### Install MRpro Python Package

Source: https://docs.mrpro.rocks/user_guide

Instructions for installing the MRpro framework using pip, including options for basic installation, additional dependencies for notebooks, and direct installation from the GitHub repository.

```Shell
pip install mrpro
```

```Shell
pip install mrpro[notebooks]
```

```Shell
pip install "git+https://github.com/PTB-MR/mrpro"
```

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

### Install mrpro library if not present

Source: https://docs.mrpro.rocks/_downloads/cc676d8b9b1ce252393c34d574f85f5e/tv_minimization_reconstruction_pdhg

Checks for the presence of the 'mrpro' library and installs it along with its 'notebooks' dependencies if it's not found. This ensures the environment is set up correctly to run the reconstruction examples.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### Install MRpro with Notebooks Dependencies

Source: https://docs.mrpro.rocks/_downloads/0bd3aa53b488f539fdf23fa4cdddd899/cartesian_reconstruction

This snippet checks if the 'mrpro' library is installed and, if not, installs it along with its notebook-specific dependencies. This ensures that the necessary environment is set up for running the examples.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### Install mrpro library and dependencies

Source: https://docs.mrpro.rocks/_downloads/08ecca3652a95795ea26ce68ca461824/comparison_trajectory_calculators

This snippet checks for the presence of the 'mrpro' library. If not found, it proceeds to install 'mrpro' along with its 'notebooks' dependencies using pip, ensuring the environment is prepared for running the provided examples.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### mrpro.data.QHeader API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.prewhiten_kspace

API documentation for the `QHeader` class, detailing its constructors from various sources, and properties related to resolution, position, orientation, and device status.

```APIDOC
mrpro.data.QHeader:
  Methods:
    __init__()
    from_dicom()
    from_iheader()
    from_kheader()
  Properties:
    resolution
    position
    orientation
    device
    is_cpu
    is_cuda
    ndim
```

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

### Install mrpro library if not found

Source: https://docs.mrpro.rocks/_downloads/0de23edf8dc53925d1e62f7ca38a8b00/direct_reconstruction

This snippet checks for the presence of the 'mrpro' library and installs it along with its 'notebooks' dependencies if it's not already installed. This ensures that all necessary components for running the examples are available in the environment.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### API Documentation for DensityCompensationOp

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.models.EPG.initial_state

Provides detailed API reference for the `DensityCompensationOp` class, specifically detailing its constructor.

```APIDOC
DensityCompensationOp:
  Methods:
    __init__()
```

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

### mrpro.data.QHeader Class API

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.models.EPG.initial_state

API documentation for the `QHeader` class, detailing its initialization methods from different sources, and properties for accessing resolution, position, orientation, and device information.

```APIDOC
mrpro.data.QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### Install mrpro library if not found

Source: https://docs.mrpro.rocks/_downloads/6785dbff279df6db8fed91f8dfedbee5/iterative_sense_reconstruction_with_regularization

Checks if the 'mrpro' library is available in the current environment. If not found, it proceeds to install 'mrpro' along with its 'notebooks' dependencies using pip, ensuring the necessary environment for running the examples.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### API Reference for mrpro.data.QHeader

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.slice_profiles.SliceGaussian

API documentation for the `QHeader` class, detailing its initialization methods from various sources and properties for accessing resolution, position, orientation, and device information.

```APIDOC
mrpro.data.QHeader:
  Methods:
    __init__()
    from_dicom()
    from_iheader()
    from_kheader()
  Properties:
    resolution
    position
    orientation
    device
    is_cpu
    is_cuda
    ndim
```

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

### mrpro.data.QData API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.prewhiten_kspace

API documentation for the `QData` class, covering its initialization, data access, device handling, shape information, and various data manipulation methods.

```APIDOC
mrpro.data.QData:
  Methods:
    __init__()
    from_single_dicom()
    apply()
    apply_()
    clone()
    concatenate()
    cpu()
    cuda()
    double()
    half()
    items()
    rearrange()
    single()
    split()
    to()
    __eq__()
    __new__()
  Properties:
    data
    header
    device
    is_cpu
    is_cuda
    ndim
    shape
```

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

### mrpro.data.QData Class Methods and Properties

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.models.WASABITI

Documentation for the `QData` class in `mrpro.data`, covering its initialization, data access properties, device management methods, and various data manipulation functions like apply, clone, and concatenate.

```APIDOC
class mrpro.data.QData:
  __init__()
  from_single_dicom()
  data
  header
  device
  is_cpu
  is_cuda
  ndim
  shape
  apply()
  apply_()
  clone()
  concatenate()
  cpu()
  cuda()
  double()
  half()
  items()
  rearrange()
  single()
  split()
  to()
  __eq__()
  __new__()
```

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

### Install mrpro library if not found

Source: https://docs.mrpro.rocks/_downloads/a80110c83292567ac7d3d5b6da5eedef/qmri_sg_challenge_2024_t1

Checks if the 'mrpro' library is installed and installs it along with 'notebooks' dependencies if it's not found. This ensures the necessary environment for running the notebook.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### API Documentation for mrpro.operators.EinsumOp

Source: https://docs.mrpro.rocks/user_guide

Documents the `EinsumOp` class within the `mrpro.operators` module, outlining its constructor, properties, and various operator methods.

