Traditionally, the histogram had been used to provide a visual clue for the general shape of the probability density function (pdf) [25]. For example, in multivariate density estimation, the following assumption has been used extensively throughout the literature: the observed histogram of any image is the summation of histograms from multiple underlying objects “hidden” in the observed histogram. Objects in the example histogram, shown in Fig. 3, can be approximated by the location of the valleys between peaks, with some inherit uncertainty in the areas of overlap [24]. Based on this, our proposed method assumes that a peak in the histogram corresponds to a relatively more homogeneous region in the image; it is very likely that a peak involves only one class. The justification behind this assumption is that the histogram of objects, in medical images, are typically thought of as the summation of Gaussian curves, which implies a peak corresponds to a homogeneous region in the image(s).

Due to the nature of medical images, histograms tend to be very noisy with large variability. This makes the optimal threshold selection for separating objects of interest burdensome. First, the histogram of the image needs to be estimated in a robust fashion such that an estimated histogram is less sensitive to local peculiarities in the image data. Second, the estimated histogram should be more sensitive to the clustering of sample values such that data clumping in certain regions and data sparseness in others–particularly the tails of the histogram–should be locally smoothed. To avoid all these problems and provide reliable signatures about objects within the images, herein we propose a framework for smoothing the histogram of PET images through diffusion-based KDE [26]. KDE via diffusion deals well with boundary bias and is much more robust for small sample sizes, as compared to traditional KDE. We detail the steps of the KDE as follows.

1) Traditional KDE uses the Gaussian kernel density estimator [26], but it lacks local adaptation; therefore, it is sensitive to outliers. To improve local adaptation, an adaptive KDE was created in [26] based on the smoothing properties of linear diffusion processes. The kernel was viewed as the transition density of a diffusion process, hence named as KDE via diffusion. For KDE, given N independent realizations, X_{u∈{1,…,N}}, the Gaussian kernel density estimator is conventionally defined as (1)g(x;t)=1N∑u=1Nϕ(x,Xu;t),x∈R where ϕ(x,Xu;t)=12πte−(x−Xu)2∕(2t) is a Gaussian pdf at scale t, usually referred to as the bandwidth. An improved kernel via diffusion process was constructed by solving the following diffusion equation with the Neumann boundary condition [26]: (2)gdiff(x,Xu;t)=∑z=−∞∞(ϕ(x,2z+Xu;t)+ϕ(x,2z−Xu;t))x∈[0,1].

2) After KDE via diffusion, an exponential smoothing was applied to further reduce the noise; the essential shape of the histogram was preserved throughout this process (see Fig. 4 for an example smooth KDE conducted on the PET image histogram). Data clumping and sparseness in the original histogram (left) were removed (middle), and any noise remaining after KDE via diffusion was reduced (right) considerably while still preserving the shape of the pdf. The resultant histogram can now serve as a proficient platform for the segmentation of the objects, as long as an effective clustering approach can locate the valleys in the histogram.

Affinity propagation was first proposed by Frey and Dueck [27] for partitioning datasets into clusters, based on the similarities between data points. AP is useful because it is efficient, insensitive to initialization, and produces clusters at a low error rate. Basically, AP partitions the data based on the maximization of the sum of similarities between data points such that each partition is associated with its exemplar (namely its most prototypical data point) [28]. Unlike other exemplarbased clustering methods such as k-centers clustering [21] and k-means [29], performance of AP does not rely on a “good” initial cluster/group. Instead, AP obtains accurate solutions by approximating the NP-hard problems in a much more efficient and accurate way [27], [28]. AP can use arbitrarily complex affinity functions since it does not need to search or integrate over a parameter space. Due to the flexibility of the AP method regarding the affinity function definition, we explored a novel affinity function that best suited PET image segmentation effectively, where the radiotracer uptake regions were distributed widespread over the lungs.

Ranging from preinfection to 38 weeks postinfection (0, 5, 10, 15, 20, 30, and 38 weeks), 70 PET scans from 10 rabbits were evaluated. The rabbits were aerosol infected with Mycobacterium TB (H37Rv strain) in a Madison chamber, implanting 3 × 10³ bacilli into the lungs. The rabbits were injected via the marginal ear vein, with 1-2 mCi of 18F-FDG PET and imaged 45-min postinjection with 30 min static PET acquisition. All CT and PET imaging was performed without respiratory gating on the NeuroLogica CereTom and the Philips Mosaic HP scanner, respectively.

