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DOC: Move radon example to gallery.
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Stefan van der Walt
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"""
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===============
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Radon transform
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===============
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The radon transform is a technique widely used in tomography to
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reconstruct an object from different projections. A projection is, for
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example, the scattering data obtained as the output of a tomographic
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scan.
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For more information see:
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- http://en.wikipedia.org/wiki/Radon_transform
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- http://www.clear.rice.edu/elec431/projects96/DSP/bpanalysis.html
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This script performs the radon transform, and reconstructs the
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input image based on the resulting sinogram.
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"""
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import matplotlib.pyplot as plt
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from scikits.image.io import imread
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from scikits.image import data_dir
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from scikits.image.transform import radon, iradon
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from scipy.ndimage import zoom
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image = imread(data_dir + "/phantom.png", as_grey=True)
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image = zoom(image, 0.4)
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plt.figure(figsize=(9, 8.5), dpi=75)
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plt.subplot(221)
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plt.title("Original");
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plt.imshow(image, cmap=plt.cm.Greys_r)
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plt.subplot(222)
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projections = radon(image, theta=[0, 45, 90])
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plt.plot(projections);
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plt.title("Projections at\n0, 45 and 90 degrees")
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plt.xlabel("Projection axis");
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plt.ylabel("Intensity");
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projections = radon(image)
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plt.subplot(223)
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plt.title("Radon transform\n(Sinogram)");
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plt.xlabel("Projection axis");
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plt.ylabel("Intensity");
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plt.imshow(projections)
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reconstruction = iradon(projections)
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plt.subplot(224)
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plt.title("Reconstruction\nfrom sinogram")
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plt.imshow(reconstruction, cmap=plt.cm.Greys_r)
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plt.subplots_adjust(hspace=0.4, wspace=0.5)
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plt.show()
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@@ -1,116 +0,0 @@
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***************
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Radon transform
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***************
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The radon transform is a technique widely used in tomography, where you
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reconstruct an object from its different projections. A projection for example
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the scattering data obtained as the output of a tomographic scan.
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For more information:
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http://en.wikipedia.org/wiki/Radon_transform
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http://www.clear.rice.edu/elec431/projects96/DSP/bpanalysis.html
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Forward transform
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=================
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First we load the Schepp-Logan phantom, a classic test image representing a
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tomographic scan.
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.. ipython::
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In [1]: from scikits.image.io import imread
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In [1]: from scikits.image import data_dir
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In [2]: from scikits.image.transform import radon, iradon
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In [3]: from scikits.image.color import rgb2gray
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In [4]: import matplotlib.pyplot as plt
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In [5]: import matplotlib.cm as cm
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In [6]: image = rgb2gray(imread(data_dir + "/phantom.png"))
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In [7]: plt.title("original image");
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In [8]: plt.imshow(image, cmap=cm.Greys_r)
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@savefig radon_original_image.png width=4in
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In [9]: plt.show()
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Let us illustrate the transform by looking at projections taken at specific
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angles.
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.. ipython::
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In [10]: projections = radon(image, theta=[0, 45, 90])
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In [11]: plt.plot(projections);
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In [12]: plt.title("radon projections");
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In [13]: plt.xlabel("projection axis");
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In [14]: plt.ylabel("intensity");
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@savefig radon_projection_plot1.png width=4in
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In [15]: plt.show()
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We are going to reconstruct an image from 180 of these projections (the
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default).
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.. ipython::
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In [16]: projections = radon(image)
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In [17]: plt.figure()
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In [18]: plt.title("radon projections");
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In [19]: plt.xlabel("projection axis");
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In [20]: plt.ylabel("intensity");
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In [21]: plt.plot(projections)
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@savefig radon_projection_plot2.png width=4in
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In [22]: plt.show()
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We have now constructed various projections, line integrals of an image, at
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specific angles. This image is called a sinogram.
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.. ipython::
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In [23]: plt.figure()
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In [24]: plt.title("sinogram");
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In [25]: plt.xlabel("projection axis");
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In [26]: plt.ylabel("intensity");
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In [27]: plt.imshow(projections)
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@savefig radon_sinogram.png width=4in
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In [28]: plt.show()
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Inverse transform
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=================
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To reconstruct the image from this sinogram, we apply the inverse transform.
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.. ipython::
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In [29]: reconstruction = iradon(projections)
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In [30]: plt.title("reconstructed image");
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In [31]: plt.imshow(reconstruction, cmap=cm.Greys_r)
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@savefig radon_reconstructed_image.png width=4in
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In [32]: plt.show()
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