2. Methodologies
3.1 The MicroPET Scanner
The microPET R4 scanner in Taipei Veterans General Hospital is shown in Figure 3.1-1. The configuration of this microPET R4 has 32 rings, 6144 detectors, 7.3 cm field-of-view (FOV), and spatial resolution of 1.85 mm in the center. It can collect both prompt and delay sinograms. Transaxial projection bin size was 1.213 mm, and axial slice thickness was 1.2115 mm. Coincidence timing window was set at 6×10-9 seconds. The lower and upper level energy thresholds were 350 and 750 keV, respectively. Span of the data set was 3 (Appendix B.5), and maximum ring difference (MRD) of the data set was 31 (Appendix B.6). The target images were reconstructed using 128×128 pixels.
Fig. 3.1-1: The microPET R4 scanner in Taipei Veterans General Hospital is displayed.
A sinogram uses the polar coordinate system to store the response of a projection line (or a line of response, LOR) at a specific orientation with a radial distance from the FOV central axis as displayed in Figure 3.1-2.
Horizontal experiment Sinogram Oblique experiment Sinogram
Fig. 3.1-2: Two images of microPET sinograms from physical experiments of six point sources are displayed. In every sonogram, the horizontal axis represents the ordering of line of responses (LORs) and the vertical axis represents different angles from 0 degree to 180 degree. The pixel intensity in the sinogram records total gamma rays detected in a scanning time window.
During scan acquisition, raw data are stored in sinograms, which are then used to reconstruct images. The prompt sinogram records coincidence events when two detectors receive two gamma rays within a specific time window (e.g., 6×10-9 second).
The coincidence events in prompt sinograms include true, random, and scatter coincidence events. The delay sinogram records coincidence events when two detectors receive two gamma rays outside another specific time window (e.g., 3×6×10-9 second). The coincidence events in delay sinograms can be used to estimate random coincidence events.
The data matrices are described as follows (as in Fig. 3.1-3 and 3.1-4). First, list mode data were histogramed into the 3D data with a span of 3 and MRD of 31, which are sized 2×703×96×84 (that is, 2 sinograms of prompt and delay windows ×703 slices × 96 angular views × 84 projection lines (LORs) as in Fig. 3.1-3) and stored as floating type data. The second data were obtained using random pre-correction and were sized 1×703×96×84 (as in Fig. 3.1-4). These 3D data were rebinned into 2D sinograms using the FORE method with dead time and decay corrections. The attenuation, normalization, scattering, and arc corrections were not performed for simplicity as this study only focused on evaluation of random correction. These further corrections for PDEM reconstruction will need more investigation in future studies.
Two matrices were constructed using the software embedded in microPET R4 (that is, microPET manager V1.6.4). The first matrix was 2×63×96×84 (that is, 2 sinograms of prompt and delay windows × 63 slices × 96 angular views × 84 projection lines (LORs) as in Fig. 3.1-3) and stored as floating type data. This matrix was reconstructed using the PDEM. The PDEM was compared with the built-in reconstruction schemes, such as 2D FBP and OSEM methods, in the microPET R4 system. The second matrix was obtained using the FORE and on-line random pre-corrected data (as in Fig. 3.1-4). The 2D OSEM, using 16 subsets with four iterations, and the 2D FBP, using ramp filtering, were applied to reconstruct the microPET images for comparison with the PDEM results. The reconstructed images were not smoothened.
Fig. 3.1-3: The above data matrices are used for reconstruction by the PDEM and segmentation by the GMM.
Fig. 3.1-4 The above data matrices are used for reconstruction by the FBP and OSEM with random pre-correction.
3.2 Reconstruction with Random Correction
This section shows the reconstruction results obtained from the PDEM, FBP and OSEM on simulation, phantoms, and real mouse microPET data.
3.2.1 Simulation Study
This study utilized the modified Shepp-Logan’s head phantom as the simulated object to assess and compare the reconstruction images using the PDEM, FORE+FBP, and FORE+OSEM. We assumed that the simulated diameter of a ring was 28.28 mm and the FOV diameter was 20 mm. Target image was 128×128 pixels (20×20 mm2) and rescaled intensity was 100. For each pixel intensity, (≤ λt(b)) was simulated and then input into λ*t(d). Then, λ*r(d) is set to the multiplication of a given noise
ratio to λ*t(d). At the end, n and *p n can be simulated using the Poisson d*
distribution with parameters λ*t(d) and λ*r(d) as in Eqs. (1) and (2). Total counts (sum of prompt and delayed counts) were 276794, 316383 and 342407 with 5%, 10%
and 30% noise levels, respectively, for the three slices simulated. The prompt and delayed sinograms had the same matrix size of 96×84 (that is, 96 angular views × 84 projection lines (LORs)) with floating type data. Three random noise ratios of random to true coincidence counts, 5%, 10% and 30% were simulated. The quality of images obtained using the PDEM, FBP and OSEM were compared (Fig. 3.2.1-1). The simulated results demonstrate that the quality of images reconstructed by the PDEM is superior to that of images reconstructed by the FORE+FBP and FORE+OSEM.
