resolving overlapping convex objects in silhouette images ...arxiv:1906.01049v1 [cs.cv] 3 jun 2019....

Noname manuscript No.(will be inserted by the editor)

Resolving Overlapping Convex Objects in Silhouette Images byConcavity Analysis and Gaussian Process

Sahar Zafari ∗1 · Mariia Murashkina1 · Tuomas Eerola1 · Jouni Sampo2 · HeikkiKalviainen1 · Heikki Haario2

Received: date / Accepted: date

Abstract Segmentation of overlapping convex objects hasvarious applications, for example, in nanoparticles and cellimaging. Often the segmentation method has to rely purelyon edges between the background and foreground makingthe analyzed images essentially silhouette images. There-fore, to segment the objects, the method needs to be ableto resolve the overlaps between multiple objects by utilizingprior information about the shape of the objects. This paperintroduces a novel method for segmentation of clustered par-tially overlapping convex objects in silhouette images. Theproposed method involves three main steps: pre-processing,contour evidence extraction, and contour estimation. Con-tour evidence extraction starts by recovering contour seg-ments from a binarized image by detecting concave points.After this, the contour segments which belong to the sameobjects are grouped. The grouping is formulated as a com-binatorial optimization problem and solved using the branchand bound algorithm. Finally, the full contours of the objectsare estimated by a Gaussian process regression method. Theexperiments on a challenging dataset consisting of nanopar-ticles demonstrate that the proposed method outperformsthree current state-of-art approaches in overlapping convexobjects segmentation. The method relies only on edge in-formation and can be applied to any segmentation problemswhere the objects are partially overlapping and have a con-vex shape.

[1]Machine Vision and Pattern Recognition Laboratory (MVPR), Schoolof Engineering Science, Lappeenranta-Lahti University of TechnologyLUT, P.O. Box 20, 53851 LappeenrantaE-mail: [email protected]

[2]Mathematics Laboratory, School of Engineering Science,Lappeenranta-Lahti University of Technology LUT, P.O. Box 20,53851 Lappeenranta E-mail: [email protected]

Keywords segmentation · overlapping objects · convexobjects · image processing · computer vision · Gaussianprocess · Kriging · concave points · branch and bound.

1 Introduction

Segmentation of overlapping objects aims to address the is-sue of representation of multiple objects with partial views.Overlapping or occluded objects occur in various applica-tions, such as morphology analysis of molecular or cellularobjects in biomedical and industrial imagery where quanti-tative analysis of individual objects by their size and shapeis desired [53, 29, 59, 20]. In these applications for furtherstudies and analysis of the objects in the images, it is funda-mental to represent the full contours of individual objects.Deficient information from the objects with occluded oroverlapping parts introduces considerable complexity intothe segmentation process. For example, in the context ofcontour estimation, the contours of objects intersecting witheach other do not usually contain enough visible geometricalevidence, which can make contour estimation problematicand challenging. Frequently, the segmentation method hasto rely purely on edges between the background and fore-ground, which makes the processed image essentially a sil-houette image (see Fig. 1). Furthermore, the task involvesthe simultaneous segmentation of multiple objects. A largenumber of objects in the image causes a large number ofvariations in pose, size, and shape of the objects, and leadsto a more complex segmentation problem.

In this paper, a novel method for the segmentation ofpartially overlapping convex objects is proposed. The pro-posed method follows three sequential steps namely, pre-processing, contour evidence extraction, and contour esti-mation. The contour evidence extraction step consists of twosub-steps of contour segmentation and segment grouping.

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Fig. 1: Overlapping nanoparticles.

In the contour segmentation, contours are divided into sepa-rate contour segments. In the segment grouping, contour ev-idences are built by joining the contour segments that belongto the same object. Once the contour evidence is obtained,contour estimation is performed by fitting a contour modelto the evidences.

We, first, revise the comprehensive comparison of con-cave points detection methods for the contour segmenta-tion and the branch and bound (BB) based contour segmentgrouping method from our earlier articles [55] and [56], re-spectively. Then, we propose a novel Gaussian Process re-gression (GP) model for contour estimation. The proposedGP contour estimation method succeeded in obtaining themore accurate contour of the objects in comparison with theearlier approaches. Finally, we utilize these sub-methods toconstruct a full segmentation method for overlapping con-vex objects.

