# -*- coding: utf-8 -*-
#  Copyright 2023 United Kingdom Research and Innovation
#
#  Licensed under the Apache License, Version 2.0 (the "License");
#  you may not use this file except in compliance with the License.
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#      http://www.apache.org/licenses/LICENSE-2.0
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#  distributed under the License is distributed on an "AS IS" BASIS,
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#
#   Authored by: Ida Puggaard, DTU, s204211@dtu.dk
#   Photo provided (with permission) by: Mohamad Khalil, Jan Kehres, and Wail Mustafa
#   Edited by:   Margaret Duff, STFC-UKRI
#                Laura Murgatroyd, STFC-UKRI
#   Reviewed by: Jakob Jorgensen, DTU

Reconstruction and regularisation for a hyperspectral dataset

This notebook will be working with a hyperspectral dataset where 5 materials have been scanned. The materials are: Sugar, H2O2, H2O, Aluminium (10 mm), and PVC (7.8 mm).

The notebook considers 3 different regularisation methods for reconstruction:

• TV reconstruction for each energy channel,

• TV reconstruction across both energy and space dimensions

• Tikhonov regularisation across the energy channels and TV reconstruction in the spatial dimensions.

Data can be found at: https://doi.org/10.5281/zenodo.8307932

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A picture from when the materials were scanned.

Tested on CIL version 23.1.0

import cil
print(cil.__version__)

23.1.0

Imports from CIL:

import numpy as np
import matplotlib.pyplot as plt
import h5py


from cil.optimisation.algorithms import  FISTA, PDHG
from cil.optimisation.functions import LeastSquares, TotalVariation, L2NormSquared, MixedL21Norm, IndicatorBox, BlockFunction

from cil.framework import AcquisitionGeometry


from cil.optimisation.operators import GradientOperator, BlockOperator
from cil.plugins.astra.operators import ProjectionOperator
from cil.utilities.display import show2D, show_geometry
from cil.utilities.jupyter import islicer

Define the scan geometry

#detector parameters
ndet=256                                 # Number of detector pixels, excluding gaps
nElem=2                                  # Number of detector modules
pixel_size=0.077                         # Pixels' size (Pitch)

Sep=0.153                                # Pixels' gap length (cm)
det_space=(ndet)*pixel_size+Sep          # Size of detector in cm (pixels*pixel_size), including the gap

# acquisition parameters
range_angle=360                          # Angular span of projections
nproj=370                                # Number of projections
SDD=115.0                                # Source-Detector distance
sourceCentShift=0                        # Vertical source shift from perfect placement
detectCentShift=0                        # Vertical detector shift from perfect placement
SAD=57.5                                 # Source-AxisOfRotation distance

rot_axis_x = 0                           # x-position of AxisOfRotation
rot_axis_y = 0.0                         # y-position of AxisOfRotation


# Define acquisition geometry
ag = AcquisitionGeometry.create_Cone2D(
    source_position=[sourceCentShift, -SAD],
    detector_position=[detectCentShift, SDD-SAD],
    rotation_axis_position=[rot_axis_x, rot_axis_y])
ag.set_angles(angles=((np.linspace(range_angle,0,nproj, endpoint=False))), angle_unit='degree')
ag.set_panel(num_pixels=[ndet, 1], pixel_size=[pixel_size, pixel_size])
ag.set_channels(128) # Set energy channels
ag.set_labels(['angle', 'horizontal', 'channel'])

<cil.framework.framework.AcquisitionGeometry at 0x7f7593ee2fe0>

Visualising the geometry:

show_geometry(ag)

<cil.utilities.display.show_geometry at 0x7f747800bb50>

Load the sinogram data from the sinogram. The sinograms have already been pre-processed removing dead/hot pixels and dealing with gaps in the detector.

# Load sinogram using h5 reader
f = h5py.File('sino25_interpol_line.h5','r')
data_set = np.array(f["data"][:])

# The sinograms have been preprocessed and more pixels have been added to compensate
# for a gap in the detector. So the panel size has to be increased.
ag.set_panel(num_pixels=[ndet+2, 1], pixel_size=[pixel_size, pixel_size])

# Change labels to astra ordering.
ag.set_labels(['channel', 'angle', 'horizontal'])

# Look at the data dimensions <channel><angle><horizontal>
print(data_set.shape)

(128, 370, 258)

Putting the data into the CIL class and linking it with the defined geometry.

data = ag.allocate() #allocate space
data.fill(data_set)

# Look at the data info
print(data)

Number of dimensions: 3
Shape: (128, 370, 258)
Axis labels: ('channel', 'angle', 'horizontal')

Obtaining the image geometry

ig=ag.get_ImageGeometry()

# Look at the image geometry info
print(ig)

Number of channels: 128
channel_spacing: 1.0
voxel_num : x258,y258
voxel_size : x0.0385,y0.0385
center : x0,y0

Look at the sinogram for one channel at a time

show2D(data.get_slice(channel=64))

<cil.utilities.display.show2D at 0x7f75904835b0>

We can also visualise the data using the islice function. Note that all the sinograms are all scaled in the same range and therefore lower energy channel sinograms appear brighter than the ones with higher energy levels. This is because the materials have higher attenuation coefficients for lower energy channels.

islicer(data)

The attenuation coefficients for the 5 materials can be found using the python library: ‘XRay DB’ in the energy span: 20.21-153.53 KeV and these are plotted below. In the plot glass is also included since Sugar, H2O2, and H2O are contained inside glass bottles.