Evaluation of the segmentation was conducted on a subset of randomly selected slices among different rabbits and time points since manual segmentation of the diffuse uptake regions of all the images was too laborious and time consuming. The random selection was done in order to minimize any slice selection bias and introduce a representative segmentation evaluation framework. A total of 168 slices were selected to be used in the TB metabolic volume quantification. In this process, two experts manually segmented the significant uptake regions using 2-D manual thresholding, namely the current clinical standard for spatially diffuse uptake. Necessary manual correction was also conducted. In order to further assess our method’s performance on various lesion sizes, the lesions were split into three groups (i.e., small, medium, and large) according to their areas. These cutoffs were carefully selected to ensure that the same number of lesions was in each group, and the cutoffs were similar to the ones found in the literature [35]. The main premise behind this approach was to show that our method was not overestimating smaller lesions, as opposed to the state-of-the-art methods, and not underestimating large uptake regions due to the diffuse nature of the uptake. The groups were defined as small (0 – 3.45 cm²), medium (3.45 – 6.84 cm²), and large (6.84 – 30.67 cm²), with 56 lesions per group. Fig. 6 summarizes the distribution of lesion areas and the group cutoffs selected.

The Dice Similarity Coefficient (DSC)–a coefficient that measures the overlap between two segmentation results–and the “sensitivity (TPVF),” and “specificity (100-FPVF)” (i.e., TPVF: true positive volume fraction, FPVF: false positive volume fraction) were calculated between the segmented region that was found by the proposed method and the expert delineations. Table I lists all evaluation results with above mentioned measures. The average DSC was found to be 91.25 ± 8.01% for all lesions when comparing the proposed method to the segmentation result using the average of the two expert defined delineations. A sensitivity of 88.80 ± 12.59% and a specificity of 96.01 ± 9.20% were achieved. Overall, the DSC, sensitivity, and specificity rates increased with larger lesion area, as expected.

Fig. 7 shows four linear regression graphs for segmentation evaluation of the proposed method with respect to the observers. An inter-observer agreement between observers was also reported in the same figure. The correlation between the proposed method versus the average between the observers was reported as R² = 0.906, (p < 0.01). Similarly, Bland–Altman plots were constructed in Fig. 8 to show an excellent agreement between the proposed method and expert delineations, where outliers are the points outside the 95% confidence interval (i.e., dotted lines).

For a qualitative comparison, the segmentation results from 12 example PET slices, all from infected rabbit lungs, are shown in Fig. 9. Columns A and D show the original PET image of the lung, while columns B and E show the boundary information of the various groups on the original PET image. Columns C and F color each group for a better visualization, and it can be readily seen that each group is homogeneous to some extent.

We compared our approach with the commonly used PET image segmentation techniques: RG [36], k-means, various adaptive thresholding-based methods, as well as the Region of interest Visualization, Evaluation, and image Registration (ROVER) software (ABX GmbH, Radeberg, Germany). Fig. 10 shows the accuracy comparison of segmentation results with these state-of-the-art methods. Our proposed method had the highest DSC, sensitivity, and specificity, while supervised k-means performed the second best.

For the thresholding-based methods, we first compared them to the current clinical standard for thresholding malignant tumors on PET images as SUV_max > 2.5 [5], which performed poorly for the small animal TB model. Two other thresholding methods were also compared to the proposed method: Otsu Thresholding [11], 73.27 ± 19.42%, and the Iterative Thresholding Method (ITM) [10], 40.76 ± 23.01%. While the Otsu thresholding method performed similar to the RG method, ITM and SUV-based fixed thresholding methods did not perform well due to the diffuse and multifocal uptake nature of the uptake regions. Our proposed method consistently resulted in higher DSC values than the thresholding-based methods.

ROVER uses an adaptive thresholding and background subtraction algorithm to identify lesion volume within a mask and is already in clinical use in Europe [37]. It performed very similar to the Otsu thresholding method with a DSC of 73.14 ± 14.86%. This software is designed for 3-D ROI analysis and volume determination. We found that in many cases the ROVER automatic segmentation agreed with the proposed method, but in some cases it included the areas of the PET images that the expert delineations had previously determined as nonsignificant uptake. For diffuse uptake, it is a very difficult task to include only the significant uptake and to disregard the other areas which is needed for proper quantification of the disease severity in small animal models. We conclude that ROVER may not be suited for the segmentation of small animal TB models as the supervised k-means outperformed it.

