Batenburg, K.J. (Joost)

Roelandts, T. (Tom)

Sijbers, J. (Jan)

English

CWI's Institutional Repository

Visualizing the Segmentation Error of a Tomogramusing the Residual Projection ErrorTom Roelandts, Kees Joost Batenburg and Jan SijbersAbstract—Tomographic reconstructions are often segmentedto quantify structure parameters. However, to our knowledge,no efforts have yet been made to visualize the accuracy of thesegmentation result. This paper introduces a way to visualize thesegmentation error, based on the residual projection error, whichis the difference between the recorded data and the forwardprojection of the segmented tomogram. From the residual projec-tion error, a segmentation error tomogram is reconstructed. Thiserror tomogram allows to detect errors in the gray levels of thesegmented tomogram, or to discriminate between reconstructionartifacts and actual features of the scanned object. The proposedtechnique is independent of the algorithms that were used tocreate and segment the tomogram.Index Terms—Image segmentation, error reconstruction.I. INTRODUCTIONIn many applications of tomography, the final tomographicreconstruction (the tomogram) must be segmented before theresults can be analyzed. Segmentation amounts to the classifi-cation of image pixels into distinct classes, based on similaritywith respect to some characteristic. Image segmentation isa well established field, and a range of methods has beendeveloped, using diverse techniques such as global or localthresholding, region growing, and clustering [1], [2].Most image segmentation methods are not specific towardsthe modality that was used to acquire the image. As aresult, such methods do not exploit the raw data (X-rayradiographs in case of CT imaging) from which the image wasreconstructed. Recently, global and local thresholding methodswere proposed that do use the projection data to improvethe selection of threshold parameters [3], [4]. In addition,reconstruction methods were recently developed in which thesegmentation was directly incorporated into the reconstructionalgorithm [5]–[8].In this paper, the projection data is used to examine thequality of the segmentation. The proposed method assumesthat the scanned object consists of homogeneous regions, andthen determines how well a given segmentation represents thatobject. Moreover, the method can also be used to test forobject homogeneity, since it generates large errors for non-homogeneous objects. The technique is independent of thereconstruction and segmentation algorithms.This work was financially supported by the FWO (The Research Founda-tion - Flanders, Belgium) and by the NWO (the Netherlands Organisation forScientific Research - The Netherlands, research programme 639.072.005).Tom Roelandts, Kees Joost Batenburg and Jan Sijbers are with the IBBT-Vision Lab, University of Antwerp, Universiteitsplein 1, 2610 Wilrijk, Bel-gium.Kees Joost Batenburg is with the Centrum Wiskunde & Informatica (CWI),Science Park 123, 1098 XG Amsterdam, The Netherlands.Corresponding author: Tom Roelandts, E-mail: tom.roelandts@ua.ac.be.The structure of this paper is as follows. In Section II,the method is introduced. Section III describes the simulationexperiments that were performed to validate it. The results arediscussed and a conclusion is reached in Section IV.II. METHODThe projection process in tomography can be modeled as alinear operator that is determined by the projection geometry.This leads to a system of linear equations,Wx = p, (1)where p ∈ Rm contains the projection data and x ∈ Rncorresponds to the image. The linear operator is representedby the m×n matrix W , the projection matrix. An approximatesolution xˆ ∈ Rn of (1) can then be computed, in practice oftenby minimizing some norm ‖Wx−p‖. The image xˆ can thenbe segmented.A segmentation method essentially partitions the pixels ofan image into sets Y1, . . . , Yd, where d is the number of classes(or gray levels) in the segmented image. Since we assumethat the scanned object consists of homogeneous regions, thesegmented image should have the same gray level for all pixelsin a set Yk. From xˆ, we create a segmented image s ∈ Rnby assigning a gray level ρk ∈ R to all pixels in a set Yk, foreach k ∈ {1, . . . , d}. The values of ρ1, . . . , ρd are not known apriori. Moreover, most of the segmentation algorithms that arecommonly used, do not have these gray levels as an output.In such case, we use the mean of all pixel values in a classas an estimate,ρk =1|Yk|∑yˆ∈Ykyˆ, for each k ∈ {1, . . . , d}. (2)We now define the residual projection error. The segmentedimage s is forward projected to give ps ∈ Rm, so ps =Ws.The residual projection error e ∈ Rm