The work makes two main contributions to the studyof object segmentation. The first contribution of this workis the proposed GP model for contour estimation. The GPmodel predicts the missing parts of the evidence and esti-mates the full contours of the objects accurately. The pro-posed GP model for contour estimation is not limited to spe-cific object shapes, similar to ellipse fitting, and estimatesthe missing contours based on the assumption that both ob-served contour points and unobserved contour points can berepresented as a sample from a multivariate Gaussian dis-tribution. The second contribution is the full method for thesegmentation of partially overlapping convex objects. Theproposed method integrates the proposed GP based con-tour estimation method with the previously proposed con-tour segmentation and segment grouping methods, enablingimprovements compared to existing segmentation methodswith higher detection rate and segmentation accuracy. Thepaper is organized as follows: Related work is reviewed inSec. 2. Sec. 3 introduces our method for the segmentation ofoverlapping convex objects. The proposed method is appliedto a dataset of nanoparticle images and compared with fourstate-of-the-art methods in Sec. 4. Conclusions are drawn inSec. 5.

2 Related work

Several approaches have been proposed to address the seg-mentation of overlapping objects in different kinds of im-ages [29, 35, 31, 59, 20]. The watershed transform is oneof the commonly used approaches in overlapping cell seg-mentation [35, 10, 19]. Exploiting specific strategies for theinitialization, such as morphological filtering [35] or theadaptive H-minima transforms [10], radial-symmetry-basedvoting [45], supervised learning [25] and marker cluster-ing [48] the watershed transform may overcome the over-segmentation problem typical for the method and can beused for segmentation of overlapping objects. A stochasticwatershed is proposed in [3] to improve the performance ofwatershed segmentation for segmentation of complex im-ages into few regions. Despite all improvement, the methodsbased on the watershed transform may still experience dif-ficulties with segmentation of highly overlapped objects inwhich a sharp gradient is not present and obtaining the cor-rect markers is always challenging. Moreover, the watershedtransform does not provide the full contours of the occludedobjects.

Graph-cut is an alternative approach for segmentation ofoverlapping objects [1, 13]. Al-Kofahi et al. [1] introduceda semi-automatic approach for detection and segmentationof cell nuclei where the detection process is performed bygraph-cuts-based binarization and Laplacian-of-Gaussianfiltering. A generalized normalized cut criterion [51] is ap-plied in [6] for cell segmentation in microscopy images.Softmax-flow/min-cut cutting along through Delaunay tri-angulation is introduced in [8] for segmenting the touch-ing nuclei. The problem of overlapping cell separation wasaddressed by introducing shape’s priors to the graph-cutframework in [13]. Graph partitioning based methods be-come computational expensive and fail to keep the globaltheir optimality [23] when applied to the task of overlappingobjects. Moreover, overlapping objects usually have similarimage intensities in the overlapping region, which can makethe graph-partition methods a less ideal approach for over-lapping objects segmentation.

Several approaches have resolved the segmentationof overlapping objects within the variational frameworkthrough the use of active contours. The method in [22] incor-porates a physical shape model in terms of modal analysisthat combines object contours and a prior knowledge aboutthe expected shape into an Active Shape Model (ASM) todetect the invisible boundaries of a single nucleus. The orig-inal form of ASM is restricted to single object segmentationand cannot deal with more complex scenarios of multipleoccluded objects. In [58] and [2], ASM was extended to ad-dress the segmentation of multiple overlapping objects si-multaneously. Accurate initialization and computation com-

Resolving Overlapping Convex Objects in Silhouette Images by Concavity Analysis and Gaussian Process 3

plexity limit the efficiency of the active contour-based meth-ods for segmentation of overlapping objects.

