For the reconstructions shown in , the target was constructed of silicone rubber and shaped in the form of the letters “DOT” and “PENN”. In the first experiment, we placed the letters “DOT” one centimeter from the source plane and the letters “PENN” one centimeter from the detector plane, directly behind the “DOT” letters. The reconstruction is shown in . The letters are clearly visible. Note that the central slice from the middle of the tank is empty, as expected. In a second experiment we placed the letters “DOT” in the center of the tank (the letters “PENN” were not present) three centimeters from source and detector planes. The reconstruction is shown in . The letters are clearly reconstructed.
Fig. 1 Slices from three dimensional image reconstructions of the relative absorption coefficient
for targets suspended in a 6 cm thick slab filled with highly scattering fluid. The three slices shown for each reconstruction correspond to depths of 1 cm (left), (more ...)
In we show the data used to reconstruct the images. Only data corresponding to a single source beam position is shown. For each reconstruction we show, from left to right, the reference intensity I0 when the target is not present (scattering fluid only), the intensity I when the target is present, and the Rytov data, ϕ = −log(I/I0), which is used in the reconstruction algorithm (see Methods). Note that the letters cannot be identified by simply inspecting images of the transmitted light. Structure is visible in the Rytov data when the letters “PENN” are close to the detector plane (). However, when the letters “DOT” are in the center of the tank () their shape is completely blurred.
Fig. 2 Representative CCD data for the image reconstructions shown in . Each image corresponds to the measured light intensity for a single source beam position. The left column shows the reference intensity I0 when the target is not present (scattering (more ...)
In order to quantify the transverse resolution of the reconstructed images, we prepared several bar targets from the same material described above. The bars were 7 mm to 9 mm thick and placed consecutively (one at a time) in the center of the slab. shows the corresponding reconstructions. As the bar widths decrease, the modulation depth between bars decreases. As can be seen, all but the 7 mm bar target are well resolved. shows reconstructions for two experiments in which the 7 mm bar target was positioned 1 cm from the source and detector planes, respectively. The bars in this figure are well resolved and the images are smoother and have fewer artifacts. As can be expected, the image is better resolved when the target is closer to the detector plane. This is because the detectors are sampled on a finer grid than the sources. Experiments were also performed with bar targets having an absorption contrast of 2:1. The resulting images were very similar to those acquired with a 4:1 absorption contrast bar targets, but they contained less contrast and more noise.
Fig. 3 Reconstructed images of bar targets. Only slices drawn at the depth of the actual target are shown. (a) 7 mm to 9 mm bar targets located in the center of the tank. Here d denotes the width of the individual bars in the targets. (b) The 7 mm bar target (more ...)
The images in were reconstructed using approximately 107 measurements. The effect of changing the size of the data set was investigated by sampling the detectors on a grid with a step size of 2 mm, five times larger than the minimum experimentally available detector spacing. We found that increasing the number of detectors up to the experimentally available maximum did not improve image quality. That is, reconstructions performed using 2 mm source separations or with a denser sampling of CCD pixels were visually indistinguishable from those in . However, decreasing the number of data points does result in poorer image quality, as is illustrated in . From left to right, three separate reconstructions of the 8 mm bar target are shown. The data for these reconstructions were taken from a single experiment with the target positioned in the center of the tank. All reconstruction parameters are kept constant, except for the number of measurements used. The reconstruction on the left uses 8 × 106 measurements with sources and detectors sampled with 4 mm and 2 mm steps, respectively. In the center reconstruction, we use 4 mm spacing for both the sources and detectors, which corresponds to 2 × 106 data points. In the reconstruction on the right, the sampling is 8 mm for sources and 4 mm for detectors, or, approximately, 5 × 105 data points. It can be seen that as the number of measurements decreases, image artifacts become more prominent and resolution is lost. We note that the optimal number measurements for a given experiment will vary depending on factors such as experimental geometry and noise level. For example, smaller/larger data sets would be optimal if the reconstruction field of view was smaller/larger.
Fig. 4 Images of the 8 mm bar target from a single experiment. All reconstruction parameters are held fixed except for the number of measurements used. From left to right, correspond to N = 8 × 106, 2 × 106, and 5 × 105 measurements were (more ...)
To investigate the capability for quantitative reconstruction of the absorption coefficient, we have performed a titration experiment. A clear plastic cylinder was positioned in the center of the tank, and laboratory tubing was used to flow fluid through the cylinder. During the first scan, the cylinder contained scattering fluid identical to the fluid in the tank. Twelve titrations were then performed in which the ratio of ink concentration in the cylinder, to that in the tank, gave an expected absorption contrast ranging from 2:1 to 64:1. The absorption contrast, i.e., the ratio of the absorption coefficient in the cylinder to that in the surrounding fluid, was taken from the corresponding reconstructed value for a single voxel located inside the cylinder. This voxel was chosen to be the voxel with the maximum reconstructed value of the absorption coefficient for the tenth titration. In , the reconstructed contrast
is plotted against the expected contrast. It can be seen that the absorption is quantitatively reconstructed with a linear dependence on ink concentration over nearly a decade in absorption contrast. Deviation from linearity occurs at higher concentrations, as expected.
Fig. 5 Reconstructed contrast of the absorption coefficient
between the cylinder and the tank vs. the expected contrast.