Fig. 3.2.1-1. The modified Shepp-Logan head image was used for simulation studies. The line profiles of PDEM were less noisy than those of FBP and OSEM.
The PDEM technique reconstructed better images than the FBP and OSEM did with 5%, 10% and 30% random noise. All images were rescaled using their own maximum values.
3.2.2 Phantom Study
The first physical phantom was 28 homogenous line-sources with an outer diameter of 1.27 mm for each line. This phantom was utilized to assess the performance and accuracy of reconstruction quality between the FBP, OSEM and PDEM (as in Fig.
3.2.2-1). The spatial resolution was measured using the FWHMs from vertical and horizontal line profiles (in Table 3.2.2-1). The average and standard deviation of FWHMs in reconstruction images using the PDEM were smaller than those obtained by the FBP and OSEM.
Fig. 3.2.2-1. The 31st slice of the 28 line-source phantom reconstructed using the three methods are displayed. Both FBP and OSEM are reconstruction methods built into the microPET R4 system. All images were rescaled using their own maximum values. The images are shown in the rectangular window. Table 3.2.2-1 presents the comparisons of their FWHMs.
Table 3.2.2-1: Average and standard deviation (in mm) of 28 FWHMs for horizontal and vertical line profiles measured for comparing the spatial resolutions of PDEM, OSEM and FBP. Those values are measured at the 31st slice.
Horizontal profile Vertical profile Methods Average Standard deviation Average Standard deviation
PDEM 1.779 0.324 1.790 0.311
OSEM 1.890 0.527 1.863 0.548
FBP 3.641 0.595 3.663 0.624
The second phantom was a uniform cylinder of 7.6 cm high with an inner radius of 20 mm. This phantom was also utilized to compare image quality obtained using the FBP, OSEM and PDEM. Imaging scan time was 1200 s using the microPET R4 after injection of 276 μCi F-18 FDG. Three reconstruction techniques were applied to reconstruct the 40th slice (in Fig. 3.2.2-2). Reconstruction images were presented with the associated central line profiles. Reconstruction images obtaining using the PDEM had better quality than those generated by the FBP and OSEM on their line profiles. A circular region of interest (ROI) was employed to measure the noise level for the different reconstruction methods. The lowest value for coefficient of variation (CV), which is the ratio of standard deviation to mean, was obtained by using the PDEM reconstruction (in Table 3.2.2-2).
Fig. 3.2.2-2. The reconstructed 40th slice from a uniform phantom was used to investigate noise level generated by the three approaches. The white line indicates the position of the investigated line profile. All images were rescaled using their own maximum values. The images are shown in the rectangular window with enlarged central parts. Table 3.2.2-2 presents the comparisons of their CVs.
Table 3.2.2-2: A circular ROI with a radius of 9 pixels to the center of the uniform phantom was utilized to compare noise levels between the PDEM, FBP and OSEM.
Those values were measured at the 40th reconstructed slice.
PDEM FBP OSEM
Average 1146.51 1107.66 1105.36
Standard deviation 36.26 46.82 65.56
Coefficient of variation (%) 3.16 4.23 5.93
The PDEM reconstructed better quality images with lower noise levels than the reconstructed approaches built into the microPET R4 system during investigations of line and uniform phantoms. Notably, in all reconstruction processing, there was no attenuation, scatter, normalization, or arc correction. However, dead time and decay correction were applied when rebinning 3D sinograms into 2D data.
3.2.3 Real Mouse Study
The PDEM method was applied to real data for small mice to compare the quality of reconstructed images with those reconstructed using the FBP and OSEM. These two real normal mice weighed 20 g. Imaging scan time was 600 second using the microPET R4 following an injection of 0.226 μCi F-18 FDG for the first mouse and an injection of 240.5 μCi F-18 FDG for the second mouse. The first mouse was utilized to investigate reconstruction performance under a weak amount of F-18 FDG activity. The second mouse was used to investigate image quality under a normal amount of F-18 FDG.
All 63 slices after the FORE were reconstructed using the PDEM, FBP and OSEM.
