The Watershed Algorithm: A Method to Segment Noisy PET Transmission Images

Transcription

1 The Watershed Algorithm: A Method to Segment Noisy PET Transmission Images C. Riddell, P. Brigger, R.E. Carson and S.L. Bacharach National Institutes of Health, Bldg. 10 Room 1C401, Bethesda, MD Abstract Attenuation correction is essential to PET imaging but often requires impractical acquisition times. Segmentation of short noisier transmission scans has been proposed as a solution. We report that a 3D morphological tool - the watershed algorithm - is well adapted for segmenting even 2- minute PET transmission images. The technique is noniterative, fast, and fully 3-D. It inherently ensures class continuity and eliminates outliers. Pre-filtering the data induced smoother class edges, showing that a multi-resolution approach could be used to deal with partial volume effect and excessive noise in the data. The algorithm was tested on 2- minute scans of a torso phantom and on a human study. I. INTRODUCTION Attenuation correction (AC) is essential to PET imaging for both qualitative and quantitative purposes. However, the additional scanning time can be intolerable in the case of whole-body exams that require multiple bed-positions and repeated transmission scans. Patient throughput and comfort could be considerably increased if very short (e.g. 2 minutes) transmission scans could be used. Unfortunately, the transmission data then becomes a significant source of noise in the reconstructed emission images. Noise is suppressed if the transmission scan can be segmented into its three expected attenuation values: tissue (water equivalent: 0.095Êcm -1 ), background (air equivalent: 0. cm -1 ) and lung (mix of water and air» cm -1 ). Histogram-based segmentation and theoretical attenuation derived from the detection of body contours in the emission studies have been proposed [1-4], but such methods have not been widely adopted yet, partly because it is difficult to accurately segment short transmission scans, due to their excessive noise. We describe here the application of morphological segmentation - the watershed algorithm - which we find well suited for segmenting even very short PET transmission images. The mathematical concept on which it is based allows a more robust segmentation while maintaining computation times compatible with routine clinical use. After introducing the watershed algorithm, we describe our data processing for the accurate segmentation of low-count PET transmission scans. It is based on iterative reconstruction of the attenuation data, followed by a multi-resolution application of the watershed algorithm. We report results on both phantom and clinical data. II. WATERSHED ALGORITHM Mathematical morphology treats gray-level images as topographies. The gray levels define altitudes, with maxima at the top of peaks and minima at the bottom of valleys. The watershed line is the basic tool for segmenting images in mathematical morphology [5]. One definition states that a drop falling on the watershed line has equal probability of falling into one or the other of the two regions it separates called catchment basins. Finding the watershed line (i.e. the maxima) of the norm of an image gradient is the basis of the separation of an image into classes, since it is equivalent to finding the "edges" separating those classes. As will be shown, the resulting watershed lines are quite good despite the high noise present in the image itself. The most efficient algorithm so far for computing watershed lines of a gray-scale image has been proposed by Vincent and SoilleÊ[6]. It is based on the simulation of an immersion process. The topography defined by the imagegradient is flooded through holes pierced at the pixels of lowest altitude, although an alternative strategy (used here) involves choosing the holes through user-defined labeled seed points. The seeds are chosen to select (and label) some of the flat areas of the image. The flooding progresses at constant speed from each seed upwards and the catchment basins containing the seeds are flooded. At the point where waters (i.e. labels) would mix, a dam is built to avoid mixing waters coming from different catchment basins. If a catchment basin has no seed, no flooding is started. The basin will be submersed by a flood coming from the neighboring catchment basin, as soon as the water reaches the saddle point between the two basins. The flooding is stopped at the highest altitude in the image, and the resulting dams correspond to the watershed of the gradient image. The dams define closed edges with all enclosed pixels labeled with their seed label. The