Finally, k-means, a commonly used clustering method [38], [39], was also compared to the proposed method. A major drawback of k-means and most clustering methods is that they require the user to specify the number of groups to cluster, which is traditionally two, i.e., a foreground and background. However, given the diffuse and multifocal nature of the TB images, an expert manually specified the number of groups based on the appearance of the images instead of just using the traditional two groups, and found a DSC of 80.40 ± 15.31% (see Fig. 10). Nevertheless, the overall performance of supervised k-means was inferior to our proposed method. Fig. 11 gives a qualitative comparison of the state-of-the-art PET image segmentation methods that were compared to, excluding the SUV > 2.5 thresholding.

In order to further assess how robust the determined thresholding levels were, we conducted additional experiments by altering the thresholding values using the proposed method through a percent change (±0, 2, 4, 6, 8, 10%) and exploring how DSC rates were being affected with respect to the given reference truths. Based on each altered threshold level, we resegmented the PET images and recalculated the DSC values. Results are summarized in Fig. 12. As can be depicted from the results, a small change in the thresholding level (around ±2%) decreased the DSC values; however, these changes can be regarded as non-significant. More than ±2% changes in the threholding levels can cause further decrease in the segmentation results due to high variation in the diffuse uptake regions. Together with the results obtained from robustness experiments, and considering the high inter-observer variation when determining near-optimal thresholding, it can be concluded that the automatically determined threshold levels found through the proposed method gives the best possible threshold levels for grouping diffuse uptake regions.

When evaluating segmentation algorithms, it is often desired to have ground truths instead of surrogate truths. In PET imaging, various phantoms were designed for this purpose; however, most of such phantoms are not realistic due to inaccurate noise assumptions and limited spatial resolutions of the PET scans. Another shortcoming is that digital phantoms include spherical lesions (due to cylindirical set up in CT base) and they do not reflect the underlying uptake pattern commonly seen in pulmonary infections (multifocal and diffuse). Nevertheless, we evaluated the efficacy of our proposed algorithm by using IEC image quality phantoms (NEMA standard) [40] to demonstrate its feasibility when the ground truth is known. The IEC phantoms contained six spherical lesions of size 10, 13, 17, 22, 28, 37 mm in diameter with two different signal to background ratios (SBRs) (i.e., 4:1 and 8:1) and the voxel size were 2 mm × 2 mm × 2 mm. Resulting PET segmentations by the proposed algorithm were compared with the ground truth, which was simulated from CT, and the following dice scores were obtained: 90.6 ± 2.9% for SBR = 4:1, and 96.1 ± 3.2% for SBR = 8:1. Further details of the phantoms and their appearance can be found in [23] and [40]. As pointed out earlier, we believe that the use of the phantoms with spherical uptake realizations (i.e., focal uptake) is suboptimal for evaluating our proposed method’s success in non-spherical uptake realizations. This limitation is revisited in the discussion section.

The proposed affinity function has two weighting parameters n and m (see equation 5). We set these parameters based on histograms of the PET images used in a training step (training images were not used in testing). Segmentation evaluations presented in the quantitative and qualitative evaluation section were based on the optimal parameters we found in this step. In order to demonstrate how the novel affinity function behaved for different n and m values, we conducted additional experiments using various parameter pairs. For simplicity, m was selected to be an integer between 1 and 4 to reflect the contribution of geodesic distance-based similarities, while n was ranged from 0 to 3. Results without contribution of the geodesic distance-based similarity metric (i.e., when m = 0) are discussed in the next section.

Different combinations of these constraints are possible; however, due to the fast convergence of power series and the intuitive meaning of the local Euclidean distances, we prefer to retain the combined affinity formulation in linear form. Note that with the current implementation of the affinities, the similarity function is still symmetric and elements of the similarity matrix are kept smooth with appropriately chosen weight parameters n and m. For different combinations of these parameters, PET images were resegmented and the resultant DSC rates are given in Fig. 13. We observed the best performance when n = 3 and m = 1; however, it would be different for other imaging datasets such as human patients. For m > 2, the variability between different n values was small, most likely because the geodesic term in the novel affinity function was much greater and had a stronger effect on the affinity value than the gradient term for these parameterizations.

Fig. 14 gives a qualitative view on the strength of affinities in an example PET image using the proposed affinity function with the original image shown in the top right. After the histogram was estimated using the framework described previously, three example intensity levels were selected (named high, medium, and low) and the voxels with these intensities (specified by red dots overlaid onto the original image) are shown in the top row of Fig. 14. For each of the three example intensity levels, the affinity to all other voxels on the image was calculated. The strength of the affinities, shown in the bottom row, gives an intuition on how the affinity function behaves in a qualitatively manner (i.e., red represents a stronger affinity strength, while blue represents a weaker strength). Finally, the affinities between all the intensities, not just the three example intensities, were calculated and then clustered using AP. The resulting segmentation is given on the bottom right.

