This article outlines a procedure for speeding up segmentation of images using active mesh systems. Active meshes and other deformable models are very popular in image segmentation due to their ability to capture weak or missing boundary information; however, where strong edges exist, computations are still done after mesh nodes have settled on the boundary. This can lead to extra computational time whilst the system continues to deform completed regions of the mesh. We propose a local termination procedure, reducing these unnecessary computations and speeding up segmentation time with minimal loss of quality

Dixon, Martin

Laissue, P. Philippe

Nelson, Carl J.

Obara, Boguslaw

Reyes-Aldasoro, Constantino Carlos

Slabaugh, Greg

English

Durham Research Online

NELSON ET AL.: LOCAL TERMINATION OF ACTIVE MESHES 1Speeding Up Active Mesh Segmentationby Local Termination of NodesCarl J. Nelson1carl.nelson@dur.ac.ukMartin Dixon2martin.dixon@durham.ac.ukP. Philippe Laissue3plaissue@essex.ac.ukBoguslaw Obara1boguslaw.obara@dur.ac.uk1 School of Engineering & ComputerScienceDurham University2 School of Biological and BiomedicalSciencesDurham University3 School of Biological SciencesUniversity of EssexAbstractThis article outlines a procedure for speeding up segmentation of images using activemesh systems. Active meshes and other deformable models are very popular in imagesegmentation due to their ability to capture weak or missing boundary information; how-ever, where strong edges exist, computations are still done after mesh nodes have settledon the boundary. This can lead to extra computational time whilst the system continuesto deform completed regions of the mesh. We propose a local termination procedure, re-ducing these unnecessary computations and speeding up segmentation time with minimalloss of quality.1 IntroductionDeformable models [6] are a group of algorithms used to segment images, particularly im-ages with weak or missing boundaries. Such deformable models, also known as active mod-els and snakes, are used heavily in biomedical imaging segmentation where low contrast,high noise or obscured edges make other segmentation techniques unsuitable. Deformablemodels come in several forms and have been used for multiple modalities and scenarios(see [5] for a comprehensive review). Here we introduce and briefly describe the field lim-ited to active meshes, three-dimensional polygonal-faced meshes that can be deformed in adiscrete manner to segment objects of interest [7].1.1 Active Meshes: a Quick BackgroundActive meshes are a subset of deformable models that commonly use a triangular-faced meshto segment an object of interest from an image with weak or missing boundaries (see [4] foran early example). The use of a mesh with distinct nodes, or vertices, and a known connectiv-ity, through face or edge information, allows computationally simple and speedy calculationsto be performed over the whole mesh. To find the segmentation shape the mesh is deformedin one of several ways. The use of physics-based forces allows a new position to be calcu-lated at each iteration by Euler-Lagrange mechanics [8]; however, there have been concernsc© 2014. The copyright of this document resides with its authors.It may be distributed unchanged freely in print or electronic forms.2 NELSON ET AL.: LOCAL TERMINATION OF ACTIVE MESHESthat optimality of such systems is not guaranteed [1]. A more common system is to provideeach vertex with an energy and, for each iteration, identify some trial points for the vertexto move to; one of these points is then accepted by some optimisation method, the simplestof which is a greedy algorithm [10]. Many of these systems require a balloon force [3] oranother force (e.g. [11]) to attract the shape towards the edges from any initialisation. Aswith many techniques each variant has its advantages and drawbacks and the optimal choicefor a particular case is dependent on the experiment and researcher.1.2 Termination of MeshesDeformable models tend to work on an iterative system with each iteration deforming themodel to minimise energies or balance forces, eventually the mesh reaches an optimal stateand the desired object is segmented. With many optimisation techniques there may be nodefinite and/or optimal termination; as such, many systems are provided with an explicitstopping criterion in order to terminate the process. One key feature of termination criteriathat are mentioned in the literature is that they focus on the overall mesh, i.e. they terminatein a global fashion, e.g. the percentage of ’inactive’ vertices [8] or a maximum number ofiterations [9]. We have been unable to find any articles in the literature that use a