wassname/scikit-image
1
0
Fork 0
mirror of https://github.com/wassname/scikit-image.git synced 2026-07-24 13:20:43 +08:00
Code Issues Packages Projects Releases Wiki Activity

Build gallery with sphinx-gallery

Browse Source 
Modified comments in some gallery examples for compatibility with
sphinx-gallery parsing. Also modified some links in the narrative doc
since image file names have changed.
This commit is contained in:
emmanuelle
2016-05-10 21:49:15 +02:00
parent ef9bcfb778
commit 844858f01b
23 changed files with 688 additions and 945 deletions
Download Patch File Download Diff File Expand all files Collapse all files
doc/examples/transform/plot_radon_transform.py
+58 -69
View File
@@ -76,23 +76,20 @@ ax2.imshow(sinogram, cmap=plt.cm.Greys_r,
fig.tight_layout()
plt.show()
"""
.. image:: PLOT2RST.current_figure
Reconstruction with the Filtered Back Projection (FBP)
======================================================
The mathematical foundation of the filtered back projection is the Fourier
slice theorem [2]_. It uses Fourier transform of the projection and
interpolation in Fourier space to obtain the 2D Fourier transform of the image,
which is then inverted to form the reconstructed image. The filtered back
projection is among the fastest methods of performing the inverse Radon
transform. The only tunable parameter for the FBP is the filter, which is
applied to the Fourier transformed projections. It may be used to suppress
high frequency noise in the reconstruction. ``skimage`` provides a few
different options for the filter.
"""
######################################################################
#
# Reconstruction with the Filtered Back Projection (FBP)
# ======================================================
#
# The mathematical foundation of the filtered back projection is the Fourier
# slice theorem [2]_. It uses Fourier transform of the projection and
# interpolation in Fourier space to obtain the 2D Fourier transform of the
# image, which is then inverted to form the reconstructed image. The filtered
# back projection is among the fastest methods of performing the inverse
# Radon transform. The only tunable parameter for the FBP is the filter,
# which is applied to the Fourier transformed projections. It may be used to
# suppress high frequency noise in the reconstruction. ``skimage`` provides a
# few different options for the filter.
from skimage.transform import iradon
@@ -110,42 +107,39 @@ ax2.set_title("Reconstruction error\nFiltered back projection")
ax2.imshow(reconstruction_fbp - image, cmap=plt.cm.Greys_r, **imkwargs)
plt.show()
"""
.. image:: PLOT2RST.current_figure
Reconstruction with the Simultaneous Algebraic Reconstruction Technique
=======================================================================
Algebraic reconstruction techniques for tomography are based on a
straightforward idea: for a pixelated image the value of a single ray in a
particular projection is simply a sum of all the pixels the ray passes through
on its way through the object. This is a way of expressing the forward Radon
transform. The inverse Radon transform can then be formulated as a (large) set
of linear equations. As each ray passes through a small fraction of the pixels
in the image, this set of equations is sparse, allowing iterative solvers for
sparse linear systems to tackle the system of equations. One iterative method
has been particularly popular, namely Kaczmarz' method [3]_, which has the
property that the solution will approach a least-squares solution of the
equation set.
The combination of the formulation of the reconstruction problem as a set
of linear equations and an iterative solver makes algebraic techniques
relatively flexible, hence some forms of prior knowledge can be incorporated
with relative ease.
``skimage`` provides one of the more popular variations of the algebraic
reconstruction techniques: the Simultaneous Algebraic Reconstruction Technique
(SART) [1]_ [4]_. It uses Kaczmarz' method [3]_ as the iterative solver. A good
reconstruction is normally obtained in a single iteration, making the method
computationally effective. Running one or more extra iterations will normally
improve the reconstruction of sharp, high frequency features and reduce the
mean squared error at the expense of increased high frequency noise (the user
will need to decide on what number of iterations is best suited to the problem
at hand. The implementation in ``skimage`` allows prior information of the
form of a lower and upper threshold on the reconstructed values to be supplied
to the reconstruction.
"""
######################################################################
#
# Reconstruction with the Simultaneous Algebraic Reconstruction Technique
# =======================================================================
#
# Algebraic reconstruction techniques for tomography are based on a
# straightforward idea: for a pixelated image the value of a single ray in a
# particular projection is simply a sum of all the pixels the ray passes
# through on its way through the object. This is a way of expressing the
# forward Radon transform. The inverse Radon transform can then be formulated
# as a (large) set of linear equations. As each ray passes through a small
# fraction of the pixels in the image, this set of equations is sparse,
# allowing iterative solvers for sparse linear systems to tackle the system
# of equations. One iterative method has been particularly popular, namely
# Kaczmarz' method [3]_, which has the property that the solution will
# approach a least-squares solution of the equation set.
#
# The combination of the formulation of the reconstruction problem as a set
# of linear equations and an iterative solver makes algebraic techniques
# relatively flexible, hence some forms of prior knowledge can be
# incorporated with relative ease.
#
# ``skimage`` provides one of the more popular variations of the algebraic
# reconstruction techniques: the Simultaneous Algebraic Reconstruction
# Technique (SART) [1]_ [4]_. It uses Kaczmarz' method [3]_ as the iterative
# solver. A good reconstruction is normally obtained in a single iteration,
# making the method computationally effective. Running one or more extra
# iterations will normally improve the reconstruction of sharp, high
# frequency features and reduce the mean squared error at the expense of
# increased high frequency noise (the user will need to decide on what number
# of iterations is best suited to the problem at hand. The implementation in
# ``skimage`` allows prior information of the form of a lower and upper
# threshold on the reconstructed values to be supplied to the reconstruction.
from skimage.transform import iradon_sart
@@ -176,19 +170,14 @@ ax4.set_title("Reconstruction error\nSART, 2 iterations")
ax4.imshow(reconstruction_sart2 - image, cmap=plt.cm.Greys_r, **imkwargs)
plt.show()
"""
.. image:: PLOT2RST.current_figure
.. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic Imaging",
IEEE Press 1988. http://www.slaney.org/pct/pct-toc.html
.. [2] Wikipedia, Radon transform,
http://en.wikipedia.org/wiki/Radon_transform#Relationship_with_the_Fourier_transform
.. [3] S Kaczmarz, "Angenaeherte Aufloesung von Systemen linearer
Gleichungen", Bulletin International de l'Academie Polonaise des
Sciences et des Lettres 35 pp 355--357 (1937)
.. [4] AH Andersen, AC Kak, "Simultaneous algebraic reconstruction technique
(SART): a superior implementation of the ART algorithm", Ultrasonic
Imaging 6 pp 81--94 (1984)
"""
######################################################################
# References
#
# .. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic Imaging", IEEE Press 1988. http://www.slaney.org/pct/pct-toc.html
#
# .. [2] Wikipedia, Radon transform, http://en.wikipedia.org/wiki/Radon_transform#Relationship_with_the_Fourier_transform
#
# .. [3] S Kaczmarz, "Angenaeherte Aufloesung von Systemen linearer Gleichungen", Bulletin International de l'Academie Polonaise des Sciences et des Lettres, 35 pp 355--357 (1937)
#
# .. [4] AH Andersen, AC Kak, "Simultaneous algebraic reconstruction
# technique (SART): a superior implementation of the ART algorithm", Ultrasonic Imaging 6 pp 81--94 (1984)