The y-axis gives the attenuation coefficients and the x-axis the energy in eV.

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TV regularisation in space

Firstly, we try to reconstruct the channels separately with TV regularisation. We will use the CIL function TotalVariation:

\[\argmin_{u} \frac{1}{2} ||Au-b||_{2}^{2}+\alpha TV(u).\]

N.B the following cell will take >15mins to run!

# Allocate space for solution
recon_data=ig.allocate()

# Define projection operator
ag2D=ag.get_slice(channel=0)
ig2D=ag2D.get_ImageGeometry()
A_2D = ProjectionOperator(ig2D, ag2D, 'gpu')

# Loop over all the channels and reconstruct each channel
for j in range(ig.channels):
    f1 = LeastSquares(A_2D, data.get_slice(channel=j))
    alpha=0.001
    # Totalvariation with non-negative constraint
    TV = alpha*TotalVariation(lower=0)

    myFISTATV_TV = FISTA(f=f1,
            g=TV,
            initial=ig2D.allocate(0),
            max_iteration=100,
            update_objective_interval = 50)

    myFISTATV_TV.run(100,verbose=1)
    recon_data.fill(myFISTATV_TV.solution,channel=j)

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          0.00000e+00
       50        100         0.371          0.00000e+00
      100        100         0.326          0.00000e+00
-------------------------------------------------------
      100        100         0.326          0.00000e+00
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          0.00000e+00
       50        100         0.263          0.00000e+00
      100        100         0.193          0.00000e+00
-------------------------------------------------------
      100        100         0.193          0.00000e+00
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          0.00000e+00
       50        100         0.296          0.00000e+00
      100        100         0.315          0.00000e+00
-------------------------------------------------------
      100        100         0.315          0.00000e+00
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          0.00000e+00
       50        100         0.291          0.00000e+00
      100        100         0.246          0.00000e+00
-------------------------------------------------------
      100        100         0.246          0.00000e+00
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.60618e+06
       50        100         0.200          5.95816e+05
      100        100         0.251          5.89042e+05
-------------------------------------------------------
      100        100         0.251          5.89042e+05
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.24900e+06
       50        100         0.335          5.26519e+05
      100        100         0.276          5.19685e+05
-------------------------------------------------------
      100        100         0.276          5.19685e+05
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.65047e+06
       50        100         0.195          3.68001e+05
      100        100         0.270          3.62960e+05
-------------------------------------------------------
      100        100         0.270          3.62960e+05
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.79624e+06
       50        100         0.298          2.59262e+05
      100        100         0.319          2.55467e+05
-------------------------------------------------------
      100        100         0.319          2.55467e+05
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.11581e+06
       50        100         0.108          1.74096e+05
      100        100         0.199          1.71339e+05
-------------------------------------------------------
      100        100         0.199          1.71339e+05
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.57819e+06
       50        100         0.321          1.13145e+05
      100        100         0.307          1.11331e+05
-------------------------------------------------------
      100        100         0.307          1.11331e+05
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.13760e+06
       50        100         0.193          7.30751e+04
      100        100         0.200          7.19002e+04
-------------------------------------------------------
      100        100         0.200          7.19002e+04
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.77369e+06
       50        100         0.299          4.68125e+04
      100        100         0.318          4.60636e+04
-------------------------------------------------------
      100        100         0.318          4.60636e+04
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.47091e+06
       50        100         0.282          3.32998e+04
      100        100         0.241          3.27946e+04
-------------------------------------------------------
      100        100         0.241          3.27946e+04
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.23306e+06
       50        100         0.194          2.18660e+04
      100        100         0.239          2.15421e+04
-------------------------------------------------------
      100        100         0.239          2.15421e+04
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.04943e+06
       50        100         0.348          1.12857e+04
      100        100         0.229          1.10845e+04
-------------------------------------------------------
      100        100         0.229          1.10845e+04
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          8.98941e+05
       50        100         0.303          1.14002e+04
      100        100         0.299          1.12455e+04
-------------------------------------------------------
      100        100         0.299          1.12455e+04
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.84629e+05
       50        100         0.282          4.50712e+03
      100        100         0.268          4.40958e+03
-------------------------------------------------------
      100        100         0.268          4.40958e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.86062e+05
       50        100         0.148          6.33147e+03
      100        100         0.219          6.24967e+03
-------------------------------------------------------
      100        100         0.219          6.24967e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.11228e+05
       50        100         0.300          2.45821e+03
      100        100         0.292          2.39296e+03
-------------------------------------------------------
      100        100         0.292          2.39296e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.44308e+05
       50        100         0.314          3.75414e+03
      100        100         0.303          3.69996e+03
-------------------------------------------------------
      100        100         0.303          3.69996e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.89027e+05
       50        100         0.230          3.55074e+03
      100        100         0.222          3.50552e+03
-------------------------------------------------------
      100        100         0.222          3.50552e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.43713e+05
       50        100         0.299          7.81434e+02
      100        100         0.298          7.54510e+02
-------------------------------------------------------
      100        100         0.298          7.54510e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.02195e+05
       50        100         0.293          6.38845e+02
      100        100         0.301          6.16383e+02
-------------------------------------------------------
      100        100         0.301          6.16383e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.66031e+05
       50        100         0.107          5.28941e+02
      100        100         0.201          5.09602e+02
-------------------------------------------------------
      100        100         0.201          5.09602e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.34256e+05
       50        100         0.301          4.60835e+02
      100        100         0.300          4.44249e+02
-------------------------------------------------------
      100        100         0.300          4.44249e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.06188e+05
       50        100         0.297          4.07529e+02
      100        100         0.204          3.93056e+02
-------------------------------------------------------
      100        100         0.204          3.93056e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.81335e+05
       50        100         0.287          3.57787e+02
      100        100         0.315          3.45128e+02
-------------------------------------------------------
      100        100         0.315          3.45128e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.59136e+05
       50        100         0.282          3.18731e+02
      100        100         0.263          3.07482e+02
-------------------------------------------------------
      100        100         0.263          3.07482e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.39186e+05
       50        100         0.170          2.91401e+02
      100        100         0.237          2.81230e+02
-------------------------------------------------------
