20060804-Spatial transformations: Defining and applying custom transforms

原文：http://blogs.mathworks.com/steve/2006/08/04/spatial-transformations-defining-and-applying-custom-transforms/

You can use maketform to create a custom spatial transformation by supplying your own function handle to perform the inverse mapping. Your function handle has to take two input arguments. The first input argument is a P-by-ndims matrix of points, one per row. (imtransform applies two-dimensional spatial transformations, so I'll be using P-by-2 matrices here.) The second argument, calledtdata, can be used to pass auxiliary information to your function handle. My examples will just ignore this second argument.

Let's start with a very simple example just to illustrate the mechanics. Make an inverse mapping that just swaps the horizontal and vertical coordinates:

% inverse mapping functionf = @(x, unused) fliplr(x);% maketform argumentsndims_in  = 2;ndims_out = 2;forward_mapping = [];inverse_mapping = f;tdata = [];tform = maketform('custom', ndims_in, ndims_out, ...    forward_mapping, inverse_mapping, tdata);body  = imread('liftingbody.png');body2 = imtransform(body, tform);subplot(1,2,1)imshow(body)subplot(1,2,2)imshow(body2)

As we might have guessed, this custom transform just transposes the image.

face = imread('http://blogs.mathworks.com/images/steve/74/face.jpg');clfimshow(face)

Note that the call to imtransform below sets up the input image to be located in the square from -1 to 1 in both directions. The output image grid is set up to be the same square.

r = @(x) sqrt(x(:,1).^2 + x(:,2).^2);w = @(x) atan2(x(:,2), x(:,1));f = @(x) [sqrt(r(x)) .* cos(w(x)), sqrt(r(x)) .* sin(w(x))];g = @(x, unused) f(x);tform2 = maketform('custom', 2, 2, [], g, []);face2 = imtransform(face, tform2, 'UData', [-1 1], 'VData', [-1 1], ...    'XData', [-1 1], 'YData', [-1 1]);imshow(face2)

This example uses the square of polar radial component.

f = @(x) [r(x).^2 .* cos(w(x)), r(x).^2 .* sin(w(x))];g = @(x, unused) f(x);tform3 = maketform('custom', 2, 2, [], g, []);face3 = imtransform(face, tform3, 'UData', [-1 1], 'VData', [-1 1], ...    'XData', [-1 1], 'YData', [-1 1]);imshow(face3)

Finally, let's try the complex-plane function used in Lundmark's article. I'll construct the inverse mapping function in several steps: First, convert output-space Cartesian coordinates to complex values; square the complex values; and then produce new input-space Cartesian coordinates from the squared complex values.

f = @(x) complex(x(:,1), x(:,2));g = @(z) z.^2;h = @(w) [real(w), imag(w)];q = @(x, unused) h(g(f(x)));tform4 = maketform('custom', 2, 2, [], q, []);face4 = imtransform(face, tform4, 'UData', [-1 1], 'VData', [-1 1], ...    'XData', [-1 1], 'YData', [-1 1]);imshow(face4)