PyNUFFT

https://pynufft.readthedocs.io/en/latest/
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A Python package for non-uniform fast Fourier transform with GPU acceleration support for...
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### OpenCL Device Information Example

Source: https://pynufft.readthedocs.io/en/latest/installation/OpenCL.html

This snippet displays the output of the 'clinfo' command, showing detailed information about the detected OpenCL platform and device. It is useful for verifying OpenCL installation and understanding device capabilities.

```text
Platform Name                                   Intel(R) OpenCL
Number of devices                                 1
Device Name                                     Intel(R) Core(TM) i7-6700HQ CPU @ 2.60GHz
Device Vendor                                   Intel(R) Corporation
Device Vendor ID                                0x8086
Device Version                                  OpenCL 1.2 (Build 117)
Driver Version                                  1.2.0.117
Device OpenCL C Version                         OpenCL C 1.2
Device Type                                     CPU
Device Profile                                  FULL_PROFILE
Device Available                                Yes
Compiler Available                              Yes
Linker Available                                Yes
Max compute units                               8
Max clock frequency                             2600MHz
Device Partition                                (core)
  Max number of sub-devices                     8
  Supported partition types                     by counts, equally, by names (Intel)
  Supported affinity domains                    (n/a)
Max work item dimensions                        3
Max work item sizes                             8192x8192x8192
Max work group size                             8192
Preferred work group size multiple              128
Preferred / native vector sizes
  char                                                 1 / 32
  short                                                1 / 16
  int                                                  1 / 8
  long                                                 1 / 4
  half                                                 0 / 0        (n/a)
  float                                                1 / 8
  double                                               1 / 4        (cl_khr_fp64)
Half-precision Floating-point support           (n/a)
Single-precision Floating-point support         (core)
  Denormals                                     Yes
  Infinity and NANs                             Yes
  Round to nearest                              Yes
  Round to zero                                 No
  Round to infinity                             No
  IEEE754-2008 fused multiply-add               No
  Support is emulated in software               No
  Correctly-rounded divide and sqrt operations  No
Double-precision Floating-point support         (cl_khr_fp64)
  Denormals                                     Yes
  Infinity and NANs                             Yes
  Round to nearest                              Yes
  Round to zero                                 Yes
  Round to infinity                             Yes
  IEEE754-2008 fused multiply-add               Yes
  Support is emulated in software               No
Address bits                                    64, Little-Endian
Global memory size                              33613447168 (31.3GiB)
Error Correction support                        No
Max memory allocation                           8403361792 (7.826GiB)
Unified memory for Host and Device              Yes
Minimum alignment for any data type             128 bytes
Alignment of base address                       1024 bits (128 bytes)
Global Memory cache type                        Read/Write
Global Memory cache size                        262144 (256KiB)
Global Memory cache line size                   64 bytes
Image support                                   Yes
  Max number of samplers per kernel             480
  Max size for 1D images from buffer            525210112 pixels

```

--------------------------------

### Example nvcc Output

Source: https://pynufft.readthedocs.io/en/latest/installation/Windows.html

This is an example of the output you would expect to see when running the 'nvcc -V' command, indicating a successful installation of the CUDA toolkit.

```text
nvcc: NVIDIA (R) Cuda compiler driver
Copyright (c) 2005-2018 NVIDIA Corporation
Built on Tue_Jun_12_23:09_12_Central_Daylight_time_2018
Cuda compilation tool, release 9.2, V9.2.148


```

--------------------------------

### Install clinfo using Homebrew

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/Macbook.rst.txt

After installing Homebrew, use this command to install clinfo, a utility to display information about installed OpenCL drivers.

```bash
brew install clinfo
```

--------------------------------

### Install PyNUFFT from Git Repository

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/init.rst.txt

Clone the latest code from the GitHub repository and install it locally. This method is useful for development or when needing the most recent changes.

```bash
git clone https://github.com/jyhmiinlin/pynufft
$ cd pynufft
$ python setup.py install --user
```

--------------------------------

### Install PyNUFFT using pip

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/init.rst.txt

Use this command to install the PyNUFFT package directly from the Python Package Index.

```bash
$ pip install pynufft
```

--------------------------------

### Full 2D NUFFT script example

Source: https://pynufft.readthedocs.io/en/latest/_sources/tutor/more2D.rst.txt

A complete Python script demonstrating a 2D NUFFT transform, including planning, forward transform, and image restoration.

```python
import numpy 
import scipy.misc
import matplotlib.pyplot 

from pynufft import NUFFT
import pkg_resources


# Load the 2D trajectory
DATA_PATH = pkg_resources.resource_filename('pynufft', './src/data/')
om = numpy.load(DATA_PATH+'om2D.npz')['arr_0']

# Define NUFFT parameters
Nd = (256, 256)  # image size
Kd = (512, 512)  # k-space size
Jd = (6, 6)  # interpolation size

# Create and plan the NUFFT object
NufftObj = NUFFT()
NufftObj.plan(om, Nd, Kd, Jd)

# Load and prepare the image
image = scipy.misc.ascent()[::2, ::2]
image = image / numpy.max(image[...])

# Perform forward transform
y = NufftObj.forward(image)

# Restore image using different solvers
image0 = NufftObj.solve(y, solver='cg', maxiter=50)
image2 = NufftObj.adjoint(y)
image3 = NufftObj.solve(y, solver='L1TVOLS', maxiter=50, rho=0.1)

# Display results
matplotlib.pyplot.figure(figsize=(12, 4))

matplotlib.pyplot.subplot(1, 3, 1)
matplotlib.pyplot.title('Restored image (cg)')
matplotlib.pyplot.imshow(image0.real, cmap=matplotlib.cm.gray, norm=matplotlib.colors.Normalize(vmin=0.0, vmax=1))

matplotlib.pyplot.subplot(1, 3, 2)
matplotlib.pyplot.title('Adjoint transform')
matplotlib.pyplot.imshow(image2.real, cmap=matplotlib.cm.gray, norm=matplotlib.colors.Normalize(vmin=0.0, vmax=5))

matplotlib.pyplot.subplot(1, 3, 3)
matplotlib.pyplot.title('L1TV OLS')
matplotlib.pyplot.imshow(image3.real, cmap=matplotlib.cm.gray, norm=matplotlib.colors.Normalize(vmin=0.0, vmax=1))

matplotlib.pyplot.tight_layout()
matplotlib.pyplot.show()

# Display the spectrum of the restored image (cg)
matplotlib.pyplot.figure()
spectrum = numpy.fft.fftshift(numpy.fft.fft2(image0))
matplotlib.pyplot.imshow(numpy.log(numpy.abs(spectrum)), cmap=matplotlib.cm.gray)
matplotlib.pyplot.title('Spectrum of restored image (cg)')
matplotlib.pyplot.show()
```

--------------------------------

### Install PyCUDA, Reikna, and PyNUFFT

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/Windows.rst.txt

Install the necessary Python packages for PyNUFFT using pip within an Anaconda prompt. Ensure you have Anaconda3 installed and the prompt is open.

```bash
pip install pycuda
pip install reikna
pip install pynufft
```

--------------------------------

### Run pynufft tests

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/Macbook.rst.txt

Execute the pynufft tests to verify installation and performance. This snippet shows the output of the test execution, including device information and error metrics.

```python
Python 3.11.5 (main, Aug 24 2023, 15:09:45) [Clang 14.0.3 (clang-1403.0.22.14.1)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> from pynufft import tests as t
>>> t.test_init()
No cuda device found. Check your pycuda installation.
device name =  <pyopencl.Device 'Apple M1' on 'Apple' at 0x1027f00>
0.024686098098754883
0.0167756986618042
error gx2= 2.1132891e-07
error gy= 1.1100628500806993e-07
acceleration= 1.4715391946662406
7.563086748123169 4.348098039627075
acceleration in solver= 1.7394011540668561
```

--------------------------------

### Experimental Multiprocessing Example for NUFFT

Source: https://pynufft.readthedocs.io/en/latest/_sources/manu/multiple_NUFFT.rst.txt

An experimental example demonstrating the use of Python's multiprocessing module with NUFFT, specifically designed to handle mixed CUDA and OpenCL backends. This requires careful setup to avoid issues with unpicklable contexts.

```python
.. literalinclude::  ../../../tests/parallel_NUFFT.py
```

--------------------------------

### Test PyNUFFT Initialization

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/Windows.rst.txt

Run a Python script to test the initialization of the PyNUFFT library. This involves starting the Python interpreter and calling a specific test function.

```python
python
from pynufft import tests
tests.test_init()
```

--------------------------------

### Install Homebrew on Macbook

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/Macbook.rst.txt

Use this command in the terminal to install Homebrew, a package manager for macOS.

```bash
/bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/HEAD/install.sh)"
```

--------------------------------

### Troubleshooting cl.exe Not Found

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/Windows.rst.txt

This example shows the error message when the 'cl' command (Microsoft C/C++ compiler) is not recognized, indicating it's not in the system's PATH. The subsequent command demonstrates the expected output after the PATH is correctly configured.

```bash
C:\Users\User>cl
`cl` is not recognized as an internal or external command, 
operable program or batch file.
```

```bash
C:\Users\User>cl
Microsoft (R) C/C++ Optimizing Compiler Version 19.0024215.1 for x86
Copyright (C) Microsoft Corpooration. All rights reserved.

usage: cl [ option... ] filename... [ /link linkoption... ]
```

--------------------------------

### Run pynufft Tests on Macbook

Source: https://pynufft.readthedocs.io/en/latest/installation/Macbook.html

Execute the pynufft tests to verify the installation and check performance metrics on your Macbook. This snippet shows the expected output, including device information and error rates.

```python
from pynufft import tests as t
t.test_init()

