pytomography.algorithms.statistical_iterative#

This module contains classes that implement statistical iterative reconstruction algorithms.

Module Contents#

Classes#

StatisticalIterative

Parent class for all statistical iterative algorithms. All child classes must implement the __call__ method to perform reconstruction.

OSEMOSL

Implementation of the ordered subset expectation algorithm using the one-step-late method to include prior information: \(\hat{f}^{n,m+1} = \left[\frac{1}{H_m^T 1 + \beta \frac{\partial V}{\partial \hat{f}}|_{\hat{f}=\hat{f}^{n,m}}} H_m^T \left(\frac{g_m}{H_m\hat{f}^{n,m}+s}\right)\right] \hat{f}^{n,m}\).

BSREM

Implementation of the block-sequential-regularized (BSREM) reconstruction algorithm: \(\hat{f}^{n,m+1} = \hat{f}^{n,m} + \alpha_n D \left[H_m^T \left(\frac{g_m}{H_m \hat{f}^{n,m} + s} -1 \right) - \beta \nabla_{f^{n,m}} V \right]\). The implementation of this algorithm corresponds to Modified BSREM-II with \(U=\infty\), \(t=0\), and \(\epsilon=0\) (see https://ieeexplore.ieee.org/document/1207396). There is one difference in this implementation: rather than using FBP to get an initial estimate (as is done in the paper), a single iteration of OSEM is used; this initialization is required here due to the requirement for global scaling (see discussion on page 620 of paper).

OSEM

Implementation of the ordered subset expectation maximum algorithm \(\hat{f}^{n,m+1} = \left[\frac{1}{H_m^T 1} H_m^T \left(\frac{g_m}{H_m\hat{f}^{n,m}+s}\right)\right] \hat{f}^{n,m}\).

KEM

Implementation of the KEM reconstruction algorithm given by \(\hat{\alpha}^{n,m+1} = \left[\frac{1}{K^T H_m^T 1} K^T H_m^T \left(\frac{g_m}{H_m K \hat{\alpha}^{n,m}+s}\right)\right] \hat{\alpha}^{n,m}\) and where the final predicted object is \(\hat{f}^{n,m} = K \hat{\alpha}^{n,m}\).

DIPRecon

Implementation of the Deep Image Prior reconstruction technique (see https://ieeexplore.ieee.org/document/8581448). This reconstruction technique requires an instance of a user-defined prior_network that implements two functions: (i) a fit method that takes in an object (\(x\)) which the network f(z;\theta) is subsequently fit to, and (ii) a predict function that returns the current network prediction \(f(z;\theta)\). For more details, see the Deep Image Prior tutorial.

Functions#

get_projection_subset(projections, subset_indices, device)

pytomography.algorithms.statistical_iterative.get_projection_subset(projections, subset_indices, device)[source]#
class pytomography.algorithms.statistical_iterative.StatisticalIterative(projections, system_matrix, object_initial=None, scatter=None, prior=None, precompute_normalization_factors=True, device=pytomography.device)[source]#

Parent class for all statistical iterative algorithms. All child classes must implement the __call__ method to perform reconstruction.

Parameters:
  • projections (torch.Tensor) – photopeak window projection data \(g\) to be reconstructed

  • system_matrix (SystemMatrix) – system matrix that models the imaging system. In particular, corresponds to \(H\) in \(g=Hf\).

  • object_initial (torch.tensor[batch_size, Lx, Ly, Lz]) – the initial object guess \(f^{0,0}\). If None, then initial guess consists of all 1s. Defaults to None.

  • scatter (torch.Tensor) – estimate of scatter contribution \(s\). Defaults to 0.

  • prior (Prior, optional) – the Bayesian prior; used to compute \(\beta \frac{\partial V}{\partial f}\). If None, then this term is 0. Defaults to None.

