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fanbeam

Fan-beam transform

Description

example

F = fanbeam(I,D) calculates the fan-beam projection data (sinogram) F from the image I. Each column of F contains fan-beam projection data at one rotation angle. D is the distance from the fan-beam vertex to the center of rotation.

F = fanbeam(I,D,Name,Value) uses name-value arguments to specify the rotation increment and sensor spacing.

example

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

Examples

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Set the IPT preference to make the axes visible.

iptsetpref('ImshowAxesVisible','on')

Create a sample image and display it.

ph = phantom(128);
imshow(ph)

Figure contains an axes object. The axes object contains an object of type image.

Calculate the fanbeam projections and display them.

[F,Fpos,Fangles] = fanbeam(ph,250);
figure
imshow(F,[],'XData',Fangles,'YData',Fpos,...
            'InitialMagnification','fit')
axis normal
xlabel('Rotation Angles (degrees)')
ylabel('Sensor Positions (degrees)')
colormap(gca,hot), colorbar

Figure contains an axes object. The axes object contains an object of type image.

Compute fan-beam projections for 'arc' geometry.

I = ones(100);
D = 200;
dtheta = 45;
[Farc,FposArcDeg,Fangles] = fanbeam(I,D,...
                    'FanSensorGeometry','arc',...
                 'FanRotationIncrement',dtheta);

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,...
           'FanSensorGeometry','line',...
        'FanRotationIncrement',dtheta);

Compute the corresponding Radon transform.

[R,Rpos]=radon(I,Fangles);

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.

figure
idx = find(Fangles==45);
plot(Rpos,R(:,idx),...
           FposArc,Farc(:,idx),...
          FposLine,Fline(:,idx))
legend('Radon','Arc','Line')

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent Radon, Arc, Line.

Input Arguments

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Input image, specified as a 2-D numeric matrix or 2-D logical matrix.

Distance in pixels from the fan beam vertex to the center of rotation, specified as a positive number. 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.

Fan-beam image geometry highlighting the distance D between the fan-beam vertex and the center of rotation

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: fanbeam(I,D,FanRotationIncrement=5)

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: fanbeam(I,D,"FanRotationIncrement",5)

Fan-beam rotation angle increment in degrees, specified as a positive scalar.

Fan-beam image geometry highlighting the angular separation of two successive fan-beam vertices by FanRotationIncrement degrees.

Data Types: double

Fan-beam sensor positioning, specified as "arc" or "line".

Value

Meaning

Diagram

"arc"

Sensors are spaced at equal angles along a circular arc. The center of the arc is the fan-beam vertex.

FanSensorSpacing defines the angular spacing in degrees.

Fan-beam image geometry highlighting the constant angular spacing between sensors in an arc

"line"

Sensors are spaced at equal distances along a line that is parallel to the x' axis. The closest sensor is distance D from the center of rotation.

FanSensorSpacing defines the distance between fan-beams on the x' axis, in pixels.

Fan-beam image geometry highlighting the constant distance between sensors in a line

Fan-bean sensor spacing, specified as a positive scalar.

  • If FanSensorGeometry is "arc", then FanSensorSpacing defines the angular spacing in degrees.

  • If FanSensorGeometry is "line", then FanSensorSpacing defines the linear distance between fan-beams, in pixels. Linear spacing is measured on the x' axis.

Data Types: double

Output Arguments

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Fan-beam projection data, returned as a numSensors-by-numAngles numeric matrix. numSensors is the number of fan-beam sensors and numAngles is the number of fan-beam rotation angles. Each column of F contains the fan-beam sensor samples at one rotation angle.

fanbeam determines numAngles by calculating how many rotation angles are required to span 360 degrees using an angular spacing of FanRotationIncrement.

fanbeam determines numSensors by calculating how many beams are required to cover the entire image for any rotation angle. Fewer sensors are required to cover the image when the distance D between the fan-beam vertex and the center of rotation is large.

Data Types: double

Location of fan-beam sensors, returned as a numSensors-by-1 numeric vector.

  • If FanSensorGeometry is "arc" (the default), then fanSensorPos contains the fan-beam spread angles.

  • If FanSensorGeometry is "line", then fanSensorPos contains the fan-beam sensor positions along the x' axis. See FanSensorSpacing for more information.

Data Types: double

Rotation angle of fan-beam sensors, returned as a 1-by-numAngles numeric vector.

Data Types: double

Tips

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.

References

[1] Kak, Avinash C., and Malcolm Slaney. Principles of Computerized Tomographic Imaging. New York: IEEE Press, 1988.. pp. 92-93.

Version History

Introduced before R2006a