Morphological operations have also been applied tooverlapping object segmentation. Park et al. [29] proposedan automated morphology analysis coupled with a statisti-cal model for contour inference to segment partially over-lapping nanoparticles. Ultimate erosion modified for convexshape objects is used for particle separation, and the problemof object inference and shape classification are solved simul-taneously using a Gaussian mixture model on B-splines. Themethod may be prone to under-segmentation with highlyoverlapped objects. In [53], a method for segmentation ofoverlapping objects with close to elliptical shape was pro-posed. The method consists of three steps: seed point ex-traction using radial symmetry, contour evidence extractionvia an edge to seed point matching, and contour estimationthrough ellipse fitting.

Several papers have addressed the problem of over-lapping objects segmentation using concave points detec-tion [59, 5, 56, 55]. The concave points in the objects con-tour divide the contour of overlapping objects into differentsegments. Then the ellipse fitting is applied to separate theoverlapping objects. As these approaches strongly rely onellipse fitting to segment object contours, they may haveproblems with either object’s boundaries containing largescale fluctuations or objects whose shape deviate from theellipse.

Supervised learning methods have also been applied fortouching cell segmentation [38]. In [9], a statistical modelusing PCA and distance-based non-maximum suppressionis learned for nucleus detection and localization. In [4], agenerative supervised learning approach for the segmenta-tion of overlapping objects based on the maximally stableextremal regions algorithm and a structured output supportvector machine framework [18] along dynamic program-ming was proposed. The Shape Boltzmann Machine (SBM)for shape completion or missing region estimation was intro-duced in [15]. In [14], a model based on the SBM was pro-posed for object shape modeling that represents the physicallocal part of the objects as a union of convex polytopes. AMulti-Scale SBM was introduced in [49] for shape model-ing and representation that can learn the true binary distribu-tions of the training shapes and generate more valid shapes.The performance of the supervised method highly dependson the amount and quality of annotated data and the learningalgorithm.

Recently, convolutional neural networks (CNN) [57]have gained particular attention due to its success in manyareas of research. In [37] a multiscale CNN and graph-partitioning-based framework for cervical cytoplasm andnuclei detection was introduced. In [43] CNN based onHough-voting method was proposed for cell nuclei local-ization. The method presented in [44] combined a CNN and

iterative region merging method for nucleus segmentationand shape representation. In [36], a CNN framework is co-operated with a deformation model to segment individualcell from the overlapping clump in pap smear images.

Although CNN based approaches have significantly con-tributed to the improvement of object segmentation, theycome along with several issues. CNN approaches requireextensive annotated data to achieve high performance, andthey do not perform well when the data volumes are small.Manually annotating large amount of object contour is verytime-consuming and expensive. This is also the case with thenanoparticle dataset used in this study limiting the amountof data that could be used for training. Moreover, end-to-end CNN based approaches solve the problem in a black-box manner. This is often not preferred in industrial appli-cations as it makes it difficult to debug the method if it isnot working correctly and lacks possibility to transfer or togeneralize the method to different measurement setups (e.g.,different microscopes for nanoparticle imaging) and to otherapplications (e.g., cell segmentation) without retraining thewhole model. Finally, the segmentation of overlapping ob-jects from silhouette images can be divided into separatesubproblems with clear definitions. These include separat-ing the object from the background (binarization), findingthe visible part of contour for each object, and estimating themissing parts. Each of these subproblems can be solved indi-vidually making it unnecessary to utilize end-to-end modelssuch as CNNs.

3 Overlapping object segmentation

Fig. 2 summarizes the proposed method. The method is amodular framework where the performance of each mod-ule directly impacts the performance of the next module.Given a grayscale image as an input, the segmentation pro-cess starts with pre-processing to build an image silhouetteand the corresponding edge map. In our method, the bina-rization of the image is obtained by background suppressionbased on the Otsu’s method [28] along with the morpholog-ical opening to smooth the object boundaries. For computa-tional reasons, the connected components are extracted andfurther analysis is performed for each connected componentseparately. The edge map is constructed using the Cannyedge detector [7]. However, it should be noted that thepre-processing steps are highly application specific. In thiswork, we consider backlit microscope images of nanopar-ticles where the objects of interest are clearly distinguish-able from the background. For other tasks such as cell im-age analysis, the binarization using the thresholding-basedmethods often does not provide good results due to variousreasons such as noise, contrast variation, and non-uniformbackground. In [52] a machine learning framework based onthe convolutional neural networks (CNN) was proposed for


binarization of nanoparticle images with non-uniform back-ground.

