The 1D example

Import pynufft module

In python environment, import pynufft module:

from pynufft import NUFFT

Create a pynufft object NufftObj:

NufftObj = NUFFT()

Planning

The locations of the non-uniform samples () must be provided:

import numpy
om = numpy.random.randn(1512,1)
# om is an M x 1 ndarray: locations of M points. *om* is normalized between [-pi, pi]
# Here M = 1512

In addition, the size of time series (), oversampled grid (), and interpolatro size () are:

Nd = (256,)
Kd = (512,)
Jd = (6,)

Now provide NufftObj with these parameters:

NufftObj.plan(om, Nd, Kd, Jd)

Forward transform

Now NufftObj has been prepared and is ready for computations. Continue with an example.:

import numpy
import matplotlib.pyplot as pyplot
time_data = numpy.zeros(256, )
time_data[96:128+32] = 1.0
pyplot.plot(time_data)
pyplot.ylim(-1,2)
pyplot.show()

This generates a time series Fig. 2.

Fig. 2 A box function time series

NufftObj transform the time_data to non-Cartesian locations:

nufft_freq_data =NufftObj.forward(time_data)
pyplot.plot(om,nufft_freq_data.real,'.', label='real')
pyplot.plot(om,nufft_freq_data.imag,'r.', label='imag')
pyplot.legend()
pyplot.show()

This displays the non-Cartesian spectrum Fig. 3.

Fig. 3 Non-Cartesian spectrum of box function in Fig. 2. Note the non-uniform density.

Signal restoration through “solve()”

The signal can be solved by the solve() method

restore_time = NufftObj.solve(nufft_freq_data,'cg', maxiter=30)
restore_time2 = NufftObj.solve(nufft_freq_data,'L1TVOLS', maxiter=30,rho=1)

Now display the restored signals:

im1,=pyplot.plot(numpy.abs(time_data),'r',label='original signal')
im3,=pyplot.plot(numpy.abs(restore_time2),'k--',label='L1TVOLS')
im4,=pyplot.plot(numpy.abs(restore_time),'r:',label='conjugate_gradient_method')
pyplot.legend([im1, im3,im4])
pyplot.show()

Fig. 4 Signals restored by “solve()”. L1TVOLS and L1TVOLS are close to Fig. 2, whereas cg is subject to distortion.

The complete code is: