Fast image reconstruction for Compton camera using stochastic origin ensemble approach
The principle of image formation in a Compton camera.
The central slice through the initial density matrix for the point source. Acceptance rules will work in the following way: (a) A new position will be accepted with 100% probability as in this case the event density increases; (b) the new position will be accepted with low probability due to the sharp decrease in event density.
The difference between (a) SOE for Compton camera and (b) SOE for PET. The small dots indicate the current photon origin candidates; the small stars indicate the true source locations. Events are “moved” by the algorithm on (a) half-cone surfaces for Compton camera or on (b) lines for PET.
(a) A location on the conical surface in SOE algorithm is parametrized by distance z and angle . (b) Forward and backprojection realized by multiple ray tracing through the surface of the cone was used for list-mode ML-EM.
3D phantoms (left column) and corresponding initial event density images (right column) for (a) single point source, (b) 12 spheres, (c) ellipsoidal phantom with hot and cold regions, and (d) same phantom as in (c) with Poisson noise added. In each case the central slice of the phantom is shown. Note that the right column images have much lower intensity than in the left column.
Point source reconstructions: (a) SOE 1000 iterations and (b) SOE 50 000 iterations. The central slice of the image with 2 mm voxel size is presented (please note the difference in intensity scale).
Number of correctly assigned events versus iteration number. Ideally, in this case, the total number of events in the central voxel should be equal to 100 000, but this number is never reached.
Central slice through the reconstructed images of the 3D phantom (experiment B) obtained with (a) realistic resolution SOE 100 iterations, (b) realistic resolution SOE 2000 iterations, and (c) realistic resolution ML-EM 50 iterations. Images were scaled to the same maximum.
Central slice through the reconstructed images of the 3D phantom (experiment B) obtained with (a) perfect resolution SOE 100 iterations, (b) perfect resolution SOE 2000 iterations, and (c) perfect resolution ML-EM 50 iterations. Images were scaled to the same maximum.
Central slice through the reconstructed images of the 3D phantom (experiment C) obtained with (a) SOE 100 iterations, (b) SOE 2000 iterations, (c) SOE 2000 iterations, postfiltered with Gaussian of 2.5 mm FWHM, (d) ML-EM 50 iterations (not filtered), and (e) profiles drawn through the centers of the large hot and small cold spheres in images (c) and (d). All images are scaled to the same maximum.
Central slice through a reconstructed image of 3D phantom with noise, of counts [Fig. 5(d)]: (a) 100 SOE iterations, (b) 2000 SOE iterations, (c) 2000 SOE iterations, postfiltered with Gaussian of 3 mm FWHM, and (d) 50 ML-EM iterations (not filtered). All images are scaled to the same maximum.
(a) Bias versus variance plot comparing the first 50 iterations (in one iteration increments) of ML-EM and the first 5000 iterations (in 100 iterations increments) of SOE. Also shown are (b) the result of 50 iterations of ML-EM, (c) 5000 SOE iteration, and (d) same 5000 SOE iterations postsmoothed with Gaussian filter of 1.5 voxels (3 mm) FWHM. The images are with 2 mm voxels.
Average values (avg) and standard deviation (std) of the reconstructed intensity for the selected ROIs in the ellipsoidal phantom (experiment C).
Average values (avg) and standard deviation (std) of the reconstructed intensity for the selected ROIs in the ellipsoidal phantom with noise (experiment D).
NMSE values measured in % for the reconstructed phantoms.
The reconstructions times of the algorithms investigated in this paper.
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