The next step called contour evidence extraction in-volves two separate tasks: contour segmentation and seg-ment grouping. In contour segmentation the contour seg-ments are recovered from a binarized image using concavepoints detection. In segment grouping, the contour segmentswhich belong to the same objects are grouped by utilizingthe branch and bound algorithm [56]. Finally, contour evi-dences are utilized to estimate the full contours of the ob-jects. Once the contour evidence has been obtained contourestimation is carried out to infer the missing parts of theoverlapping objects.

3.1 Concave point detection

Concave points detection has a pivotal role in contour seg-mentation and provides an important cue for further objectsegmentation (contour estimation). The main goal in con-cave points detection (CPD) is to find the concavity loca-tions on the objects boundaries and to utilize them to seg-ment the contours of overlapping objects in such way thateach contour segment contains edge points from one objectonly. Although a concave point has a clear mathematicaldefinition, the task of concave points detection from noisydigital images with limited resolution is not trivial. Severalmethods have been proposed for the problem. These meth-ods can be categorized into three groups: concave chord,skeleton and polygonal approximation based methods.

3.1.1 Concave chord

In concave chord methods, the concave points are the deep-est points on the concave region of contours that has thelargest perpendicular distance from the corresponding con-vex hull. The concave regions and their corresponding con-vex hull chords can be obtained by two approaches as fol-lows. In the method proposed by Farhan et al. [16], seriesof lines are fitted on object contours and concave region arefound as a region of contours such that the line joins the twocontour points does not reside inside an object. The con-vex hull chord corresponding to that concave region is a lineconnecting the sixth adjacent contour point on either side ofthe current point on the concave region of the object con-tours. In the method proposed by Kumar et al. [21], the con-cave points are extracted using the boundaries of concaveregions and their corresponding convex hull chords. In thisway, the concave points are defined as points on the bound-aries of concave regions that have maximum perpendiculardistance from the convex hull chord. The boundaries andconvex hull are obtained using a method proposed in [33].

3.1.2 Skeleton

Skeletonization is a thinning process in the binary image thatreduces the objects in the image to skeletal remnant such thatthe essential structure of the object is preserved. The skele-ton information along with the object’s contour can also beused to detect the concave points. Given the extracted objectcontours C and skeleton SK the concave points are deter-mined in two ways. In the method proposed by Samma etal. [34] the contour points and skeleton are detected by themorphological operations and then the concave points areidentified as the intersections between the skeleton and con-tour points. In the method proposed by Wang et al. [39],the objects are first skeletonized and contours are extracted.Next, for every contour points ci the shortest distance fromthe skeleton is computed. The concave points are detected asthe local minimum on the distance histogram whose value ishigher than a certain threshold.

3.1.3 Polygonal approximation

Polygonal approximation is a well-known method for dom-inant point representation of the digital curves. It is usedto reduce the complexity, smooth the objects contours, andavoid detection of false concave points in CPD methods. Themajority of the polygonal approximation methods are basedon one of the following two approaches: 1) finding a subsetof dominant points by local curvature estimation or 2) fittinga curve to a sequence of line segments

The methods that are based on finding a subset of dom-inant points by local curvature estimation work as fol-lows. Given the sequence of extracted contour points C =

c1, c2, ..., the dominants points are identified as pointswith extreme local curvature. The curvature value k for ev-ery contour points ci = (xi, yi) can be obtained by:

k = (x′iy′′i − y′ix′′i )/(x′2i + y′2i )

3/2. (1)

The detected dominant points cd,i ∈ Cdom may locatein both concave and convex regions of the objects contours.To detect the dominant points that are concave, various ap-proaches exist. In the method proposed by Wen et al. [40]the detected dominant points are classified as concave if thevalue of maximum curvature is larger than a preset thresholdvalue. In the method proposed by Zafari et al. [54], the dom-inant points are concave points if the line connecting cd,i+1

to cd,i−1 does not reside inside the object. In the methodproposed by Dai et al. [12], the dominant points are quali-fied as concave points if the value of the triangle area S forthe points cd,i+1, cd,i, cd,i−1 is positive and the points arearranged in order of counter-clockwise.