```

--------------------------------

### Experimental Multiprocessing with NUFFT (PyCUDA Backend)

Source: https://pynufft.readthedocs.io/en/latest/manu/multiple_NUFFT.html

An example demonstrating the experimental use of Python's multiprocessing module with PyNUFFT, utilizing a PyCUDA backend. Each process creates and executes a NUFFT instance on the GPU, with CPU affinity controlled via taskset.

```python

"""
An example of multiprocessing with NUFFT using PyCUDA backend, 
wrapped inside the atomic_NUFFT wrapper class.  
The two processes are running on two CPU cores.
Each process creates one NUFFT and offloads the computations to GPU.
nvidia-smi confirms that two python programs are using the GPU.
"""

import numpy
from pynufft import NUFFT, helper
import scipy.misc
import matplotlib.pyplot
import multiprocessing
import os 
    


class atomic_NUFFT:
    def __init__(self, om, Nd, Kd, Jd, device_indx):
        """
        This caches the parameters only.
        Any other GPU related stuffs are carried out in run()
        """
        self.om = om
        self.Nd = Nd
        self.Kd = Kd
        self.Jd = Jd
#         self.API = API
        self.device_indx = device_indx
        
    def run(self, x, cpu_cores):
        """
        In this method, the NUFFT are created and executed on a fixed CPU core.
        """
        pid= os.getpid()
        print('pid=', pid)
        os.system("taskset -p -c %d-%d %d" % (cpu_cores[0], cpu_cores[1], pid))
        """
        Control the CPU affinity. Otherwise the process on one core can be switched to another core.
        """

        # create NUFFT
#         NUFFT = NUFFT(self.API, )
        
        # plan the NUFFT
        
        device_list = helper.device_list()  
        self.NUFFT = NUFFT(device_list[self.device_indx])
        self.NUFFT.plan(self.om, self.Nd, self.Kd, self.Jd)
        # send the image to device
        gx = self.NUFFT.to_device(x)
        
        # carry out 10000 forward transform
        for pp in range(0, 10000):
            gy = self.NUFFT._forward_device(gx)

        # return the object
        return gy.get()

Nd = (256,256)
Kd = (512,512)
Jd = (6,6)
om = numpy.random.randn(35636, 2)
x = scipy.misc.ascent()[::2,::2]
om1 = om[om[:,0]>0, :]
om2 = om[om[:,0]<=0, :]

# create pool
pool = multiprocessing.Pool(2)

# create the list to receive the return values
results = []

# Now enter the first process
# This is the standard multiprocessing Pool
D = atomic_NUFFT(om1, Nd, Kd, Jd, 0)

```

--------------------------------

### diagnose()

Source: https://pynufft.readthedocs.io/en/latest/index.html

Performs diagnostic checks on the system or PyNUFFT installation.

```APIDOC
## diagnose()

### Description
Runs diagnostic tests to check the status and configuration of PyNUFFT.

### Function
`diagnose()`
```

--------------------------------

### Check CUDA Compiler Version

Source: https://pynufft.readthedocs.io/en/latest/_sources/installation/Windows.rst.txt

Verify that the NVIDIA CUDA compiler (nvcc) is installed and accessible by checking its version. This command should be run in a Windows command prompt.

```bash
nvcc -V
```

--------------------------------

### Plan Each NUFFT Instance Immediately After Creation

Source: https://pynufft.readthedocs.io/en/latest/_sources/manu/multiple_NUFFT.rst.txt

Create and plan each NUFFT instance individually as soon as it is initialized. This method allows for immediate execution of forward or backward transforms for each instance.

```python
# Create the first NUFFT
NufftObj1 = NUFFT()
NufftObj1.plan(om1, Nd, Kd, Jd)

# Create the second NUFFT
NufftObj2 = NUFFT()
NufftObj2.plan(om2, Nd, Kd, Jd)  

y1 = NufftObj1.forward(x)
y2 = NufftObj2.forward(x)
```

--------------------------------

### Plan NUFFT Object (Basic)

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Initializes the NUFFT object and plans it with the provided geometry. Use this for standard planning without specifying Fourier transform axes.

```python
>>> from pynufft import NUFFT
>>> NufftObj = NUFFT()
>>> NufftObj.plan(om, Nd, Kd, Jd)
```

--------------------------------

### Initialize NUFFT Device Constructor

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Constructor for initializing NUFFT on a specific heterogeneous device. Requires API type, platform, and device numbers.

```python
>>> from pynufft import NUFFT
>>> NufftObj = NUFFT_device(API='cuda', platform_number=0,
                                 device_number=0, verbosity=0)
```

--------------------------------

### Initiate PyNUFFT Object

Source: https://pynufft.readthedocs.io/en/latest/_sources/tutor/basic_use.rst.txt

Import the NUFFT class and create an instance of the NufftObj object. This object is initially empty and requires planning before use.

```python
# import NUFFT class
from pynufft import NUFFT

# Initiate the NufftObj object
NufftObj = NUFFT()
```

--------------------------------

### Multiple NUFFT_hsa Instances with Specific Devices

Source: https://pynufft.readthedocs.io/en/latest/_sources/manu/multiple_NUFFT.rst.txt

Demonstrates creating and planning multiple NUFFT instances using specific devices, likely for HSA-compatible hardware. Each instance is planned immediately after creation, followed by forward transformations.

```python
from pynufft import NUFFT, helper
# Create the first NUFFT(device)
NufftObj1 = NUFFT(helper.device_list()[0])
NufftObj1.plan(om1, Nd, Kd, Jd)

# Create the second NUFFT(device)
NufftObj2 = NUFFT(helper.device_list()[0])
NufftObj2.plan(om2, Nd, Kd, Jd)

y1 = NufftObj1.forward(x)
y2 = NufftObj2.forward(x)
```

--------------------------------

### Initialize NUFFT Object with Device

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Instantiate the NUFFT class specifying a device index for acceleration. This requires the helper module to list available devices.

```python
>>> from pynufft import NUFFT, helper
>>> device = helper.device_list()[0]
>>> NufftObj = NUFFT(device) # for first acceleration device in the system
```

--------------------------------

### Successful cl.exe Execution

Source: https://pynufft.readthedocs.io/en/latest/installation/Windows.html

After correctly setting the system path for Visual Studio, running 'cl.exe' should now execute without the 'not recognized' error, displaying its usage information.

```text
C:\Users\User>cl
Microsoft (R) C/C++ Optimizing Compiler Version 19.0024215.1 for x86
Copyright (C) Microsoft Corpooration. All rights reserved.

usage: cl [ option... ] filename... [ /link linkoption... ]