      100        100         0.237          2.81230e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.21061e+05
       50        100         0.277          2.60856e+02
      100        100         0.247          2.51678e+02
-------------------------------------------------------
      100        100         0.247          2.51678e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.04576e+05
       50        100         0.281          2.38070e+02
      100        100         0.278          2.29701e+02
-------------------------------------------------------
      100        100         0.278          2.29701e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.89722e+05
       50        100         0.142          2.17544e+02
      100        100         0.214          2.09862e+02
-------------------------------------------------------
      100        100         0.214          2.09862e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.76708e+05
       50        100         0.313          1.95117e+02
      100        100         0.304          1.87999e+02
-------------------------------------------------------
      100        100         0.304          1.87999e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.65758e+05
       50        100         0.223          1.68387e+02
      100        100         0.176          1.61761e+02
-------------------------------------------------------
      100        100         0.176          1.61761e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.56902e+05
       50        100         0.289          1.46484e+02
      100        100         0.282          1.40248e+02
-------------------------------------------------------
      100        100         0.282          1.40248e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.49882e+05
       50        100         0.292          1.29167e+02
      100        100         0.289          1.23222e+02
-------------------------------------------------------
      100        100         0.289          1.23222e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.44254e+05
       50        100         0.115          1.16580e+02
      100        100         0.208          1.10905e+02
-------------------------------------------------------
      100        100         0.208          1.10905e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.39496e+05
       50        100         0.305          1.11038e+02
      100        100         0.299          1.05551e+02
-------------------------------------------------------
      100        100         0.299          1.05551e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.35102e+05
       50        100         0.306          1.07558e+02
      100        100         0.205          1.02267e+02
-------------------------------------------------------
      100        100         0.205          1.02267e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.30659e+05
       50        100         0.299          1.07003e+02
      100        100         0.303          1.01927e+02
-------------------------------------------------------
      100        100         0.303          1.01927e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.25950e+05
       50        100         0.297          1.03291e+02
      100        100         0.234          9.85156e+01
-------------------------------------------------------
      100        100         0.234          9.85156e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.20939e+05
       50        100         0.218          1.02353e+02
      100        100         0.253          9.78379e+01
-------------------------------------------------------
      100        100         0.253          9.78379e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.15831e+05
       50        100         0.324          1.02242e+02
      100        100         0.256          9.79494e+01
-------------------------------------------------------
      100        100         0.256          9.79494e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.10962e+05
       50        100         0.212          9.87620e+01
      100        100         0.297          9.47055e+01
-------------------------------------------------------
      100        100         0.297          9.47055e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.06608e+05
       50        100         0.295          9.39068e+01
      100        100         0.302          9.00545e+01
-------------------------------------------------------
      100        100         0.302          9.00545e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          1.02866e+05
       50        100         0.147          8.98091e+01
      100        100         0.203          8.61318e+01
-------------------------------------------------------
      100        100         0.203          8.61318e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          9.96412e+04
       50        100         0.326          8.69387e+01
      100        100         0.218          8.34198e+01
-------------------------------------------------------
      100        100         0.218          8.34198e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          9.68024e+04
       50        100         0.215          8.41843e+01
      100        100         0.210          8.08075e+01
-------------------------------------------------------
      100        100         0.210          8.08075e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          9.42058e+04
       50        100         0.235          8.37718e+01
      100        100         0.219          8.04940e+01
-------------------------------------------------------
      100        100         0.219          8.04940e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          9.17331e+04
       50        100         0.162          8.17475e+01
      100        100         0.169          7.86455e+01
-------------------------------------------------------
      100        100         0.169          7.86455e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          8.93175e+04
       50        100         0.280          8.04634e+01
      100        100         0.248          7.74747e+01
-------------------------------------------------------
      100        100         0.248          7.74747e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          8.69299e+04
       50        100         0.179          7.77902e+01
      100        100         0.195          7.49055e+01
-------------------------------------------------------
      100        100         0.195          7.49055e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          8.45918e+04
       50        100         0.137          7.72986e+01
      100        100         0.152          7.45210e+01
-------------------------------------------------------
      100        100         0.152          7.45210e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          8.23640e+04
       50        100         0.240          7.65372e+01
      100        100         0.263          7.38748e+01
-------------------------------------------------------
      100        100         0.263          7.38748e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          8.03238e+04
       50        100         0.341          7.74555e+01
      100        100         0.290          7.48950e+01
-------------------------------------------------------
      100        100         0.290          7.48950e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.84777e+04
       50        100         0.160          7.82257e+01
      100        100         0.232          7.57197e+01
-------------------------------------------------------
      100        100         0.232          7.57197e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.67965e+04
       50        100         0.288          7.72996e+01
      100        100         0.296          7.48658e+01
-------------------------------------------------------
      100        100         0.296          7.48658e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.52393e+04
       50        100         0.107          7.66482e+01
      100        100         0.204          7.42576e+01
-------------------------------------------------------
      100        100         0.204          7.42576e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.38339e+04
       50        100         0.341          7.47841e+01
      100        100         0.320          7.24686e+01
-------------------------------------------------------
      100        100         0.320          7.24686e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.26185e+04
       50        100         0.260          7.32199e+01
      100        100         0.224          7.09500e+01
-------------------------------------------------------
      100        100         0.224          7.09500e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.16501e+04
       50        100         0.294          7.06395e+01
      100        100         0.300          6.84600e+01
-------------------------------------------------------
      100        100         0.300          6.84600e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.09074e+04
       50        100         0.282          7.04050e+01