The methods that are based on fitting a curve to a se-quence of line segments work as follows. First, a series oflines are fitted to the contour points. Then, the distances


Fig. 2: The workflow of the proposed segmentation method includes three major steps: pre-processing, contour evidenceextraction and contour segmentation. In pre-processing, the region of the image that is covered by the object of interestis separated from the background. Contour evidence extraction infers the visible parts of the object contours and contourestimation resolves the full contours of an individual object.

from all the points of the object contour located in betweenthe starting and ending points of the line to the fitted linesare calculated. The points whose distance are larger than apreset threshold value considered as the dominant point.

After polygon approximation and dominant points de-tection, the concave points can be obtained using methodsproposed by Bai et al. [5] and Zhang et al. [59]. In themethod proposed by Bai et al. [5], the the dominant pointcd,i is defined as concave if the concavity value of cd,i iswithin the range of a1 to a2 and the line connecting cd,i+1

and cd,i−1 does not pass through the inside the objects. Theconcavity value of cd,i is defined as the angle between lines(cd,i−1, cd,i) and (cd,i+1, cd,i) as follows:

| γ1 − γ2 | if |γ1 − γ2| < π

π − |γ1 − γ2| otherwise,(2)

where, γ1 is the angle of the line (cd,i−1, cd,i) and γ2 is theangle of the line (cd,i+1, cd,i) defined as:

γ1 = tan−1((yd,i−1 − yd,i)/(xd,i−1 − xd,i)),γ2 = tan−1((yd,i+1 − yd,i)/(xd,i+1 − xd,i)).

In the method proposed by Zhang et al. [59], the domi-nant point cd,i ∈ Cdom is considered to be a concave pointif # »cd,i−1cd,i × # »cd,icd,i+1 is positive:

Ccon = cd,i ∈ Cdom : # »cd,i−1cd,i × # »cd,icd,i+1 > 0. (3)

3.2 Proposed method for concave point detection

Most existent concave point detection methods require athreshold value that is selected heuristically [59, 5]. In fact,finding the optimal threshold value is very challenging, anda unique value may not be suitable for all images in a dataset.

We propose a parameter free concave point detectionmethod that relies on polygonal approximation by fittingthe curve with a sequence of line segments. Given the se-quence of extracted contour points C = c1, c2, ..., first,the dominant points are determined by co-linear suppres-sion. To be specific, every contour point ci is examined forco-linearity, while it is compared to the previous and the nextsuccessive contour points. The point ci is considered as thedominant point if it is not located on the line connectingci−1 = (xi−1, yi−1) and ci+1 = (xi+1, yi+1) and the dis-tance di from ci to the line connecting ci−1 to ci+1 is biggerthan a pre-set threshold di > dth.

The value of dth is selected automatically using the char-acteristics of the line as in [30]. In this method, first the lineconnecting ci−1 to ci+1 is digitized by rounding the value(xi−1, yi−1) and (xi+1, yi+1) to their nearest integer. Next,the angular difference between the numeric tangent of lineci−1ci+1 and the digital tangent of line c′i−1c

′i+1 is com-

puted as (see Fig. 3):

δφ = tan−1(m)− tan−1(m′), (4)

where, m and m′ are slop of actual and digital line respec-tively. Then, value of dth is given by:

dth = Sδφ, (5)


where S is length of actual line.

Fig. 3: Representation of the actual and the digitized lines.

Finally, the detected dominant points are qualified asconcave if they met the criterion by Zhang (Eq. 3) forconcave point detection. We referred the proposed concavepoint detection as Prasad+Zhang [55].

3.3 Segment grouping

Due to the overlap between the objects and the irregularitiesin the object shapes, a single object may produce multiplecontour segments. Segment grouping is needed to merge allthe contour segments belonging to the same object.