system oflocal vertex termination, that is to say, each vertex is deformed as part of the mesh until itreaches some independent termination criteria. In this article we propose a heuristic systemfor local vertex termination during deformation. We show that this method of terminationcan speed up the overall process of segmentation by testing the algorithm on synthetic dataand also demonstrating the system on real 3D images.2 MethodsFor the purpose of this article we have used a simple balloon-driven, energy minimisationmesh (based on [2]). The mesh is initiated as an icosahedron of user-defined size. For each it-eration, every vertex is assigned a new ‘more optimal’ position selected, by a simple greedyalgorithm, from a 3D neighbourhood. A steepest gradient descent method is employed tolink the image gradient to the size of the steps available. After each iteration of deformationthe mesh undergoes local resampling, again based on [2]. During each iteration, all verticesare checked to see if they should be terminated (described fully below). Deformation contin-ues until a termination criterion is met; this criterion will be described for each experimentin Results. All codes were written in MATLAB 2012b (The MathWorks, Inc., US) on aWindows 7 64-bit PC running an Intel Core i5-2320 CPU (3.0 GHz) with 6GB RAM.2.1 Local Termination of VerticesIn order to terminate vertices we identify, at the end of each deformation, a list of verticesto be considered ‘terminated’. During the next iteration we only deform vertices that are notterminated, decreasing computation time per iteration. Our system checks whether or nota vertex should be considered terminated after every iteration; this prevents a vertex beingturned off due to, for example, a large noise signal, and being prevented from re-initiation asthe mesh around it deforms. We define whether or not a vertex should be terminated usingthe following cost function,Cv = αCI +βCG + γCT , (1)where α,β and γ are user-defined parameters such that α + β + γ = 1. The first twoterms, CI and CG, are the value of the normalised image and gradient magnitude image,NELSON ET AL.: LOCAL TERMINATION OF ACTIVE MESHES 3respectively, calculated through trilinear interpolation. Finally, CT is one if the vertex wasterminated during the last iteration and zero otherwise. We note that more advanced costfunctions could look at the history or neighbourhood of the vertex in more detail but wehave, so far, only looked at basic, faster cost functions. In order to make this cost-basedmethod more robust we use a combination of the cost at the vertex and the average cost ofits direct neighbourhood. So Tv, the choice to terminate a vertex or not, is given by,Tv ={TRUE if µCv +(1−µ)∑ j C jNv> c,FALSE otherwise;(2)here, Nv is the size of the neighbourhood around the vertex, v, C j is the cost at a neigh-bour, j, and µ is a user-defined parameter. The user-defined cost-level, c, is a thresholdabove which a point is terminated.2.2 Measure of QualityTo determine segmentation ‘quality’, Q, of synthetic images we have used a simple metriccomparing the segmented mesh to the binary ‘ground truth’ image, G. The resultant meshis voxelised, B, and the measure of quality is calculated as Q = B∩G/B∪G. If the meshextends outside of the ground truth the denominator increases and if the mesh has not fullyextended to the ground truth boundaries then the numerator decreases. An ideal segmentationwill always have a quality of one. Any segmentation that over or under segments will have aquality less than one; over segmentation and under segmentation in different regions cannotbalance out using this metric.3 ResultsIn order to clearly show when and where local termination may be effective for active meshsegmentation we have run experiments on five noise-free synthetic shapes. We also com-pare the results of globally terminated and locally terminated segmentation on real data ofdehisced mature pollen grains.3.1 Synthetic DataWe have chosen five synthetic shapes to represent various scenarios found in biological andbiomedical images: a sphere, a cube, a hollow, rectangular cuboid, a tube and a weeble (acone placed on top a hemisphere). We have chosen to demonstrate on noise-free data toshow the differing effect of local termination on different shape scenarios, where we feelthe difference is most significant. User-chosen parameters for local termination were usedfor each shape. The origin for segmentation was user-chosen based on manual selection ofan approximate centroid. For all the control segmentations, i.e. segmentations using globaltermination, we terminated the procedure based upon the Haussdorf distance before andafter each iteration. For each