is then defined ase = p− ps. (3)This projection error is then reconstructed by solving thesystem Wy = e, where y ∈ Rn corresponds to the (unknown)error image. This results in an approximate solution yˆ ∈ Rn,the reconstructed residual error.As mentioned in the introduction, this procedure does notdepend on any particular reconstruction or segmentation algo-rithm. To create the segmented reconstructions, we use filteredbackprojection (FBP), with a Ram-Lak filter, as an exampleof an analytical method. We use the simultaneous iterativereconstruction technique (SIRT) [9] as an example of anThe second international conference on image formation in X-ray computed tomography Page 293iterative method. The reconstructions are globally thresholdedusing Otsu’s method [10]. The residual projection error isreconstructed using SIRT.III. SIMULATION EXPERIMENTSAll simulation experiments were performed on a squarereconstruction grid of 512×512 pixels. Three phantom imageswere created, as shown in Fig. 1. The size of each phantom is2048 × 2048 pixels. The phantoms are much larger than thereconstruction grid, to reduce the effect of the pixelation onthe reconstructions.(a) Phantom 1 (b) Phantom 2 (c) Phantom 3Fig. 1: Phantom images: (a) two gray levels, (b) three graylevels, (c) continuous grayscale.A. Optimally Segmented ReconstructionsIn the first simulation experiment, the residual projectionerror was reconstructed for a dataset with no noise and alarge number of projections. This demonstrates the result of theproposed technique with minimal hindrance from reconstruc-tion artifacts. From Phantom 1, which is a binary phantom,a synthetic dataset was created using 360 parallel beamprojections, evenly spaced at 0.5◦ intervals. A detector with512 pixels was used, to simulate the practical situation wherethe detector pixel size equals the width of the reconstructiongrid. The “ideal” segmented reconstruction was approximatedby binning Phantom 1 to a 512×512 grid, and then segmentingit using a threshold of 0.5.The “ideal” segmented reconstruction was then forwardprojected, and the difference with the original projectionscomputed. This results in the residual projection error, whichwas then reconstructed using 300 iterations of SIRT. Fig. 2ashows the result. The true error (Fig. 2b) was computed asthe difference between the original phantom (Fig. 1a) and anupscaled version of the ideal segmented reconstruction. Forcomparison, the true error is shown downscaled to the sameresolution as Fig. 2a. The true error is only nonzero at theedges of the different structures, where errors are unavoidable,since the original phantom is not pixelated at the size of thereconstruction. The reconstructed residual error (Fig. 2a) isvery close to the true error.B. Reconstructions with Homogeneous RegionsThe second simulation experiment is based on Phantom 2(Fig. 1b), which contains three gray levels. Reconstructions ofPhantom 2 should consist of homogeneous regions, since theground truth object is divided into several such regions (com-pare with Phantom 3 (Fig. 1c), where this is clearly not the  −0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25(a)  −0.3−0.2−0.100.10.20.3(b)Fig. 2: (a) Reconstructed residual error for Phantom 1. (b) Trueerror for Phantom 1, at the resolution of Fig. 2a.case). From Phantom 2, a synthetic dataset was created using90 parallel beam projections, evenly spaced at 2◦ intervals.A detector with 512 pixels was again used. This syntheticdataset was then reconstructed twice, once using FBP andonce using 300 iterations of SIRT. These reconstructions werethen segmented using Otsu’s method. The final segmentedreconstructions are shown in Fig. 3. Due to the relatively lownumber of projection angles, a number of streaks are visible,especially in the FBP reconstruction of Fig. 3a.(a) FBP (b) SIRTFig. 3: Reconstructions of Phantom 2, segmented using Otsu’smethod.The segmented reconstructions were then forward projected,and the difference with the original projections computed,resulting in the residual projection error. This was then recon-structed using 300 iterations of SIRT. Figs. 4a (FBP) and 5a(SIRT) show the result. For comparison, the true error is alsoshown, in Figs. 4b and 5b. As before, there seems to be aclose correspondence between the reconstructed error and thetrue error. The streaks from the FBP reconstruction are clearlyvisible in the reconstructed residual error (Fig. 4a). Hence, theerror tomogram can be used to discriminate between this typeof artifact and actual features of the scanned object. Apart fromthese streaks, the largest errors are again situated at the edgesof the different structures. Figs. 4a and 5a further