Tomographic reconstruction is a type of multidimensional
inverse problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
where the challenge is to yield an estimate of a specific system from a finite number of
projections. The mathematical basis for tomographic imaging was laid down by
Johann Radon
Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna).
Life
RadonBrigitte Bukovics: ''Biography of Johan ...
. A notable example of applications is the
reconstruction
Reconstruction may refer to:
Politics, history, and sociology
*Reconstruction (law), the transfer of a company's (or several companies') business to a new company
*'' Perestroika'' (Russian for "reconstruction"), a late 20th century Soviet Unio ...
of
computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the
Radon transform
In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
and its inverse used for tasks related to realistic object insertion required for testing and evaluating
computed tomography use in
airport security
Airport security includes the techniques and methods used in an attempt to protect passengers, staff, aircraft, and airport property from malicious harm, crime, terrorism, and other threats.
Aviation security is a combination of measures and hum ...
.
This article applies in general to reconstruction methods for all kinds of
tomography
Tomography is imaging by sections or sectioning that uses any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, astrophysics, quantu ...
, but some of the terms and physical descriptions refer directly to the
reconstruction of X-ray computed tomography.
Introducing formula
The projection of an object, resulting from the tomographic measurement process at a given angle
, is made up of a set of
line integrals (see Fig. 1). A set of many such projections under different angles organized in 2D is called sinogram (see Fig. 3). In X-ray CT, the line integral represents the total attenuation of the beam of
x-rays
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
as it travels in a straight line through the object. As mentioned above, the resulting image is a 2D (or 3D) model of the
attenuation coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
. That is, we wish to find the image
. The simplest and easiest way to visualise the method of scanning is the system of
parallel projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or '' image plane'', where the ''rays'', known as '' li ...
, as used in the first scanners. For this discussion we consider the data to be collected as a series of parallel rays, at position
, across a projection at angle
. This is repeated for various angles.
Attenuation
In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable att ...
occurs
exponentially
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
* Exponential decay, decrease at a rate proportional to value
*Exp ...
in tissue:
:
where
is the attenuation coefficient as a function of position. Therefore, generally the total attenuation
of a ray at position
, on the projection at angle
, is given by the line integral:
:
Using the coordinate system of Figure 1, the value of
onto which the point
will be projected at angle
is given by:
:
So the equation above can be rewritten as
:
where
represents
and
is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. This function is known as the
Radon transform
In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
(or ''sinogram'') of the 2D object.
The
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the projection can be written as
where
represents a slice of the 2D Fourier transform of
at angle
. Using the
inverse Fourier transform In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information ...
, the inverse Radon transform formula can be easily derived.
where
is the derivative of the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
of
In theory, the inverse Radon transformation would yield the original image. The
projection-slice theorem
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal:
* Take a two-dimensional function ''f''(r), project (e.g. using the ...
tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object,
. However, there will only be a finite number of projections available in practice.
Assuming
has effective diameter
and desired resolution is
, rule of thumb number of projections needed for reconstruction is
Reconstruction algorithms
Practical reconstruction algorithms have been developed to implement the process of reconstruction of a 3-dimensional object from its projections.
[Herman, G. T., Fundamentals of computerized tomography: Image reconstruction from projection, 2nd edition, Springer, 2009] These
algorithms
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing c ...
are designed largely based on the mathematics of the
Radon transform
In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
, statistical knowledge of the data acquisition process and geometry of the data imaging system.
Fourier-Domain Reconstruction Algorithm
Reconstruction can be made using interpolation. Assume
-projections of
are generated at equally spaced angles, each sampled at the same rate. The
Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...
on each projection will yield sampling in the frequency domain. Combining all the frequency-sampled projections would generate a polar raster in the frequency domain. The polar raster will be sparse so interpolation is used to fill the unknown DFT points and reconstruction can be done through
inverse Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a compl ...
.
Reconstruction performance may improve by designing methods to change the sparsity of the polar raster, facilitating the effectiveness of interpolation.
For instance, a concentric square raster in the frequency domain can be obtained by changing the angle between each projection as follow:
where
is highest frequency to be evaluated.
The concentric square raster improves computational efficiency by allowing all the interpolation positions to be on rectangular DFT lattice. Furthermore, it reduces the interpolation error.
Yet, the Fourier-Transform algorithm has a disadvantage of producing inherently noisy output.
Back Projection Algorithm
In practice of tomographic image reconstruction, often a stabilized and
discretized version of the inverse Radon transform is used, known as the
filtered back projection algorithm.
With a sampled discrete system, the inverse Radon Transform is
where
is the angular spacing between the projections and
is radon kernel with frequency response
.
The name back-projection comes from the fact that 1D projection needs to be filtered by 1D Radon kernel (back-projected) in order to obtain a 2D signal. The filter used does not contain DC gain, thus adding
DC bias
In signal processing, when describing a periodic function in the time domain, the DC bias, DC component, DC offset, or DC coefficient is the mean amplitude of the waveform. If the mean amplitude is zero, there is no DC bias. A waveform with no DC ...
may be desirable. Reconstruction using back-projection allows better resolution than interpolation method described above. However, it induces greater noise because the filter is prone to amplify high-frequency content.
Iterative Reconstruction Algorithm
Iterative algorithm is computationally intensive but it allows to include ''a priori'' information about the system
.
Let
be the number of projections,
be the distortion operator for
th projection taken at an angle
.
are set of parameters to optimize the conversion of iterations.