Radon transforms
The mathematics behind Computertomography
PD Dr. Swanhild Bernstein, Institute of Applied Analysis,
Freiberg University of Mining and Technology,
International Summer academic course 2008,
„Modelling and Simulation of Technical Processes“
Historical remarks
 C. Röntgen (1895) – X-rays
 J. Radon (1917)– Mathematical Model
 G. Grossmann (1935) – Tomography
 G. Hounsfield, McCormack (1972) –
Computerized assited tomography (CAT scan)
Why does it work?
The physical priniples.
 Tomography means slice imaging,
 Quantification of the tendency of objects to
absorb or scatter x-rays by the attunation
coefficient, involving Beer‘s law.
Model
 No refraction or diffraction: X-ray beams
travel along straight lines that are not „bent“ by
the objects they pass through.
This is a good approximation because x-rays have very
high energies, and therefore very short wavelength.
 The X-rays used are monochromatic: The
waves making up the x-ray beams are all of the
same frequency.
This is not a realistic assumption, but it is needed to
construct a linear model for the measurements.
 Beer‘s law: Each material encountered has a
characteristic linear attenuation coefficient μ
for x-rays of a given energy.
 The intensity, I of the x-ray beam satisfies Beer‘s
law:
Here, s is the arc-length along the straight line
trajectory of the x-ray beam.
Model
The failure to distinguish objects
one object two objects
same projection
Solution: more directions
Different angles lead to different projections.
The more directions from which we make
measurement, the more arrangements of objects
we can distinguish.
Analysis of a Point Source Device,
2D model, what do we measure?
2D model, what do we measure?
Beer‘s law:
First order ordinary differential equation for the
intensity I with boundary condition I at r=r0>0 equals I0.
Ex 1
2D model, what do we measure?
Analysis if a Point Source Device
The density of the developed film at a point is proportional
to the logarithm of the total energy incident at that point:
density of the film = γ × log (total energy intensity),
where γ is a constant, we obtain:
This formula expresses the measurements as
linear function of the attunation coefficient.
Oriented lines
t
t is the distant of the line from the origin,
s is the parameter of the line.
Radon transform
Radon transform
Radon transform
 The Radon transform can be defined, a priori for a function, f
whose restriction to each line is locally integrable and
 This is really two different conditions:
1. The function is regular enough so that restricting it to any
line gives a locally integrable function,
2. The function goes to zero rapidly enough for the improper
integrals to converge.
In applications functions of interest are usually piecewise
continuous and zero outside of some disk.
Ex 2
Properties of the Radon transform
 The Radon transform is linear:
 The Radon transform of f is an even function:
 The Radon transform is monotone: if f is a non-negative
function then
Pencilgeometry (Nadelstrahlgeometrie)
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Back-projection formula
 It is difficult to use the line integrals of a function
directly to reconstruct the function:
 Results of the recontruction by back-projection
What is that??
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Fourier transform in 1D
 The Fourier transform of an absolutely integrable
function f, defined on the real line, is
 Suppose that the Fourier transform of f is again an
absolutely integrable function then
Square integrable functions
Ex 3
 A (complex-valued) function f, defined on , is square
integrable if
 Examples: The function is not
absolutely integrable but square integrable, the
function
is absolutely integrable but not square integrable.
Fourier transform in nD
 The Fourier transform of an absolutely integrable
function is defined by
 Let and define
then
 Parseval formula: If f is square integrable then
 Let f be an absolutely integrable function. For any real
number r and unit vector , we have the identity
 For a given vector the inner product ,
is constant along any line perpendicular to the direction
. The central slice theorem interprets the compution
of the Fourier transform of as a two-step process:
1. First, integrate the function along lines perp. to .
2. Compute the one-dimensional Fourier transform of
this function of the affine parameter.
Central Slice Theorem
Ex 4
Inverse Radon Transform and
Central Slice Theorem
3. Compute the one dimensional
Fourier transform of
1. Choose a line L, determined
by the direction (Cartesian
coord.) or by the angle γ .
Then the coorinate axis ξ shows
in the same direction.
2. Integrate along all lines perp.
to (those lines are parallel
to the cood. axis η. We obtain
the Radon transform .
4. With u=q cos γ and v=q sin γ,
we get F(u,v) = and
f(x,y) is equal to the 2D inverse
Fourier transform of F(u,v).
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Radon Transform
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Radon Transform
In Cartesian coordinates.
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Radon Transform
In Polar coordinates.
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Radon Transform
Abdomen, Radon transform in Cartesian coord.
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Reconstruction
based on 1 projection.
Now, we can try to do some reconstruction by the
before mentioned procdure
E. Meyer, www1.am.uni-erlangen.de/~bause/Seminar/seminar.html
based on 4 projections.
Reconstruction
E. Meyer, www1.am.uni-erlangen.de/~bause/Seminar/seminar.html
based on 8 projections.
Reconstruction
E. Meyer, www1.am.uni-erlangen.de/~bause/Seminar/seminar.html
based on 30 projections.