suggest thatthe gray levels of the different objects are underestimated,since the error is clearly positive and relatively uniform insidethe objects. This is also confirmed by the true error.Table I shows this quantitatively. It contains an overviewof a number of parameters that were determined from thereconstructions, while the true gray levels were taken fromPhantom 2. The computed gray levels were determined fromthe segmented reconstructions (Fig. 3), from which they werePage 294 The second international conference on image formation in X-ray computed tomography  −0.2−0.15−0.1−0.0500.050.10.150.2(a)  −0.2−0.15−0.1−0.0500.050.10.150.2(b)Fig. 4: (a) Reconstructed residual error for the FBP recon-struction of Phantom 2. (b) True error for that reconstruction,at the resolution of Fig. 4a. The scales have been limited to[−0.2, 0.2] to keep the smaller errors from disappearing.  −0.2−0.15−0.1−0.0500.050.10.150.2(a)  −0.2−0.15−0.1−0.0500.050.10.150.2(b)Fig. 5: (a) Reconstructed residual error for the SIRT recon-struction of Phantom 2. (b) True error for that reconstruction,at the resolution of Fig. 5a. The scales have been limited to[−0.2, 0.2] to keep the smaller errors from disappearing.computed using (2). The gray level errors were computedfrom Figs. 4a and 5a, using the same procedure. To verifythe accuracy of the gray level errors, the last lines for eachalgorithm in Table I show the corrected gray levels, i.e., thesum of the computed gray levels and the gray level errors. Thelast lines show that the gray level error is a good estimate ofthe difference between the true gray level and the computedgray level, and that it can be used to correct the computed graylevel. This result could potentially be improved even furtherby recreating the reconstructed residual error, starting with asegmented reconstruction that uses the corrected gray levels.TABLE I: Estimated and true gray levels, noiseless dataset.Algorithm Parameter ρ1 ρ2 ρ3FBP True gray level 0.000 0.502 1.000Computed gray level −0.004 0.476 0.979Gray level error 0.003 0.018 0.017Corrected gray level −0.001 0.494 0.996SIRT True gray level 0.000 0.502 1.000Computed gray level 0.002 0.491 0.994Gray level error −0.002 0.010 0.009Corrected gray level −0.000 0.502 1.002We also ran these experiments with Poisson noise appliedto the synthetic dataset. The results are shown in Figs. 6and 7, and in Table II. The quality of the FBP reconstruction(Fig. 6a) has suffered more from the noise than that of theSIRT reconstruction (Fig. 7a). In the reconstructed residualerror (Figs. 6b and 7b), the noise from the original projectionsis apparent. The results in Table II, however, are still largelycomparable with those from Table I. This means that the graylevel errors in Figs. 6b and 7b, which seem drowned by thenoise, are still quite accurate when averaged. This implies thatwe can decide, from the reconstructions of the residual errorin Figs. 6b and 7b, that the segmentation from Fig. 7a is moreaccurate than that from Fig. 6a.(a)  −0.2−0.15−0.1−0.0500.050.10.150.2(b)Fig. 6: (a) FBP Reconstruction of Phantom 2, segmented usingOtsu’s method. Poisson noise was applied to the syntheticdataset. (b) Reconstructed residual error. The scale has beenlimited to [−0.2, 0.2] to make it the same as in Fig. 4.(a)  −0.2−0.15−0.1−0.0500.050.10.150.2(b)Fig. 7: (a) SIRT Reconstruction of Phantom 2, segmented us-ing Otsu’s method. Poisson noise was applied to the syntheticdataset. (b) Reconstructed residual error. The scale has beenlimited to [−0.2, 0.2] to make it the same as in Fig. 5.TABLE II: Estimated and true gray levels, dataset with noise.Algorithm Parameter ρ1 ρ2 ρ3FBP True gray level 0.000 0.502 1.000Computed gray level −0.006 0.467 0.979Gray level error 0.002 0.022 0.020Corrected gray level −0.004 0.489 0.999SIRT True gray level 0.000 0.502 1.000Computed gray level 0.002 0.491 0.994Gray level error −0.002 0.010 0.009Corrected gray level −0.000 0.502 1.003C. Reconstructions with Non-Homogeneous RegionsThe last simulation experiment is based on Phantom 3(Fig. 1c), in which some of the homogeneous objects ofThe second international conference on image formation in X-ray computed tomography Page 295Phantom 2 are replaced by objects with a continuously varyingintensity. Reconstructions of Phantom 3 are not expectedto consist of homogeneous regions, since the ground truthobject is not a collection of such regions. From Phantom 3, asynthetic dataset was again created using 90 parallel beamprojections, evenly spaced at 2◦ intervals. This syntheticdataset was reconstructed using FBP and SIRT, and segmentedusing Otsu’s method. The result is shown in Fig. 8.(a) FBP (b) SIRTFig. 