Figures 3.2.3-1 and 3.2.3-2 present coronal and sagittal images of the two mice reconstructed by the PDEM. These images are less noisy and have clearer boundaries than those reconstructed by the FBP and OSEM. These results demonstrate that the PDEM reconstructed images with better contrast and clearer boundaries than those reconstructed with the FBP and OSEM.
Fig. 3.2.3-1: Sagittal (top) and coronal (middle) images of the first mouse image reconstructed by PDEM (left), FBP (middle) and OSEM (right). The images reconstructed by PDEM have less noise than those reconstructed using FBP and OSEM with comparison by line profiles. The images are shown in the rectangular window with enlarged central parts.
Fig. 3.2.3-2. Coronal and sagittal images of the second mouse image reconstructed using PDEM (left), FBP (middle) and OSEM (right). The images reconstructed by PDEM have less noise than those reconstructed by FBP and OSEM, as shown in the respective line profile near the heart. The images are shown in the rectangular window with enlarged central parts.
3.3 Segmentation of 3D MicroPET Images
This section introduces the use of GMM to segment 3D microPET images from the reconstruction images by the PDEM.
3.3.1 3D Images
Figure 3.1-3 shows the data matrices used for reconstruction and segemtation.
There are 703 sinograms obtained by the 3D microPET with span 3 and MRD 31. We applied the FORE on prompt and delay sinograms to obtain 63 sinograms. Each 2D sinogram was reconstructed by the PDEM. In this study, each slice has the size of 96
x 84 and each reconstructed images has the size of 128 x 128. A matrix with the dimension of 63 x 128 x 128 was used for segmentation by the GMM and K-means algorithms. The analysis flow chart to segment 3D images is displayed in Fig. 3.3.1-1.
The PDEM is the reconstruction process of microPET images. The GMM or K-means are applied algorithms for segmentation. The phantoms for simulation studies are displayed as Fig. 3.3.1-2. There are 63 reconstructed slices with the size of 128 x 128.
2x703 sinograms
Obtain 2x63 sinograms by the FORE
Reconstruction by the PDEM
Smoothing in each slice
Segmentation by K-means and GMM
3D segmented images Fig. 3.3.1-1. The flow chart of 3D segmentation is plotted.
Slice 1
……. …….
Slice 63
Figure 3.3.1-2: 3D microPET images for 63 slices are illustrated. Every image has the pixels size of 128 by 128.
3.3.2 Simulation Study
The simulated phantom study with 457932 total counts is displayed in Figure 3.3.2-1. This simulated study is focused on testing and evaluating the performance of GMM. Figure 3.3.2-1A shows target image with five ROIs. Figure 3.3.2-1B displays target image with 50% noise added. Figure 3.3.2-1C presents the clustering results by GMM. The number of clusters is decided by the KDE and is shown in Figure 3.3.2-1D. There are four local high peaks that are regarded as the means of four clusters.
Fig. 3.3.2-1. A) Simulation image of five clusters is displayed. B) Adding 50% noise into panel A. C) Clustering results using GMM. D) Kernel density curve using C with values of high and low peaks. Four peaks are identified on the density curve.
Hence, the number of groups is set as four.
Figure 3.3.2-1 shows the indices of ROIs by the GMM. Figure 3.3.2-3 presents the accuracy comparison between the simulated results obtained from K-means and GMM.
Fig. 3.3.2-2. Target ROIs are marked.
Fig. 3.3.2-3. The results of A) by K-means and B) by GMM clustering are shown.
It is observed that the GMM has a clearer segmentation result than the K-means method. Results of ROIs are shown in details in Table 3.3.2-1. The total accuracy of GMM is 92.1% and that of K-means is 66.6%.
Table 3.3.2-1: Comparisons of the clustering results by K-means and GMM in Fig.
3.3.2-1A.
Exact Counts Accuracy (%) ROI # True Pixel
Count K-means GMM K-means GMM ROI 1 6604 4128 6206 62.5% 94.0%
ROI 2 702 488 522 69.5% 74.4%
ROI 3 748 700 745 93.6% 99.6%
ROI 4 350 280 285 80.0% 81.4%
ROI 5 88 58 59 65.9% 67.0%
Total 8492 5654 7817 66.6% 92.1%
Another simulated volume data based on the modified Shepp-Logan's head phantom image is shown in Figure 3.3.2-4A and 3.3.2-4B. Fifty percentage of noise ratio to phantom images are added. In order to compare the effects of variation between slices, different image levels and shapes of ROIs are considered in slice 1 and 2. First, we use the MLEM reconstruction, the result is shown in Figure 3.3.2-4C and 3.3.2-4D. Meanwhile, the GMM is also applied to segment two images without slice normalization as shown in Figure 3.3.2-4E and 3.3.2-4F. It is observed that the boundaries of ROIs are difficult to distinguish. Therefore, slice normalization is applied to the volume data and then the GMM is used to segment images as shown in Figure 3.3.2-4G and 3.3.2-4H. The boundaries of these segmentations become clearer after slice normalization. Figure 3.3.2-4I plots the estimated kernel density curve of volume data for finding the number of clusters and initialized values.
