gaussian_blur.m

function [ blurred_image ] = gaussian_blur( image, pixels_per_sigma_gaussian )
%% gaussian_blur
% SAM 4/8/19

size_of_image = size( image );

pad_lengths = 2 * ceil( pixels_per_sigma_gaussian * 3 / 2 ) + mod( size_of_image, 2 );

padded_image = padarray( image, pad_lengths, 'post' );

image_dft = fftn( padded_image );

size_of_image_dft = size_of_image + pad_lengths ;

% mesh generation for building the gaussian blurring filter in the Fourier domain
[ y_pixel_freq_mesh, x_pixel_freq_mesh, z_pixel_freq_mesh ]                                                       ...
    = ndgrid([ 0 : size_of_image_dft( 1 ) / 2 - 1, - size_of_image_dft( 1 ) / 2 : - 1 ] / size_of_image_dft( 1 ), ...
             [ 0 : size_of_image_dft( 2 ) / 2 - 1, - size_of_image_dft( 2 ) / 2 : - 1 ] / size_of_image_dft( 2 ), ...
             [ 0 : size_of_image_dft( 3 ) / 2 - 1, - size_of_image_dft( 3 ) / 2 : - 1 ] / size_of_image_dft( 3 )  );

sigma_freq_mesh_squared = ( y_pixel_freq_mesh * pixels_per_sigma_gaussian( 1 )) .^ 2 ...
                        + ( x_pixel_freq_mesh * pixels_per_sigma_gaussian( 2 )) .^ 2 ...
                        + ( z_pixel_freq_mesh * pixels_per_sigma_gaussian( 3 )) .^ 2 ;
                           
gaussian_kernel_dft = exp( - pi ^ 2 * 2 * ( sigma_freq_mesh_squared ));

%         % uncomment to inspect the gaussian blurring kernel in the spatial domain
% 
%         gaussian_kernel_image = ifftn( gaussian_kernel_dft, 'symmetric' );
% 
%         sum( gaussian_kernel_image( : ))

blurred_image_dft = image_dft .* gaussian_kernel_dft ;

% inverse fourier transforms to compute blurring operation
padded_blurred_image = ifftn( blurred_image_dft, 'symmetric' );

blurred_image = padded_blurred_image( 1 : size_of_image( 1 ), ...
                                      1 : size_of_image( 2 ), ...
                                      1 : size_of_image( 3 )  );

end