```

--------------------------------

### Generate 2D Random Coordinates

Source: https://pynufft.readthedocs.io/en/latest/_sources/tutor/basic_use.rst.txt

Generate random non-Cartesian coordinates for planning the NUFFT object. This example creates 100 random samples across a 2D plane.

```python
# generating 2D random coordinates
import numpy
om = numpy.random.randn(100, 2)
```

--------------------------------

### Generate Cartesian k-space Trajectories

Source: https://pynufft.readthedocs.io/en/latest/manu/realistic_om.html

This snippet generates Cartesian k-space trajectories for NUFFT. It's primarily for testing purposes as FFT is generally preferred for Cartesian data. Ensure PyNUFFT and SciPy are installed.

```python
# Generating trajectories for Cartesian k-space
import numpy
import matplotlib.pyplot
matplotlib.pyplot.gray()

def fake_Cartesian(Nd):
    dim = len(Nd) # dimension
    M = numpy.prod(Nd)
    om = numpy.zeros((M, dim), dtype = numpy.float)
    grid = numpy.indices(Nd)
    for dimid in range(0, dim):
        om[:, dimid] = (grid[dimid].ravel() *2/ Nd[dimid] - 1.0)*numpy.pi
    return om

import scipy.misc

from pynufft import NUFFT


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

image = scipy.misc.ascent()[::2,::2]
om = fake_Cartesian(Nd)


print('Number of samples (M) = ', om.shape[0])
print('Dimension = ', om.shape[1])
print('Nd = ', Nd)
print('Kd = ', Kd)
print('Jd = ', Jd)

NufftObj = NUFFT()
NufftObj.plan(om, Nd, Kd, Jd)
y = NufftObj.forward(image)

y2 = y.reshape(Nd, order='C')
x2 = numpy.fft.ifftshift(numpy.fft.ifftn(numpy.fft.ifftshift(y2)))
matplotlib.pyplot.subplot(1,3,1)
matplotlib.pyplot.imshow(image.real, vmin = 0, vmax = 255)
matplotlib.pyplot.title('Original image')
matplotlib.pyplot.subplot(1,3,2)
matplotlib.pyplot.imshow(x2.real, vmin = 0, vmax = 255)
matplotlib.pyplot.title('Restored image')
matplotlib.pyplot.subplot(1,3,3)
matplotlib.pyplot.imshow(abs(image - x2), vmin = 0, vmax = 255)
matplotlib.pyplot.title('Difference map')
matplotlib.pyplot.show()

```

--------------------------------

### Initialize and Plan NUFFT Object

Source: https://pynufft.readthedocs.io/en/latest/tutor/more2D.html

Creates a NUFFT object and plans it with specified image dimensions (Nd), k-space dimensions (Kd), and interpolation grid size (Jd).

```python
NufftObj = NUFFT()

Nd = (256, 256)  # image size
print('setting image dimension Nd...', Nd)
Kd = (512, 512)  # k-space size
print('setting spectrum dimension Kd...', Kd)
Jd = (6, 6)  # interpolation size
print('setting interpolation size Jd...', Jd)

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

--------------------------------

### Nvidia OpenCL Device Information

Source: https://pynufft.readthedocs.io/en/latest/installation/OpenCL.html

This snippet displays the output of 'clinfo' for an Nvidia GPU, showing device name, vendor, OpenCL version, driver version, and compute capabilities. It is useful for verifying OpenCL installation on Nvidia hardware.

```text
Platform Name                                   NVIDIA CUDA
Number of devices                                 1
  Device Name                                     GeForce GTX 1060
  Device Vendor                                   NVIDIA Corporation
  Device Vendor ID                                0x10de
  Device Version                                  OpenCL 1.2 CUDA
  Driver Version                                  415.18
  Device OpenCL C Version                         OpenCL C 1.2
  Device Type                                     GPU
  Device Topology (NV)                            PCI-E, 01:00.0
  Device Profile                                  FULL_PROFILE
  Device Available                                Yes
  Compiler Available                              Yes
  Linker Available                                Yes
  Max compute units                               10
  Max clock frequency                             1670MHz
  Compute Capability (NV)                         6.1
  Device Partition                                (core)
    Max number of sub-devices                     1
    Supported partition types                     None
    Supported affinity domains                    (n/a)
  Max work item dimensions                        3
  Max work item sizes                             1024x1024x64
  Max work group size                             1024
  Preferred work group size multiple              32
  Warp size (NV)                                  32
  Preferred / native vector sizes
    char                                                 1 / 1
    short                                                1 / 1
    int                                                  1 / 1
    long                                                 1 / 1
    half                                                 0 / 0        (n/a)
    float                                                1 / 1
    double                                               1 / 1        (cl_khr_fp64)
  Half-precision Floating-point support           (n/a)
  Single-precision Floating-point support         (core)
    Denormals                                     Yes
    Infinity and NANs                             Yes
    Round to nearest                              Yes
    Round to zero                                 Yes
    Round to infinity                             Yes
    IEEE754-2008 fused multiply-add               Yes
    Support is emulated in software               No
    Correctly-rounded divide and sqrt operations  Yes
  Double-precision Floating-point support         (cl_khr_fp64)
    Denormals                                     Yes
    Infinity and NANs                             Yes
    Round to nearest                              Yes
    Round to zero                                 Yes
    Round to infinity                             Yes
    IEEE754-2008 fused multiply-add               Yes
    Support is emulated in software               No

```

--------------------------------

### _create_kspace_sampling_density() (CPU Solvers)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Helper function to create k-space sampling density for CPU solvers.

