< X, Y > = < [x1; x2; ...; xn], [y1; y2; ...; yn] >, = [x1; x2; ...; xn]' * [y1; y2; ...; yn], = SUM_(i=1)^(n) xi * yi, = (x1 * y1) + (x2 * y2) + ... + (xn * yn). (a + ib)' = (a - ib). If A satisfies the following relation,
< A * X, Y > = < X, AT * Y >, then,
AT is transpose of A. (1) 2D matrix
If A is defined as follow,
A in R ^ (M, N), then,
AT in R ^ (N, M). If A(x) is defined as follow,
A(x) = x(i+1) - x(i), then AT(y) is that,
AT(y) = y(i) - y(i+1). If A(x) is Fourier transform,
A(x) = fftn(x)/numel(x), then AT(y) is Inverse Fourier transform,
AT(y) = ifftn(y). (4) Radon transform
If A(x) is Radon transform called by 'Projection',
A(x) = radon(x, THETA) where, THETA is degrees vector. then AT(y) is Inverse Radon transform without Filtration called by 'Backprojection',
AT(y) = iradon(y, THETA, 'none', N)/(pi/(2*length(THETA))). where, 'none' is filtration option and N is image size.