Budapest University of Technology and Economics. Partial Volume Effect Correction using Segmented CT Images with Distance Mapping

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1 Budapest University of Technology and Economics Faculty of Electrical Engineering and Informatics Department of Control Engineering and Information Technology Internship Report Partial Volume Effect Correction using Segmented CT Images with Distance Mapping Author: Zsolt Márta Supervisor: Dr. László Szirmay-Kalos December 13, 2011

2 1 Introduction Positron Emission Tomography allows functional analysis of physiological processes. However, its poor spatial resolution limits accurate quantitative measurements particularly in small structures, partially occupying the pointspread function (PSF ) of the scanner [1]. The effect partially caused by this poor spatial resolution is commonly known as the Partial Volume Effect (PVE). The aim of my internship was to present a general approach for PVE correction using registered and segmented Computed Tomography (CT) images. Utilizing the high resolution anatomical image of CT, our method enhances region borders in PET images based on the assumption that region boundaries present in both images are co-located. The proposed algorithm produces a corrected output image without any a priori information about the scanner s properties, such as the PSF. 2 Partial Volume Effect PVE actually refers to two phenomena that affect image intensities, the blurring effect and sampling error of the finite discrete grid. The sampling error is due to the finite voxel size, so multiple tissue types are included in most voxels. The main purpose of the proposed algorithm is to correct the blurring effect. The blurring effect and the poor resolution of PET imaging comes from detector design and physical properties of the methodology used. As its name implies, PET works by detecting positron emissions from radioactive decays. The procedure begins with injecting fast-decaying radiopharmaceuticals determined by the biological properties of tissues being examined. The aim is to monitor positron emissions from radiotracer decays, though they can only be detected indirectly. Generated positrons pair up with electrons in nearby tissue and annihilation occurs with photon emission which can be detected by certain crystals in the scanner. Physical reasons of image blurring include time and space offset of annihilation and decay origin (called positron range), acollinearity of the two generated photons, photon scattering (both in tissues and detector crystals), finite size of crystal detectors, and the deficiencies of the detector electronics [2]. Mathematically, the PET image is the result of a low-pass filter, the PSF of the scanner, with some reconstruction noise added. PVE severely affects tissue boundaries as blurring reduces high-frequency details. This results in the underestimation of uptake values in regions of interest (ROIs). More specifically, regions with higher activity are distorted in a 1

3 way that their total activity is spread across a large area. Due to reasons earlier described, PVE does not change the total activity in regions, only spreads them, possibly leading to lower maximum values in smaller areas. This spreading results in a spillover effect, i.e. surrounding tissues with lower activity seem to have gained uptake. The smaller the region is, the dimmer it looks resulting in erroneous qualitative assessment especially with small lesions [3]. 2.1 Partial Volume Effect Correction Due to its high medical relevancy great efforts have been made to correct PVE, though there is no generally accepted method to date. Partial volume correction (PVC) methods were first examined for brain scans, due to the numerous small structures it contains. A family of algorithms deal with the problem on regional level, resulting in regional uptake values instead of corrected images. In addition, a priori information is required, for example, the PSF for deconvolution methods, or the assumption that the original activity is constant in specific regions. The PSF, however, is usually not available and may depend on the measured object as well, which restricts the availability of this approach. Another family of algorithms deals with the problem on pixel/voxel level. These produce PVE corrected images enabling visual evaluation. Particularly popular pixel level methods are based on image fusion techniques, which exploit simultaneous and automatically registered PET and CT acquisition and fuse functional and anatomical information. The main assumption is that tissue boundaries appearing in both images enables correction relying on anatomical data. This assumption seems realistic as different tissue types have different density (thus different intensity in CT scan), and also the radiotracer density depends on the tissue type. Therefore on the boundaries of different tissues, it is assumed that both CT and PET values change and thus CT and PET boundaries are co-located. In 2006, Boussion et al. [4] published a novel multiresolution approach based on wavelet transform and image fusion. Their method extracts the highresolution components from the anatomical image, and incorporates in the low-resolution PET image using a simple intensity scaling transform between CT and PET wavelet coefficients [5, 6]. In their paper, Boussin et al. used a simple pixel-by-pixel division to find the appropriate global scaling factor. This produces uneven edges due to PET fluctuations, and introduces CT noise in addition to PET noise. Moreover, in homogeneous anatomical areas where high frequency components are near zero it results in extreme corrected values. Global scaling ignores local image features and when CT and PET activity 2

