Image quality improvement via spatial deconvolution in optical tomography: time-series imaging

Yong Xu, Yaling Pei, Harry L. Graber, Randall L. Barbour
2005 Journal of Biomedical Optics  
Image quality is one of key factors that determines the practicality of an imaging scheme. Experience with diffuse optical tomography (DOT) research and applications has indicated that most image reconstruction algorithms yield blurred images because localized information from the object domain is mapped to more than one position in the image domain. To reduce the blurring in reconstructed images and improve image quality, as measured by parameters such as quantitative spatial and temporal
more » ... l and temporal accuracy of recovered optical coefficients, a linear deconvolution strategy was proposed [1, 2] . An illustration depicting this strategy is shown in Figure 1 . As shown in the figure, the function of the deconvolution operator/filter is to reduce the mixing of information and to make the recovered image as nearly as possible a one-to-one correspondence between object and image pixels. The original idea of the deconvolution scheme is to borrow the concept of frequency encoding of spatial information from MR imaging and to use this strategy to label information that is "transferred" from the object to image space [3] . As discussed below, in practice we find that the method works best when applied in time domain directly, rather than in the frequency domain [1] . In this report, we continue our investigation of image quality improvement via the spatial deconvolution scheme in DOT. In contrast to our provious work [2, 4] , in which we have demonstrated that the deconvolution method brings about substantial qualitative improvement in spatial resolution and spatial accuracy for 2D[4] and 3D[2] static images reconstructed from steady-state (cw) DOT measurement data, we now investigate the effect of the spatial deconvolution method on the dynamical features of time-series images [5, 6] in dense-scattering media by quantitative assessments of spatial and temporal accuracy of the reconstructed images. METHODS Spatial Deconvolution Algorithm The reasoning that underlay our linear deconvolution strategy, and the mathematical details of its implementation, are given in Refs. 1, 2 and 4. Here, we only briefly introduce the method in an intuitive way which is compared to the procedure of calibration of a measurement system. discretized by an n-node mesh for numerical computations. Taking the known distribution as the input for the imaging system, we can obtain the reconstructed distribution [X r (r)] = [x r1 , x r2 ,...,x rn ] T . So calibration of the imaging system can be performed by computing [X 0 (r)] = [F] [X r (r)], where the calibration coefficient [F] is an n×n matrix and is called deconvolution operator or image-correcting filter. In practice, the basic steps to generate an image-correcting filter are as follows: (1) Generate N independently known optical coefficient distributions by computer: where N≥n; (2) Use the forward model [7, 8] to simulate the detector readings from the known distributions; (3) Reconstruct the optical coefficient distributions from the simulated detector readings by use of the inverse model: (4) Solve the matrix equation to determine the image-correcting filter: Finally, any image [Y r (r)] that is recovered using the same numerical mesh and sourcedetector geometry as used in the generation of filter [F] can be corrected by computing the matrix product The test medium geometry and source-detector configuration used for the filter generation and image reconstructions that are reported here are shown in Figure 3 . For all computations considered in this report, the absorption coefficient of the test medium's background is µ a =0.06 cm -1 , and the medium has spatially homogeneous and temporally invariant scattering, with µ s =10 cm -1 . Dynamic features of inclusions To explore dynamic characteristics of time-series images under deconvolution operation, as shown in Figure 4 , the following four time-varying functions are assigned to the absorption coefficients of the test medium's inclusions: (a) sinusoidal time series: (1) (b) amplitude-modulated time series: (2) (c) constant-amplitude time series with time-dependent frequency: (3) (d) time series with time-dependent frequency and amplitude modulation: Where parameters µ a0 =0.12 cm -1 , ∆µ a =0.024 cm-1, f 0 =0.1 Hz, f a =0.03 Hz, f m =0.03 Hz, φ 0 =0, φ a =0 and φ m =0 have been used for the calculation of time-series curves in Figure 4 . 