Image Reconstruction from Fan-Beam Projection Data

To reconstruct an image from fan-beam projection data, use theifanbeamfunction. 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 imageIfrom the projection dataPand distanceD.

I = ifanbeam(P,D);

By default, theifanbeamfunction 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 thefanbeamfunction, you can useifanbeam参数s 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, seeifanbeam.

Theifanbeamfunction converts the fan-beam projection data to parallel-beam projection data with thefan2parafunction, and then calls theiradonfunction to perform the image reconstruction. For this reason, theifanfeamfunction supports certainiradon参数s, which it passes to theiradonfunction. SeeThe Inverse Radon Transformationfor more information about theiradonfunction.

Reconstruct Image using Inverse Fanbeam Projection

This example shows how to usefanbeamandifanbeamto form projections from a sample image and then reconstruct the image from the projections.

Generate a test image and display it. The test image is the Shepp-Logan head phantom, which can be generated by thephantomfunction. The phantom image illustrates many of the qualities that are found in real-world tomographic imaging of human heads.

P = phantom(256); imshow(P)

Compute fan-beam projection data of the test image, using theFanSensorSpacing参数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,fanbeamrotates 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);

Plot the projection dataF3. Becausefanbeamcalculates 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.

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

Reconstruct the image from the fan-beam projection data usingifanbeam. In each reconstruction, match the fan sensor spacing with the spacing used when the projection data was created previously. The example uses theOutputSize参数约束的输出大小of each reconstruction to be the same as the size of the original imageP. In the output, 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.

output_size = max(size(P)); Ifan1 = ifanbeam(F1,D,...'FanSensorSpacing',dsensor1,'OutputSize',output_size); figure, imshow(Ifan1) title('Ifan1')

Ifan2 = ifanbeam(F2,D,...'FanSensorSpacing',dsensor2,'OutputSize',output_size); figure, imshow(Ifan2) title('Ifan2')

Ifan3 = ifanbeam(F3,D,...'FanSensorSpacing',dsensor3,'OutputSize',output_size); figure, imshow(Ifan3) title('Ifan3')