The symme
try in
k-space is a fundamental property of Fourier
transformations. For a two-dimensional example, let g(x,y) be a complex function, i.e. the value of g at any (x,y) is a complex number. If nothing is known about the function g, data throughout all of
k-space is needed to fully characterize it.
If the function g is 'real', meaning that at every (x,y) the
imaginary component of g(x,y) is zero, then you only need half as much data to characterize g. The result is redundancy between the data on one half of
k-space and the other. Specifically, if G(kx,ky) is the
Fourier transformation of g(x,y), and g(x,y) is
real, then G(kx,ky)=G*(- kx,- ky), where * indicates a
complex conjugate. The data in mirrored positions in
k-space, i.e. (kx,ky) versus (- kx,- ky), are conjugates of each other.
See
Imaginary Numbers and
Complex Conjugate.