4 (a) (b) (c) (d) Figure 1: (a) Original PET image (b) Corrected PET image using global scaling and homogeneous CT image (c) Line profile of original PET image (d) Line profile of corrected PET image Hyperintense edges are present due to uncorrelated images, and neglected local features. is less correlated, it causes visual artifacts and quantitative errors at boundaries. Since the correction is a function of global PET features and local CT features only, it inherently cannot adapt to local dissimilarities between PET and CT intensity. A specific issue is when the functional image already contains high-frequency information and additional values are added leading to hyperintensive edges in corrected images. Furthermore, extreme values affect the whole image as global scale calculations involve them as well. On the other side, low-intensity features become less significant as they weigh little in averages. Moreover, it may happen that CT and PET intensities are negatively correlated in some part of the image, thus the method amplifies PVE instead of correcting it. Another possible approach may involve using local scales. For each point the scaling between wavelet coefficients is determined by a function of nearby 3

5 wavelet values. In this case, the window size greatly affects the results: using a too small window results in too little information to define scaling and thus leads to a visually and quantitatively incorrect image due to PET fluctuations and extremes. Therefore the correction becomes unstable. On the other hand, using larger window blurs local image features. This window size parameter greatly depends on the actual images, or even different regions require different window widths. Consequently no optimal window width is guaranteed and choosing one must be done manually, rendering this method less effective. The main problem of image fusion type PVE correction algorithms is the assumption that original PET and CT features are highly correlated. This, however, is unrealistic in whole-body scans [3] for technological differences, as PET image is essentially a functional information while CT is structural. PET intensities depend on radiotracer amount in specific regions, depending on blood density, metabolism, etc. CT, however, reveals anatomical structures, depending on tissue density. Thus, instead of the correlation of boundaries, a more relaxed connection seems feasible that boundaries being present in both modalities are co-located. 3 Proposed Method The main objectives of our proposed method are summarized in the following lists of requirements: The algorithm should require no user interaction, except for some parameter setting. To allow visual evaluation as expected in most medical applications, the algorithm should output a corrected image instead of just determining correct region uptake values. The algorithm should be fully 3D as PVE is essentially a 3D effect. The aim of any correction is to enhance region boundaries. In other words, it should make steep edges with increased intensity gradient between regions. PVE correction should involve boundary enhancing, although inner parts of tissues should remain unchanged as they are not affected by PVE. Therefore modifications are limited to edges of certain size, determined by the PSF of the scanner. Of course small regions may be affected completely, obviously the algorithm should correct accordingly. 4

6 Since PET allows quantitative examination of certain physiological functions, corrected images are expected to correctly represent regional uptakes. More specifically, output images should contain regions with total intensities equivalent to original ones. This includes the avoidance of hyperintensive corrected edges as seen in Figure 1d. As for the qualitative assessment, correction output should contain sharp, near-homogeneous regions to enable more accurate medical evaluation. An ideal PVC algorithm that rely on anatomical data should account for any offset between the functional and anatomical images. This offset may be due to different acquisition time and patient motion. [7] A moderate computational complexity is required to speed up medical evaluation, and the PVC should not take longer than the PET reconstruction. With massively parallel reconstructions available, this time limit around a few minutes for a common voxel array. The inputs of our PVC are the PET data and the registered CT volume. Since CT technology have larger spatial resolution and moderate noise, especially compared to emission tomography, segmentation can be applied to determine CT regions. Using the region information, the structure of anatomical parts can be taken into consideration. Numerous image segmentation algorithms exist [8, 9, 10] for noisy grayscale images including those for medical applications. In most cases, CT images have larger spatial resolution, thus the PET image is first scaled to the size of the CT image using a simple trilinear interpolation. The overview of our PVC algorithm is shown by Figure 2. The steps are discussed in the following subsections. 3.1 Boundary map generation To fulfil the requirement of only changing the image where necessary, a 3D binary boundary map is created, which indicates where the PET volume is worth being corrected. The algorithm starts with an initial binary map that is 1 where CT data contain boundaries and 0 elsewhere. This boundary map is then filtered by using a standard box-filter and a threshold based on global average PET intensity, leaving voxels where the mean surrounding PET gradient intensity is large enough. This way the image is altered only where PET signal is changing, i.e. where the boundaries are also present in PET image. 5