```APIDOC
## _create_kspace_sampling_density() (CPU Solvers)

### Description
Generates the sampling density in k-space for CPU-based computations.

### Function
`_create_kspace_sampling_density()`
```

--------------------------------

### Initialize NUFFT Object (Default)

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Instantiate the NUFFT class with default settings. This is suitable for basic CPU operations.

```python
>>> from pynufft import NUFFT
>>> NufftObj = NUFFT()
```

--------------------------------

### NUFFT Class Initialization

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Initializes the NUFFT class with optional device index and legacy settings.

```APIDOC
## NUFFT Class Initialization

### Description
Initializes the NUFFT class with optional device index and legacy settings.

### Method
__init__

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Parameters
- **_device_indx** (int) - Optional - The index of the device to use.
- **_legacy** (bool) - Optional - Flag for legacy behavior.

### Request Example
```python
from pynufft import NUFFT

nufft_instance = NUFFT(_device_indx=0, _legacy=False)
```

### Response
#### Success Response (200)
Initializes the NUFFT object.

#### Response Example
```python
# No explicit response object, initialization is successful if no exception is raised.
```
```

--------------------------------

### device_list()

Source: https://pynufft.readthedocs.io/en/latest/index.html

Returns a list of available devices.

```APIDOC
## device_list()

### Description
Retrieves a list of available computational devices (e.g., CPU, GPU).

### Function
`device_list()`
```

--------------------------------

### device_list Function

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Returns a tuple of available devices for acceleration. Returns an empty tuple if no devices are available.

```APIDOC
## device_list Function

### Description
Returns a tuple of available devices for acceleration. Returns an empty tuple if no devices are available.

### Returns
- `devices` (tuple): A tuple containing information about available acceleration devices.

### Usage
```python
# Example usage
available_devices = pynufft.src._helper.helper.device_list()
```
```

--------------------------------

### NUFFT Class - _precompute_sp_cpu

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Private method to precompute the adjoint (gridding) and Toepitz interpolation matrix on the CPU.

```APIDOC
## NUFFT Class - _precompute_sp_cpu

### Description
Private: Precompute adjoint (gridding) and Toepitz interpolation matrix.

### Method
`_precompute_sp_cpu()`

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Parameters
**None** (_Python Nonetype_)

### Returns
self: instance

### Return type
instance
```

--------------------------------

### Plan Multiple NUFFT Instances After Creation

Source: https://pynufft.readthedocs.io/en/latest/_sources/manu/multiple_NUFFT.rst.txt

Create multiple NUFFT objects and plan them after all instances have been initialized. This approach is suitable when you need to set up several NUFFT operations before executing any.

```python
# Create the first NUFFT
NufftObj1 = NUFFT()
# Create the second NUFFT
NufftObj2 = NUFFT()
# Plan the first instance
NufftObj1.plan(om1, Nd, Kd, Jd)
NufftObj2.plan(om2, Nd, Kd, Jd)
```

--------------------------------

### plan()

Source: https://pynufft.readthedocs.io/en/latest/index.html

Plans or sets up a computation, likely for NUFFT/NUDFT.

```APIDOC
## plan()

### Description
Sets up and plans a computation, typically for NUFFT or NUDFT.

### Function
`plan()`
```

--------------------------------

### Plan NUFFT Object (with ft_axes)

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Initializes the NUFFT object and plans it with geometry and optional Fourier transform axes. This allows for specifying which axes to use for the Fourier transform.

```python
>>> NufftObj.plan(om, Nd, Kd, Jd, ft_axes)
```

--------------------------------

### Plan NUFFT using Legacy Method

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Plans the NUFFT object using a legacy method, typically for CPU-based operations. This is an alternative to the standard CPU planning.

```python
>>> import pynufft
>>> NufftObj = pynufft.NUFFT_cpu()
>>> NufftObj.plan(om, Nd, Kd, Jd)
```

--------------------------------

### Import PyNUFFT and Dependencies

Source: https://pynufft.readthedocs.io/en/latest/tutor/more2D.html

Import necessary libraries including numpy, scipy.misc, matplotlib.pyplot, and the NUFFT class from pynufft.

```python
import numpy
import scipy.misc
import matplotlib.pyplot

from pynufft import NUFFT
```

--------------------------------

### CPU NUFFT Forward Transform and Comparison

Source: https://pynufft.readthedocs.io/en/latest/manu/multiple_NUFFT.html

Plans and executes a NUFFT forward transform on the CPU using the `NUFFT` class. It then compares the result with a previously computed one (e.g., from GPU or multiprocessing).

```python
NUFFT_cpu1 = NUFFT()
NUFFT_cpu1.plan(om1, Nd, Kd, Jd)
y1 = NUFFT_cpu1.forward(x)
print('norm = ', numpy.linalg.norm(y1 - result1) / numpy.linalg.norm(y1))
```

```python
NUFFT_cpu2 = NUFFT()
NUFFT_cpu2.plan(om2, Nd, Kd, Jd)
y2 = NUFFT_cpu2.forward(x)
print('norm = ', numpy.linalg.norm(y2 - result2) / numpy.linalg.norm(y2))
```

--------------------------------

### cCopy() (Metaprogramming)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Copies complex data.

```APIDOC
## cCopy() (Metaprogramming)

### Description
Copies complex data from one location to another.

### Function
`cCopy()`
```

--------------------------------

### cSqrt() (Metaprogramming)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Computes the square root of complex numbers.