  • precompute_normalization_factors (bool) –

  • device (str) – The device correpsonding to the tensors output by the system matrix. In some cases, although the system matrix implementation uses pytomography.device in its internal computation, it will output tensors on the CPU due to their size (such as in listmode PET). Defaults to pytomography.device.

abstract __call__(n_iters, n_subsets, callback=None)[source]#

Abstract method for performing reconstruction: must be implemented by subclasses.

Parameters:
  • n_iters (int) – Number of iterations

  • n_subsets (int) – Number of subsets

  • callbacks (Callback, optional) – Callbacks to be evaluated after each subiteration. Defaults to None.

  • callback (pytomography.callbacks.Callback | None) –

Return type:

None

_compute_callback(n_iter)[source]#
Parameters:

n_iter (int) –

class pytomography.algorithms.statistical_iterative.OSEMOSL(projections, system_matrix, object_initial=None, scatter=None, prior=None, precompute_normalization_factors=True, device=pytomography.device)[source]#

Bases: StatisticalIterative

Implementation of the ordered subset expectation algorithm using the one-step-late method to include prior information: \(\hat{f}^{n,m+1} = \left[\frac{1}{H_m^T 1 + \beta \frac{\partial V}{\partial \hat{f}}|_{\hat{f}=\hat{f}^{n,m}}} H_m^T \left(\frac{g_m}{H_m\hat{f}^{n,m}+s}\right)\right] \hat{f}^{n,m}\).

Parameters:
  • projections (torch.Tensor) – photopeak window projection data \(g\) to be reconstructed

  • system_matrix (SystemMatrix) – system matrix that models the imaging system. In particular, corresponds to \(H\) in \(g=Hf\).

  • object_initial (torch.tensor[batch_size, Lx, Ly, Lz]) – the initial object guess \(f^{0,0}\). If None, then initial guess consists of all 1s. Defaults to None.

  • scatter (torch.Tensor) – estimate of scatter contribution \(s\). Defaults to 0.

  • prior (Prior, optional) – the Bayesian prior; used to compute \(\beta \frac{\partial V}{\partial f}\). If None, then this term is 0. Defaults to None.

  • precompute_normalization_factors (bool) –

  • device (str) – The device correpsonding to the tensors output by the system matrix. In some cases, although the system matrix implementation uses pytomography.device in its internal computation, it will output tensors on the CPU due to their size (such as in listmode PET). Defaults to pytomography.device.

_set_recon_name(n_iters, n_subsets)[source]#

Set the unique identifier for the type of reconstruction performed. Useful when saving reconstructions to DICOM files

Parameters:
  • n_iters (int) – Number of iterations

  • n_subsets (int) – Number of subsets

_compute_normalization_factors()[source]#

Computes normalization factors \(H_m^T 1\) for all subsets \(m\).

__call__(n_iters, n_subsets, n_subset_specific=None, callback=None)[source]#

Performs the reconstruction using n_iters iterations and n_subsets subsets.

Parameters:
  • n_iters (int) – Number of iterations

  • n_subsets (int) – Number of subsets

  • n_subset_specific (int) – Iterate only over the subset specified. Defaults to None

  • callback (Callback, optional) – Callback function to be evaluated after each subiteration. Defaults to None.

Returns:

reconstructed object

Return type:

torch.tensor[batch_size, Lx, Ly, Lz]

class pytomography.algorithms.statistical_iterative.BSREM(projections, system_matrix, object_initial=None, scatter=0, prior=None, relaxation_function=lambda x: ..., scaling_matrix_type='subind_norm', precompute_normalization_factors=True, device=pytomography.device)[source]#

Bases: StatisticalIterative

Implementation of the block-sequential-regularized (BSREM) reconstruction algorithm: \(\hat{f}^{n,m+1} = \hat{f}^{n,m} + \alpha_n D \left[H_m^T \left(\frac{g_m}{H_m \hat{f}^{n,m} + s} -1 \right) - \beta \nabla_{f^{n,m}} V \right]\). The implementation of this algorithm corresponds to Modified BSREM-II with \(U=\infty\), \(t=0\), and \(\epsilon=0\) (see https://ieeexplore.ieee.org/document/1207396). There is one difference in this implementation: rather than using FBP to get an initial estimate (as is done in the paper), a single iteration of OSEM is used; this initialization is required here due to the requirement for global scaling (see discussion on page 620 of paper).