image pixels are first sorted by altitude, and then the image is flooded. The immersion process is efficient because only a restricted group of pixels is processed by altitude. The flooding step is a recursive procedure that classifies the pixels at altitude h+1 knowing the classification at altitude h. A pixel at altitude h+1 can have: - One neighbor with a label: the pixel is put in the same class. - Several neighbors, at least one with a different label: water with different labels would mix at this pixel, so a dam is built. - No neighbors with a label: it is a new unseeded catchment basin. Classification is postponed until the pixel is reached by a neighboring flood. The flooding of non-seeded catchment basins by a neighboring flood is a very elegant and powerful way to get rid of irrelevant edges always present in noisy gradient images. It also inherently ensures class continuity over all dimensions and eliminates outliers, since no pixel can belong to a class if it is not a neighbor of another pixel of the class. The

2 Êmm Fig. 1: Effect of image smoothness over watershed segmentation algorithm does not rely on any particular connection scheme, and therefore, is able to handle many dimensions. Our implementation is limited to three. The watershed technique is of potential interest for segmentation of short transmission PET images because the technique deals well with noisy data and because computation times are reasonable (< 3Êminutes for 35 slices of 256x256 pixels on a Hewlett Packard, Apollo Series 735, computer). III. PET IMAGE SEGMENTATION: A. Maximum-likelihood reconstruction Transmission reconstruction is usually performed by applying the filtered backprojection (FBP) algorithm to the log-ratio of a blank scan (transmission with no object in the field of view) and the object transmission scan. However, FBP is ill suited to very short acquisition times, where the Poisson nature of the data becomes more important and should be included in the reconstruction model. One problem is that, for short acquisition times, there may be some lines of response (among the most heavily attenuated ones) for which no transmitted photon is recorded. With FBP, this corresponds to an infinitely attenuating medium, resulting in streak artifacts in the reconstructed image. Such artifacts are particularly difficult to deal with when applying segmentation algorithms. If the reconstruction algorithm takes into account the Poisson nature of the data, this effect can be avoided. In this formulation, zeroes represent lines of response of unknown, rather than infinite, attenuation. We reconstructed the attenuation data with a maximumlikelihood (ML) gradient ascent reconstruction algorithm [7, 8], with under-relaxation [9] and an ordered-subsets scheme [10-12]. An equivalent of about 40 iterations was computed by performing 2 iterations with 21 subsets. Reconstruction of low-count data yielded high noise in the most attenuated areas, but lower noise in the less attenuated areas such as the lungs. Images were free of streak artifacts and did not require pre-filtering of the raw data, maintaining the original resolution. Although the attenuation data contained too much noise to be used as is for attenuation correction, the ML reconstructed attenuation images were better suited for segmentation than FBP reconstructions. B. Multi-resolution segmentation Because the watershed is applied to the norm of the image gradient, which is very sensitive to noise, unexpected passageways between classes can occur at high noise levels, with catchment basins submerged by the wrong flood. Typically, one class may "leak" into another in these high noise situations (e.g. the lung class might "leak" into the tissue class). Post-reconstruction smoothing is applied to regularize the image gradient and its watershed lines. This means that the watershed algorithm produces edges that are more regular with increasing smoothing. To illustrate this, fig. 1 shows a noisy image filtered with increasing gaussian kernels (0, 4, 8, 12, and 16Êmm) and below, the corresponding watershed segmentation of each image. We see how one class "leaked" into the other because the watershed lines of the noisiest image did not correspond to the edges of interest. At 8Êmm most of the leakage artifact was removed because the watershed lines of this image are a good fit to the edges we are interested in. Further filtering regularized those edges while preserving their position. In order to have a robust procedure without using too much pre-filtering, we took advantage of the versatility of the watershed procedure, which offers the additional possibility of segmenting