```APIDOC
mrpro.operators.EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
```

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

### Install mrpro library and dependencies

Source: https://docs.mrpro.rocks/_downloads/5c9b574f51852d92d9336bdf4d7a2616/qmri_cardiac_fingerprinting

Checks if the 'mrpro' library is installed in the current environment. If not found, it proceeds to install 'mrpro' along with its notebook-specific dependencies using pip. This ensures all necessary packages are available for executing the notebook's code.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### Install mrpro library if not found

Source: https://docs.mrpro.rocks/_downloads/6ae3823ec28b1274464026f5ea725b96/qmri_t1_mapping_with_grad_acq

This snippet checks if the `mrpro` library is installed in the current environment. If it's not found, it proceeds to install `mrpro` along with its notebook-specific dependencies using `pip`. This ensures the necessary packages are available for executing the rest of the notebook.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### EinsumOp Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.functionals

API documentation for the `EinsumOp` class, including its initialization, properties, and various operational methods.

```APIDOC
mrpro.operators.EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### Python: Examples for Rearranging Dataclass Dimensions

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.CsmData

Illustrative Python code examples demonstrating how to use the `rearrange()` method to split and reorder data dimensions.

```Python
# Split the phase encode lines into 8 cardiac phases
data.rearrange('batch coils k2 (phase k1) k0 -> batch phase coils k2 k1 k0', phase=8)
# Split the k-space samples into 64 k1 and 64 k2 lines
data.rearrange('... 1 1 (k2 k1 k0) -> ... k2 k1 k0', k2=64, k1=64, k0=128)
```

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

### mrpro.operators.DensityCompensationOp API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.pgd

API documentation for the `DensityCompensationOp` class, specifically detailing its constructor (`__init__`).

```APIDOC
DensityCompensationOp:
  __init__()
```

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

### EinsumOp Class API

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.models.WASABITI

API documentation for the `EinsumOp` class, including its initialization, properties, and various operator methods.

```APIDOC
EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### Python: Example Usage of mrpro.data.KHeader.rearrange

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.KHeader

Illustrates how to use the `rearrange` method of the `mrpro.data.KHeader` class with practical examples, demonstrating splitting phase encode lines and k-space samples.

```Python
# Split the phase encode lines into 8 cardiac phases
data.rearrange('batch coils k2 (phase k1) k0 -> batch phase coils k2 k1 k0', phase=8)
# Split the k-space samples into 64 k1 and 64 k2 lines
data.rearrange('... 1 1 (k2 k1 k0) -> ... k2 k1 k0', k2=64, k1=64, k0=128)
```

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

### mrpro.data.QHeader API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.traj_calculators.KTrajectoryRpe

API documentation for the `QHeader` class, which encapsulates quantitative header information. It provides methods for initialization from various sources and properties for accessing resolution, position, orientation, and device status.

```APIDOC
QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### Parameters and Examples for Data Splitting

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.AcqInfo

Defines parameters such as dimension, size, overlap, and dilation for splitting data, along with illustrative examples demonstrating their effect on the resulting data segments.

```APIDOC
Parameters:
  dim (int): dimension to split along.
  size (int, default: 1): size of the splits.
  overlap (int, default: 0): overlap between splits. The stride will be `size - overlap`. Negative overlap will leave spaces between splits.
  dilation (int, default: 1): dilation of elements in each split.

Examples:
If the dimension has 6 elements:
* split with size 2, overlap 0, dilation 1 -> elements (0,1), (2,3), and (4,5)
* split with size 2, overlap 1, dilation 1 -> elements (0,1), (1,2), (2,3), (3,4), (4,5), and (5,6)
* split with size 2, overlap 0, dilation 2 -> elements (0,2), and (3,5)
* split with size 2, overlap -1, dilation 1 -> elements (0,1), and (3,4)

Returns:
  A tuple of the splits.
```

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

### API Documentation for DensityCompensationOp Class

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.broadcasted_rearrange

API documentation for the `DensityCompensationOp` class, specifically detailing its constructor.

```APIDOC
DensityCompensationOp:
  __init__()
```

