This video presents a new approach
to reconstruction of Computed Tomography
data driven by geometric techniques.
Computer tomography is one
of the most popular modalities
because its capability to visualize
internal organs of the body
with high resolution.
In Computed tomography an unknown
object is reconstructed from projection images.
To acquire the projection images,
we have an x-ray source
and an image detector.
Then, data are obtained by
exposing a patient or an
object to a beam of x-rays,
which can be understood as
a set of rays penetrating
a weighted region.
Then we move our imaging system
to a different position and
generate an additional projection image.
We follow this rotation and
acquire many projection images
up to a thousand different positions.
Here you see an example of
projection images of a rabbit.
We used 220 positions
to capture these images.
These projection data are then
reconstructed using standard techniques
based on the Radon Transform.
The volume is reconstructed by
back projecting the acquired data.
The summation is performed
all projection angles.
Here you see how a
slice of rabbit data is reconstructed.
The Nyquist sampling criteria limit
techniques based on the Radon transform.
Therefore, these techniques provide
reduced image quality when
using a reduced number of
projections. But using fewer
projections reduces radiation dose
and reduces incidence of
cancer in patient population.
The literature shows that
high radiation dose as used
in standard CT can lead to cancer.
Compressed sensing presents a new
method to capture and represent compressible
signals at a sampling rate significantly
below the Nyquist rate. The main
idea behind this new approach can
be summarized as an alternative to
directly reconstructing a signal
or an image. The projection
data are transformed into a sparse version.
Then the signal or image is
reconstructed in a dimensionality-reduced
space by minimizing the L1 - distance.
To use this method, the
signal to-be-reconstructed needs
to have sparse-like attributes,
which standard medical images do not have.
We observed that standard CT
techniques converge quickly when the
intensity of all voxels is similar.
This occurs because the value
placed in each voxel along each
penetrating ray will be close to its
actual intensity when all voxels
have similar intensity.
Additionally, we observed that when
plotting the entropy as a function of the
number of projections used in the
reconstruction a plateau is visible
even after a small number of projections.
Thus, it should be possible to
reconstruct high quality images from
a limited number of projections.
Based on these observations, our
reconstruction strategy resembles a
geometric compressed sensing approach.
As the sparse transform, we
are handling each region of the
to-be-reconstructed image as separate entity.
Let's review our approach step by step.
We have an unknown object.
A radon-based technique is used
to generate a poor quality
reconstruction. To obtain a rough
segmentation of our object-of-interest,
we make use of models or priors.
In order to make use of our earlier
observation, we have to classify the rays
that penetrate the object.
As you see here, the rays
that penetrate the object are
shown in red, and the complement
of these rays is shown in green.
To classify our rays, we reduce
the problem from 3D to 2D.
Each plane of rays is handled separately.
Now lets take a closer look
at one of our 2D arrangements.
In the 2D plane, we have
a set of objects obtained from
the rough segmentation, surrounded
by lower density regions called
complements. To separate the reconstruction
of the objects from that of
the complements, we first approximate
the rough boundary of the objects by
simple polygons and then treat the polygons as obstacles.
To efficiently classify all the rays,
we use a point-line duality transformation
on the set of boundary segments
of the polygons. Each vertex of
the polygons transforms to a dual
line and each ray maps to a point in
the dual space. The set of dual lines
forms an arrangement, which partitions the
space into a set of convex cells.
Each cell contains the set of rays
intersecting the same subset of boundary
segments of the polygons, that is,
all rays traveling through the hourglass
defined by the set of segments. All
types of rays can be identified by sweeping the
arrangement in an online fashion.
To report all the cells and rays in
a more efficient way, we use
the topological peeling algorithm due to its
optimal time and space complexities.
Let's go back into 3D.
We have shown how to optimally classify the
rays for one plane by using geometric techniques.
In our algorithm, we perform
this for all projection images.
For the remainder of this presentation,
two projections will continue to represent
the vast number of images that are actually used.
We reconstruct from the outside
inward in a layer-by-layer fashion.
First, we reconstruct the outer region
by using the green rays. Next,
we project the reconstructed outer region.
These resulting data are subtracted
from the original projections.
Second, we reconstruct the outer object
by using the red rays. We project
the reconstructed object by
using the remaining rays. These
resulting data are subtracted from the projection space.
Finally we reconstruct the remaining
object by using the yellow rays.
In the end, we have produced a
reliable reconstruction from
a reduced number of projections.
Now to our results:
Here you see real data from our
rabbit study. To the left is
a rough segmentation of the object-of-interest.
The center object shows the reconstruction
from a reduced number of projections obtained
using standard techniques.
On the right you see our reconstruction.
Compared to the center image that is
calculated from the same number of projections,
our result on the right has a significantly improved image quality.
We thank you for your time and
interest from the University at Buffalo.