an image knowing its segmentation at a previous resolution. This approach, which we call multi-resolution segmentation, is as follows: The first step consists of the segmentation of the whole volume at a low resolution. We expect no leakage but edges that may be too simple. Therefore, in a second step, each slice is refined by stacking both the original and the low resolution images, and segmenting them again as a two-slice volume, with resolution as the z-axis. The low-resolution segmentation labels (obtained in the first step) are taken as seeds for this second segmentation. Fig. 2 compares the direct segmentation approach to the multiresolution one. By directly applying the watershed algorithm to the noisy image at the top left of the figure (image a), the segmentation exhibits strong "leakage" - e.g. patches of lung intensities have "leaked" into soft tissue portions of the image (image b). In the multi-resolution approach, the original noisy image shown in (a) (reproduced again as B) is stacked together with a filtered version of itself (image A). The low-resolution a A C direct & direct through resolution Fig. 2: Direct vs. multi-resolution segmentation of a noisy image. B D b

3 image, A, can be directly segmented without leakage into image C. Taking the two images A and B as though they were a volume, and segmenting this volume using image C as the seed, we obtain the segmented volume made of images C and D. By so doing, the original noisy image (a or B) could be successfully segmented (image D) while avoiding nearly all of the leakage artifacts (b vs D). From C, it would seem that the watershed method could be applied directly to the smoothed image. However, we show below that doing so often results in loss of detailed structure, which may (or may not) be important. The multi-resolution segmentation (D) avoids this problem. In addition, since two segmentations (corresponding to the "low" and "high" resolution slices) are generated, they could be combined (not shown): The pixels having different labels at different resolutions could then be set apart in a new class that presumably represents an area of partial volume, i.e. the interface between tissue and lung or background. Such an area could then be treated differently than the main classes. C. Attenuation correction To perform attenuation correction of the PET emission scan with a segmented map, attenuation values must be attributed to the pixels according to their class. For humans, the division of the attenuation map into three uniform segments is known to produce biases because the hypothesis of uniformity is only an approximation. For the tissue class, attenuation is nearly uniform except for the spine and possible intestinal air bubbles, which were neglected (although an airbubble could be isolated with an additional seed). Lungs are not uniform, due in part to the effects of gravity. Finally, partial volume effects can cause additional heterogeneities. It was not our goal to investigate such biases, which can be introduced by any segmentation technique. Rather we wished to investigate whether the watershed method could be used to perform equally accurate segmentation of short noisy transmission scans and longer nearly noise free scans. None the less, we tried to minimize some of the expected biases by using the following post-processing of the segmentation maps: For the tissue class, attenuation was assumed uniform and a mean attenuation value was computed over the entire 3D-class. The background was set to zero, and a high-count image of the bed was inserted into the attenuation map. For the interface classes (background/tissue and lung/tissue), mean class attenuation values were computed for each slice, to take into account that the effects of partial volume are different from slice to slice. Following [13], we replaced the lung class with the original lung data. Since segmented images have infinite resolution, they were also filtered with a 3D-gaussian to match the emission data resolution. Averaging and filtering maintained the total amount of attenuation of a class. The processed volumes were reprojected and exponentiated to obtain the attenuation correction factors. IV. EXPERIMENTS The watershed technique was first evaluated on a human torso phantom (Data Spectrum Inc) with liver, heart and lung inserts. Experiments were carried on a GE Advance PET scanner [14], which has 18 detector rings yielding 35 slices at 4.25Êmm center-to-center slice separation. Transmission measurements were obtained with two rotating rod sources containing 5 and 10 mci of GeÊ68 each. The images were reconstructed on a 256x256 grid (pixel size 