step the Haussdorf distance between the set of vertices beforedeformation and resampling and after were compared. If the Haussdorf distance was belowa user-defined value then the shape was considered to be terminated.Table 1 shows the time taken to segment and final segmentation quality for these shapesunder global and local termination procedures. As can be seen there is a dramatic decreasein segmentation time with only a moderate loss of quality. We note that shapes that arefairly round or with a surface fairly evenly distributed from the point of initiation, i.e. theweeble and sphere, reap the smallest benefit in terms of time. The very slight increase in4 NELSON ET AL.: LOCAL TERMINATION OF ACTIVE MESHESShape (Vertices) Global Termination Local Termination ReductionTime(s) Quality Time(s) Quality Mean Time QualitySphere (4,342) 186.1(0.1) 0.9564 70.0(0.4) 0.9595 62.4% -0.3%Cube (4,281) 336.0(0.7) 0.9241 69.8(0.2) 0.8730 79.2% 5.5%Weeble (6,606) 251.4(0.6) 0.9589 118.2(0.5) 0.9647 53.0% -0.6%Tube (6,617) 265.5(0.7) 0.9580 40.2(0.07) 0.9287 84.9% 3.1%Cuboid (3,694) 124.8(0.3) 0.9142 59.3(0.03) 0.8738 52.5% 4.4%Table 1: Table showing reduction in time and quality of segmentation using local terminationprocedures. Times are taken from an arithmetic mean of ten full segmentations; the numberin standard brackets is the standard error upon the mean to one significant figure. Number ofvertices, taken from global termination results, are shown in brackets after shape names.quality seen in some shapes is caused by the ’wiggle’ of vertices across the surfaces due tothe multidirectional nature of our method: as the Haussdorf distance measures the minimumdistance between one vertex before deformation and its closest match after, it is conceivablethat, as the mesh settles on the surface of the shape, vertices ’wiggle’ in a Brownian motion-like manner across the surface, leading to resampling of particular regions of the surface,hence keeping the Haussdorf distance high. We should also note that we tracked the qualityof the segmentation for each iteration (data not shown) and found that the maximum qualitywas not at the point of termination but a few iterations before termination due to the samereason. Figure 1 shows distance maps from the segmentation mesh of the hollow cuboid.The colouring at that area of the surface shows the average distance between that region ofthe mesh and the true boundary of the synthetic shape.3.2 Noisy DataIn order to test our local termination data on images more similar to those found in biologicaland biomedical modalities we applied pseudo-random noise to our hollow cuboid image.Under these noisy conditions the local termination method lead to a decrease of 50 (±10)%in run time with only a 7.9 (±0.5)% decrease in quality compared to the global terminationmethod. The average peak signal to noise ratio for the global termination trials (n=10) was73.1685 (±0.0002) and for the local termination trials (n=10) was 73.1688 (±0.0003); peaksignal to noise ratio was calculated as maximum(Is)/variance(N), where Is is the noise-free image and N the noise applied to the image.(a) (b)Figure 1: Local Termination of Vertices Reduces Segmentation Time with a Small Reduc-tion in Quality. (a) Distance difference map of segmentation using global termination. Thesegmented mesh is compared to a ground truth image and the colours represent the distancebetween the mesh and ground truth. (b) Distance difference map of local termination-basedsegmentation. The scale bar applies to both subfigures. Units are in voxels.NELSON ET AL.: LOCAL TERMINATION OF ACTIVE MESHES 5Local Termination... Time(s) Quality Reduction from GlobalMean Time Qualitywith Random Benefit 54(2) 0.9737(0.008) 79.4% -1.6%with Double Take 49.54(0.08) 0.9733 81.3% -1.6%Table 2: Table showing the results of variations upon the basic cost-based local terminationmethod. Note that the Quality of Local Termination with Random Benefit segmentation isstated as a mean (standard error) of ten repeats due to the use of a random value.3.3 Improving Segmentation QualityAs shown above, local termination of active mesh segmentation can lead to a dramatic re-duction in computing time; however, segmentation within biological and biomedical imagingrequires as high a precision as possible. Here we show two slight modifications of our localtermination scheme that help to improve the quality of segmentation. The results of thesemethods are shown in Table 2 for the synthetic tube; we have chosen this shape as there wasthe largest effect on reduction in time and a small decrease in quality. Tube-like structuresare also heavily prevalent in biomedical images.The first variation subtracts a