**The Radon Transform**

The Radon Transform
- Theory and Implementation, PhD Theis by
Peter Toft, IMM, DTU, 1996.
## Background

In recent years the Hough transform and the related Radon transform
have received much attention. These two transforms are able to transform
two dimensional images with lines into a domain of possible line parameters,
where each line in the image will give a peak positioned at the corresponding
line parameters. This have lead to many line detection applications within
image processing, computer vision, and seismics.

Several definitions
of the Radon transform exists, but the are related, and a very popular
form expresses lines in the form rho=x*cos(theta)+y*sin(theta), where theta
is the angle and rho the smallest distance to the origin of the coordinate
system. As shown in the two foloowing definitions (which are
identical), the Radon transform for a set of parameters (rho,theta) is
the line integral through the image g(x,y), where the line is
positioned corresponding to the value of (rho,theta). The delta() is
the Dirac delta function which is infinite for argument 0 and zero for
all other arguments (it integrates to one), and in digital versions the
Kronecker delta is used.

or the identical expression
Using this definition an image containing two lines are transformed
into the Radon transform shown to the right

It can be seen that two very bright spots are found in the Radon
transform, and the positions shown the parameters of the lines in the
original image. A simple thresholding algorithm could then be used to
pick out the line parameters, and given that the transform is linear
many lines will just give rise to a a set of distinct point in the
Radon domain. In my Ph.D. thesis the relationship with the Hough
transform is investigated, and it is shown that the Radon transform
and the Hough transform are related but NOT the same.

## Noise

The very strong property of the Radon transform is the ability to extract
lines (curves in general) from very noise images as shown
below. Theoretically results regarding the influence of noise can be
found in Chapter 5 of my Ph.D. thesis.

In general many lines hidden in an image can be transformed into a
set of peaks, where the value in the Radon domain (to the right)
reflect the value on the individual lines. From the Radon transform,
shown to the right, it can be seen that crossing lines makes no
problem.

## The Generalized Radon transform

It is possible to generalize
the Radon- (and Hough) transform in order to detect parameterized
curves with non-linear behaviour. In chapter 4 of my Ph.D. thesis a
fast algorithm can be found that used the generalized Hough transform
to create irregular regions in the parameter domain corresponding to
the parameter regions of interest. Subsequently, it uses the
generalized Radon transform within these regions in order to estimate
the curve parameter with high resolution.
Last modified: Apr. 7, 1997.

You are the visitor since September 6 1996.

Peter Toft *pto@imm.dtu.dk*
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