1.97Êmm) with 2 iterations and 21 subsets of our ordered-subsets implementation of the gradient ascent algorithm. No prefiltering was required for the blank and transmitted scans, but a 6Êmm 3D gaussian filtering was applied after reconstruction, defining the "high" resolution case. For each study, a set of four seeds was chosen on a slice sampling heart and lungs. We took one seed per lung (with label 32), one seed in the heart area (with label 95), and the top left pixel for the background (with label 1). The multi-resolution segmentation of the 6-mm attenuation map was performed as described above, using the high-resolution image and an image filtered to 16Êmm resolution as the low-resolution image. Two new classes were obtained by averaging the segmentations obtained at low and high resolutions at the lung/tissue and background/tissue interfaces (with labels 63 and 48). To compare attenuation correction with measured and segmented data, attenuation factors were computed in the following way for all transmission data sets and acquisition times: For the measured AC, the maximum-likelihood reconstruction of the attenuation volume with 6Êmm resolution was reprojected and exponentiated. For the segmentation based AC, the segmented map were processed as described above. (3D average for the tissue class, 2D averages for the interface classes, original data for the lung class, and a final 6Êmm 3D gaussian filter to match the resolution of the emission data). The resulting volume was reprojectedandexponentiated. A high-count emission distribution was used to evaluate AC. Identical activity concentrations were put in the liver and in the heart wall (but none in the cavity). Activity was added to the background to yield a 4:1 tissue/background activity ratio. Sinogram total counts ranged from 1.3 M for the heart slices to 2.8ÊM for the liver slices. The emission data was also pre-filtered by 6Êmm (in-plane and across slices) before applying the OSEM algorithm with 2 iterations and 21 subsets [10]. The same procedure for segmentation, attenuation correction and emission reconstruction was applied to an FDG human study for which both a short (2-min) scan and an 8-min scan (the standard clinical transmission time in our laboratory) were available. Both transmission scans were acquired prior to administration of 20-mCi of FDG. V. RESULTS AND DISCUSSION: A. Segmentation: Fig. 3 compares multi-resolution watershed segmentations of the human torso phantom between short and long acquisition times. The slices shown are not contiguous but are approximately 9Êmm apart. The top two rows show the maximum likelihood reconstruction of the 32-min attenuation scan, and, below, the corresponding segmentation. The bottom rows show in a similar manner the results using the 2-min scan. The watershed algorithm was reliable at handling the

4 strong noise difference between a 32-min scan and a 2-min scan, producing very similar segmentations. Only very little leakage from the lungs into the tissue was visible in the 2-min segmentation with the multi-resolution approach (last row of fig. 3, slice C). In addition, this leakage was not put in the lung class but in an interface class. With respect to the 32-min case (taken as "truth") the number of pixels classified in the tissue class was reduced by 2% in the 2-min case, whereas the number of pixels in the lung class was increased by 1.4%. We computed the average attenuation value over a class from the 32-min attenuation map. The tissue class had values of Êcm -1 and Êcm -1 for the 32-min and the 2-min scan respectively, yielding a 0.5% difference. The lung class had values of Êcm -1 and Êcm -1, yielding a 2% difference. The watershed algorithm segmented the data into three classes at the points of highest gradient. However, the data itself may not be adequately represented by this limited number of classes, particularly at the lung/tissue interface, which is affected by partial volume effects. This is best seen on fig. 3, slice C of the third row, where the finite z-axis resolution causes low-intensity liver values appear within the lung. For computing accurate attenuation factors, the post-filtering of the segmented map described above will reduce the mismatch in resolution and to some extent recreate partial volume effects and the associated intermediate attenuation values. Fig. 4 displays in a similar manner the results from a patient. Slices shown were taken 25Êmm apart. The top two rows show the maximum likelihood reconstruction of the 8- min scan and the corresponding segmentation. The bottom rows show the 2-min scan. Again, the watershed procedure was reliable at