Improve the text in and add references to the Radon example.

Based on comments from Emmanuelle.

doc/examples/plot_radon_transform.py

@@ -4,38 +4,36 @@ Radon transform
 ===============
 
 In computed tomography, the tomography reconstruction problem is to obtain
-a tomographic slice image from a set of projections. A projection is formed
+a tomographic slice image from a set of projections [1]_. A projection is formed
 by drawing a set of parallel rays through the 2D object of interest, assigning
 the integral of the object's contrast along each ray to a single pixel in the
 projection. A single projection of a 2D object is one dimensional. To
 enable computed tomography reconstruction of the object, several projections
-must be acquired, each of them with the rays making a different angle with
-the axes of the object. A collection of projections at several angles is
-called a sinogram.
+must be acquired, each of them corresponding to a different angle between the
+rays with respect to the object. A collection of projections at several angles
+is called a sinogram, which is a linear transform of the original image.
 
 The inverse Radon transform is used in computed tomography to reconstruct
-a 2D image from can hence be used to reconstruct an object from the measured
-projections (the sinogram). A practical, exact implementation of the inverse
-Radon transform does not exist, but there are several good approximate
-algorithms available.
+a 2D image from the measured projections (the sinogram). A practical, exact
+implementation of the inverse Radon transform does not exist, but there are
+several good approximate algorithms available.
 
 As the inverse Radon transform reconstructs the object from a set of
 projections, the (forward) Radon transform can be used to simulate a
 tomography experiment.
 
-For more information see:
-
-  - AC Kak, M Slaney, "Principles of Computerized Tomographic Imaging",
-    http://www.slaney.org/pct/pct-toc.html
-  - http://en.wikipedia.org/wiki/Radon_transform
-
 This script performs the Radon transform to simulate a tomography experiment
 and reconstructs the input image based on the resulting sinogram formed by
 the simulation. Two methods for performing the inverse Radon transform
-and reconstructing the original image will be used: The Filtered Back
+and reconstructing the original image are compared: The Filtered Back
 Projection (FBP) and the Simultaneous Algebraic Reconstruction
 Technique (SART).
 
+.. seealso::
+
+    - AC Kak, M Slaney, "Principles of Computerized Tomographic Imaging",
+      http://www.slaney.org/pct/pct-toc.html
+    - http://en.wikipedia.org/wiki/Radon_transform
 
 The forward transform
 =====================
@@ -85,14 +83,14 @@ Reconstruction with the Filtered Back Projection (FBP)
 ======================================================
 
 The mathematical foundation of the filtered back projection is the Fourier
-slice theorem (http://en.wikipedia.org/wiki/Projection-slice_theorem). 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 is needed to suppress high frequency noise in the
-reconstruction. ``skimage`` provides a few different options for the filter.
+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.
 
 """
 
@@ -117,15 +115,16 @@ 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
+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, which has the property
-that the solution will approach a least-squares solution of the equation set.
+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
@@ -134,12 +133,15 @@ with relative ease.
 
 ``skimage`` provides one of the more popular variations of the algebraic
 reconstruction techniques: the Simultaneous Algebraic Reconstruction Technique
-(SART). It uses Kaczmarz' method as the iterative solver. A good
+(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
-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.
+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.
 
 """
 
@@ -174,4 +176,17 @@ 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, "Angenäherte auflösung 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)
+
 """