8: Reconstructions of Phantom 3, segmented using Otsu’smethod.As before, the residual projection error was computed andreconstructed using 300 iterations of SIRT. The result is shownin Figs. 9a (FBP) and 10a (SIRT). Both figures show that thereconstructed residual error is useful to discriminate betweenobjects that can be reconstructed as a homogeneous region (thehomogeneous objects from Phantom 3), and those that cannot(the objects with continuously varying intensity, which havelarge structured errors in Fig. 9a and 10a). The reconstructedresidual error is again quite close to the true error (Figs. 9band 10b).  −0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25(a)  −0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25(b)Fig. 9: (a) Reconstructed residual error for the FBP recon-struction of Phantom 3. (b) True error for that reconstruction,at the resolution of Fig. 9a.IV. DISCUSSION AND CONCLUSIONIf tomograms are segmented without exploiting the projec-tion data, which is often the case, the reconstructed residualerror can still provide information on the quality of thesegmentation.The result for the ideal segmented reconstruction fromSection III-A shows that the reconstructed residual error is agood approximation of the true error. For objects that consistof homogeneous regions, the experiments from Section III-Bshow that the proposed method can detect errors in the gray  −0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25(a)  −0.25−0.2−0.15−0.1−0.0500.050.10.150.20.25(b)Fig. 10: (a) Reconstructed residual error for the SIRT recon-struction of Phantom 3. (b) True error for that reconstruction,at the resolution of Fig. 10a.levels of the segmented reconstruction, even if those are quitesmall (Fig. 5), and in the presence of noise. The experimentfrom Section III-C shows that the technique can discriminatebetween homogeneous and non-homogeneous objects, withoutthis knowledge being available a priori. Several experimentsalso show that artifacts that are visible in the segmented re-construction, are also visible in the reconstructed residual error(e.g., Figs. 4a and 9a). Together, the simulation experimentsdemonstrate that the reconstructed residual error can be usedas a visual map of the errors in the segmentation.Reconstructing the residual projection error is a simpleway to visualize the segmentation error of a tomogram. Thetechnique is trivial to implement, since the only necessarytools are a forward projector and a reconstruction algorithm.The computational cost is modest, since only a single forwardprojection and a single reconstruction are needed.REFERENCES[1] N. R. Pal and S. K. Pal, “A review on image segmentation techniques,”Pattern Recogn., no. 9, pp. 1277–1294, 1993.[2] D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical imagesegmentation,” Annu. Rev. Biomed. Eng., pp. 315–337, 2000.[3] K. J. Batenburg and J. Sijbers, “Optimal threshold selection for tomo-gram segmentation by projection distance minimization,” IEEE Trans.Med. Imag., vol. 28, no. 5, pp. 676–686, 2009.[4] W. van Aarle, K. J. Batenburg, and J. Sijbers, “Optimal threshold se-lection for segmentation of dense homogeneous objects in tomographicreconstructions,” IEEE Trans. Med. Imag., vol. 30, no. 4, pp. 980–989,2011.[5] K. J. Batenburg, S. Bals, J. Sijbers, C. Ku¨bel, P. A. Midgley, J. C.Hernandez, U. Kaiser, E. R. Encina, E. A. Coronado, and G. Van Ten-deloo, “3D imaging of nanomaterials by discrete tomography,” Ultrami-croscopy, vol. 109, no. 6, pp. 730–740, 2009.[6] K. J. Batenburg and J. Sijbers, “DART: A practical reconstructionalgorithm for discrete tomography,” IEEE Trans. Image Process., vol. 20,no. 9, pp. 2542–2553, 2011.[7] T. Roelandts, K. J. Batenburg, and J. Sijbers, “PDART: A partiallydiscrete algorithm for the reconstruction of dense particles,” in Proc.of the 11th International Meeting on Fully Three-Dimensional ImageReconstruction in Radiology and Nuclear Medicine (Fully 3D), Potsdam,Germany, 2011.[8] T. Roelandts, K. J. Batenburg, E. Biermans, C. Ku¨bel, S. Bals,and J. Sijbers, “Accurate segmentation of dense nanoparticles bypartially discrete electron tomography,” Ultramicroscopy, In Press,doi:10.1016/j.ultramic.2011.12.003.[9] J. Gregor and T. Benson, “Computational analysis and improvement ofSIRT,” IEEE Trans. Med. Imag., vol. 27, no. 7, pp. 918–924, 2008.[10] N. Otsu, “A threshold selection method from gray-level histograms,”IEEE Trans. Syst., Man, Cybern., vol. 9, no. 1, pp. 62–66, 1979.Page 296 The second international conference on image formation in X-ray computed tomography

Visualizing the Segmentation Error of a Tomogram using the Residual Projection Error

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