% UPWARDCONTINUE Upward continuation for magnetic or gravity potential field data
%
% Usage: [up, psf] = upwardcontinue(im, h, dx, dy, padding)
%
% Arguments: im - Input potential field image
% h - Height to upward continue to (+ve)
% dx, dy - Grid spacing in x and y. The upward continuation height
% is computed relative to the grid spacing. If omitted dx =
% dy = 1, that is, the value of h is in grid spacing units.
% If dy is omitted it is assumed dy = dx.
% padding - Width of tapered padding to apply to the image to reduce
% edge effects. Defaults to 0.
%
% Returns: up - The upward continued field image
% psf - The point spread function corresponding to the upward
% continuation height.
%
% Upward continuation filtering is done in the frequency domain whereby the
% Fourier transform of the upward continued image F(Up) is obtained from the
% Fourier transform of the input image F(U) using
% F(Up) = e^(-2*pi*h * sqrt(u^2 + v^2)) * F(U)
% where u and v are the spatial frequencies over the input grid.
%
% To minimise edge effect problems Moisan's Periodic FFT is used. This avoids
% the need for data tapering.
%
% Reference:
% Richard Blakely, "Potential Theory in Gravity and Magnetic Applications"
% Cambridge University Press, 1996, pp 315-319
% Copyright (c) 2012-2017 Peter Kovesi
% Centre for Exploration Targeting
% The University of Western Australia
% peter.kovesi at uwa edu au
%
% Permission is hereby granted, free of charge, to any person obtaining a copy
% of this software and associated documentation files (the "Software"), to deal
% in the Software without restriction, subject to the following conditions:
%
% The above copyright notice and this permission notice shall be included in
% all copies or substantial portions of the Software.
%
% The Software is provided "as is", without warranty of any kind.
%
% June 2012 - Original version.
% June 2014 - Tidied up and documented.
% August 2014 - Smooth, non periodic component of orginal image added back to
% filtered result so that calculation of residual against the
% original image is facilitated.
% October 2017 - Changed to use filtergrid(), decided against the use of
% perfft2 and added padding option instead.
function [up, psf] = upwardcontinue(im, h, dx, dy, padding)
if ~exist('dx', 'var'), dx = 1; end
if ~exist('dy', 'var'), dy = dx; end
if ~exist('padding', 'var')
padding = 0;
else
im = impad(im, padding, 'taper');
end
[rows,cols,chan] = size(im);
assert(chan == 1, 'Image must be single channel');
mask = ~isnan(im);
IM = fft2(fillnan(im));
% Generate horizontal and vertical frequency grids that vary from
% -0.5 to 0.5
[~, u1, u2] = filtergrid(rows,cols);
% Divide by grid size in each dimension to get correct spatial frequencies.
u1 = u1/dx;
u2 = u2/dy;
freq = sqrt(u1.^2 + u2.^2); % Matrix values contain spatial frequency
% values as a radius from centre (but
% quadrant shifted).
% Continuation filter in the frequency domain.
W = exp(-2*pi*h*freq);
% Apply filter to obtain upward continuation and apply mask corresponding to
% NaN values. Also reconstruct the spatial representation of the point
% spread function corresponding to the upward continuation height.
if padding
up = imtrim(real(ifft2(IM.*W)) .* double(mask), padding);
psf = imtrim(real(fftshift(ifft2(W))), padding);
else
up = real(ifft2(IM.*W)) .* double(mask);
psf = real(fftshift(ifft2(W)));
end