scipy.signal._signaltools:hilbert2

Notes

The "single-orthant" transform, as defined in ^[2], is calculated by performing the following steps:

• Calculate the two-dimensional FFT of the input, i.e.,

  $$X[p,q]=\sum _{k,l=0}^{N_0,N_1}x[k,l]e^{-2j\pi kp/N_0}{e}^{-2j\pi lq/N_1}$$

• Zero negative frequency bins and double their positive counterparts, i.e.,

  $$X_a[p,q]=(1+s_{N_0}(p))(1+s_{N_1}(q))X[p,q]$$

  with $s_N(.)$ being a modified sign function defined as

  $$\begin{array}{r}{s}_N(p):=\{\begin{array}{ll}-1& \text{ for }p<0,\\ 0& \text{ for }p=0,\\ +1& \text{ for }1\le p<(N+1)//2,\\ 0& \text{ elsewhere.}\end{array}\end{array}$$

  The limitation of the "$+1$" case to the range of [1:(N+1)//2] accounts for the unpaired Nyquist frequency bin at $N/2$ for even $N$. Note that $X_a[p]=(1+s_N(p))X[p]$ is the one-dimensional Hilbert function (as in hilbert) in Fourier space.

• Produce the analytic signal by performing the inverse FFT, i.e.,

  $$x_a[k,l]=\frac{1}{N_0{N}_1}\sum _{p,q=0}^{N_0,N_1}{X}_a[p,q]e^{2j\pi kp/N_0}{e}^{2j\pi lq/N_1}$$

The "single-orthant" transform is not the only possible definition of an analytic signal in multiple dimensions (as noted in ^[1]). Consult ^[3] for a description of properties that this 2-D transform does and does not share with the 1-D transform. The second example below shows one of the downsides of this approach.