20060228-Spatial transformations: Where is the input image?

原文：http://blogs.mathworks.com/steve/2006/02/28/spatial-transformations-where-is-the-input-image/

First we have to answer a question that is often overlooked: Where is the input image? In other words, the input u-v space is a plane. Where does the image lie on the plane?

Image location in the plane

With the default spatial coordinates for images in MATLAB, the center of the upper-left image pixel is at (1, 1). Each pixel is a square with unit area, so the corner of the upper-left image pixel is at (0.5, 0.5). A 4-by-4 image looks like this on the plane:

pixel_centers_x = [1 4 4 1 1];
pixel_centers_y = [1 1 4 4 1];
pixel_edges_x   = [0.5 4.5 4.5 0.5 0.5];
pixel_edges_y   = [0.5 0.5 4.5 4.5 0.5];

plot(pixel_centers_x, pixel_centers_y);
hold on
plot(pixel_edges_x, pixel_edges_y, ':')

% Plot axes lines
c = [.7 .7 .7];
plot([-50 50], [0 0], 'color', c);
plot([0 0], [-50 50], 'color', c);

axis ij, axis equal
axis([-5 5 -5 5]);

legend({'Pixel centers', 'Pixel edges'})
title('Default image bounding rectangle location')
xlabel('u')
ylabel('v')
hold off

Rotation

A "pure" affine rotation rotates about the origin. This means it can move an image into a completely different quadrant from where it started.

theta = 3*pi/4;
A1 = [ cos(theta)  sin(theta)  0
      -sin(theta)  cos(theta)  0
       0           0           1];

tform1 = maketform('affine', A1);

[pixel_centers_x1, pixel_centers_y1] = tformfwd(tform1, ...
   pixel_centers_x, pixel_centers_y);
[pixel_edges_x1, pixel_edges_y1] = tformfwd(tform1, ...
   pixel_edges_x, pixel_edges_y);

plot(pixel_centers_x1, pixel_centers_y1);
hold on
plot(pixel_edges_x1, pixel_edges_y1, ':')

% Plot axes lines
c = [.7 .7 .7];
plot([-50 50], [0 0], 'color', c);
plot([0 0], [-50 50], 'color', c);

axis ij, axis equal
axis([-5 5 -5 5]);

legend({'Pixel centers', 'Pixel edges'})
title('Rotated image')
xlabel('x')
ylabel('y')
hold off

Scaling

A "pure" affine scaling also operates with respect to the origin. Since the image corner isn't exactly at the origin (by default), scaling with a scale factor greater than 1 not only increases the size of the pixels, but it also moves the corner of the image away from the origin.

A2 = [ 3  0  0
       0  3  0
       0  0  1];

tform2 = maketform('affine', A2);

[pixel_centers_x2, pixel_centers_y2] = tformfwd(tform2, ...
   pixel_centers_x, pixel_centers_y);
[pixel_edges_x2, pixel_edges_y2] = tformfwd(tform2, ...
   pixel_edges_x, pixel_edges_y);

plot(pixel_centers_x2, pixel_centers_y2);
hold on
plot(pixel_edges_x2, pixel_edges_y2, ':')

% Plot axes lines
c = [.7 .7 .7];
plot([-50 50], [0 0], 'color', c);
plot([0 0], [-50 50], 'color', c);

axis ij, axis equal
axis([-5 5 -5 5]);

legend({'Pixel centers', 'Pixel edges'})
title('Scaled image')
xlabel('x')
ylabel('y')
hold off

The center of the upper-left pixel is now at (3, 3), and the corner of the upper-left pixel is now at (1.5, 1.5).
When I discuss imtransform, I'll explain how it automatically takes care of these details so that most users don't have to worry about it.