```APIDOC
## cSqrt() (Metaprogramming)

### Description
Calculates the square root of complex numbers.

### Function
`cSqrt()`
```

--------------------------------

### pynufft.src._helper.helper.plan

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Plans the NUFFT object, preparing necessary structures for computation based on input parameters.

```APIDOC
## pynufft.src._helper.helper.plan

### Description
Plans the NUFFT object, preparing necessary structures for computation based on input parameters.

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
*   **om** (numpy.float) - Coordinate
*   **Nd** (tuple of int) - Image shape
*   **Kd** (tuple of int) - Oversampled grid shape
*   **Jd** (tuple of int) - Interpolator size
*   **ft_axes** (tuple of int, optional) - Axes where FFT takes place
*   **format** (string, optional, default='CSR') - Output format of the interpolator. ‘CSR’: the precomputed Compressed Sparse Row (CSR) matrix. ‘pELL’: partial ELLPACK which precomputes the concatenated 1D interpolators.
*   **radix** (int, optional) - Radix for computation

### Request Example
None

### Response
#### Success Response
*   **dictionary** - Dictionary containing NUFFT plan information

#### Response Example
None
```

--------------------------------

### create_kernel_sets() (Metaprogramming)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Creates sets of kernels for metaprogramming tasks.

```APIDOC
## create_kernel_sets() (Metaprogramming)

### Description
Generates sets of kernels used in metaprogramming operations.

### Function
`create_kernel_sets()`
```

--------------------------------

### NUFFT Constructor

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Initializes a NUFFT object. It can be initialized without arguments for default settings, or with a device index to specify a particular acceleration device.

```APIDOC
## NUFFT Constructor

### Description
Initializes a NUFFT object. It can be initialized without arguments for default settings, or with a device index to specify a particular acceleration device.

### Method
__init__

### Parameters
- **_device_indx** (NoneType) - Optional - The index of the device to use.
- **_legacy** (NoneType) - Optional - Legacy mode flag.

### Returns
- **NUFFT**: The pynufft.NUFFT instance.

### Return Type
- **NUFFT**: The pynufft.NUFFT class

### Example
```python
>>> from pynufft import NUFFT
>>> NufftObj = NUFFT()
```

```python
>>> from pynufft import NUFFT, helper
>>> device = helper.device_list()[0]
>>> NufftObj = NUFFT(device) # for first acceleration device in the system
```
```

--------------------------------

### NUFFT Class - plan_legacy

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Designs the min-max interpolator for legacy execution. It requires the off-grid locations, image size, oversampled frequency grid size, and interpolator size as input.

```APIDOC
## NUFFT Class - plan_legacy

### Description
Designs the min-max interpolator for legacy execution.

### Method
`plan_legacy(_om_, _Nd_, _Kd_, _Jd_, _ft_axes=None_)

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Parameters
* **om** (_numpy.float array_, matrix size = M * ndims_) – The M off-grid locations in the frequency domain. Normalized between [-pi, pi]
* **Nd** (_tuple_, ndims integer elements.) – The matrix size of equispaced image. Example: Nd=(256,256) for a 2D image; Nd = (128,128,128) for a 3D image
* **Kd** (_tuple_, ndims integer elements.) – The matrix size of the oversampled frequency grid. Example: Kd=(512,512) for 2D image; Kd = (256,256,256) for a 3D image
* **Jd** (_tuple_, ndims integer elements.) – The interpolator size. Example: Jd=(6,6) for 2D image; Jd = (6,6,6) for a 3D image
* **ft_axes** (_None_ or _tuple with optional integer elements._) – (Optional) The axes for Fourier transform. The default is all axes if ‘None’ is given.

### Returns
0

### Return type
int, float

### Request Example
```python
>>> import pynufft
>>> NufftObj = pynufft.NUFFT_cpu()
>>> NufftObj.plan(om, Nd, Kd, Jd)
```
```

--------------------------------

### NUFFT_cupy

Source: https://pynufft.readthedocs.io/en/latest/index.html

Documentation for the NUFFT_cupy class, designed for NUFFT operations using CuPy for GPU acceleration.

```APIDOC
## NUFFT_cupy

### Description
NUFFT implementation utilizing CuPy for GPU computations.

### Class
`NUFFT_cupy`
```

--------------------------------

### _create_kspace_sampling_density() (HSA Solvers)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Helper function to create k-space sampling density for HSA solvers.

```APIDOC
## _create_kspace_sampling_density() (HSA Solvers)

### Description
Generates the sampling density in k-space for HSA-based computations.

### Function
`_create_kspace_sampling_density()`
```

--------------------------------

### cSpmv() (Metaprogramming)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Performs sparse matrix-vector multiplication for complex numbers.

```APIDOC
## cSpmv() (Metaprogramming)

### Description
Computes the product of a sparse matrix and a complex vector.

### Function
`cSpmv()`
```

--------------------------------

### Check OpenCL Driver Information

Source: https://pynufft.readthedocs.io/en/latest/installation/Macbook.html

Execute clinfo to display details about your system's OpenCL platform and devices. This is used to verify OpenCL compatibility for GPU acceleration.