7 Figure 2: Overview of the proposed PVC algorithm The filtered boundary map is then extended by a morphological dilation [11] operator. The dilation operator is implemented by a filter, that takes the maximum value of the edge map in a cube around each voxel. The dilation is required to assure symmetry and constant width at region boundaries. Using directly a box-filter with thresholding, asymmetries may occur depending on PET intensity changes and kernel size. More specifically, the created boundary map may have different widths in segments, possibly leading to erroneous correction. This extended binary map is used as a boundary map later, only marked voxels are to be corrected. 3.2 Distance map generation Considering the fact that partial volume effect occurs at the edge of ROI-s, the basic principle behind the algorithm is that the intensity resembles the original most in the inner part of each region. The general assumption is that CT and PET regions are co-located, thus the PET image can be corrected using CT region structure information. With the help of a CT distance map, PET activities can be labeled based on their proximity to region boundaries. The algorithm repeats the following procedure for every CT region. The distance map is computed by iteratively applying a morphological erosion [11] operator, starting with an initial binary image that is 1 for each voxel corre- 6

8 (a) (b) Figure 3: Illustration of the distance mapping. (a) Segmented CT image, a circle (red) in a square (b) Resulting distance map with three iterations. Darker colors mean higher labels thus higher distance to boundary (red). sponding to the CT region, and 0 elsewhere. By using an increasing width kernel, the region shrinks in each iteration while preserving the shape. The distance map is completed by assigning to each voxel the last iteration number in which the voxel still belonged to the actual (shrunk) region. Clearly, the iteration number determines how far the voxel lies from the nearest region boundary. It is only necessary to iterate until the region is empty, or for practical reasons as PVE has limited range until an iteration limit has been reached. The erosion operator is implemented the same way as the dilation, but taking the minimum value instead of maximum. Obviously, the result is only 1, where the image contains the whole kernel. This way finding the inner parts of each segment is simple, voxels with the highest label number are the innermost ones. 3.3 PET Correction In the distance map, the larger the value of a voxel, the further it is from the boundary, thus applying the basic idea the less affected by PVE. One simple idea is then, to set every boundary voxel to the innermost average PET intensity. To exclude extreme values while preserving local features the mean value of the surrounding innermost (highest labeled) voxels is taken. A proper kernel size must be chosen as large kernel disrupts local features, while a too small kernel may not reach inner regions correctly. However, failing to do so has no severe consequences as averaging is a smooth function of voxels involved, and not reaching innermost values only leaves more resemblance 7

9 with the uncorrected image. This method effectively sets boundary voxels to the inner region intensity, regardless of original PET activity, which is in less accordance with basic PVE properties earlier mentioned. Recall that a basic property of PVE is that higher activity regions cause spillovers to neighboring lower activity regions. In order to maintain average intensity, the algorithm must not change the total intensity, only reallocation should occur. The total PET activity in the boundary region is distributed over a larger volume, reaching neighboring CT segments. The basic idea is then that to regain the original activity, these spillovers must be removed and their additional (residual) intensity must be added to their corresponding origin. To do so, spillover regions must be distinguished first Identifying Spillover Regions A simple approach is to simply check if the innermost mean intensity is less than the outermost (voxels with smallest labels) average. However, local features may cause false classification. For example, if the outer intensity average is near the inner one, spillover areas would be incorrectly detected due to intensity fluctuations (the result of noise in PET reconstruction). This may be solved with applying a threshold to the difference, identifying spillover areas where the outer intensity is larger than the inner one s multiplied with some predefined constant. Instead, another approach is used to avoid additional parameters. One step in the à trous [12] wavelet algorithm gives the high-frequency components of the CT image. Due to the method itself, the result is negative at areas with lower activity than its surroundings, exactly where spillover effect occurs. Applying the above method to the segmented CT image (containing segment labels), non-zero values are ensured as segment labels are discrete integers, and boundary map is created in a way that corresponds to the low-pass filter of the algorithm, ensuring that positions marked by the boundary map have positive absolute value in this gradient-like map. Consequently, classifying based on the sign of CT high frequency components perfectly identifies spillover areas. Note that the segment labels must be in order according to average PET intensity for this method to work, but that can be easily achieved during the segmentation process or by a simple initial reordering Reallocation Having identified spillover areas, the next step is to reallocate intensities. In 1D it is a fairly simple task. At each boundary area, spillover residue values 8

