Image Reconstruction from Projections

This demonstration illustrates the mechanics of
reconstructing the cross section of an object, given
projections through the object from multiple
directions, using the method of "filtered
back-projection".  This demonstration allows the
selection of various filters and reconstruction
parameters. The results of the various techniques
may then be simply compared.

X-ray CT (computerized tomography) scanners,
ECT (emission computed tomography), PET (positron 
emission tomography), MRI (magnetic resonance
imaging), all share this problem of reconstructing
a slice of an object given projections through
cross sections.  Although filtered back-projection
is not the only reconstruction method, it is the
most widely used.

This demonstration begins with the actual cross
section of a simulated object, called the phantom.
A projection contains M points. There is a total
of N projections that have an equal angular distance
from each another. The N projections are constructed
by summing along lines drawn through the object.
Stacking the N projections to form an M column by N
row matrix forms a "sinogram" which is shown in the
lower left corner. This simulates the actual data 
that would be acquired by an actual instrument.

From the sinogram, each row (projection) is convolved
with the selected filter. Then they are summed 
(back-projected) into a matrix. The reconstructed
image, after all projections have been summed, 
is shown in the upper right corner. The difference
between the reconstructed and the original slices
is shown in the lower right corner.

This demonstration is interactive in that the "ray"
through the original image corresponding to each
point of the sinogram is drawn by clicking the
mouse on the sinogram image.  Conversely, all
points of the sinogram that pass through a given
point of the original (phantom) image are
illustrated by clicking on that image.  Clicking on
the error image displays a plot of the horizontal
profiles through the original and reconstructed
images.

The IDL RIEMANN function is used to compute both
the projections and to sum the back-projections.


                OPERATION

Operation of this demonstration is quite simple.
First select the reconstruction parameters,
using the buttons at the lower left, and then
press the "Reconstruct" button. After two or more
reconstructions have been computed, they may be
compared by pressing the "Compare" button.


MENU OPTIONS
------------

File Menu:

    Select one of the various simulated objects (including
    Shepp-Logan Phantom, Circles, Squares, or Computerized
    Tomography scanning slice) from the "Objects" sub-menu.

    Select "Reconstruct" to perform the numerical
    reconstruction and display the images.

    Select "Quit" to exit the demo.

Edit Menu:
    
    Select "Color Palette" to activate a palette chooser
    and manipulator tool for fine-tuning of the color
    palette and display contrast.

View Menu:

    Select "Redraw" to redraw the images minus the line
    overlays.

    Select "Compare" to display the four most recent
    reconstructions.

About Menu:
  
    Select "About Reconstruction" to display this help text.


FEATURES OF THE RECONSTRUCTION DEMO
-----------------------------------

<<Number of angles droplist>>
    Select the number of views or angles. Note that on
    small screen monitors, the image size might be smaller
    than the number of angles and the sinogram views will
    not show the image in its entirety.
   
<<Filter type droplist>> 
    Selects the type of filter (or none) used to convolve
    each projection before back-projecting.

<<Kernel size droplist>>    
    Select the half-width of the filter convolution kernel.
    Larger values give better results.

<<Interpolation droplist>>  
    Select either nearest neighbor, linear, or cubic
    interpolation. Cubic interpolation provides the best
    results, but takes longest to compute.

<<Left Mouse Button>>

    In the original display (top left viewing area):
        Selecting a point in the original
        image draws a line through the sinogram
        over the points whose rays are derived
        from the selected point.

    In the Sinogram (Lower left viewing area):
        Selecting a point in the sinogram draws
        the corresponding ray in the Phantom
        image and the profile of the ray sum
        (line integral) all parallel rays.

    In the Error window (Bottom right viewing area):
        Selecting a point makes a plot showing
        horizontal profiles through the selected
        row of the original phantom (solid line),
        and the reconstructed image (points).