`scipy.signal._signaltools:hilbert2`

source: /scipy/signal/_signaltools.py :2533

Signature

def hilbert2 ( x , N = None , axes = (-2, -1) )

Summary

Compute the '2-D' analytic signal of x.

Extended Summary

The 2-D analytic signal is calculated as a so-called "single-orthant" transform. This is achieved by applying one-dimensional Hilbert functions (as in hilbert) to the first and to the second array axis in Fourier space.

For NumPy arrays, scipy.fft.set_workers can be used to change the number of workers used for the FFTs.

Parameters

x : array_like: Input signal. Must be at least two-dimensional.
N : int or tuple of two ints, optional: Number of output samples. x is initially cropped or zero-padded to length N along axes. Default: x.shape[i] for i in axes
axes : tuple of two ints, optional: Axes along which to do the transformation. Default: (-2, -1).
versionchanged 1.17
Added axes parameter

Returns

xa : ndarray: Analytic signal of x taken along given axes.

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.,
$X [p, q] = k, l = 0 \sum N_{0}, N_{1} x [k, l] e^{- 2 j π k p / N_{0}} e^{- 2 j π l q / N_{1}}$
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
$s_{N} (p) := ⎩ ⎨ ⎧ - 1 - 0 + 1 - 0 for p < 0, for p = 0, for 1 \leq p < (N + 1) //2, elsewhere.$
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.,
$x_{a} [k, l] = \frac{1}{N _{0} N _{1}} p, q = 0 \sum N_{0}, N_{1} X_{a} [p, q] e^{2 j π k p / N_{0}} e^{2 j π l q / N_{1}}$

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.

Array API Standard Support

hilbert2 has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ⛔                     ⛔                   
Dask                  ✅                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

The following example calculates the two-dimensional analytic signal from a single impulse with an added constant offset. The impulse produces an FFT where each bin has a value of one and the constant offset component produces only a non-zero component at the ``(0,0)`` bin.

import numpy as np
from scipy.fft import fft2, fftshift, ifftshift
from scipy.signal import hilbert2
x = np.ones((5, 5)) / 5
x[0, 0] += 1
X = fftshift(fft2(x))  # Zero frequency bin is at center
print(X)
x_a = hilbert2(x)
X_a = fftshift(fft2(x_a))

✓

print(np.round(X_a, 3))

✗

The FFT of the result illustrates that the values of the lower right quadrant (or orthant) with purely positive frequency bins have been quadrupled. The values at its borders, where only one frequency component is zero, are doubled. The zero frequency bin ``(0, 0)`` has not been altered. All other quadrants have been set to zero. This second example illustrates a problem with the "single-orthant" convention. A purely real signal can produce an analytic signal which is completely zero:

from scipy.fft import fft2, fftshift, ifft2, ifftshift
from scipy.signal import hilbert2
Z = np.array([[0, 0, 0, 0, 0],
              [0, 0, 0, 1, 0],
              [0, 0, 0, 0, 0],
              [0, 1, 0, 0, 0],
              [0, 0, 0, 0, 0]]) * 25
z = ifft2(ifftshift(Z))
np.allclose(z.imag, 0)  # z is a real signal
np.sum(z.real**2)  # z.real is non-zero
z_a = hilbert2(z.real)
np.allclose(z_a, 0)  # analytic signal is zero

✓

Aliases

scipy.signal.hilbert2

Referenced by

This package

release:1.17.0-notes