3D Detector Noise Model In most demonstrated cases of this report the white Gaussian noise is added to simulated detector readings for investigating the robustness of our deconvolution method. The noiseto-signal ratio of our 3D detector noise model can be expressed by [9] (5) where d ij is the distance between the i-th source and the j-th detector; W is the maximal distance between sources and detectors, i.e. W=max(d ij ); K 0 is the noise-to-signal ratio at the co-located point of source and detector; and K w stands for the noise-to-signal ratio when the distance between source and detector equals W. This noise model is in agreement with usual experimental and clinical expeirence. To quantitatively analyze the effect of noise on spatial and temporal accuracy of reconstructed images, we, here, define six noise levels: Level 1: K 0 =0.5% and K w =5%; Level 2: K 0 =1% and K w =10%; Level 3: K 0 =2% and K w =20%; Level 4: K 0 =3% and K w =30%; Level 5: K 0 =4% and K w =40%; Level 6: K 0 =5% and K w =50%. Quantitative Assessments of Spatial and Temporal Accuracy In this report, we select the spatial and temporal correlations between target medium and reconstructed images as the measurements of spatial and temporal accuracy of recovered images, respectively, for whose numerical values can be precisely evaluated [9] . The spatial correlation is defined as (6) Where is accurate values, is reconstructed values, and are the mean values of u and v , and and are their standard deviations. The sum runs over all (N d ) mesh nodes. The temporal correlation is defined as (7) Where is accurate values, is reconstructed values, and the sum runs over all (N t ) time points. RESULTS Qualitative and quantitative assessments of the effectiveness of the linear deconvolution method when applied to time series of images are presented in this section. In the first example, in which noise-free data were used, a comparison between convolved and deconvolved images, for selected time frames within the image sequence, are shown in Figure 6 . These results demonstrate that the spatial accuracy of the images is markedly improved by use of the linear deconvolution correction, and that there is no comcomitant degradation of temporal information. An important, logical next step is to determine the effect of noise in the detector data on the spatial and temporal accuracy. Figure 7 shows a case with added noise, in which the level-2 white Gaussian noise is added to detector measurements. Comparing Figures 7(a) and 7(b) , it can be seen that using deconvolution the spatial accuracy of time-series images is improved, but the temporal accuracy of the images is degraded due to the additive detector noise. However, when a simple temporal low-pass filter is used to denoise the deconvolved time-series images, the quality of the images is additionally enhanced, as shown in Figure 7 (c). In next three cases, we have investigated three simple denoising techniques: temporal low-pass filtering, spatial low-pass filtering and optimizing regularization factors. The corresponding results of reconstructed images are presented in Figures 8, 9 and 10, respectively. These results show that the three simple denoising methods can all enhance the performance of deconvolution. To quantitatively assess the spatial and temporal accuracy of reconstructed time-series images, we make use of definitions (6) and (7) to calculate the spatial and temporal correlations of reconstructed images under different conditions. The contrast dependence of spatial correlations of recovered images with two different noise levels is shown in Figure 11 . Figure 12 gives the noise dependence of spatial correlations of reconstructed results. The quantitative results indicate that the spatial accuracy is clearly improved by deconvolution, even for high noise levels. The amplitude dependence and noise dependence of temporal correlations of reconstructed time-series images are plotted in Figures 13 and 14 , respectively. Figures 13 and 14 show that the temporal accuracy increases with the increase in amplitude of time series and is degraded by the deconvolution procedure. Finally, the comparisons of temporal correlations of time-series images between four different dynamic features of inclusions are listed in Table 1 , which clearly indicates that simple time series is easier to recover than complex time series. CONCLUSIONS In this report, we have investigated effectiveness of the linear deconvolution method applied to reconstructed time-series images. The qualitative and quantitative results show that (1) For noise-free or low noise level (<0.5%) data, both spatial and temporal accuracy of time-series images are markedly improved by the deconvolution method; (2) Simple time-series features (e.g. sinusoidal) are easier to recover than complex time-series features (e.g. modulation of frequency); (3) For noisy data, deconvolution procedures can significantly improve the spatial accuracy of time series images but degrade the temporal accuracy; (4) Denoising methods (even simple techniques) can enhance the performance of the deconvolution method; (5) Combined with a temporal low-pass filter, satisfactory spatial and temporal accuracy (>60%) can be achieved by use of the deconvolution method at an experimental noise level (K 0 =1% and K w =10%). , "Signal source separation and localization in the analysis of dynamic near-infrared optical tomographic time series," in
doi:10.1117/1.2103747 pmid:16292953 fatcat:7xbdopvum5d3rcdcs4hwfdvwte