Parameters:
  • projections (torch.Tensor) – projection data \(g\) to be reconstructed.

  • system_matrix (SystemMatrix) – System matrix \(H\) used in \(g=Hf\).

  • object_initial (torch.tensor[batch_size, Lx, Ly, Lz]) – represents the initial object guess \(f^{0,0}\) for the algorithm in object space

  • scatter (torch.Tensor) – estimate of scatter contribution \(s\).

  • prior (Prior, optional) – the Bayesian prior; computes \(\beta \frac{\partial V}{\partial f}\). If None, then this term is 0. Defaults to None.

  • relaxation_function (Callable, optional) – Sequence \(\alpha_n\) used for relaxation. Defaults to \(\alpha_n=1/(n+1)\).

  • scaling_matrix_type (str, optional) – The form of the scaling matrix \(D\) used. If subind_norm (sub-iteration independent + normalized), then \(D=\left(S_m/M \cdot H^T 1 \right)^{-1}\). If subdep_norm (sub-iteration dependent + normalized) then \(D = \left(H_m^T 1\right)^{-1}\). See section III.D in the paper above for a discussion on this.

  • device (str) – The device correpsonding to the tensors output by the system matrix. In some cases, although the system matrix implementation uses pytomography.device in its internal computation, it will output tensors on the CPU due to their size (such as in listmode PET). Defaults to pytomography.device.

  • precompute_normalization_factors (bool) –

_compute_normalization_factors()[source]#

Computes normalization factors \(H_m^T 1\) for all subsets \(m\).

_set_recon_name(n_iters, n_subsets)[source]#

Set the unique identifier for the type of reconstruction performed. Useful for saving to DICOM files

Parameters:
  • n_iters (int) – Number of iterations

  • n_subsets (int) – Number of subsets

_scale_prior_gradient(gradient)[source]#

Used to scale gradient to avoid divisional errors in null regions when using CutOffTransform

Parameters:

gradient (torch.tensor) – Gradient returned by prior function

Returns:

New gradient tensor where values are set to 0 outside the cutoff region.

Return type:

torch.tensor

__call__(n_iters, n_subsets, n_subset_specific=None, callback=None)[source]#

Performs the reconstruction using n_iters iterations and n_subsets subsets.

Parameters:
  • n_iters (int) – Number of iterations

  • n_subsets (int) – Number of subsets

  • n_subset_specific (int) – Iterate only over the subset specified. Defaults to None

  • callback (Callback, optional) – Callback function to be called after each subiteration. Defaults to None.

Returns:

reconstructed object

Return type:

torch.tensor[batch_size, Lx, Ly, Lz]

class pytomography.algorithms.statistical_iterative.OSEM(projections, system_matrix, object_initial=None, scatter=0, precompute_normalization_factors=True, device=pytomography.device)[source]#

Bases: OSEMOSL

Implementation of the ordered subset expectation maximum algorithm \(\hat{f}^{n,m+1} = \left[\frac{1}{H_m^T 1} H_m^T \left(\frac{g_m}{H_m\hat{f}^{n,m}+s}\right)\right] \hat{f}^{n,m}\).

Parameters:
  • projections (torch.Tensor) – photopeak window projection data \(g\) to be reconstructed

  • system_matrix (SystemMatrix) – system matrix that models the imaging system. In particular, corresponds to \(H\) in \(g=Hf\).

  • object_initial (torch.tensor[batch_size, Lx, Ly, Lz]) – the initial object guess \(f^{0,0}\). If None, then initial guess consists of all 1s. Defaults to None.

