix that depends on 4 parameters plus an optional reflection.

% Try varying these parameters. scale = 1.5; % scale factor angle = 10*pi/180; % rotation angle tx = 0; % x translation ty = 0; % y translation a = -1; % -1 -> reflection, 1 -> no reflection sc = scale*cos(angle); ss = scale*sin(angle); T = [ sc -ss 0; a*ss a*sc 0; tx ty 1];

Since similarities are a subset of affine transformations, create an `affine2d`

object using:

t_sim = affine2d(T); % As in the translation example above, retrieve the output spatial % referencing object |RI| from the |imwarp| function, and pass |RI| to % |imshow| to reveal the reflection. [I_similarity,RI] = imwarp(I,t_sim,'FillValues',.3); figure, imshow(I_similarity,RI) axis on title('similarity')

In an affine transformation, the x and y dimensions can be scaled or sheared independently and there may be a translation, a reflection, and/or a rotation. Parallel lines remain parallel. Straight lines remain straight. Similarities are a subset of affine transformations.

For an affine transformation, the equation is the same as for a similarity and nonreflective similarity:

`T`

is 3-by-3 matrix, where all six elements of the first and second columns can be different. The third column must be [0;0;1].

% Try varying the definition of T. T = [1 0.3 0; 1 1 0; 0 0 1]; t_aff = affine2d(T); I_affine = imwarp(I,t_aff,'FillValues',.3); figure, imshow(I_affine) title('affine')

In a projective transformation, quadrilaterals map to quadrilaterals. Straight lines remain straight. Affine transformations are a subset of projective transformations.

For a projective transformation:

T is a 3-by-3 matrix, where all nine elements can be different.

The above matrix equation is equivalent to these two expressions:

Try varying any of the nine elements of `T`

.

T = [1 0 0.008; 1 1 0.01; 0 0 1 ]; t_proj = projective2d(T); I_projective = imwarp(I,t_proj,'FillValues',.3); figure, imshow(I_projective) title('projective')

In a polynomial transformation, polynomials in x and y define the mapping.

For a second-order polynomial transformation:

Both u and v are second-order polynomials of x and y. Each second-order polynomial has six terms.

fixedPoints = reshape(randn(12,1),6,2); movingPoints = fixedPoints; t_poly = fitgeotrans(movingPoints,fixedPoints,'polynomial',2); I_polynomial = imwarp(I,t_poly,'FillValues',.3); figure, imshow(I_polynomial) title('polynomial')

In a piecewise linear transformation, affine transformations are applied separately to triangular regions of the image. In this example the triangular region at the upper-left of the image remains unchanged while the triangular region at the lower-right of the image is stretched.

movingPoints = [10 10; 10 30; 30 30; 30 10]; fixedPoints = [10 10; 10 30; 40 35; 30 10]; t_piecewise_linear = fitgeotrans(movingPoints,fixedPoints,'pwl'); I_piecewise_linear = imwarp(I,t_piecewise_linear); figure, imshow(I_piecewise_linear) title('piecewise linear')

This example and the following two examples show how you can create an explicit mapping `tmap_b`

to associate each point in a regular grid (xi,yi) with a different point (u,v). This mapping `tmap_b`

is used by `tformarray`

to transform the image.

% locally varying with sinusoid [nrows,ncols] = size(I); [xi,yi] = meshgrid(1:ncols,1:nrows); a1 = 5; % Try varying the amplitude of the sinusoids. a2 = 3; imid = round(size(I,2)/2); % Find index of middle element u = xi + a1*sin(pi*xi/imid); v = yi - a2*sin(pi*yi/imid); tmap_B = cat(3,u,v); resamp = makeresampler('linear','fill'); I_sinusoid = tformarray(I,[],resamp,[2 1],[1 2],[],tmap_B,.3); figure, imshow(I_sinusoid) title('sinusoid')

Barrel distortion perturbs an image radially outward from its center. Distortion is greater farther from the center, resulting in convex sides.

% radial barrel distortion xt = xi(:) - imid; yt = yi(:) - imid; [theta,r] = cart2pol(xt,yt); a = .001; % Try varying the amplitude of the cubic term. s = r + a*r.^3; [ut,vt] = pol2cart(theta,s); u = reshape(ut,size(xi)) + imid; v = reshape(vt,size(yi)) + imid; tmap_B = cat(3,u,v); I_barrel = tformarray(I,[],resamp,[2 1],[1 2],[],tmap_B,.3); figure, imshow(I_barrel) title('barrel')

Pin-cushion distortion is the inverse of barrel distortion because the cubic term has a negative amplitude. Distortion is still greater farther from the center but it results in concave sides.

% radial pin cushion distortion xt = xi(:) - imid; yt = yi(:) - imid; [theta,r] = cart2pol(xt,yt); a = -.0005; % Try varying the amplitude of the cubic term. s = r + a*r.^3; [ut,vt] = pol2cart(theta,s); u = reshape(ut,size(xi)) + imid; v = reshape(vt,size(yi)) + imid; tmap_B = cat(3,u,v); I_pin = tformarray(I,[],resamp,[2 1],[1 2],[],tmap_B,.3); figure, imshow(I_pin) title('pin cushion')

figure subplot(5,2,1),imshow(I),title('original') subplot(5,2,2),imshow(I_nonreflective_similarity),title('nonreflective similarity') subplot(5,2,3),imshow(I_similarity),title('similarity') subplot(5,2,4),imshow(I_affine),title('affine') subplot(5,2,5),imshow(I_projective),title('projective') subplot(5,2,6),imshow(I_polynomial),title('polynomial') subplot(5,2,7),imshow(I_piecewise_linear),title('piecewise linear') subplot(5,2,8),imshow(I_sinusoid),title('sinusoid') subplot(5,2,9),imshow(I_barrel),title('barrel') subplot(5,2,10),imshow(I_pin),title('pin cushion')

Note that `subplot`

changes the scale of the images being displayed.

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