segmenting the images at this high level of noise. With respect to the 8-min case, the tissue class was reduced by 2% at 2 min. Half of those missing pixels were classified instead in the lung class, causing the lung class to grow by 5%. The other half of the pixels removed from the tissue class were classified in the additional lung/tissue class. The mean values of the tissue class at 2 and 8 min differed by only 0.2%, but by 5% in the lung class. Again, the effect of finite z-axis resolution was also visible on the real data (slice C), requiring post-filtering for the computation of the attenuation factors. Slice E shows the base of the pulmonary artery segmented in an intermediate class because of its small size. This means that the structure was lost on the low-resolution segmentation, but recovered during the refinement step. In both cases, the watershed algorithm did not introduce visual artifacts, because the reconstruction itself although noisy, was free of artifacts such as streaks. In fact, we speculate that other segmentation algorithms would also benefit from being applied to ML reconstructed attenuation data. However, for short duration transmission scans, the ML reconstructed attenuation map, while free of streak artifacts, is still quite noisy. This noise might well cause many segmentation algorithms to produce incorrect classifications. Our data show that the watershed algorithm, especially when used in the "multi-resolution" manner described above, is quite effective at segmenting 2-min noisy attenuation images. In addition, the watershed algorithm appears to have many other advantages: Its ability to yield continuous classes, and more generally, the possibility of constraining the classification of a pixel by the result of the classification of its neighbors (the "neighbor" being near in space or in resolution, or even in time) makes the watershed particularly robust and easy to use. For example, the amount of filtering used at the lower resolution was not critical to the procedure. Direct application of the watershed algorithm to the reconstructed images would require some optimal pre-filtering (as shown on fig. 1) which might be problematic if different images required different filtering. This is avoided with the multi-resolution approach, because filtering was set high enough to always avoid leakage, but edge details that were filtered out were recovered in the refinement step. The "perfect" resolution of the segmented images of course requires further post-processing (matching emission and transmission resolutions) in order to perform accurate attenuation correction. B. Attenuation correction: Fig. 5 shows one slice at the heart level of the emission distribution of the human torso phantom. Attenuation correction was performed with the attenuation maps shown on the bottom row. From left to right, the map used was the measured 32-min scan (A), the 32-min scan segmented and post-processed as described (B), the 2-min scan (C) and the 2- min scan segmented and post-processed (D). From visual inspection, we see that using the 2-min measured scan led to a strong increase in the noise of the emission image (C). As expected, segmentation suppressed this noise propagation, and the three other images (1 st row A, B & D) had equivalent noise characteristics. As mentioned above, the segmentation of the 2-min and 32-min scans differed by only 0.5% in their mean tissue attenuation value. The lungs in the 2-min segmented attenuation image were, however, much noisier than in the 32- min map. Those differences resulted in less than 2% mean differences in attenuation corrected emission data regardless of whether one used the 2 minute segmented or the 32 minute measured data to perform the attenuation correction. This finding was true in the background, heart areas, and liver - in fact anywhere except the lungs (which had no activity). Therefore, whereas a 2-min measured scan was a major source of noise in attenuation corrected emission images, noise was suppressed by segmenting the tissue at the cost of a very small bias. Fig. 6 illustrates the potential impact of segmentation on the noise content of an attenuation corrected emission distribution from a human FDG study. Two slices of this study are shown: one through the liver (1 st row), the other through the heart (3 rd row). Below each emission reconstruction is shown the attenuation map that was used for attenuation correction. Attenuation maps were, from left to right: 8-min measured scan (A), 8-min scan segmented and post-processed (B), 2-min scan (C) and 2-min scan segmented and post-processed (D). Again, the 2-min scan was a major source of noise in the reconstructed emission distribution (3 rd col., 1 st and 3 rd rows, fig.ê6), but this noise was suppressed after segmentation (4 th col., 1 st and 3 rd rows). Both the segmented 8-min scan and the segmented 2-min scan had equivalent noise levels, both producing half the noise obtained with the 8-min measured scan. However, the hypothesis of three uniform attenuation regions proved less correct for the human case than it was for