Let S = S1, S2, ..., SN be an ordered set of N con-tour segments in a connected component of an image. Theaim is to group the contour segments into M subsets suchthat the contour segments that belong to individual objectsare grouped together andM ≤ N . Let ωi be the group mem-bership indicator giving the group index to which the con-tour segment Si belongs to. Denote Ω be the ordered set ofall membership indicators: ω1, ω2, ...., ωN.

The grouping criterion is given by a scalar function J(·)of Ω which maps a possible grouping of the given contoursegments onto the set of real numbers R. That is, J is thecost of grouping that ideally measures how the groupingΩ resembles the true contour segments of the objects. Thegrouping problem for the given set of S is to find the optimalmembership set Ω∗ such that the grouping criterion (cost ofgrouping) is minimized:

Ω∗ = argminΩ

J(Ω;S). (6)

We formulated the grouping task as a combinatorial op-timization problem and solved using the branch and bound(BB) algorithm [56]. The BB algorithm can be applied withany grouping criterion J . However, the selection of thegrouping criterion significantly affects the overall method

performance. In this work the grouping criterion is a hy-brid cost function consisting of two parts: 1) generic part(Jconcavity) that encapsulate the general convexity proper-ties of the objects and 2) specific part that encapsulates theproperties of objects that are exclusive to a certain applica-tion, e.g., symmetry (Jsymmetry) and ellipticity (Jellipticity).The cost function is defined as follows:

J = Jconcavity︸︷︷︸Generic

+αJellipticity + βJsymmetry︸︷︷︸Specific

, (7)

where α, β are the weighting parameters. The generic partencourages the convexity assumption of the objects in orderto penalize the grouping of the contour segments belong-ing to different objects. This is achieved by incorporatinga quantitative concavity measure. Given two contour seg-ments Si and Sj the generic part of the cost function is de-fined as follows:

Jconcavity =

1, concave point between Si and Sj

(1−ASi∪Sj

Ach,Si∪Sj) otherwise,

(8)

where ASi∪Sj is the area of a region bounded by Si, Sj ,and the lines connecting the endpoint of Sj to the startingpoint of Si and the end point of Si to the starting point ofSj .Ach,Si∪Sj is the upper bound for area of any convex hullwith boundary points Si and Sj . The specific part is adaptedto consider the application criteria and certain object proper-ties. Considering the object under examination, several func-tions can be utilized.

Here, the specific term is formed by the ellipticity andsymmetry cost functions. The ellipticity term is defined bymeasuring the discrepancy between the fitted ellipse and thecontour segments [59]. Given the contour segment Si con-sisting of n points, Si = (xk, yk)nk=1, and the correspond-ing fitted ellipse points, Sf,i = (xf,k, yf,k)nk=1, the ellip-ticity term is defined as follows:

Jellipticity =1

n

n∑k=1

√(xk − xf,k)2 + (yk − yf,k)2. (9)

The symmetry term penalizes the resulting objects thatare non-symmetry. Let oi and oj be the center of symmetryof the contour segments Si and Sj obtained by aggregat-ing the normal vector of the contour segments. The proce-dure is similar to fast radial symmetry transform [24], butthe gradient vectors are replaced by the contour segments’normal vector. This transform is referred to as normal sym-metry transform (NST).

In NST, every contour segment point gives a vote forthe plausible radial symmetry at some specific distance fromthat point. Given the distance value n of the predefined range


[Rmin Rmax], for every contour segment point (x, y), NSTdetermines the negatively affected pixels P−e by

P−ve(x, y) = (x, y)− round( n(x, y)

‖n(x, y)‖n), (10)

and increment the corresponding point in the orientationprojection image On by 1. The symmetry contribution Snfor the radius n ∈ [Rmin, Rmax] is formulated as

Sn(x, y) = ( |On(x, y)|kn

), (11)

where kn is the scaling factor that normalizes On acrossdifferent radii. On is defined as

On(x, y) =

On(x, y), if On(x, y) < kn,

kn, otherwise.(12)

The full NST transform S, by which the interest symmetricregions are defined, is given by the average of the symmetrycontributions over all the radii n ∈ [Rmin, Rmax] consid-ered as

S =1

|N |∑

n∈[Rmin,Rmax]

Sn. (13)

The center of symmetry oi and oj of the contour segmentsare estimated as the average locations of the detected sym-metric regions in S. The symmetry term Jsymmetry is de-fined as the Euclidean distance between oi and oj as

Jsymmetry = |oi − oj |. (14)

The distance is normalized to [0,1] according to the maxi-mum diameter of the object.