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

### mrpro.utils.broadcasted_rearrange Python Examples

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.broadcasted_rearrange

Illustrative Python examples demonstrating the usage of `mrpro.utils.broadcasted_rearrange` with different input tensors and parameters, showcasing basic rearrangement, explicit broadcasting, and view reduction behavior.

```Python
import torch
from mrpro.utils import broadcasted_rearrange

tensor = torch.randn(1, 16, 1, 768, 256)
shape_result = broadcasted_rearrange(tensor, "... (phase k1) k0 -> phase ... k1 k0", phase=8, reduce_views=False).shape
# Expected output: torch.Size([8, 1, 16, 1, 96, 256])
```

```Python
import torch
from mrpro.utils import broadcasted_rearrange

tensor = torch.randn(1, 1, 1, 768, 1)
shape_result = broadcasted_rearrange(tensor, "... (phase k1) k0 -> phase ... k1 k0",
   broadcasted_shape=(1, 16, 1, 768, 256), phase=8, reduce_views=False).shape
# Expected output: torch.Size([8, 1, 16, 1, 96, 256]) # Behaves as-if the tensor was of shape (1, 16, 1, 768, 256)
```

```Python
import torch
from mrpro.utils import broadcasted_rearrange

tensor = torch.randn(1, 1, 1, 768, 1)
shape_result = broadcasted_rearrange(tensor, "... (phase k1) k0 -> phase ... k1 k0",
   broadcasted_shape=(1, 16, 1, 768, 256) phase=8, reduce_views=True).shape
# Expected output: torch.Size([8, 1, 1, 1, 96, 1]) # Dimensions that are stride-0 are reduced to singleton dimensions
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.cg

API documentation for the `QHeader` class in `mrpro.data`, covering its initialization, methods for creating headers from DICOM, iheader, or kheader, and properties for accessing resolution, position, orientation, and device information.

```APIDOC
class QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution: property
  position: property
  orientation: property
  device: property
  is_cpu: property
  is_cuda: property
  ndim: property
```

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

### mrpro Split Function Parameters and Examples

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.KData

Documents the parameters for a data splitting function, including dimension, size, overlap, and dilation, along with illustrative examples demonstrating how these parameters influence the resulting splits.

```APIDOC
Parameters:
  dim: int – dimension to split along.
  size: int (default: 1) – size of the splits.
  overlap: int (default: 0) – overlap between splits. The stride will be `size - overlap`. Negative overlap will leave spaces between splits.
  dilation: int (default: 1) – dilation of elements in each split.

Examples (If the dimension has 6 elements):
  split with size 2, overlap 0, dilation 1 → elements (0,1), (2,3), and (4,5)
  split with size 2, overlap 1, dilation 1 → elements (0,1), (1,2), (2,3), (3,4), (4,5), and (5,6)
  split with size 2, overlap 0, dilation 2 → elements (0,2), and (3,5)
  split with size 2, overlap -1, dilation 1 → elements (0,1), and (3,4)

Returns:
  A tuple of the splits.
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.ConstraintsOp

API documentation for the `QHeader` class in `mrpro.data`, detailing its initialization from various sources, and properties for accessing resolution, position, orientation, and device information.

```APIDOC
mrpro.data.QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### Example Output Log from PDHG Algorithm

Source: https://docs.mrpro.rocks/_notebooks/tv_minimization_reconstruction_pdhg

This is an example output log showing the objective function's value at various iterations during the PDHG algorithm's execution, demonstrating the convergence of the optimization process.

```Text
Iteration   0: Objective = 1.721e-06
Iteration  16: Objective = 1.031e-06
Iteration  32: Objective = 8.298e-07
Iteration  48: Objective = 7.132e-07
Iteration  64: Objective = 6.370e-07
Iteration  80: Objective = 5.819e-07
Iteration  96: Objective = 5.397e-07
Iteration 112: Objective = 5.071e-07
Iteration 128: Objective = 4.808e-07
Iteration 144: Objective = 4.592e-07
Iteration 160: Objective = 4.414e-07
Iteration 176: Objective = 4.257e-07
Iteration 192: Objective = 4.118e-07
Iteration 208: Objective = 4.002e-07
Iteration 224: Objective = 3.904e-07
Iteration 240: Objective = 3.807e-07
Iteration 256: Objective = 3.721e-07
```

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

### Conceptual Example for mrpro.utils.split_idx Logic

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.split_idx

An illustrative example demonstrating the conceptual splitting logic of the `mrpro.utils.split_idx` function, showing how an input index array is divided into blocks based on specified block size, overlap, and cyclic parameters.

```Python
# idx = [1,2,3,4,5,6,7,8,9], np_per_block = 5, np_overlap = 3, cycle = True
#
# idx: 1 2 3 4 5 6 7 8 9
# block 0: 0 0 0 0 0
# block 1: 1 1 1 1 1
# block 2: 2 2 2 2 2
# block 3: 3 3 3 3 3
```

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

### Usage and Examples for mrpro.data.Dataclass.to Method

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.AcqInfo

Explains the core functionality of the `to` method for `mrpro.data.Dataclass`, detailing how it performs dtype and device conversions. It clarifies its relationship with `torch.Tensor.to` and provides specific examples of how different data types are handled during conversion.

