File:Bump2D illustration.png

% illustration of a bump function in two dimensions
function main()

   % the number of data points. More points means prettier picture.
   N = 300;

   % a function close to what we want, but not smooth
   Z = get_step_function (N);

   % a smooth function with small support, that will serve as mollifier
   W = get_mollifier     (N);

% get the convolution of the two, so a mollified step function
   S = conv2(Z, W);

% truncate S at the edges, and scale it
   p=0.2;
   [m, n] = size(S);
   m1 = floor(p*m)+1; m2=floor((1-p)*m)-1;
   n1 = floor(p*n)+1; n2=floor((1-p)*n)-1;
   S = S(m1:m2, n1:n2);
   S = 100*S/max(max(S));
   
% plot the surface
   figure(2); clf; hold on; axis equal; axis off;
   surf(S);
   
% make the surface beautiful
   shading interp;
   colormap autumn;

% add in a source of light
   camlight (-50, 54);
   
% viewing angle
   view(-40, 38);

   % save as png
  print('-dpng', '-r400', 'Bump2D_illustration_uncropped.png');

  % optionally, crop the margins
  !convert -trim Bump2D_illustration_uncropped.png Bump2D_illustration.png
  
% get a function which is 1 on a set, and 0 outside of it
function Z = get_step_function(N)
   XX = linspace(-1.5, 4, N);
   YY = linspace(-4, 4, N);
   [X, Y] = meshgrid(XX, YY);
   
   c = 2;
   k=1.2;
   shift=10;
   Z = (c^2-X.^2-Y.^2).^2 + k*(c-X).^3-shift;
   
   Z =1-max(sign(Z), 0);

function W = get_mollifier(N)
% now try to get a function with compact support
% as a mollifier
% We will cheat by using a gaussian

   a = 4;
   XX = linspace(-a, a, N);
   YY = linspace(-a, a, N);
   [X, Y] = meshgrid(XX, YY);
   
   K = 4;
   W = exp(-K*(X.^2+Y.^2));

   % truncate the Gaussian to make it with compact support
   trunc = 1e-2;
   W = max(W-trunc, 0);