10 Figure 4: Illustration of the step in à trous creating the gradient-like map. At a CT segment boundary the filtered (dashed line) signal is subtracted from the original. The result is positive where the intensity was higher (blue), and negative where the intensity was lower (red). The former is considered as the source region, while the latter is the spillover region. must be added to the other source side. Since spillover values are determined by the whole boundary region, aggregation must include every value between the two innermost-flagged position. Residue values are calculated by subtracting the innermost value from spillover values. The sum of residues and the sum of values in the higher-intensity part gives the sum of original PET activity in the higher-intensity region. Dividing the sum by the number of positions in the source area results in the original activity. This is the corrected value of each position in the source boundary area. On the other side, every boundary position in spillover regions gets the value of the innermost one. This way the spillover values are reallocated to their source, and both segment is corrected to their original intensity. Observe that distance map assignments are symmetrical to the innermost position. When neighboring positions are grouped by their distance labels it may occur that values from both side in the source region are summed, hence distorting the result, especially when the region acts as source (has higher intensity) at the current edge, but gains spillover values at the other side (having lower intensity). For this reason an increasing size window is used. The iteration stops when the innermost label does not change within this square window. It means then that the labels are not increasing, the other side of the segment is in the window. For practical reasons, a minimum number of innermost labeled positions are required to stop to ensure that enough information is collected to determine the innermost average. This method works even when the source region consists of only one label, one position. In higher dimensions (2D and with the same concept in 3D), it is hard to find corresponding spillover areas and sources, due to the interference of 9

11 (a) (b) Figure 5: Illustration of the 1D correction algorithm in a general case (a). Positions are labeled by distance map assignments (numbers at the bottom), the innermost is marked by green. The correction transfers spillover residue values (red striped area) to the source. CT segment boundaries are displayed as double lines, and the current boundary is red. Gained values in the result (b) are colored purple. 10

12 Figure 6: Illustration of how pairing may be done on a symmetrical axial configuration (the window is green). Sources are blue, and spillover pixels are red. The pair has darker color. neighboring source pixels (voxels). So instead an approximation is used. For every pixel, the neighboring average original activity is calculated. The idea is that in a square (cube) area (volume) the sum of spillover residue values is the same as the source pixels corresponding spillover sum. Source pixels on the edge of this window cause spillover on pixels outside, though this effect can be neglected for the following reasons. Assume now that the current pixel to be corrected lies on a long straight axial boundary line (plane), where the two segment is homogeneous, and let the neighboring area be symmetrical to the pixel and the boundary line. Here every source pixel can be paired with a spillover pixel according to the window s symmetry on the boundary line. The number of spillover pixels related to the source pixel of the pair (meaning the pixels affected by spillover) is the same as vica-versa; the amount of additional intensity of the spillover pixel can be paired with the loss of intensity related to the source pixel of the chosen pair. If the window is not too large, the assumption is not far from reality, and the summing method is correct. Of course the aggregation window must contain the innermost values just as in the 1D case, and interference with the other side of the segment must be avoided. Therefore the same iterative window size method is used. Another non-trivial task is to calculate the residual spillover value for each pixel. It is much simpler if instead of summing, we use the mean intensity for each distance label. This way the problem is reduced to the 1D case. The correctness can be seen with the assumption that PET activity does not change abruptly within CT segments, so that pixels in the window can be replaced with 11

13 the mean of their distance label. Recall that the distance map assigns labels to pixels according to their distance to the nearest boundary. This way, mean pixel intensities within each distance labels are near constant as signal loss/gain is gradual at edges. Therefore distance labels determine loss of signal/spillover residual value. In addition, this method resolves the problem with non-axial configuration, where different amount of source and spillover pixels may be found. With the averaging, the number of pixels each side are eliminated, and this way they are not affecting the result. For the same reasons as before, it s hard to determine which pixels should be set to the calculated mean activity. In 1D it was trivial that every position in the source area is set to the calculated intensity, as no interference could occur. In this case, however, values are calculated for a square window around a point on the boundary line (surface in 3D). One could think that applying the same method for inner pixels works the same way, but the position of the window is fixed by the boundary line as values between the two innermost regions are essential for the algorithm. This essentially disallows using the same method. Therefore the idea is that the above method is used only for pixels on the very edge of CT segments, and the values of inner pixels are an interpolation of edge values. Interpolation is done by a distance weighted average, avoiding interference between spillover and source areas with counting them separately. This way, spillover origins are determined evenly without loss of correctness as PET activity is assumed to change non-abruptly in regions. 3.4 Multiple Boundaries So far only boundaries between two segments were discussed. Handling multiple boundaries within the aggregation window requires some alteration of the algorithm. First of all, multiple spillover areas are affected by one source, possibly in several segments. In order to correct the source value, aggregation should involve all of these spillover areas. Another difference is that the same segment may have both spillover and source areas, depending on its relative intensity to surrounding other segments. Therefore grouping by distance labels must be done separately for spillover and source pixels not to interfere with each other. To ensure that no unrelated boundary interferes with calculations, the iterative window size method must be stopped when the innermost label does not change for segments involved, determining innermost labels separately. Different window sizes for segments does not pose a problem, since the correction is reduced to the 1D case because of the label-based grouping. In this multiple boundary case, multiple sources jointly determine spillover values nearby. This means that the surrounding residual values must be dis- 12