```APIDOC
Description:
Perform dtype and/or device conversion of data.
A torch.dtype and torch.device are inferred from the arguments args and kwargs. Please have a look at the documentation of torch.Tensor.to for more details.
A new instance of the dataclass will be returned.
The conversion will be applied to all Tensor- or Module fields of the dataclass, and to all fields that implement the Dataclass.
The dtype-type, i.e. float or complex will always be preserved, but the precision of floating point dtypes might be changed.

Examples:
If called with `dtype=torch.float32` OR `dtype=torch.complex64`:
* A `complex128` tensor will be converted to `complex64`
* A `float64` tensor will be converted to `float32`
* A `bool` tensor will remain `bool`
* An `int64` tensor will remain `int64`
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.pgd

API documentation for the `QHeader` class, detailing its constructor, methods for creating instances from DICOM, iheader, or kheader, and properties for resolution, position, orientation, device information, and dimensionality.

```APIDOC
QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### QData Class API Documentation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.unit_conversion.ms_to_s

API documentation for the `mrpro.data.QData` class, covering its initialization, data access properties, device management methods, and various data manipulation functions.

```APIDOC
mrpro.data.QData:
  __init__()
  from_single_dicom()
  data
  header
  device
  is_cpu
  is_cuda
  ndim
  shape
  apply()
  apply_()
  clone()
  concatenate()
  cpu()
  cuda()
  double()
  half()
  items()
  rearrange()
  single()
  split()
  to()
  __eq__()
  __new__()
```

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

### API Reference & Example: mrpro.data.DcfData.rearrange Method

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.DcfData

Documents the `rearrange` method, which allows flexible rearrangement of tensor dimensions within a `DcfData` object, similar to `einops.rearrange`. Includes a Python example demonstrating its usage for splitting phase encode lines.

```APIDOC
rearrange(*pattern: str, **axes_lengths: int) -> Self
  Rearrange the data according to the specified pattern.
  Similar to `einops.rearrange`, allowing flexible rearrangement of data dimensions.
  Parameters:
    pattern (str): The rearrangement pattern string.
    axes_lengths (int): Keyword arguments specifying lengths for new axes.
```

```python
# Split the phase encode lines into 8 cardiac phases
data.rearrange('batch coils k2 (phase k1) k0 -> batch phase coils k2 k1 k0', phase=8)
```

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

### APIDOC: EinsumOp Class Reference

Source: https://docs.mrpro.rocks/_notebooks/qmri_cardiac_fingerprinting

API documentation for the `EinsumOp` class, detailing its initialization, adjoint, gramian, forward, and operator norm methods, along with various arithmetic and bitwise dunder methods.

```APIDOC
EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### Download Raw Data for mrpro Example from Zenodo

Source: https://docs.mrpro.rocks/_notebooks/comparison_trajectory_calculators

This Python code snippet facilitates the download of raw MR data from a specified Zenodo record into a temporary directory. It leverages the `zenodo_get` library to fetch the data, which is a prerequisite for running the subsequent image reconstruction examples.

```python
# Download raw data from Zenodo
import tempfile
from pathlib import Path

import mrpro
import torch
import zenodo_get

tmp = tempfile.TemporaryDirectory()  # RAII, automatically cleaned up
data_folder = Path(tmp.name)
zenodo_get.download(record='14617082', retry_attempts=5, output_dir=data_folder)
```

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

### QHeader Class API Documentation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.unit_conversion.ms_to_s

API documentation for the `mrpro.data.QHeader` class, detailing its constructors from various sources, and properties for accessing resolution, position, orientation, and device information.

```APIDOC
mrpro.data.QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### Install mrpro Library if Not Found

Source: https://docs.mrpro.rocks/_downloads/9cf9875eec8da88abc45b6a8b64afbff/pgd_wavelet_reconstruction

This Python snippet checks for the presence of the `mrpro` library. If the library is not found in the current environment, it proceeds to install `mrpro` along with its notebook-specific dependencies using `pip`. This ensures that all necessary modules for the reconstruction process are available before execution.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### mrpro.operators.models Module API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.EinsumOp

Comprehensive API documentation for the `mrpro.operators.models` module, listing key classes and their associated methods. This includes initialization, call, forward pass, and various operator overloading methods.

