Information content of data types in time-domain optical tomography

Abstract

The information content of data types in time-domain optical tomography is quantified by studying the detectability of signals in the attenuation and reduced scatter coefficients. Detection in both uniform and structured backgrounds is considered, and our results show a complex dependence of spatial detectability maps on the type of signal, data type, and background. In terms of the detectability of lesions, the mean time of arrival of photons and the total number of counts effectively summarize the information content of the full temporal waveform. A methodology for quantifying information content prior to reconstruction without assumptions of linearity is established, and the importance of signal and background characterization is highlighted.

Fig. 1

Schematic of the experimental setup. The 32 detectors and 32 sources are uniformly placed in the boundary of a circular domain with a radius of 50 mm. One source and 32 detectors are shown. The 10 detectors nearest to the source (shown with dotted lines) are not used in our calculations.

Fig. 2

Sample lumpy background (in μ_a [mm⁻¹]) and signal half way from the center to the boundary of the domain. The correlated structure of the background confounds the task of detecting the signal.

Fig. 3

Detectability of μ_a,+ signal in a flat background showing only a small angular dependence away from the boundary. The maxima in the boundary occur near the detectors. The arrow pointing into the domain shows the location of one of the sources, and the arrow pointing outward shows the location of a detector. As expected, the detectability is lower in the center of the domain.

Fig. 4

Detectability of μ_a,+ signal1 in a flat background. Note that the normalized waveform and τ plots overlap. For an attenuating inclusion in a flat background, the majority of the information is encoded by the total counts. Peak detectability occurs close to the boundary.

Fig. 5

Detectability of ${\mu }_{s,-}^{\prime }$ signal in a flat background. For a scattering inclusion the total counts contain the majority of the information, and peak detectability occurs at the boundary.

Fig. 6

Detectability of ${\mu }_{a,+},{\mu }_{s,-}^{\prime }$ signal in a flat background. For an inclusion that has an increase in attenuation and scatter, the mean time contains most of the information, and we see that the behavior near the boundary depends on the data type.

Fig. 7

Detectability for μ_a,+ lesion and the E data type in a flat background. Each curve represents exclusion of a different number of detectors at either side of each source. Note that using all detectors [fan(0)] and only excluding one [fan(1)] produces plots that lie on top of each other.

Fig. 8

Stability plot for four sets of 500 lumpy backgrounds each. We see that for the random backgrounds our detectability estimate is biased high for a small number of samples but converges as we increase our sample size.

Fig. 9

Detectability of μ_a,+ signal in a lumpy background. The randomness in the background reduces the detectability of the inclusions and affects the behavior close to the boundary. We see that the mean time has a lower but more uniform detectability than the total counts in the presence of random fluctuations in the background. The overall decrease in detectability for the mean time was less than that for the total counts when compared to the flat background.

Fig. 10

Histogram of standardized Hotelling test statistic. The approximate Gaussianity of the test statistic justifies using ${\mathrm{SNR}}_{\mathrm{Hot}}^2$ as our measure of detectability.