14 tributed in accordance with their sources contribution. More specifically, as the amount of intensity transferred from one higher-intensity to a lower-intensity segment depends on the intensity difference, the total residual value must be split among contributing sources with their average intensity taken into account. Source pixels are determined the same way as earlier, by checking the sign of the high-frequency components of the segmented CT image. This way, the source pixels relative frequency in the aggregation window weighted by region intensities gives a good approximation of the amount of contribution to spillover areas. Spill-out and spill-in at window edges are neglected as they compensate each other the same label-grouping method is used as earlier. Residual values are also determined with the same concept, but with the difference that innermost average of spillover values are used to avoid interference with other sources in the same segment. In addition, when determining spillover areas, only other segments are considered, as the segment is assumed to be homogeneous in this small window, thus no intra-region spilling could occur. 4 Results Results were obtained by GATE simulation of the micro Derenzo phantom with Mediso s NanoPET/CT [13] geometry at 128 x 144 x 144 voxel resolution. Segmented CT image was provided: distinct segments for tubes, the container, and the background. Correction results are presented for both algorithms as well as line profiles (generated via Amide [14]). Correction time was nearly half a minute for the simple algorithm, and approximately a minute for the enhanced version on a Core i5 2.27GHz processor. Simple correction may seem visually more satisfying, as it seems more homogeneous. However due to reasons explained earlier it is less relevant in qualitative measurements. 13

15 (a) (b) (c) (d) Figure 7: The Derenzo phantom, YZ and XZ slices (a) 1s GATE simulation (b) Input CT image (c) Boundary map (d) Distance labels 14

16 (a) (b) Figure 8: Corrected Derenzo phantom (a) Simple correction by taking innermost averages (b) Enhanced correction that preserves average intensities (a) Figure 9: Line profiles of correction outputs. Magenta refers to the original PET activity, yellow represents the simple correction output and the enhanced version is drawn with teal. Notice the different scales, the average intensity is significantly lower than the PET spikes, and innermost values. 15

17 This work has been supported by the scientific program of the Development of quality-oriented and harmonized R+D+I strategy and functional model at BME (Project ID: TÁMOP-4.2.1/B-09/1/KMR ). References [1] O. G. Roussel, Y. Ma, and A. C. Evans, Correction for partial volume effects in pet: Principle and validation, Journal of Nuclear Medicine, vol. 39, pp , [2] J. M. Ollinger and J. A. Fessler, Positron-emission tomography, IEEE Signal Processing Magazine, pp , [3] M. Soret, S. L. Bacharach, and I. Buvat, Partial-volume effect in pet tumor imaging, Journal of Nuclear Medicine, [4] N. Boussion, M. Hatt, F. Lamare, Y. Bizais, A. Turzo, C. C.-L. Rest, and D. Visvikis, A multiresolution image based approach for correction of partial volume effects in emission tomography, in Physics In Medicine And Biology, pp , Institue Of Physics Publishing, [5] M. Shidahara, C. Tsoumpas, A. Hammers, N. Boussion, D. Visvikis, T. Suhara, I. Kanno, and F. E. Turkheimer, Functional and structural synergy for resolution recovery and partial volume correction in brain pet, Elsevier NeuroImage, [6] F. P. Figueiras, X. Jimenez, D. Pareto, and J. D. Gispert, Partial volume correction using an energy multiresolution analysis, in IEEE Nuclear Science Symposium Conference Record, pp , [7] W. P. Segars, B. M. W. Tsui, A. J. D. Silva, and L. Shao, Ct-pet image fusion using the 4d ncat phantom with the purpose of attenuation correction, in 2002 IEEE Nuclear Science Symposium Conference Record. [8] P. Kaur, D. I. M. S. Lamba, and D. A. Gosain, A robust method for image segmentation of noisy digital images, in 2011 IEEE International Conference on Fuzzy Systems, IEEE, [9] Z. Kato, Markovian Image Models and their Application in Image Segmentation. PhD thesis, University of Szeged, Hungary, [10] D.-Q. Zhang and S.-C. Chen, A novel kernelized fuzzy c-means algorithm with application in medical image segmentation, Elsevier Artificial Intelligence in Medicine, pp , [11] G. X. Ritter and J. N. Wilson, Handbook of Computer Vision Algorithms in Image Algebra. CRC Press, second ed., [12] J.-L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis, The Multiscale Approach. Cambridge University Press, [13] Mediso, Nanopet/ct in-vivo preclinical imager. molecular-imaging/nanopet-ct, [14] Amide, A medical imaging data examiner. 16