```APIDOC
mrpro.operators.models.MOLLI
  __mul__()
  __radd__()
  __rmul__()

mrpro.operators.models.MonoExponentialDecay
  __init__()
  __call__()
  forward()
  __add__()
  __matmul__()
  __mul__()
  __radd__()
  __rmul__()

mrpro.operators.models.SaturationRecovery
  __init__()
  __call__()
  forward()
  __add__()
  __matmul__()
  __mul__()
  __radd__()
  __rmul__()

mrpro.operators.models.SpoiledGRE
  __init__()
  __call__()
  forward()
  __add__()
  __matmul__()
  __mul__()
  __radd__()
  __rmul__()

mrpro.operators.models.TransientSteadyStateWithPreparation
  __init__()
  __call__()
```

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

### Visualize M0 and T1 Starting Values

Source: https://docs.mrpro.rocks/_notebooks/qmri_sg_challenge_2024_t1

Plots the calculated starting values for `M0` (equilibrium magnetization) and `T1` (longitudinal relaxation time) using `matplotlib`. This visualization provides a crucial visual inspection of the initial parameters derived from the dictionary matching, helping to assess their quality before proceeding with the non-linear optimization.

```Python
fig, axes = plt.subplots(1, 2, figsize=(6, 2), squeeze=False)

im = axes[0, 0].imshow(m0_start[0, 0])
axes[0, 0].set_title('$M_0$ start values')
axes[0, 0].set_axis_off()
fig.colorbar(im, ax=axes[0, 0], label='a.u.')

im = axes[0, 1].imshow(t1_start[0, 0], vmin=0, vmax=2.5, cmap=Colormap('lipari').to_mpl())
axes[0, 1].set_title('$T_1$ start values')
axes[0, 1].set_axis_off()
fig.colorbar(im, ax=axes[0, 1], label='s')

plt.show()
```

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

### QData Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.ConstraintsOp

API documentation for the `QData` class in `mrpro.data`, covering its initialization, data access properties, device management methods, and various data manipulation functions.

```APIDOC
mrpro.data.QData:
  __init__()
  from_single_dicom()
  data
  header
  device
  is_cpu
  is_cuda
  ndim
  shape
  apply()
  apply_()
  clone()
  concatenate()
  cpu()
  cuda()
  double()
  half()
  items()
  rearrange()
  single()
  split()
  to()
  __eq__()
  __new__()
```

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

### Determine Initial Value for Iterative Reconstruction

Source: https://docs.mrpro.rocks/_downloads/3f8ef0c72e8d9b31624d32f0f6e83cef/iterative_sense_reconstruction_radial2D

This code block calculates the initial value for the iterative reconstruction. It starts with the density-compensated adjoint reconstruction (A^H W y) and then applies a scaling factor to account for normalization differences, ensuring a well-conditioned starting point.

```python
(u,) = (acquisition_operator.H @ dcf_operator)(kdata.data)
scale = vdot(u, u) / vdot(u, *(acquisition_operator.H @ dcf_operator @ acquisition_operator)(u))
initial_value = u * scale
```

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

### Install mrpro Library if Not Found

Source: https://docs.mrpro.rocks/_downloads/3f8ef0c72e8d9b31624d32f0f6e83cef/iterative_sense_reconstruction_radial2D

This Python snippet checks for the presence of the 'mrpro' library. If it's not found in the current environment, it proceeds to install 'mrpro' along with its notebook-specific dependencies using the pip package manager, ensuring all required components are available for execution.

```python
import importlib

if not importlib.util.find_spec('mrpro'):
    %pip install mrpro[notebooks]
```

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

### API Reference for MOLLI Class

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.prewhiten_kspace

Provides the API documentation for the MOLLI class, specifically detailing its constructor.

```APIDOC
MOLLI:
  __init__():
```

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

### Download Example MRI Data from Zenodo

Source: https://docs.mrpro.rocks/_downloads/9cf9875eec8da88abc45b6a8b64afbff/pgd_wavelet_reconstruction

This Python code block downloads example MRI k-space data from a specified Zenodo record. It uses `tempfile` to create a temporary directory for storing the downloaded files, ensuring a clean environment. The `zenodo_get` library handles the download process, including retries for robustness.

```python
import tempfile
from pathlib import Path

import zenodo_get

tmp = tempfile.TemporaryDirectory()  # RAII, automatically cleaned up
data_folder = Path(tmp.name)
zenodo_get.download(record='14617082', retry_attempts=5, output_dir=data_folder)
```

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

### API Reference for EinsumOp

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils

API documentation for the `EinsumOp` class, providing details on its initialization, properties, and methods for Einstein summation and related functional and arithmetic operations.

```APIDOC
EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### API Documentation for ZeroOp Class

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.summarize_object

Details the API for the `ZeroOp` class, outlining its constructor, adjoint, forward methods, and properties related to zero operations.

```APIDOC
ZeroOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
```

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

### Initialize M0 and T1 Start Values from Dictionary

Source: https://docs.mrpro.rocks/_downloads/a80110c83292567ac7d3d5b6da5eedef/qmri_sg_challenge_2024_t1

This line initializes the starting values for M0 (equilibrium magnetization) and T1 (longitudinal relaxation time) by looking up the closest values in a pre-computed dictionary based on cosine similarity for each voxel. These values serve as initial guesses for the subsequent optimization process.

```python
m0_start, t1_start = dictionary(idata_multi_ti.data.real)
```

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

### APIDOC: EinsumOp Class Methods

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.QData

API documentation for the EinsumOp class, detailing its available methods and properties.

```APIDOC
EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### Get Identity Rotation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Retrieves the identity rotation, where a single rotation is generated.