Reconstruction
E. Meyer, www1.am.uni-erlangen.de/~bause/Seminar/seminar.html
based on 60 projections.
Reconstruction
What is the difference to the back-projection formula?
E. Meyer, www1.am.uni-erlangen.de/~bause/Seminar/seminar.html
Radon Inversion Formula
 If f is an absolutely integrable function and ist Fourier
transform is absolutely integrable too, then
Radon Inversion Formula
 If f is an absolutely integrable function and its Fourier
transform is absolutely integrable too, then
 Filtered Back-Projection
1. The radial integral is interpreted as a filter applied to
the Radon transform. The filter acts only the affine
parameter; is output of the filter is denoted
2. The angular integral is then interpreted as the backprojection
of the filtered Radon transform.
Back-Projection vs.
Filtered Back-Projection
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Back-Projection vs.
Filtered Back-Projection
back-projection filtered back-projection
based on 1 projection
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Back-Projection vs.
Filtered Back-Projection
back-projection filtered back-projection
based on 3 projections
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Back-Projection vs.
Filtered Back-Projection
back-projection filtered back-projection
based on 10 projections
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Back-Projection vs.
Filtered Back-Projection
back-projection filtered back-projection
based on 180 projections
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Back-Projection vs.
Filtered Back-Projection
back-projection filtered back-projection
based on 180 projections
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Back-Projection vs.
Filtered Back-Projection
a) back-projection and b) filtered back-projection,
based on 1, 2, 3, 10, 45 projections resp.
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Different Inversion formulas
 We already had the Radon inversion formula:
 We write |r| as sgn(r) r :
A Different Inversion formula
 We already had the Radon inversion formula:
 Where we write |r|as sgn(r) r
 denotes the 1D Fourier
transform with respect to t.
 Suppose that g is square integrable on the real line. The
Hilbert transform of g is defined by
If is also absolutely integrable, then
We obtain
Mathematical Model for CT
 We consider a two-dimensional slice of an three-dimensional
object, the physical parameters of interest is the attenuation
coefficient f of the two-dimensional slice. According to Beer‘s
law, the intensity traveling along a line is attenuated according
to the differential equation
where s is arclength along the line.
 By comparing the intensity of an incident beam of x-rays to that
emitted, we measure the Radon transform of f:
 Using the Radon inversion formula, the attenuation coefficient f
is reconstructed from the measurements
Scanner geometry
http://www.impactscan.org/slides/impactcourse/basic_principles_of_ct/
Scanner geometry
http://www.impactscan.org/slides/impactcourse/basic_principles_of_ct/
Scanner geometry
http://www.impactscan.org/slides/impactcourse/basic_principles_of_ct/
Scanner geometry
http://www.impactscan.org/slides/impactcourse/basic_principles_of_ct/
Radon transform - Polar grid
Fourier transform – Cartesian grid
Buzug, Einführung in die Computertomography, Springer Verlag , 2004
Why fan beam?
http://www.impactscan.org/slides/impactcourse/basic_principles_of_ct/
Reconstruction Algorithm for a
Parallel Beam Machine
 We assume that we can measure all the data from a finite set
of equally spaced angles. In this case data would be
 With these data we can apply the central slice theorem to
compute angular samples of the two-dimensional Fourier
transform of f,
 Using the two-dimensional Fourier inversion formula and
using a Riemann sum in the angular direction gives
Concluding remarks
 The model present here is a CT-model, there exist other
types of tomographical methods that are based on
other mathematical models.
 All mathematical models are based on so-called integral
geometry and connected with wave equations.
 Modern tomography even combines different methods:
 fusion of CT-scan (grey)
 and PET-scan (grey)
 PET = Positron Emission
 Tomography
http://www.sdirad.com/PatientInfo/pt_pet.htm
Most of the pictures are dealing with medical applications
but Computer tomography can be applied to more
applications, as for example:
Material sciences
Tomographic visualisation of a metallic foam structure
http://www2.tu-berlin.de/fak3/sem/GB_index.html
Geology
http://www.geo.cornell.edu/geology/classes/Geo101/
101images_spring.html
Seismic tomography reveals a more
complex interior structure.
Archeology
3D-Computer Tomography of
Prehispanic Sound Artifacts.
Supported by the Ethnological Museum Berlin and the St. Gertrauden Hospital, Berlin.
http://www.mixcoacalli.com/wp-content/uploads/2007/09/ct2.jpg
Concluding remarks
Bibliography
 Charles L. Epstein, Introduction to the Mathematics of
Medical Imaging, Pearson Education, Inc., 2003
 Thorsten M. Buzug, Einführung in die
Computertomographie, Springer Verlag, 2004
 Esther Meyer, Die Mathematik der Computertomogrphie,
Seminarvortrag,
www1.am.unierlangen.de/~bause/Seminar/seminar.html
 Other pictures are from
www.impactscan.org/slides/impactcourse/basic_principles_of_ct/
www.sdirad.com/PatientInfo/pt_pet.htm