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Fan-Beam Projection Data

    Note   For information about creating projection data from line integrals along parallel paths, see Radon Transform. To convert fan-beam projection data to parallel-beam projection data, use the fan2para function.

Fan-Beam Projection Data Definition

The fanbeam function computes projections of an image matrix along specified directions. A projection of a two-dimensional function f(x,y) is a set of line integrals. The fanbeam function computes the line integrals along paths that radiate from a single source, forming a fan shape. To represent an image, the fanbeam function takes multiple projections of the image from different angles by rotating the source around the center of the image. The following figure shows a single fan-beam projection at a specified rotation angle.

Fan-Beam Projection at Rotation Angle Theta

Computing Fan-Beam Projection Data

To compute fan-beam projection data, use the fanbeam function. You specify as arguments an image and the distance between the vertex of the fan-beam projections and the center of rotation (the center pixel in the image). The fanbeam function determines the number of beams, based on the size of the image and the settings of fanbeam parameters.

The FanSensorGeometry parameter specifies how sensors are aligned. If you specify the value 'arc' for FanSensorGeometry (the default), fanbeam positions the sensors along an arc, spacing the sensors at 1 degree intervals. Using the FanSensorSpacing parameter, you can control the distance between sensors by specifying the angle between each beam. If you specify the value 'line' for FanSensorGeometry parameter, fanbeam position sensors along a straight line, rather than an arc. With 'line' geometry, the FanSensorSpacing parameter specifies the distance between the sensors, in pixels, along the x´ axis.

fanbeam takes projections at different angles by rotating the source around the center pixel at 1 degree intervals. Using the FanRotationIncrement parameter you can specify a different rotation angle increment.

The following figures illustrate both these geometries. The first figure illustrates geometry used by the fanbeam function when FanSensorGeometry is set to 'arc' (the default). Note how you specify the distance between sensors by specifying the angular spacing of the beams.

Fan-Beam Projection with Arc Geometry

The following figure illustrates the geometry used by the fanbeam function when FanSensorGeometry is set to 'line'. In this figure, note how you specify the position of the sensors by specifying the distance between them in pixels along the x´ axis.

Fan-Beam Projection with Line Geometry

Reconstructing an Image from Fan-Beam Projection Data

To reconstruct an image from fan-beam projection data, use the ifanbeam function. With this function, you specify as arguments the projection data and the distance between the vertex of the fan-beam projections and the center of rotation when the projection data was created. For example, this code recreates the image I from the projection data P and distance D.

I = ifanbeam(P,D);

By default, the ifanbeam function assumes that the fan-beam projection data was created using the arc fan sensor geometry, with beams spaced at 1 degree angles and projections taken at 1 degree increments over a full 360 degree range. As with the fanbeam function, you can use ifanbeam parameters to specify other values for these characteristics of the projection data. Use the same values for these parameters that were used when the projection data was created. For more information about these parameters, see Computing Fan-Beam Projection Data.

The ifanbeam function converts the fan-beam projection data to parallel-beam projection data with the fan2para function, and then calls the iradon function to perform the image reconstruction. For this reason, the ifanfeam function supports certain iradon parameters, which it passes to the iradon function. See The Inverse Radon Transformation for more information about the iradon function.

Reconstructing a Head Phantom Image

The commands below illustrate how to use fanbeam and ifanbeam to form projections from a sample image and then reconstruct the image from the projections. The test image is the Shepp-Logan head phantom, which can be generated by the phantom function. The phantom image illustrates many of the qualities that are found in real-world tomographic imaging of human heads.

  1. Generate the test image and display it.

    P = phantom(256);
    imshow(P)

  2. Compute fan-beam projection data of the test image, using the FanSensorSpacing parameter to vary the sensor spacing. The example uses the fanbeam arc geometry, so you specify the spacing between sensors by specifying the angular spacing of the beams. The first call spaces the beams at 2 degrees; the second at 1 degree; and the third at 0.25 degrees. In each call, the distance between the center of rotation and vertex of the projections is constant at 250 pixels. In addition, fanbeam rotates the projection around the center pixel at 1 degree increments.

    D = 250;
    
    dsensor1 = 2;
    F1 = fanbeam(P,D,'FanSensorSpacing',dsensor1);
    
    dsensor2 = 1;
    F2 = fanbeam(P,D,'FanSensorSpacing',dsensor2);
    
    dsensor3 = 0.25
    [F3, sensor_pos3, fan_rot_angles3] = fanbeam(P,D,...
    'FanSensorSpacing',dsensor3);
  3. Plot the projection data F3. Because fanbeam calculates projection data at rotation angles from 0 to 360 degrees, the same patterns occur at an offset of 180 degrees. The same features are being sampled from both sides. Compare this plot to the plot of the parallel-beam projection data of the head phantom using 90 projections in Reconstructing an Image from Parallel Projection Data.

    figure, imagesc(fan_rot_angles3, sensor_pos3, F3)
    colormap(hot); colorbar
    xlabel('Fan Rotation Angle (degrees)')
    ylabel('Fan Sensor Position (degrees)')

  4. Reconstruct the image from the fan-beam projection data using ifanbeam. In each reconstruction, match the fan sensor spacing with the spacing used when the projection data was created in step 2. The example uses the OutputSize parameter to constrain the output size of each reconstruction to be the same as the size of the original image |P|.

    output_size = max(size(P));
    
    Ifan1 = ifanbeam(F1,D, ...
           'FanSensorSpacing',dsensor1,'OutputSize',output_size);
    figure, imshow(Ifan1)
    
    Ifan2 = ifanbeam(F2,D, ...
           'FanSensorSpacing',dsensor2,'OutputSize',output_size);
    figure, imshow(Ifan2)
    
    Ifan3 = ifanbeam(F3,D, ...
           'FanSensorSpacing',dsensor3,'OutputSize',output_size);
    figure, imshow(Ifan3)

    The following figure shows the result of each transform. Note how the quality of the reconstruction gets better as the number of beams in the projection increases. The first image, Ifan1, was created using 2 degree spacing of the beams; the second image, ifan2, was created using 1 degree spacing of the beams; the third image, ifan3, was created using 0.25 spacing of the beams.

    Reconstructions of the Head Phantom Image from Fan-Beam Projections

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