```APIDOC
Rotation:
  identity():
    Description: A single rotation is generated.
    Returns:
      identity: The identity rotation.
```

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

### Get Quaternion Y Component

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Retrieves the 'y' component (second vector part) of the quaternion representation of the rotation.

```APIDOC
Property: quaternion_y
  Type: Tensor
  Description: Get y component of the quaternion.
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.split_idx

API documentation for the `QHeader` class, detailing its constructors from different sources, and properties for accessing resolution, position, orientation, device information, and dimensionality.

```APIDOC
mrpro.data.QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.unit_conversion.s_to_ms

API documentation for the `QHeader` class in `mrpro.data`, outlining its constructors, properties for resolution, position, and orientation, and device-related attributes.

```APIDOC
mrpro.data.QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### Get Device of Rotation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Retrieves the PyTorch device on which the Rotation object's data is located.

```APIDOC
Property: device
  Type: device
  Description: Get the device of the Rotation.
```

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

### API Reference for EinsumOp

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms

API documentation for the `EinsumOp` class, detailing its constructor, properties, and methods for Einstein summation operations.

```APIDOC
EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### Python Example: L1 Regularized Image Reconstruction with PGD

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.pgd

This Python example demonstrates the application of the `pgd` function for L1 regularized image reconstruction. It initializes k-space data, defines the forward operator (Fast Fourier Transform), the data fidelity term (L2 norm squared), and the regularization term (L1 norm). It then calculates an appropriate step size and calls the `pgd` optimizer to find the image solution.

```Python
kspace_data = torch.randn(3, 10, 10 , dtype=torch.complex64)
fft = FastFourierOp()
l2 = L2NormSquared(target=kspace_data)
f = l2 @ fft
reg_parameter = 0.01
g = reg_parameter * L1Norm()
operator_norm = 1.
stepsize = 0.85 * 1 / operator_norm**2
initial_value = torch.ones(3,10,10)
pgd_image_solution = pgd(
    f=f,
    g=g,
    initial_value=initial_value,
    stepsize=stepsize,
    max_iterations=200,
    backtrack_factor=1.0,
)
```

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

### API Reference for EinsumOp Class

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.lbfgs

Documents the `EinsumOp` class, detailing its initialization, properties like H and gram, and various operator methods for Einstein summation.

```APIDOC
EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### Get Quaternion W Component

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Retrieves the 'w' component (scalar part) of the quaternion representation of the rotation.

```APIDOC
Property: quaternion_w
  Type: Tensor
  Description: Get w component of the quaternion.
```

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

### API Reference for mrpro.data.AcqInfo Class

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.prewhiten_kspace

Documents the `AcqInfo` class, including its constructor.

```APIDOC
AcqInfo:
  __new__()
```

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

### Get Batch Shape of Rotation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Returns the batch shape of the Rotation object, indicating the dimensions of the rotation data.

```APIDOC
Property: shape
  Type: Size
  Description: Return the batch shape of the Rotation.
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.typing

API documentation for the QHeader class, detailing its constructors from different sources, and properties for accessing resolution, position, orientation, device status, and dimensions.

```APIDOC
QHeader:
  - __init__()
  - from_dicom()
  - from_iheader()
  - from_kheader()
  - resolution
  - position
  - orientation
  - device
  - is_cpu
  - is_cuda
  - ndim
```

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

### Get Quaternion Z Component

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Retrieves the 'z' component (third vector part) of the quaternion representation of the rotation.

```APIDOC
Property: quaternion_z
  Type: Tensor
  Description: Get z component of the quaternion.
```

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

### API Documentation for EinsumOp

Source: https://docs.mrpro.rocks/_autosummary/mrpro.phantoms.brainweb

API documentation for the `EinsumOp` class, detailing its constructor, properties, and methods for Einstein summation operations.

```APIDOC
EinsumOp:
  __init__():
  H:
  gram:
  __call__():
  adjoint():
  forward():
  operator_norm():
  __add__():
  __and__():
  __matmul__():
  __mul__():
  __or__():
  __radd__():
  __rmul__():
```

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

### Get Quaternion X Component

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Retrieves the 'x' component (first vector part) of the quaternion representation of the rotation.

```APIDOC
Property: quaternion_x
  Type: Tensor
  Description: Get x component of the quaternion.
```

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

### mrpro.operators.DensityCompensationOp API

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.lbfgs

API documentation for the `DensityCompensationOp` class, detailing its initialization method.

```APIDOC
DensityCompensationOp:
  __init__()
```

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

### Get Number of Dimensions (ndim) Property

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.KTrajectory

Returns the number of dimensions of the dataclass, representing the broadcasted shape of all fields.