```text
Number of platforms                               1
  Platform Name                                   Apple
  Platform Vendor                                 Apple
  Platform Version                                OpenCL 1.2 (Jun 23 2023 20:24:12)
  Platform Profile                                FULL_PROFILE
  Platform Extensions                             cl_APPLE_SetMemObjectDestructor cl_APPLE_ContextLoggingFunctions cl_APPLE_clut cl_APPLE_query_kernel_names cl_APPLE_gl_sharing cl_khr_gl_event

  Platform Name                                   Apple
Number of devices                                 1
  Device Name                                     Apple M1
  Device Vendor                                   Apple
  Device Vendor ID                                0x1027f00
  Device Version                                  OpenCL 1.2
  Driver Version                                  1.2 1.0
  Device OpenCL C Version                         OpenCL C 1.2
  Device Type                                     GPU
  Device Profile                                  FULL_PROFILE
  Device Available                                Yes
  Compiler Available                              Yes
  Linker Available                                Yes
  Max compute units                               8
  Max clock frequency                             1000MHz
  Device Partition                                (core)
    Max number of sub-devices                     0
    Supported partition types                     None
    Supported affinity domains                    (n/a)
  Max work item dimensions                        3
  Max work item sizes                             256x256x256
  Max work group size                             256
  Preferred work group size multiple (kernel)     32
  Preferred / native vector sizes
    char                                                 1 / 1
    short                                                1 / 1
    int                                                  1 / 1
    long                                                 1 / 1
    half                                                 0 / 0        (n/a)
    float                                                1 / 1
    double                                               1 / 1        (n/a)
  Half-precision Floating-point support           (n/a)
  Single-precision Floating-point support         (core)
    Denormals                                     No
    Infinity and NANs                             Yes
    Round to nearest                              Yes
    Round to zero                                 Yes
    Round to infinity                             Yes
    IEEE754-2008 fused multiply-add               Yes
    Support is emulated in software               No
    Correctly-rounded divide and sqrt operations  Yes
  Double-precision Floating-point support         (n/a)
  Address bits                                    64, Little-Endian
  Global memory size                              11453251584 (10.67GiB)
  Error Correction support                        No
  Max memory allocation                           2147483648 (2GiB)
  Unified memory for Host and Device              Yes
  Minimum alignment for any data type             1 bytes
  Alignment of base address                       32768 bits (4096 bytes)
  Global Memory cache type                        None
  Image support                                   Yes
    Max number of samplers per kernel             32
    Max size for 1D images from buffer            268435456 pixels
    Max 1D or 2D image array size                 2048 images
    Base address alignment for 2D image buffers   256 bytes
    Pitch alignment for 2D image buffers          256 pixels
    Max 2D image size                             16384x16384 pixels
    Max 3D image size                             2048x2048x2048 pixels
    Max number of read image args                 128
    Max number of write image args                8
  Local memory type                               Local
  Local memory size                               32768 (32KiB)
  Max number of constant args                     31
  Max constant buffer size                        1073741824 (1024MiB)
  Max size of kernel argument                     4096 (4KiB)
  Queue properties
    Out-of-order execution                        No
    Profiling                                     Yes
  Prefer user sync for interop                    Yes
```

--------------------------------

### pynufft.src.re_subroutine.cMultiplyRealInplace

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Returns the kernel source of cMultiplyRealInplace.

```APIDOC
## pynufft.src.re_subroutine.cMultiplyRealInplace

### Description
Returns the kernel source of cMultiplyRealInplace.

### Method
N/A (Function call)

### Endpoint
N/A (Function call)

### Parameters
N/A

### Request Example
N/A

### Response
N/A
```

--------------------------------

### pynufft.linalg.solve_cpu.solve

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Solves the NUFFT problem using specified solvers on CPU.

```APIDOC
## pynufft.linalg.solve_cpu.solve

### Description
Solves the NUFFT problem on CPU. Supports solvers like 'cg', 'L1TVOLS', or 'L1TVLAD'.

### Parameters
* **nufft**: NUFFT_cpu object
* **y**: (M,) array, non-uniform data
* **solver**: Optional. Solver to use ('cg', 'L1TVOLS', 'L1TVLAD'). Defaults to None.
* **args**: Additional arguments for the solver.
* **kwargs**: Additional keyword arguments for the solver.

### Returns
* **x**: The reconstructed image.
```

--------------------------------

### QR_process()

Source: https://pynufft.readthedocs.io/en/latest/index.html

Processes QR decomposition or related operations.

```APIDOC
## QR_process()

### Description
Handles the processing of QR decomposition or related matrix operations.

### Function
`QR_process()`
```

--------------------------------

### Multiprocessing NUFFT Execution

Source: https://pynufft.readthedocs.io/en/latest/manu/multiple_NUFFT.html

Demonstrates using `multiprocessing.Pool` to run NUFFT computations asynchronously across multiple processes. Results are collected and compared.

```python
result = pool.apply_async(D.run, args = (x, (0,3)))
# the result is appended
results.append(result)

# Now enter the second process
# This is the standard multiprocessing Pool
D = atomic_NUFFT(om2, Nd, Kd, Jd, 0)
# Non-obstructive
result = pool.apply_async(D.run, args = (x, (4,7)))
results.append(result)

# closing the pool 
pool.close()
pool.join()

# results are appended
# Now print the outputs
result1 = results[0].get()
result2 = results[1].get()
```

--------------------------------

### pynufft.src.re_subroutine.cMultiplyVecInplace

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Returns the kernel source of cMultiplyVecInplace.

```APIDOC
## pynufft.src.re_subroutine.cMultiplyVecInplace

### Description
Returns the kernel source of cMultiplyVecInplace.

### Method
N/A (Function call)

### Endpoint
N/A (Function call)

### Parameters
N/A

### Request Example
N/A

### Response
N/A
```

--------------------------------

### cRealShrink() (Metaprogramming)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Performs real shrinkage on complex numbers.