      100        100         0.238          6.82594e+01
-------------------------------------------------------
      100        100         0.238          6.82594e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.04015e+04
       50        100         0.238          7.14765e+01
      100        100         0.258          6.94065e+01
-------------------------------------------------------
      100        100         0.258          6.94065e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.01415e+04
       50        100         0.350          7.34230e+01
      100        100         0.233          7.13896e+01
-------------------------------------------------------
      100        100         0.233          7.13896e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.01067e+04
       50        100         0.303          7.71705e+01
      100        100         0.315          7.51553e+01
-------------------------------------------------------
      100        100         0.315          7.51553e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.02370e+04
       50        100         0.281          8.02870e+01
      100        100         0.301          7.82722e+01
-------------------------------------------------------
      100        100         0.301          7.82722e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.04308e+04
       50        100         0.107          8.25912e+01
      100        100         0.202          8.05986e+01
-------------------------------------------------------
      100        100         0.202          8.05986e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.05335e+04
       50        100         0.319          8.23141e+01
      100        100         0.299          8.03205e+01
-------------------------------------------------------
      100        100         0.299          8.03205e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.04676e+04
       50        100         0.278          8.11656e+01
      100        100         0.197          7.91728e+01
-------------------------------------------------------
      100        100         0.197          7.91728e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          7.02116e+04
       50        100         0.283          8.00944e+01
      100        100         0.294          7.81070e+01
-------------------------------------------------------
      100        100         0.294          7.81070e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.97718e+04
       50        100         0.289          8.18271e+01
      100        100         0.240          7.98497e+01
-------------------------------------------------------
      100        100         0.240          7.98497e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.91272e+04
       50        100         0.248          8.17985e+01
      100        100         0.268          7.98525e+01
-------------------------------------------------------
      100        100         0.268          7.98525e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.82435e+04
       50        100         0.282          8.06958e+01
      100        100         0.227          7.87731e+01
-------------------------------------------------------
      100        100         0.227          7.87731e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.71986e+04
       50        100         0.219          7.87749e+01
      100        100         0.275          7.69018e+01
-------------------------------------------------------
      100        100         0.275          7.69018e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.60680e+04
       50        100         0.285          7.79519e+01
      100        100         0.265          7.61214e+01
-------------------------------------------------------
      100        100         0.265          7.61214e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.49046e+04
       50        100         0.163          7.87942e+01
      100        100         0.229          7.69840e+01
-------------------------------------------------------
      100        100         0.229          7.69840e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.37477e+04
       50        100         0.302          8.07714e+01
      100        100         0.277          7.90197e+01
-------------------------------------------------------
      100        100         0.277          7.90197e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.25988e+04
       50        100         0.153          8.63247e+01
      100        100         0.278          8.46017e+01
-------------------------------------------------------
      100        100         0.278          8.46017e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.15028e+04
       50        100         0.294          9.27143e+01
      100        100         0.203          9.10261e+01
-------------------------------------------------------
      100        100         0.203          9.10261e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          6.04705e+04
       50        100         0.350          9.59345e+01
      100        100         0.320          9.42775e+01
-------------------------------------------------------
      100        100         0.320          9.42775e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.94657e+04
       50        100         0.265          9.56406e+01
      100        100         0.202          9.40117e+01
-------------------------------------------------------
      100        100         0.202          9.40117e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.84551e+04
       50        100         0.292          9.22495e+01
      100        100         0.298          9.06612e+01
-------------------------------------------------------
      100        100         0.298          9.06612e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.74476e+04
       50        100         0.298          8.81796e+01
      100        100         0.303          8.66404e+01
-------------------------------------------------------
      100        100         0.303          8.66404e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.64786e+04
       50        100         0.112          8.71818e+01
      100        100         0.208          8.56700e+01
-------------------------------------------------------
      100        100         0.208          8.56700e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.55497e+04
       50        100         0.295          8.85664e+01
      100        100         0.300          8.70767e+01
-------------------------------------------------------
      100        100         0.300          8.70767e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.46376e+04
       50        100         0.325          9.09475e+01
      100        100         0.248          8.94806e+01
-------------------------------------------------------
      100        100         0.248          8.94806e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.37486e+04
       50        100         0.242          9.12174e+01
      100        100         0.293          8.97653e+01
-------------------------------------------------------
      100        100         0.293          8.97653e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.28826e+04
       50        100         0.294          9.39694e+01
      100        100         0.303          9.25440e+01
-------------------------------------------------------
      100        100         0.303          9.25440e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.20532e+04
       50        100         0.217          9.75343e+01
      100        100         0.220          9.61336e+01
-------------------------------------------------------
      100        100         0.220          9.61336e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.12575e+04
       50        100         0.348          9.96369e+01
      100        100         0.325          9.82514e+01
-------------------------------------------------------
      100        100         0.325          9.82514e+01
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          5.04792e+04
       50        100         0.323          1.04894e+02
      100        100         0.225          1.03528e+02
-------------------------------------------------------
      100        100         0.225          1.03528e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.97178e+04
       50        100         0.276          1.08254e+02
      100        100         0.306          1.06911e+02
-------------------------------------------------------
      100        100         0.306          1.06911e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.89643e+04
       50        100         0.297          1.12059e+02
      100        100         0.334          1.10740e+02
-------------------------------------------------------
      100        100         0.334          1.10740e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.82092e+04
       50        100         0.224          1.14267e+02
      100        100         0.209          1.12966e+02
-------------------------------------------------------
      100        100         0.209          1.12966e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.74503e+04
