fanbeam

Fan-beam transform

Syntax

F = fanbeam(I,D)

F = fanbeam(I,D,Name,Value)

[F,fan_sensor_positions,fan_rotation_angles] = fanbeam(___)

Description

F = fanbeam(I,D) computes 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 pairs to specify the rotation increment and sensor spacing.

[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.

Examples

Compute Fan-beam Projections for Rotation Angles Over Entire Image

Set the IPT preference to make the axes visible.

iptsetpref('ImshowAxesVisible','on')

Create a sample image and display it.

ph = phantom(128);
imshow(ph)

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

Compute Radon and Fan-beam Projections and Compare Results

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')

Input Arguments

I — Input image

2-D numeric matrix | 2-D logical matrix

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

D — Distance from fan beam vertex to center of rotation

positive number

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. The figure illustrates D in relation to the fan-beam vertex for one fan-beam geometry.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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

'FanRotationIncrement' — Fan-beam rotation angle increment

1 (default) | positive scalar

Fan-beam rotation angle increment in degrees, specified as the comma-separated pair consisting of 'FanRotationIncrement' and a positive scalar.

Data Types: double

'FanSensorGeometry' — Fan-beam sensor positioning

'arc' (default) | 'line'

Fan-beam sensor positioning, specified as the comma-separated pair consisting of 'FanSensorGeometry' and one of the following values.

Value	Meaning	Diagram
'arc'	Sensors are spaced at equal angles along a circular arc at distance D from the center of rotation. FanSensorSpacing defines the angular spacing in degrees.
'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.

'FanSensorSpacing' — Fan-bean sensor spacing

1 (default) | positive scalar

Fan-bean sensor spacing, specified as the comma-separated pair consisting of 'FanSensorSpacing' and 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

F — Fan-beam projection data

numsensors-by-numangles numeric matrix

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.

Data Types: double

fan_sensor_positions — Location of fan-beam sensors

numsensors-by-1 numeric vector

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

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

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

fanbeam determines the number of sensors 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

fan_rotation_angles — Rotation angle of fan-beam sensors

1-by-numangles numeric vector

Rotation angle of fan-beam sensors, returned as a 1-by-numangles numeric vector. numangles is 360/FanRotationIncrement.

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