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Creating a Gallery of Transformed Images

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

Overview

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.

Image 1: Create Checkerboard

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')

Image 2: Apply Nonreflective Similarity to Checkerboard

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,

$$[\begin{array}{c c}u\ v\end{array}] = [\begin{array}{c c}x\ y\ 1\end{array}]\,T$$

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.

Image 3: Apply Similarity to Checkerboard

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:

$$[\begin{array}{c c}u\ v\end{array}] = [\begin{array}{c c}x\ y\ 1\end{array}]\,T$$

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')

Image 4: Apply Affine Transformation to Checkerboard

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:

$$[\begin{array}{c c}u\ v\end{array}] = [\begin{array}{c c}x\ y\ 1\end{array}]\,T$$

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')

Image 5: Apply Projective Transformation to Checkerboard

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

For a projective transformation:

$$[\,\begin{array}{c c}up\ vp\ wp\end{array}\,] = [\,\begin{array}{c c}x\ y\ w\end{array}\,]\,T$$

$$u = \frac{up}{wp}$$

$$v = \frac{vp}{wp}$$

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

$$T = \left[\begin{array}{c c c}A\;D\;G\\B\;E\;H\\C\;F\;I\end{array}\right]$$

The above matrix equation is equivalent to these two expressions:

$$u = \frac{Ax + By + C}{Gx + Hy + I}$$

$$v = \frac{Dx + Ey + F}{Gx + Hy + I}$$

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')

Image 6: Apply Polynomial Transformation to Checkerboard

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

For a second-order polynomial transformation:

$$[\begin{array}{c c}u\ v\end{array}] = [\begin{array}{c c} 1\ x\ y\ x*y\ x^2\ y^2\end{array}]\,T$$

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')

Image 7: Apply Piecewise Linear Transformation to Checkerboard

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')

Image 8: Apply Sinusoidal Transformation to Checkerboard

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')

Image 9: Apply Barrel Transformation to Checkerboard

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')

Image 10: Apply Pin Cushion Transformation to Checkerboard

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')

Summary: Display All of the Geometric Transformations of Checkerboard

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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