       50        100         0.352          1.21636e+02
      100        100         0.327          1.20340e+02
-------------------------------------------------------
      100        100         0.327          1.20340e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.67169e+04
       50        100         0.338          1.25882e+02
      100        100         0.267          1.24607e+02
-------------------------------------------------------
      100        100         0.267          1.24607e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.60128e+04
       50        100         0.274          1.26517e+02
      100        100         0.316          1.25264e+02
-------------------------------------------------------
      100        100         0.316          1.25264e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.53388e+04
       50        100         0.297          1.27391e+02
      100        100         0.321          1.26147e+02
-------------------------------------------------------
      100        100         0.321          1.26147e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.46970e+04
       50        100         0.290          1.29231e+02
      100        100         0.253          1.28005e+02
-------------------------------------------------------
      100        100         0.253          1.28005e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.41166e+04
       50        100         0.194          1.37140e+02
      100        100         0.246          1.35932e+02
-------------------------------------------------------
      100        100         0.246          1.35932e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.36097e+04
       50        100         0.315          1.47205e+02
      100        100         0.267          1.46012e+02
-------------------------------------------------------
      100        100         0.267          1.46012e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.31626e+04
       50        100         0.233          1.57815e+02
      100        100         0.267          1.56644e+02
-------------------------------------------------------
      100        100         0.267          1.56644e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.27589e+04
       50        100         0.297          1.66803e+02
      100        100         0.308          1.65648e+02
-------------------------------------------------------
      100        100         0.308          1.65648e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.23457e+04
       50        100         0.289          1.79405e+02
      100        100         0.281          1.78250e+02
-------------------------------------------------------
      100        100         0.281          1.78250e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.19182e+04
       50        100         0.152          1.97090e+02
      100        100         0.226          1.95932e+02
-------------------------------------------------------
      100        100         0.226          1.95932e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.14391e+04
       50        100         0.319          2.21791e+02
      100        100         0.306          2.20651e+02
-------------------------------------------------------
      100        100         0.306          2.20651e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.09400e+04
       50        100         0.318          2.38727e+02
      100        100         0.245          2.37621e+02
-------------------------------------------------------
      100        100         0.245          2.37621e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.04666e+04
       50        100         0.233          2.42157e+02
      100        100         0.280          2.41053e+02
-------------------------------------------------------
      100        100         0.280          2.41053e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.00416e+04
       50        100         0.282          2.45343e+02
      100        100         0.314          2.44228e+02
-------------------------------------------------------
      100        100         0.314          2.44228e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.96328e+04
       50        100         0.228          2.45413e+02
      100        100         0.224          2.44273e+02
-------------------------------------------------------
      100        100         0.224          2.44273e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.92809e+04
       50        100         0.234          2.40772e+02
      100        100         0.264          2.39595e+02
-------------------------------------------------------
      100        100         0.264          2.39595e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.89558e+04
       50        100         0.312          2.45430e+02
      100        100         0.297          2.44250e+02
-------------------------------------------------------
      100        100         0.297          2.44250e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.87038e+04
       50        100         0.241          2.65040e+02
      100        100         0.209          2.63845e+02
-------------------------------------------------------
      100        100         0.209          2.63845e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.84393e+04
       50        100         0.289          2.96864e+02
      100        100         0.298          2.95626e+02
-------------------------------------------------------
      100        100         0.298          2.95626e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.81144e+04
       50        100         0.293          3.31374e+02
      100        100         0.263          3.30102e+02
-------------------------------------------------------
      100        100         0.263          3.30102e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.76420e+04
       50        100         0.240          3.74161e+02
      100        100         0.268          3.72781e+02
-------------------------------------------------------
      100        100         0.268          3.72781e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.70622e+04
       50        100         0.387          4.33692e+02
      100        100         0.278          4.32145e+02
-------------------------------------------------------
      100        100         0.278          4.32145e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.65262e+04
       50        100         0.256          5.02500e+02
      100        100         0.324          5.00695e+02
-------------------------------------------------------
      100        100         0.324          5.00695e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.59538e+04
       50        100         0.298          5.78041e+02
      100        100         0.322          5.75791e+02
-------------------------------------------------------
      100        100         0.322          5.75791e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.51408e+04
       50        100         0.118          6.83695e+02
      100        100         0.214          6.80955e+02
-------------------------------------------------------
      100        100         0.214          6.80955e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.41040e+04
       50        100         0.337          8.22856e+02
      100        100         0.311          8.19776e+02
-------------------------------------------------------
      100        100         0.311          8.19776e+02
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.27028e+04
       50        100         0.223          1.02173e+03
      100        100         0.248          1.01816e+03
-------------------------------------------------------
      100        100         0.248          1.01816e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          3.11100e+04
       50        100         0.307          1.27183e+03
      100        100         0.317          1.26769e+03
-------------------------------------------------------
      100        100         0.317          1.26769e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.95248e+04
       50        100         0.107          1.63073e+03
      100        100         0.204          1.62589e+03
-------------------------------------------------------
      100        100         0.204          1.62589e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.83739e+04
       50        100         0.319          2.16533e+03
      100        100         0.301          2.15876e+03
-------------------------------------------------------
      100        100         0.301          2.15876e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.78148e+04
       50        100         0.198          2.82023e+03
      100        100         0.207          2.81122e+03
-------------------------------------------------------
      100        100         0.207          2.81122e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.76767e+04
       50        100         0.284          3.43063e+03
      100        100         0.307          3.41968e+03
-------------------------------------------------------
      100        100         0.307          3.41968e+03
Stop criterion has been reached.

