Question

How can one implement the fisheye lens effect illustrated in that image:

One can use Google’s logo for a try:

BTW, what’s the term for it?

Answer 1

I believe this is typically referred to as either a “fisheye lens” effect or a “barrel transformation”. Here are two links to demos that I found:

• Sample code for how you can apply fisheye distortions to images using the 'custom' option for the function maketform from the Image Processing Toolbox.

• An image processing demo which performs a barrel transformation using the function tformarray.

Example

In this example, I started with the function radial.m from the first link above and modified the way it relates points between the input and output spaces to create a nice circular image. The new function fisheye_inverse is given below, and it should be placed in a folder on your MATLAB path so you can use it later in this example:

function U = fisheye_inverse(X, T)

  imageSize = T.tdata(1:2);
  exponent = T.tdata(3);
  origin = (imageSize+1)./2;
  scale = imageSize./2;

  x = (X(:, 1)-origin(1))/scale(1);
  y = (X(:, 2)-origin(2))/scale(2);
  R = sqrt(x.^2+y.^2);
  theta = atan2(y, x);

  cornerScale = min(abs(1./sin(theta)), abs(1./cos(theta)));
  cornerScale(R < 1) = 1;
  R = cornerScale.*R.^exponent;

  x = scale(1).*R.*cos(theta)+origin(1);
  y = scale(2).*R.*sin(theta)+origin(2);
  U = [x y];

end

The fisheye distortion looks best when applied to square images, so you will want to make your images square by either cropping them or padding them with some color. Since the transformation of the image will not look right for indexed images, you will also want to convert any indexed images to RGB images using ind2rgb. Grayscale or binary images will also work fine. Here’s how to do this for your sample Google logo:

[X, map] = imread('logo1w.png');  % Read the indexed image
rgbImage = ind2rgb(X, map);       % Convert to an RGB image
[r, c, d] = size(rgbImage);       % Get the image dimensions
nPad = (c-r)/2;                   % The number of padding rows
rgbImage = cat(1, ones(nPad, c, 3), rgbImage, ones(nPad, c, 3));  % Pad with white

Now we can create the transform with maketform and apply it with imtransform (or imwarp as recommended in newer versions):

options = [c c 3];  % An array containing the columns, rows, and exponent
tf = maketform('custom', 2, 2, [], ...  % Make the transformation structure
               @fisheye_inverse, options);
newImage = imtransform(rgbImage, tf);   % Transform the image
imshow(newImage);                       % Display the image

And here’s the image you should see:

You can adjust the degree of distortion by changing the third value in the options array, which is the exponential power used in the radial deformation of the image points.