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Fan-beam transform


F = fanbeam(I,D)
F = fanbeam(..., param1, val1, param1, val2,...)
[F, fan_sensor_positions, fan_rotation_angles] = fanbeam(...)


F = fanbeam(I,D) computes the fan-beam projection data (sinogram) F from the image I.

D is the distance in pixels from the fan-beam vertex to the center of rotation. The center of rotation is the center pixel of the image, defined as floor((size(I)+1)/2). D must be large enough to ensure that the fan-beam vertex is outside of the image at all rotation angles. See Tips for guidelines on specifying D. The following figure illustrates D in relation to the fan-beam vertex for one fan-beam geometry. See the FanSensorGeometry parameter for more information.

Each column of F contains the fan-beam sensor samples at one rotation angle. The number of columns in F is determined by the fan rotation increment. By default, the fan rotation increment is 1 degree so F has 360 columns.

The number of rows in F is determined by the number of sensors. fanbeam determines the number of sensors by calculating how many beams are required to cover the entire image for any rotation angle.

For information about how to specify the rotation increment and sensor spacing, see the documentation for the FanRotationIncrement and FanSensorSpacing parameters, below.

F = fanbeam(..., param1, val1, param1, val2,...) specifies parameters, listed below, that control various aspects of the fan-beam projections. Parameter names can be abbreviated, and case does not matter.

'FanRotationIncrement' -- Positive real scalar specifying the increment of the rotation angle of the fan-beam projections. Measured in degrees. Default value is 1.

'FanSensorGeometry' -- Positioning of sensors, specified as the value 'arc' or 'line'. In the 'arc' geometry, sensors are spaced equally along a circular arc, as shown below. This is the default value.

In 'line' geometry, sensors are spaced equally along a line, as shown below.

'FanSensorSpacing' -- Positive real scalar specifying the spacing of the fan-beam sensors. Interpretation of the value depends on the setting of 'FanSensorGeometry'. If 'FanSensorGeometry' is set to 'arc' (the default), the value defines the angular spacing in degrees. Default value is 1. If 'FanSensorGeometry' is 'line', the value specifies the linear spacing. Default value is 1.

    Note   This linear spacing is measured on the x' axis. The x' axis for each column, col, of F is oriented at fan_rotation_angles(col) degrees counterclockwise from the x-axis. The origin of both axes is the center pixel of the image.

[F, fan_sensor_positions, fan_rotation_angles] = fanbeam(...) returns the location of fan-beam sensors in fan_sensor_positions and the rotation angles where the fan-beam projections are calculated in fan_rotation_angles.

If 'FanSensorGeometry' is 'arc' (the default), fan_sensor_positions contains the fan-beam spread angles. If 'FanSensorGeometry' is 'line', fan_sensor_positions contains the fan-beam sensor positions along the x' axis. See 'FanSensorSpacing' for more information.

Class Support

I can be logical or numeric. All other numeric inputs and outputs can be double. None of the inputs can be sparse.


The following example computes the fan-beam projections for rotation angles that cover the entire image.

ph = phantom(128);
[F,Fpos,Fangles] = fanbeam(ph,250);
axis normal
xlabel('Rotation Angles (degrees)')
ylabel('Sensor Positions (degrees)')
colormap(gca,hot), colorbar

The following example computes the Radon and fan-beam projections and compares the results at a particular rotation angle.

I = ones(100);
D = 200;
dtheta = 45;

% Compute fan-beam projections for 'arc' geometry
[Farc,FposArcDeg,Fangles] = fanbeam(I,D,...
% Convert angular positions to linear distance 
% along x-prime axis
FposArc = D*tan(FposArcDeg*pi/180);

% Compute fan-beam projections for 'line' geometry
[Fline,FposLine] = fanbeam(I,D,...
% Compute the corresponding Radon transform
% Display the three projections at one particular rotation 
% angle. Note the three are very similar. Differences are 
% due to the geometry of the sampling, and the numerical  
% approximations used in the calculations. 
idx = find(Fangles==45);

More About

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As a guideline, try making D a few pixels larger than half the image diagonal dimension, calculated as follows

sqrt(size(I,1)^2 + size(I,2)^2)

The values returned in F are a numerical approximation of the fan-beam projections. The algorithm depends on the Radon transform, interpolated to the fan-beam geometry. The results vary depending on the parameters used. You can expect more accurate results when the image is larger, D is larger, and for points closer to the middle of the image, away from the edges.


[1] Kak, A.C., & Slaney, M., Principles of Computerized Tomographic Imaging, IEEE Press, NY, 1988, pp. 92-93.

Introduced before R2006a

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