At any iteration of the AP clustering, the current exemplars and clusters can be determined and can shed light on how the algorithm moves toward convergence. For this, we examined the segmentation result at several iterations and the results are demonstrated in Fig. 15. The top row shows the current segmentation in color coded groups, while the bottom row gives the segmentation boundary on the original PET image. Each column of Fig. 15 is a different iteration (in increasing order) of the AP clustering using the proposed affinity calculation. The last column shows the final segmentation result obtained when iterations are converged to a steady state.

We also investigated different ways of computing the affinity functions within the AP clustering that may serve as potential similarity metrics for delineating multifocal and diffuse uptake regions. For example, by taking into account only the geodesic distance between the data along the histogram, a Gaussian distance function can be used to define the affinity between the points as (9)s(i,j)=−exp(−dijGσ2) where σ is a constant showing variation. Here, we called this affinity construction the exponential distance function. The main advantage of this affinity calculation is the ability to vary σ along the curve to influence the size of the object found. Potentially, this method could correctly group a histogram with a very large variation in the group size. However, as noted earlier, the major drawback was how to adapt the σ value within a histogram to account for the local variations in group sizing. The approximate location of these peaks must be known beforehand and due to the significant variation and noisy nature of PET image histograms, (9) may not be a very robust metric. Therefore, to give an approximate performance of this Gaussian distance function as an affinity function, we chose a constant σ for all the images and tested the segmentation accuracy. In addition, we also evaluated the performance using the Euclidean distance between the data points as the affinity function as well as using only the geodesic distance. The results of these experiments are provided in Fig. 16. For the four definitions of the affinities, the proposed metric outperformed the others with an average DSC value of 91.25 ± 8.01%. Notably, utilizing the Euclidean distance as the affinity function segmented with an accuracy of 50.76 ± 16.96%, whereas the sole use of Geodesic distance resulted in segmentation accuracy of 78.57 ± 15.89%. Compared to Euclidean and Geodesic distance, the exponential distance had an accuracy of 68.77 ± 15.02%. Our experiments validated that the use of the proposed affinity function was well suited for diffuse and multifocal uptake regions in PET images compared to other possible affinity functions commonly used in the literature for different applications.

The parameters for AP were set as follows: in the AP algorithm, each message is set to λ times its value from the previous iteration plus 1 – λ times its prescribed updated value, where the damping factor λ is between 0 and 1. In the original AP formulation [27], the damping factor was set to 0.5; however, in our PET image segmentation experiments, it was sufficient in all cases when the damping factor was set to λ = 0.8. For the convergence of the AP clustering, the maximum number of iteration was set to t_max = 500, and the average iteration number for the convergence was found to be less than 100 in all cases. The threshold level for the diffusion-based KDE method was 10%, while a window size of 20 was used for the exponential smoothing. The proposed method can be run either in 2-D or 3-D. It took 0.66 seconds per slice (2-D computation) and around 1 min for 3-D computation on an Intel workstation with 24-GB memory running at 3.10 GHz. The computation times of the state-of-the-art methods that we compared to our proposed method were similar. After the proposed method, the fastest methods were the Otsu thresholding and Iterative thresholding which on average took 1.3 s and 1.5 s per slice, respectively. The k-means took 4.5 s, while the ROVER software took 7.5 s per slice. Finally, the region growing took about 12 s per slice due to the need for manual seed selection on all the multifocal uptake on the testing images.

Our proposed method can also be used in a semi-supervised manner, allowing a user to select certain image intensities as preferred exemplars. This interaction may be useful because it can force (or influence) the algorithm to form additional local groupings or refine the extent of the local groups to further enhance the delineation result. However, there is a drawback of a slightly longer delineation time, which is mostly consisting of waiting for the user to define the seeds, and the user input adds some inherent variation in the segmentation accuracy. Additional constraints from a user defined seed governs an allowed set of solutions in AP.

For semi-supervised AP, availability and responsibility formulations do not change but the affinity function does. Assume that data point i is similar to j and data point q is similar to t. If j and q must be in the same cluster and a user incorporates this into the proposed framework as an instance constraint, then the similarity between i and t in the original AP formulation alters from s(i, t) < s(i, j) + s(q, t) into s^(i,t)=s(i,j)+s(q,t). Finally, the updated affinity s^(i,t) is used in responsibility formulation as r(i,k)←s^(i,k)−maxk′{k′≠k}{a(i,k′)+s^(i,k′)} and in (4) until convergence.