```APIDOC
property ndim: int
  Description: Return the number of dimensions of the dataclass. This is the number of dimensions of the broadcasted shape of all fields.
```

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

### API Documentation for mrpro.data.AcqInfo Class

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.ZeroPadOp

Documents the `AcqInfo` class, primarily detailing its `__new__` method for object instantiation.

```APIDOC
mrpro.data.AcqInfo:
  __new__()
```

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

### Get number of dimensions (ndim property)

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.AcqInfo

Returns the number of dimensions of the dataclass, which is derived from the broadcasted shape of all its fields.

```APIDOC
ndim: int
  Return the number of dimensions of the dataclass.
  This is the number of dimensions of the broadcasted shape of all fields.
```

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

### Get Determinant of Rotation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Returns the determinant of the rotation matrix. This value will be 1.0 for proper rotations and -1.0 for improper rotations.

```APIDOC
Property: det
  Type: Tensor
  Description: Returns the determinant of the rotation matrix.
    Will be 1. for proper rotations and -1. for improper rotations.
```

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

### API Documentation for EinsumOp

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.models.SaturationRecovery

API reference for the `EinsumOp` class, detailing its constructor, properties, and methods, including operator overloading methods.

```APIDOC
EinsumOp:
  __init__()
  H
  gram
  __call__()
  adjoint()
  forward()
  operator_norm()
  __add__()
  __and__()
  __matmul__()
  __mul__()
  __or__()
  __radd__()
  __rmul__()
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.models.SaturationRecovery

API documentation for the `QHeader` class, detailing its constructor, methods for creating headers from various sources, and properties for accessing resolution, position, orientation, and device information.

```APIDOC
QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### Get Broadcasted Shape (shape) Property

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.KTrajectory

Returns the broadcasted shape of all tensor/data fields within the dataclass. Each field is broadcastable to this shape.

```APIDOC
property shape: Size
  Description: Return the broadcasted shape of all tensor/data fields. Each field of this dataclass is broadcastable to this shape.
  Returns:
    The broadcasted shape of all fields.
  Raises:
    InconsistentShapeError: If the shapes cannot be broadcasted.
```

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

### Get Iterator for Dataclass Fields (mrpro.data.Dataclass)

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Dataclass

Returns an iterator over the names and values of the fields contained within the Dataclass instance.

```APIDOC
items() -> Iterator[tuple[str, Any]]

Description:
  Get an iterator over names and values of fields.
```

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

### mrpro.data Module API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.lbfgs

Detailed API documentation for the `mrpro.data` module, including classes `KTrajectory`, `Limits`, `QData`, and `QHeader` and their respective methods and properties.

```APIDOC
mrpro.data.KTrajectory:
  Methods:
    concatenate()
    cpu()
    cuda()
    double()
    half()
    items()
    rearrange()
    single()
    split()
    to()
    __eq__()
    __new__()
mrpro.data.Limits:
  Methods:
    __init__()
    from_ismrmrd()
    to_ismrmrd()
    __eq__()
    __new__()
  Properties:
    min
    max
    center
    length
mrpro.data.QData:
  Methods:
    __init__()
    from_single_dicom()
    apply()
    apply_()
    clone()
    concatenate()
    cpu()
    cuda()
    double()
    half()
    items()
    rearrange()
    single()
    split()
    to()
    __eq__()
    __new__()
  Properties:
    data
    header
    device
    is_cpu
    is_cuda
    ndim
    shape
mrpro.data.QHeader:
  Methods:
    __init__()
    from_dicom()
    from_iheader()
    from_kheader()
  Properties:
    resolution
    position
    orientation
    device
    is_cpu
    is_cuda
    ndim
```

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

### Get Number of Dimensions (ndim) - mrpro.data

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.CsmData

Retrieves the total number of dimensions for the dataclass, which corresponds to the broadcasted shape of all its fields.

```APIDOC
property ndim: int

Return the number of dimensions of the dataclass.
This is the number of dimensions of the broadcasted shape of all fields.
```

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

### mrpro.data.QData Class API

Source: https://docs.mrpro.rocks/_autosummary/mrpro.operators.models.EPG.initial_state

API documentation for the `QData` class, outlining its initialization methods, data access properties, and various utility functions for data processing, device management, and array manipulation.

```APIDOC
mrpro.data.QData:
  __init__()
  from_single_dicom()
  data
  header
  device
  is_cpu
  is_cuda
  ndim
  shape
  apply()
  apply_()
  clone()
  concatenate()
  cpu()
  cuda()
  double()
  half()
  items()
  rearrange()
  single()
  split()
  to()
  __eq__()
  __new__()
```

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

### QHeader Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.TensorAttributeMixin

API documentation for the `QHeader` class, detailing its constructor, methods for creating instances from DICOM, iheader, or kheader, and properties for accessing resolution, position, orientation, device information, and dimensionality.