5 the phantom data, resulting in significant biases of the emission corrected data in the liver and the heart shown on fig. 6. Again, these biases, while important, were not the subject of this investigation. Rather, our goal was to determine if the watershed method would yield equally accurate segmentation when applied to short, noisy 2-minute attenuation scans, compared to the longer, less noisy, 8-minute scans. For those same areas of the heart and the liver, both segmentations yielded emission values within a few percent of each. The only differences larger than this occurred in areas of large partial volume effect (e.g. edges and at the dome of the liver) where quantification is problematic in any case. Therefore, segmentation of short 2-minute scans seems practicable. Further work will be needed to validate these results in a larger population, and to investigate how one might employ such segmentation to produce unbiased emission reconstructions. VI. CONCLUSION The watershed algorithm is a promising technique for reducing transmission scan acquisition time and eliminating the propagation of transmission noise in attenuation corrected emission distributions in PET. Our preliminary studies of human and phantom data showed that noisy 2 minute transmission data produced equally accurate segmentations as did 8 minute or 32 minute transmission data. This finding is a first step toward significantly reducing the total scan time in multi-level whole-body PET oncology studies. Further work is necessary to confirm these findings in a large patient population, and to investigate how to best use such segmented data to correct the emission scan in an unbiased fashion. VII. REFERENCES [1] M. Bergstršm, J. Litton, L. Eriksson, C. Bohm, and G. Blomqvist, ÒDetermination of object contour from projections for attenuation correction in cranial positron emission tomography,ó Journal of Computer Assisted Tomography, vol. 6, pp , [2] S. Siegel and M. Dahlbom, ÒImplementation and evaluation of a calculated attenuation correction for PET,Ó IEEE Transactions on Nuclear Sciences, vol. 39, pp , [3] S. R. Meikle, M. Dahlbom, and S. R. Cherry, ÒAttenuation correction using count-limited transmission data in positron emission tomography,ó Journal of Nuclear Medicine, vol. 34, pp , [4] M. Xu, P. D. Cutler, and W. K. Luk, ÒAdaptive, segmented attenuation correction for whole-body PET imaging,ó IEEE Transactions on Nuclear Sciences, vol. 43, pp , [5] F. Meyer and S. Beucher, ÒMorphological segmentation,ó Journal of Visual Communications and Image Representation, vol. 1, pp , [6] L. Vincent and P. Soille, ÒWatersheds in digital spaces: an efficient algorithm based on immersion simulations,ó IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, pp , [7] K. Lange, ÒA theoretical study of some maximum likelihood algorithms for emission and transmission tomography,ó IEEE Transactions on Medical Imaging, vol. 6, pp , [8] K. Lange and J. A. Fessler, ÒGlobally convergent algorithms for maximum a posteriori transmission tomography,ó IEEE Transactions on Image Processing, vol. 4, pp , [9] J. A. Fessler, ÒHybrid Poisson/Polynomial objective function for tomographic image reconstruction from transmission scans.,ó IEEE Transactions on Image Processing, vol. 4, pp , [10] H. M. Hudson and R. S. Larkin, ÒAccelerated image reconstruction using ordered subsets of projection data,ó IEEE Transactions on Medical Imaging, vol. 13, pp , [11] C. Kamphuis and F. J. Beekman, ÒAccelerate iterative transmission CT reconstruction using an ordered subsets convex algorithm,ó Nuclear Science Symposium & Medical Imaging Conference, Toronto, Canada, [12] H. Erdogan, G. Gualtieri, and J. A. Fessler, ÒAn ordered subsets algorithm for transmission tomography,ó Nuclear Science Symposium & Medical Imaging Conference, Toronto, Canada, [13] M. T. Madsen, ÒPET attenuation correction using mean attenuation coefficients: A simulation study,ó Nuclear Science Symposium & Medical Imaging Conference, Toronto, Canada, [14] T. R. DeGrado, T. G. Turkington, J. J. Williams, C. W. Stearns, J. M. Hoffman, and R. E. Coleman, ÒPerformance characteristics of a whole-body PET scanner,ó Journal of Nuclear Medicine, vol. 35, pp , 1994.