pytomography.algorithms.osem#

This module contains classes that implement ordered-subset maximum liklihood iterative reconstruction algorithms. Such algorithms compute \(f_i^{n,m+1}\) from \(f_i^{n,m}\) where \(n\) is the index for an iteration, and \(m\) is the index for a subiteration (i.e. for a given subset). The notation is defined such that given \(M\) total subsets of equal size, \(f_i^{n+1,0} \equiv f_i^{n,M}\) (i.e. after completing a subiteration for each subset, we start the next iteration). Any class that inherits from this class must implement the forward method. __init__ initializes the reconstruction algorithm with the image data \(g_j\), the forward and back projections used (i.e. networks to compute \(\sum_i H_{ij} a_i\) and \(\sum_j H_{ij} b_j\)), the initial object guess \(f_i^{0,0}\), the estimated scatter contribution \(s_j\), and the Bayesian Prior function \(V(f)\). Once the class is initialized, the number of iterations and subsets are specified at recon time when the forward method is called.

Module Contents#

Classes#

OSML

Abstract class for different algorithms. The difference between subclasses of this class is the method by which they include prior information. If no prior function is used, they are all equivalent.

OSEMOSL

Implements the ordered subset expectation algorithm using the one-step-late method to include prior information: \(f_i^{n,m+1} = \frac{f_i^{n,m}}{\sum_j H_{ij} + \beta \frac{\partial V}{\partial f_r}|_{f_i=f_i^{n,m}}} \sum_j H_{ij}\frac{g_j}{\sum_i H_{ij}f_i^{n,m}+s_j}\).

OSEMBSR

Implements the ordered subset expectation algorithm using the block-sequential-regularized (BSREM) method to include prior information. In particular, each iteration consists of two steps: \(\tilde{f}_i^{n,m+1} = \frac{f_i^{n,m}}{\sum_j H_{ij}} \sum_j H_{ij}\frac{g_j^m}{\sum_i H_{ij}f_i^{n,m}+s_j}\) followed by \(f_i^{n,m+1} = \tilde{f}_i^{n,m+1} \left(1-\beta\frac{\alpha_n}{\sum_j H_{ij}}\frac{\partial V}{\partial \tilde{f}_i^{n,m+1}} \right)\).

class pytomography.algorithms.osem.OSML(image, system_matrix, object_initial=None, scatter=0, prior=None)[source]#

Abstract class for different algorithms. The difference between subclasses of this class is the method by which they include prior information. If no prior function is used, they are all equivalent.

Parameters:
  • image (torch.tensor[batch_size, Lr, Ltheta, Lz]) – image data \(g_j\) to be reconstructed

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

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

  • scatter (torch.tensor[batch_size, Lr, Ltheta, Lz]) – estimate of scatter contribution \(s_j\).

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

get_subset_splits(n_subsets, n_angles)[source]#

Returns a list of arrays; each array contains indices, corresponding to projection numbers, that are used in ordered-subsets. For example, get_subsets_splits(2, 6) would return [[0,2,4],[1,3,5]].

Parameters:
  • n_subsets (int) – number of subsets used in OSEM

  • n_angles (int) – total number of projections

Returns:

list of index arrays for each subset

Return type:

list

abstract __call__(n_iters, n_subsets, callbacks=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.

Return type:

None

class pytomography.algorithms.osem.OSEMOSL(image, system_matrix, object_initial=None, scatter=0, prior=None)[source]#

Bases: OSML

Implements the ordered subset expectation algorithm using the one-step-late method to include prior information: \(f_i^{n,m+1} = \frac{f_i^{n,m}}{\sum_j H_{ij} + \beta \frac{\partial V}{\partial f_r}|_{f_i=f_i^{n,m}}} \sum_j H_{ij}\frac{g_j}{\sum_i H_{ij}f_i^{n,m}+s_j}\).

Parameters:
  • image (torch.tensor[batch_size, Lr, Ltheta, Lz]) – image data \(g_j\) to be reconstructed

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

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

  • scatter (torch.tensor[batch_size, Lr, Ltheta, Lz]) – estimate of scatter contribution \(s_j\).

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

__call__(n_iters, n_subsets, callback=None, delta=1e-11)[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 evaluated after each subiteration. Defaults to None.

  • delta (float, optional) – Used to prevent division by zero when calculating ratio, defaults to 1e-11.

Returns:

reconstructed object

Return type:

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

class pytomography.algorithms.osem.OSEMBSR(image, system_matrix, object_initial=None, scatter=0, prior=None)[source]#

Bases: OSML

Implements the ordered subset expectation algorithm using the block-sequential-regularized (BSREM) method to include prior information. In particular, each iteration consists of two steps: \(\tilde{f}_i^{n,m+1} = \frac{f_i^{n,m}}{\sum_j H_{ij}} \sum_j H_{ij}\frac{g_j^m}{\sum_i H_{ij}f_i^{n,m}+s_j}\) followed by \(f_i^{n,m+1} = \tilde{f}_i^{n,m+1} \left(1-\beta\frac{\alpha_n}{\sum_j H_{ij}}\frac{\partial V}{\partial \tilde{f}_i^{n,m+1}} \right)\).

Parameters:
  • image (torch.tensor[batch_size, Lr, Ltheta, Lz]) – image data \(g_j\) to be reconstructed

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

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

  • scatter (torch.tensor[batch_size, Lr, Ltheta, Lz]) – estimate of scatter contribution \(s_j\).

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

__call__(n_iters, n_subsets, relaxation_function=lambda x: ..., callback=None, delta=1e-11)[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

  • relaxation_function (function) – Specifies relaxation sequence \(\alpha_n\) where \(n\) is the iteration number. Defaults to \(\alpha_n=1\) for all \(n\).

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

  • delta (_type_, optional) – Used to prevent division by zero when calculating ratio, defaults to 1e-11.

Returns:

reconstructed object

Return type:

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