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          2.77227e+04
       50        100         0.166          3.70708e+03
      100        100         0.205          3.69640e+03
-------------------------------------------------------
      100        100         0.205          3.69640e+03
Stop criterion has been reached.

We can then look at the reconstruction.

show2D([recon_data.get_slice(channel=40), recon_data.get_slice(channel=60), recon_data.get_slice(channel=80)], num_cols=3,  fix_range=(0,1))

<cil.utilities.display.show2D at 0x7f74407bc700>

We can construct the energy curves from the 5 materials by extracting one pixel from each material and plot the cuvres.

solution_arr=recon_data.as_array()
plt.plot(solution_arr[:,50,125],'r',label='Alu')
plt.plot(solution_arr[:,125,150],'m',label='H2O2')
plt.plot(solution_arr[:,200,140],'g',label='pvc')
plt.plot(solution_arr[:,100,100],'c',label='Sugar')
plt.plot(solution_arr[:,160,100],'b',label='Water')

plt.legend()
plt.xlabel('Energy [keV]')
plt.ylabel('Attenuation [1/cm]')
extent=np.linspace(21.20,153.53,8)
extent = ["%.2f" % v for v in extent]
plt.xticks(ticks=np.linspace(0,127,8),labels=extent)
plt.show()

TV regularisation across both energy channels and spatial dimensions

We can also use TV regularisation in both the space and the energy dimensions by giving the input ‘SpaceChannels’ to the TotalVariation function.