```APIDOC
## cRealShrink() (Metaprogramming)

### Description
Applies real shrinkage to complex numbers.

### Function
`cRealShrink()`
```

--------------------------------

### Plan NUFFT on Device

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Plans the NUFFT object for device execution (e.g., GPU). This method is suitable for accelerated computations.

```python
>>> import pynufft
>>> device=pynufft.helper.device_list()[0]
>>> NufftObj = pynufft.NUFFT(device)
>>> NufftObj.plan(om, Nd, Kd, Jd)
```

--------------------------------

### NUFFT.plan

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Plans the NUFFT object by setting up the necessary parameters for the transformation based on the provided geometry.

```APIDOC
## NUFFT.plan

### Description
Plans the NUFFT object by setting up the necessary parameters for the transformation based on the provided geometry.

### Method
plan

### Parameters
#### Parameters
- **om** (numpy.float array) - The M off-grid locations in the frequency domain, normalized between [-pi, pi]. Matrix size is M * ndims.
- **Nd** (tuple of int) - The matrix size of the equispaced image. Example: Nd=(256,256) for a 2D image.
- **Kd** (tuple of int) - The matrix size of the oversampled frequency grid. Example: Kd=(512,512) for a 2D image.
- **Jd** (tuple of int) - The interpolator size. Example: Jd=(6,6) for a 2D image.
- **ft_axes** (tuple of int, optional) - The axes for Fourier transform. Defaults to all axes if None.

### Returns
- **int, float** - Returns 0 upon successful planning.

### Variables
- **Nd** - Initial value: Nd
- **Kd** - Initial value: Kd
- **Jd** - Initial value: Jd
- **ft_axes** - Initial value: None

### Example
```python
>>> from pynufft import NUFFT
>>> NufftObj = NUFFT()
>>> NufftObj.plan(om, Nd, Kd, Jd)
```

or

```python
>>> NufftObj.plan(om, Nd, Kd, Jd, ft_axes)
```
```

--------------------------------

### L1TVOLS() (CPU Solvers)

Source: https://pynufft.readthedocs.io/en/latest/index.html

CPU solver for L1 Total Variation optimization problems.

```APIDOC
## L1TVOLS() (CPU Solvers)

### Description
Performs L1 Total Variation optimization using CPU.

### Function
`L1TVOLS()`
```

--------------------------------

### pynufft.src.re_subroutine.cMultiplyConjVecInplace

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Returns the kernel source of cMultiplyConjVecInplace.

```APIDOC
## pynufft.src.re_subroutine.cMultiplyConjVecInplace

### Description
Returns the kernel source of cMultiplyConjVecInplace.

### Method
N/A (Function call)

### Endpoint
N/A (Function call)

### Parameters
N/A

### Request Example
N/A

### Response
N/A
```

--------------------------------

### Plan NUFFT on CPU

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Plans the NUFFT object for CPU execution using specified geometry. Use this when targeting CPU computations.

```python
>>> from pynufft import NUFFT
>>> NufftObj = NUFFT()
>>> NufftObj.plan(om, Nd, Kd, Jd)
```

```python
>>> NufftObj.plan(om, Nd, Kd, Jd, ft_axes)
```

--------------------------------

### NUFFT_torch

Source: https://pynufft.readthedocs.io/en/latest/index.html

Documentation for the NUFFT_torch class, enabling NUFFT operations with PyTorch.

```APIDOC
## NUFFT_torch

### Description
NUFFT implementation compatible with PyTorch.

### Class
`NUFFT_torch`
```

--------------------------------

### _pipe_density() (CPU Solvers)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Internal function for processing density in CPU solvers.

```APIDOC
## _pipe_density() (CPU Solvers)

### Description
Processes density information within the CPU solver pipeline.

### Function
`_pipe_density()`
```

--------------------------------

### Import NUFFT Module

Source: https://pynufft.readthedocs.io/en/latest/_sources/tutor/example.rst.txt

Import the NUFFT class from the pynufft library.

```python
from pynufft import NUFFT
```

--------------------------------

### cHypot() (Metaprogramming)

Source: https://pynufft.readthedocs.io/en/latest/index.html

Calculates the hypotenuse for complex numbers.

```APIDOC
## cHypot() (Metaprogramming)

### Description
Calculates the hypotenuse (magnitude) for complex numbers.

### Function
`cHypot()`
```

--------------------------------

### solve() (CPU Solvers)

Source: https://pynufft.readthedocs.io/en/latest/index.html

The main solve function for CPU-based optimization problems.

```APIDOC
## solve() (CPU Solvers)

### Description
Executes the solving process for optimization problems on the CPU.

### Function
`solve()`
```

--------------------------------

### Forward NUFFT (Legacy)

Source: https://pynufft.readthedocs.io/en/latest/API/init.html

Performs the forward Non-Uniform Fast Fourier Transform using legacy device support.

```APIDOC
## Forward NUFFT (Legacy)

### Description
Performs the forward Non-Uniform Fast Fourier Transform using legacy device support.

### Method
_forward_legacy

### Parameters
- **gx** (reikna gpu array) - The input gpu array, with size = Nd.
  - Type: reikna gpu array with dtype = numpy.complex64

### Returns
- **gy**: The output gpu array, with size = (M,)

### Return Type
- **reikna gpu array**: reikna gpu array with dtype = numpy.complex64
```
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