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Radon transform

`R = radon(I, theta)`

[R,xp] = radon(...)

[___]= radon(gpuarrayI,theta)

`R = radon(I, theta)`

returns
the Radon transform `R`

of the intensity image `I`

for
the angle `theta`

degrees.

The Radon transform is the projection of the image intensity
along a radial line oriented at a specific angle. If `theta`

is
a scalar, `R`

is a column vector containing the Radon
transform for `theta`

degrees. If `theta`

is
a vector, `R`

is a matrix in which each column is
the Radon transform for one of the angles in `theta`

.
If you omit `theta`

, it defaults to 0:179.

`[R,xp] = radon(...)`

returns
a vector `xp`

containing the radial coordinates corresponding
to each row of `R`

.

The radial coordinates returned in `xp`

are
the values along the *x**'*-axis,
which is oriented at `theta`

degrees counterclockwise
from the *x*-axis. The origin of both axes is the
center pixel of the image, which is defined as

floor((size(I)+1)/2)

For example, in a 20-by-30 image, the center pixel is (10,15).

`[___]= radon(gpuarrayI,theta)`

performs
the Radon transform on a GPU. The input image and the return values
are 2-D gpuArrays. `theta`

can be a `double`

or
gpuArray of underlying class `double`

. This syntax
requires the Parallel Computing Toolbox™.

`I`

can be of class `double`

, `logical`

,
or any integer class. All other inputs and outputs are of class `double`

.
Neither of the inputs can be sparse.

`gpuarrayI`

is a gpuArray with underlying class `uint8`

, `uint16`

, `uint32`

, `int8`

, `int16`

, `int32`

, `logical`

, `single`

or `double`

and
must be two-dimensional. `theta`

is a `double`

vector
or gpuArray vector of underlying class `double`

.

The Radon transform of an image is the sum of the Radon transforms of each individual pixel.

The algorithm first divides pixels in the image into four subpixels and projects each subpixel separately, as shown in the following figure.

Each subpixel's contribution is proportionally split into the two nearest bins, according to the distance between the projected location and the bin centers. If the subpixel projection hits the center point of a bin, the bin on the axes gets the full value of the subpixel, or one-fourth the value of the pixel. If the subpixel projection hits the border between two bins, the subpixel value is split evenly between the bins.

Bracewell, Ronald N., *Two-Dimensional Imaging*,
Englewood Cliffs, NJ, Prentice Hall, 1995, pp. 505-537.

Lim, Jae S., *Two-Dimensional Signal and Image Processing*,
Englewood Cliffs, NJ, Prentice Hall, 1990, pp. 42-45.

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