pseudo-random value between 0 and 0.1 from the termi-nation cost before comparison to c; this causes some vertices that should be terminated thisiteration to be left on for at least one more iteration. As this is random it should lead to aslightly improved segmentation quality at the end result. With this method there is, however,a chance that no iteration will have all vertices terminated and, as such, a global terminationcriterion must be employed; for consistency we have chosen the Haussdorf distance as usedabove. In Table 2, this method is referred to as ’with Random Benefit’. The second methodignores the first termination of a vertex, i.e. a vertex must be classified as terminated for twoiterations before it is actually terminated. The final termination criterion for this method isthat when all vertices are terminated the segmentation is considered completed. In Table 2,this method is referred to as ’with Double Take’.Table 2 shows that both variations keep the reduction in segmentation time low but im-prove the segmentation quality; in fact, both methods improve the quality beyond the orig-inal segmentation system. It is likely that this difference in quality is due to the particularstopping criterion for global termination used throughout this paper. As noted above themaximum quality for segmentation is often not the final quality when using the Haussdorfdistance.3.4 Biological DataThe proposed approach has been tested and validated on 3D CLSM images of DAPI-staineddehisced mature pollen grains obtained from wildtype Arabidopsis thaliana plants of theColumbia (Col-0) ecotype (Figure 2). On average, the proposed local termination approachhas been able to speed up the overall process of pollen grains segmentation by 51%.4 ConclusionWe have demonstrated the effect of local termination of vertices on the segmentation ofshapes using active mesh systems. We have shown how the segmentation of long thin shapesis sped up dramatically by terminating those vertices that are ’complete’ and no longer de-forming the mesh in that region. We have chosen to show a basic cost function with someinteresting improvements upon the basic idea; however, we are currently looking into moreadvanced ways of deciding whether or not a vertex is completed and, in general, the idea6 NELSON ET AL.: LOCAL TERMINATION OF ACTIVE MESHES(a) (b) (c)Figure 2: Mature pollen grain: (a) 3D volume rendering. (b) 2D contour and (c) 3D meshrepresentation of the segmentation results. Scale bar corresponds to 10µm.of a cost function can be easily extrapolated for many imaging scenarios. Scenarios wherethis system will have the greatest effect are the segmentation of long, tubular structures,e.g. blood vessels, nerves or pathways such as the branchial network of the lungs, and bycombining local termination with a topologically adaptive active mesh we could segmentcomplicated tubular systems in a fraction of the time of the traditional methods. We arecurrently expanding application of the algorithm on biological and medical datasets.References[1] AA Amini, TE Weymouth, and RC Jain. Using dynamic programming for solvingvariational problems in vision. Pattern Anal., 12(9):855–67, 1990.[2] AJ Bulpitt and ND Efford. An efficient 3D deformable model with a self-optimisingmesh. Image Vis. Comput., 14:573–80, 1996.[3] LD Cohen. On active contour models and balloons. CVGIP, 53(2):211–8, 1991.[4] H Delingette. Adaptive and deformable models based on simplex meshes. Proc. 1994IEEE Work. Motion Non-rigid Articul. Objects, pages 152–7, 1994.[5] L He, Z Peng, B Everding, X Wang, CY Han, KL Weiss, and WG Wee. A compara-tive study of deformable contour methods on medical image segmentation. Image Vis.Comput., 26(2):141–63, 2008.[6] M Kass, A Witkin, and D Terzopoulos. Snakes: Active contour models. Int. J. Comput.Vis., 331:321–31, 1988.[7] D Molloy and PF Whelan. Active-meshes. Pattern Recognit. Lett., 21:1071–80, 2000.[8] J-Y Park, T McInerney, D Terzopoulos, and M-H Kim. A non-self-intersecting adaptivedeformable surface for complex boundary extraction from volumetric images. Comput.Graph., 25(3):421–40, 2001.[9] X Wang, L He, and WG Wee. Deformable Contour Method: A Constrained Optimiza-tion Approach. Int. J. Comput. Vis., 59(1):87–108, 2004.[10] DJ Williams and M Shah. A fast algorithm for active contours and curvature estimation.CVGIP, 1992.[11] C Xu and JL Prince. Snakes, shapes, and gradient vector flow. IEEE Trans. ImageProcess., 7(3):359–69, 1998.

Speeding Up Active Mesh Segmentation by Local Termination of Nodes

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Speeding Up Active Mesh Segmentation by Local Termination of Nodes

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