Fig. 3.3.2-4. Simulated volume data including slice 1 and 2 are marked as A and B.
C and D are reconstructed images after added 50% noise ratio to A and B. E and F are segmented results without slice normalization. G and H are segmented results with slice normalization. I is the estimated kernel density curve of simulated volume data after slice normalization.
For these simulation cases, the performance and accuracy using GMM is better than those of using K-means. The KDE is adopted to decide the number of clusters and the starting values of parameters in the EM algorithm. The slice normalization is necessary when the GMM is applied to segment volume data in this study.
3.3.3 Real Mouse Study
The empirical data of a big mouse injected by F-18 isotope scanning is collected from the microPET R4 system. The acquired configurations are listed as below.
Scanner energy is between 350 and 750 keV with the total scanning of 3600 s. There are 32 rings in microPET R4 system. File format of histogram data is stored by 2 bytes for each voxel. Ten slices (from the 51st to the 60th slice) of the volume data are used for investigation and evaluation.
Figure 3.3.3-1 shows the estimated kernel density curve of volume data. Based on this KDE, four groups are determined by local high peaks and their starting values are obtained for applying the EM algorithm.
Fig.3.3.3-1. The estimated kernel density curve using the rat volume data of 10 slices is shown with values of high and low peaks. There are four peaks. Hence, the number of groups is set as four. Values of peaks are applied to compute the starting values in EM algorithm.
Figure 3.3.3-2 shows the reconstructed rat images by MLEM from the 51st to the 60th slice. Besides, Figure 3.3.3-3 and 3.3.3-4 show the segmentation results by GMM and K-means respectively. The detail segmentation from GMM is shown with the comparison to Figure 3.3.3-5. The uptake areas can be segmented by GMM from 59th and 60th images. In addition, it can segment small areas with high gene expression when compared to K-means. On the contrary, the K-means method segments big areas and ignores small uptake areas.
For this real mouse study by microPET, the GMM leads to more detail segmentation results than the K-means method does. The GMM also has better performance than K-means. The full width half maximum (FWHM) is usually used to evaluate performance of segmented results. The horizontal line profile near the center of the 60th slice is used to investigate the performance between GMM and K-means.
Figure 3.3.3-5 is plotted with four regions in this line profile and their FWHMs for Fig. 3.3.3-2.
Fig. 3.3.3-2. The reconstructed rat images are shown from the 51st (top-left) to the 60th (bottom -right) slice.
Fig. 3.3.3-3. The results of segmentation by the GMM are shown from the 51st (top-left) to the 60th (bottom -right) slice.
Fig. 3.3.3-4. The results of segmentation by the K-means are shown from the 51st (top-left) to the 60th (bottom -right) slice.
Fig. 3.3.3-5. The horizontal line profile of the 60th slice of Fig. 3.3.3-2 is shown with FWHMs. The FWHMs of region 1, 2, 3 and 4 are 3.75, 3.40, 4.14 and 4.45 pixels respectively. The top part shows the location of this line profile in the MLEM reconstruction image and the segmentation by GMM and K-means.
Table 3.3.3-1 shows that the FWHMs of segmented results by GMM are closer to target FWHMs than those by K-means. Meanwhile, the signal to noise ratio (SNR) defined by the ratio of mean value to standard deviation is used to compare the segmentation performance between GMM and K-means. The SNRs of four regions of GMM are higher than those of K-means.
Table 3.3.3-1: The FWHMs and SNRs of segmented results by GMM are better than those by K-means in Fig. 3.3.3-5.
Pixel of Boundary
Signal to Noise Ratio (SNR) Region FWHM of
Region
GMM K-means GMM K-means
1 3.75 4 7 9.83 6.82
2 3.40 4 6 8.64 6.23
3 4.14 5 7 4.05 2.13
4 4.45 5 7 3.15 2.13
4. Applications on Microarray Images
This chapter investigates the applications of segmentation methods on spotted microarray images. The GMM and KDE are both employed to segment spotted images. Furthermore, we combine GMM and KDE to form a new method, GKDE, to segment spots. The GKDE can keep advantages of KDE and refining the final results from the GMM. We will compare and evaluate the performance of three methods together with the adaptive irregular segmentation method in GenePix 6 based on spike genes, duplicated genes, and swapped arrays in real microarray data.