A = ProjectionOperator(ig, ag, 'gpu')
f1 = LeastSquares(A, data)

alpha = 0.001
GTV = alpha*TotalVariation(correlation='SpaceChannels',lower=0)

myFISTATV = FISTA(f=f1,
                  g=GTV,
                  initial=ig.allocate(0),
                  max_iteration=100,
                  update_objective_interval = 10)


myFISTATV.run(100,verbose=1)

     Iter   Max Iter     Time/Iter            Objective
                               [s]
        0        100         0.000          4.63959e+07
       10        100         6.924          2.87992e+06
       20        100         6.991          2.44187e+06
       30        100         7.125          2.35016e+06
       40        100         7.227          2.30902e+06
       50        100         7.302          2.28662e+06
       60        100         7.332          2.27356e+06
       70        100         7.347          2.26557e+06
       80        100         7.380          2.26069e+06
       90        100         7.413          2.25766e+06
      100        100         7.426          2.25574e+06
-------------------------------------------------------
      100        100         7.426          2.25574e+06
Stop criterion has been reached.

Look at the solution.

show2D([myFISTATV.solution.get_slice(channel=40), myFISTATV.solution.get_slice(channel=60), myFISTATV.solution.get_slice(channel=80)], num_cols=3, fix_range=(0,1))

<cil.utilities.display.show2D at 0x7f74402db640>

We can again look at the energy curves and see that the regularisation has had an effect on the curves.

solution_arr=myFISTATV.solution.as_array()
plt.plot(solution_arr[:,50,125],'r',label='Alu')
plt.plot(solution_arr[:,125,150],'m',label='H2O2')
plt.plot(solution_arr[:,200,140],'g',label='pvc')
plt.plot(solution_arr[:,100,100],'c',label='Sugar')
plt.plot(solution_arr[:,160,100],'b',label='Water')

plt.legend()
plt.xlabel('Energy [keV]')
plt.ylabel('Attenuation [1/cm]')
extent=np.linspace(21.20,153.53,8)
extent = ["%.2f" % v for v in extent]
plt.xticks(ticks=np.linspace(0,127,8),labels=extent)
plt.show()

TV regularisation in spatial dimensions and Tikhonov regularisation across the energy channels

The regularisation used in the space dimension and the energy dimension does not have to be the same. We can therefore try to use 2-norm for regularisation of the energy dimension of the gradient instead of a sparse norm, as in TV. The reasoning is that Tikhonov is good for smoothing compared to TV which is good for maintaining sharp edges.

\[\argmin_{u} \frac{1}{2} ||Au-b||_{2}^{2}+\frac{\beta}{2} ||\nabla_{energy}u ||_2^2 +\alpha TV_{spatial}(u)\]

Here, $ :nbsphinx-math:`beta `$ is the additional regularisation parameter for Tikhonov.

In order to set this up, define a gradient operator:

\[Du=[D_{x_0}u, [D_{x_1}u,D_{x_2}u ]]\]

Also define a block operator:

\[\begin{split} K= \begin{bmatrix} A\\ D \end{bmatrix}\end{split}\]

and block functions

\[h(z_1,z_2)=\frac{\beta}{2}||z_1 ||_2^2+\alpha||z_2||_{2,1}\]

and

\[f(z_0, [z_1,z_2])=\frac{1}{2}||z_0-y||_2^2+ h(z_1,z_2).\]

Thus

\[f(K(u))=f([Au, [D_{x_0}u, [D_{x_1}u,D_{x_2}u ]] ])= \frac{1}{2}||Au-y||_2^2+ h(D_{x_0}u,[D_{x_1}u,D_{x_2}u ])\]

and so

\[f(K(u))=\frac{1}{2}||Au-y||_2^2+ \frac{\beta}{2}||D_{x_0}u||_2^2+\alpha||[D_{x_1}u,D_{x_2}u]||_{2,1}.\]

Where the final term is the TV-norm, as required.