```APIDOC
QHeader:
  __init__()
  from_dicom()
  from_iheader()
  from_kheader()
  resolution
  position
  orientation
  device
  is_cpu
  is_cuda
  ndim
```

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

### Get Iterator for mrpro.data.AcqInfo Fields

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.AcqInfo

Documents the `items()` method, which provides an iterator over the names and values of fields within the data class.

```APIDOC
items() -> Iterator[tuple[str, Any]]
  Description: Get an iterator over names and values of fields.
```

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

### mrpro.data.traj_calculators Module Classes

Source: https://docs.mrpro.rocks/_autosummary/mrpro.phantoms.brainweb

This section outlines the classes within the `mrpro.data.traj_calculators` module, which are responsible for computing k-space trajectories. An example class is `KTrajectoryCalculator`.

```APIDOC
mrpro.data.traj_calculators:
  KTrajectoryCalculator
```

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

### AcqInfo Class API Documentation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.csm.inati

API documentation for the `AcqInfo` class, detailing its constructor.

```APIDOC
AcqInfo:
  __new__()
```

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

### Get Magnitude of mrpro.data.Rotation

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

Documents the `magnitude` method, which returns the angular magnitude(s) of the rotation(s) in radians, always within the range [0, pi].

```APIDOC
magnitude() -> torch.Tensor
  Get the magnitude(s) of the rotation(s).

  Returns:
    magnitude (torch.Tensor): Angles in radians. The magnitude will always be in the range [0, pi].
```

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

### API Documentation for DensityCompensationOp

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.unit_conversion.deg_to_rad

API documentation for the `DensityCompensationOp` class, specifically detailing its constructor.

```APIDOC
DensityCompensationOp:
  __init__()
```

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

### QData Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.algorithms.optimizers.pgd

API documentation for the `QData` class, covering its initialization, methods for creating instances from DICOM, properties for data, header, device information, and methods for applying operations, cloning, concatenation, device transfer, data type conversion, item access, rearrangement, and splitting.

```APIDOC
QData:
  __init__()
  from_single_dicom()
  data
  header
  device
  is_cpu
  is_cuda
  ndim
  shape
  apply()
  apply_()
  clone()
  concatenate()
  cpu()
  cuda()
  double()
  half()
  items()
  rearrange()
  single()
  split()
  to()
  __eq__()
  __new__()
```

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

### Get Identity Rotation(s) (identity)

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.Rotation

This class method generates one or more identity rotation objects. Composition with an identity rotation has no effect on other rotations.

```APIDOC
classmethod identity(
  shape: int | None | tuple[int, ...] = None
) -> Self

Description: Get identity rotation(s). Composition with the identity rotation has no effect.

Parameters:
  shape (int | None | tuple[int, ...], default: None): Number of identity rotations to generate. If None, then a single identity rotation is returned.

Returns:
  Self: Object containing the identity rotation(s).
```

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

### Get number of dimensions

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.QHeader

Returns the total number of dimensions for the dataclass. This value represents the number of dimensions of the broadcasted shape across all fields.

```APIDOC
property ndim: int
  Returns:
    The number of dimensions of the dataclass. This is the number of dimensions of the broadcasted shape of all fields.
```

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

### QData Class API Reference

Source: https://docs.mrpro.rocks/_autosummary/mrpro.utils.typing

API documentation for the QData class, covering its initialization, data access properties, device management methods, and various data manipulation functions.

```APIDOC
QData:
  - __init__()
  - from_single_dicom()
  - data
  - header
  - device
  - is_cpu
  - is_cuda
  - ndim
  - shape
  - apply()
  - apply_()
  - clone()
  - concatenate()
  - cpu()
  - cuda()
  - double()
  - half()
  - items()
  - rearrange()
  - single()
  - split()
  - to()
  - __eq__()
  - __new__()
```

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

### API Documentation for EinsumOp

Source: https://docs.mrpro.rocks/_autosummary/mrpro.phantoms.brainweb.BrainwebTissue

Provides an API reference for the `EinsumOp` class, detailing its constructor, properties, and methods for Einstein summation operations.

```APIDOC
EinsumOp:
  __init__():
  H:
  gram:
  __call__():
  adjoint():
  forward():
  operator_norm():
  __add__():
  __and__():
  __matmul__():
  __mul__():
  __or__():
  __radd__():
  __rmul__():
```

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

### Get Number of Dimensions for Dataclass

Source: https://docs.mrpro.rocks/_autosummary/mrpro.data.KNoise

Retrieves the number of dimensions for the dataclass. This value is derived from the broadcasted shape of all fields contained within the dataclass.

```APIDOC
property ndim: int
  Description: Return the number of dimensions of the dataclass.
  Details: This is the number of dimensions of the broadcasted shape of all fields.
```