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,
number of views, 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.  N projections, each
containing M points are constructed by summing along lines
drawn through the object, at N equally spaced angles,
(j*Pi/N, (0 <= j < N)), at a distance of m from the origin,
which is in the center.  Stacking the N projections, to from
an M column by N row matrix, forms a "sinogram", 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, and then summed (back-projected) into a
matrix.  The reconstructed image results after all
projections have been summed, and is shown in the upper
right corner.  Finally, 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 reconstructed 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 "Go" button.  After two
or more reconstructions have been computed, they may be
compared by pressing the "Compare" button.  

	Upper Command Buttons

Done - exits this demo.
Help - displays this text.
Palette - activates the palette chooser/manipulator tool for
	fine-tuning of the color palette and display
	contrast.
Redraw - redraws the images minus the line overlays.
Compare - displays the four most recent reconstructions.
Go - performs a reconstruction using the selected
	parameters.


	Lower Parameter Buttons

Views - Selects the number of views, from 4 to 256.  It will
	be quickly seen that more views give better results.
Filter - Selects the type of filter (or none) used to
	convolve each projection before back-projecting.
	Selecting a filter also causes a graph of the
	available filters to be drawn.  
Interpolation - selects either linear or nearest neighbor
	interpolation.  Linear interpolation gives better
	results. 
Kernel size - the half-width of the filter convolution
	kernel, larger values give better results.
Object - allows the selection of various simulated objects.

	
	Mouse / Buttons

Original display: selecting a point in the original image
	draws a line through the sinogram over the points whose rays
	are derived from the selected point.
Sinogram: 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.
Reconstruction: selecting a point makes a plot showing
	horizontal profiles through the selected row of the original
	phantom (solid line), and the reconstructed image
	(points).