# Use same alpha
alpha = 0.001

beta = 0.5

# Projection operator
A = ProjectionOperator(ig, ag, 'gpu')

# Use the gradient operator to calculate the gradient. "Correlation='Space" ensures that the gradient is computed only in the spatial dimensions.
#'split=True' returns a BlockDataContainer with grouped spatial domains
D = GradientOperator(ig, correlation='SpaceChannels',split=True)
K = BlockOperator(A, D)
h=BlockFunction(beta*0.5*L2NormSquared(), alpha*MixedL21Norm())
f = BlockFunction(0.5*L2NormSquared(b=data),h)
# Non-negative constraint
G = IndicatorBox(lower=0)

# Use PDHG as solver
num_iter=100
PDHG_Tik_TV = PDHG(f=f, g=G, operator=K, max_iteration=num_iter, update_objective_interval = 10)

PDHG_Tik_TV.run(num_iter,verbose=2)

     Iter   Max Iter     Time/Iter        Primal          Dual     Primal-Dual
                               [s]     Objective     Objective             Gap
        0        100         0.000    2.31980e+07  -0.00000e+00    2.31980e+07
       10        100         3.390    5.42617e+06  -1.28427e+06    6.71044e+06
       20        100         3.332    9.55424e+06          -inf            inf
       30        100         3.324    2.71634e+06          -inf            inf
       40        100         3.327    1.86622e+06          -inf            inf
       50        100         3.327    2.03287e+06          -inf            inf
       60        100         3.320    1.71434e+06          -inf            inf
       70        100         3.314    1.62615e+06          -inf            inf
       80        100         3.326    1.62649e+06          -inf            inf
       90        100         3.322    1.44775e+06          -inf            inf
      100        100         3.319    1.40322e+06          -inf            inf
----------------------------------------------------------------------------
      100        100         3.319    1.40322e+06          -inf            inf
Stop criterion has been reached.

Show the solution for each channel.

show2D([PDHG_Tik_TV.solution.get_slice(channel=40), PDHG_Tik_TV.solution.get_slice(channel=60),PDHG_Tik_TV.solution.get_slice(channel=80)], num_cols=3, fix_range=(0,1))

<cil.utilities.display.show2D at 0x7f742c3e1090>

We can once again look at the energy curves for each material.

Tik_TV_sol=PDHG_Tik_TV.solution.as_array()

plt.plot(Tik_TV_sol[:,50,125],'r',label='Alu')
plt.plot(Tik_TV_sol[:,125,150],'m',label='H2O2')
plt.plot(Tik_TV_sol[:,200,140],'g',label='pvc')
plt.plot(Tik_TV_sol[:,100,100],'c',label='Sugar')
plt.plot(Tik_TV_sol[:,160,100],'b',label='Water')

plt.legend()
plt.xlabel('Energy [keV]')
plt.ylabel('Attenuation [1/cm]')
extent=np.linspace(21.20,153.53,8)
extent = ["%.2f" % v for v in extent]
plt.xticks(ticks=np.linspace(0,127,8),labels=extent)
plt.show()

It can be seen that the energy curves looks more smooth and closer to the table value curves from xraydb. Further work could be done to optimise the choice of regularisation parameters.

Summary - Comparison of Reconstruction by each algorithm

show2D([recon_data.get_slice(channel=40),recon_data.get_slice(channel=60),recon_data.get_slice(channel=80), myFISTATV.solution.get_slice(channel=40),myFISTATV.solution.get_slice(channel=60),myFISTATV.solution.get_slice(channel=80), PDHG_Tik_TV.solution.get_slice(channel=40), PDHG_Tik_TV.solution.get_slice(channel=60), PDHG_Tik_TV.solution.get_slice(channel=80)], title=['FISTA-TV','FISTA-TV','FISTA-TV', 'FISTA-TV+TV','FISTA-TV+TV','FISTA-TV+TV', 'PDHG-TIK+TV','PDHG-TIK+TV','PDHG-TIK+TV'], num_cols=3, fix_range=(0,1))

def plot_materials(solution_arr):
    plt.plot(solution_arr[:,50,125],'r',label='Alu')
    plt.plot(solution_arr[:,125,150],'m',label='H2O2')
    plt.plot(solution_arr[:,200,140],'g',label='pvc')
    plt.plot(solution_arr[:,100,100],'c',label='Sugar')
    plt.plot(solution_arr[:,160,100],'b',label='Water')
    plt.legend()
    plt.xlabel('Energy [keV]')
    plt.ylabel('Attenuation [1/cm]')
    extent=np.linspace(21.20,153.53,8)
    extent = ["%.2f" % v for v in extent]
    plt.xticks(ticks=np.linspace(0,127,8),labels=extent)


# Plot all 3 graphs next to each other:
plt.figure(figsize=(15,5))
plt.subplot(131)
solution_arr=recon_data.as_array()
plot_materials(solution_arr)
plt.title('FISTA-TV')

plt.subplot(132)
solution_arr=myFISTATV.solution.as_array()
plot_materials(solution_arr)
plt.title('FISTA-TV+TV')

plt.subplot(133)
solution_arr=PDHG_Tik_TV.solution.as_array()
plot_materials(solution_arr)
plt.title('PDHG-TV+TV')

Text(0.5, 1.0, 'PDHG-TV+TV')