  • scatter (torch.Tensor) – estimate of scatter contribution \(s\). Defaults to 0.

  • precompute_normalization_factors (bool) –

  • device (str) –

class pytomography.algorithms.statistical_iterative.KEM(projections, system_matrix, kem_transform, object_initial=None, scatter=0)[source]#

Bases: OSEM

Implementation of the KEM reconstruction algorithm given by \(\hat{\alpha}^{n,m+1} = \left[\frac{1}{K^T H_m^T 1} K^T H_m^T \left(\frac{g_m}{H_m K \hat{\alpha}^{n,m}+s}\right)\right] \hat{\alpha}^{n,m}\) and where the final predicted object is \(\hat{f}^{n,m} = K \hat{\alpha}^{n,m}\).

Parameters:
  • projections (torch.Tensor) – projection data \(g\) to be reconstructed

  • system_matrix (SystemMatrix) – System matrix \(H\) used in \(g=Hf\).

  • kem_transform (KEMTransform) – The transform corresponding to the matrix \(K\).

  • object_initial (torch.tensor[batch_size, Lx, Ly, Lz]) – represents the initial object guess \(f^{0,0}\) for the algorithm in object space

  • scatter (torch.Tensor) – estimate of scatter contribution \(s\).

_compute_callback(n_iter)[source]#

Computes callback for KEM transform; this is reimplemented here because the self.object_prediction corresponds to the \(\alpha\) value and not \(f\). As such, the KEMTransform needs to be applied before the object is input to the callback.

Parameters:

n_iter (int) – _description_

__call__(n_iters, n_subsets, callback=None)[source]#

Performs the reconstruction using n_iters iterations and n_subsets subsets.

Parameters:
  • n_iters (int) – Number of iterations

  • n_subsets (int) – Number of subsets

  • callback (Callback, optional) – Callback function to be called after each subiteration. Defaults to None.

Returns:

reconstructed object

Return type:

torch.tensor[batch_size, Lx, Ly, Lz]

class pytomography.algorithms.statistical_iterative.DIPRecon(projections, system_matrix, prior_network, rho=0.003, scatter=0, precompute_normalization_factors=True)[source]#

Bases: StatisticalIterative

Implementation of the Deep Image Prior reconstruction technique (see https://ieeexplore.ieee.org/document/8581448). This reconstruction technique requires an instance of a user-defined prior_network that implements two functions: (i) a fit method that takes in an object (\(x\)) which the network f(z;\theta) is subsequently fit to, and (ii) a predict function that returns the current network prediction \(f(z;\theta)\). For more details, see the Deep Image Prior tutorial.

Parameters:
  • projections (torch.tensor) – projection data \(g\) to be reconstructed

  • system_matrix (SystemMatrix) – System matrix \(H\) used in \(g=Hf\).

  • prior_network (nn.Module) – User defined prior network that implements the neural network f(z;\theta)

  • rho (float, optional) – Value of \(\rho\) used in the optimization procedure. Defaults to 1.

  • scatter (torch.tensor | float, optional) – Projection space scatter estimate. Defaults to 0.

  • precompute_normalization_factors (bool, optional) – Whether to precompute \(H_m^T 1\) and store on GPU in the OSEM network before reconstruction. Defaults to True.

__call__(n_iters, subit1, n_subsets_osem=1, callback=None)[source]#

Implementation of Algorithm 1 in https://ieeexplore.ieee.org/document/8581448. This implementation gives the additional option to use ordered subsets. The quantity SubIt2 specified in the paper is controlled by the user-defined prior_network class.

Parameters:
  • n_iters (int) – Number of iterations (MaxIt in paper)

  • subit1 (int) – Number of OSEM iterations before retraining neural network (SubIt1 in paper)

  • n_subsets_osem (int, optional) – Number of subsets to use in OSEM reconstruction. Defaults to 1.

Returns:

Reconstructed image

Return type:

torch.Tensor