3.4 Contour estimation

Segment grouping results in, a set of contour points for eachobject. Due to the overlaps between objects, the contours arenot complete and the full contours need to be estimated. Sev-eral methods exist to address the contour estimation prob-lem. A widely known approach is Least Square Ellipse Fit-ting (LESF) [17]. The purpose of LESF is to computing theellipse parameters by minimizing the sum of squared alge-braic distances from the known contour evidences points tothe ellipse. The method is highly robust and easy to imple-ment, however, it fails to accurately approximate compli-cated non-elliptical shapes.

Closed B-splines is another group of methods for thecontour estimation. In [29], the B-Splines with ExpectationMaximization (BS) algorithm was used to infer the full con-tours of predefined possible shapes. The reference shapeand the parameters of the B-spline of the object are com-prised in the Gaussian mixture model. The inference of the

curve parameters, namely shape properties and the parame-ters of B-spline, from this probabilistic model is addressedthrough the Expectation Conditional Maximization (ECM)algorithm [27]. The accuracy of the method depends on thesize and diversity of the set of reference shapes and gets lowif the objects highly overlap.

3.5 Proposed method for contour estimation

To address the issues with the existing methods, we proposea Gaussian process based method for the contour estimation.Gaussian processes (GP) are nonparametric kernel-basedprobabilistic models that have been used in several ma-chine learning applications such as regression, prediction,and contour estimation [26, 32]. In [47] a GP model basedon charging curve is proposed for state-of-health (SOH) es-timation of lithium-ion battery. In [42] GP is applied forpattern discovery and extrapolation. In [46] a regressionmethod based on the GP is proposed for prediction of windpower.

The primary assumption in GP is that the data can berepresented as a sample from a multivariate Gaussian dis-tribution. GP can probabilistically estimate the unobserveddata point, f∗, based on the observed data, f , assuming thatthe both observed data point and unobserved data can berepresented as a sample from a multivariate Gaussian distri-bution(f

f∗

)= N

((µx

µx∗

),

(Kxx kTx∗kx∗ Kx∗x∗

)), (15)

where K is the n × n covariance matrix and kx∗ is ann × 1 vector containing the covariances between x∗ andx1, x2, . . . , xn given by

[K]ij = κ(xi, xj), (16)

and

kx∗ = [κ(x1, x∗), κ(x2, x

∗), . . . , κ(xn, x∗)]T . (17)

κ(., .) is defined by the Matern covariance function [41] asfollows:

κ(xi, xj) =21−v

Γ (v)(

√2v|d|l

)vKv(

√2v|d|l

), (18)

where xi, xj , i 6= j, i = 1, 2, . . . , n are a pair of data point, lis the characteristic length scale, d is the Euclidean distancebetween the points xi and xj and Kv is the modified Besselfunction of order v. The most common choices for v arev = 3/2 and v = 5/2 that are denoted by Matern 3/2 andMatern 5/2.

Let D = (xi, yi)ni=1 be the input data, x =

[x1, . . . , xn]T be the input vector and y = [y1, . . . , yn]

T

be the corresponding observations vector given by a latent


function f(.). GP assumes the latent function comes from azero-mean Gaussian and the observations are modeled by

yi = f(xi) + εi, εi ∼ N (0, σ2). (19)

Given the D, f and a new point x∗, the GP defines the pos-terior distribution of the latent function f(.) at the new pointf(x∗) as Gaussian and obtains the best estimate for x∗ bythe mean and variance of the posterior distribution as fol-lows:

p(f∗|x∗, D) = N (f∗|µx∗ , σ2x∗), (20)

where

µx∗ = kTx∗K−1xxy, (21)

and

σ2x∗ = Kx∗x∗ − kTx∗K

−1xxkx∗ . (22)

To exploit GP for the task of contour estimation and tolimit the interpolation interval to a finite domain, the polarrepresentation of contour evidence is needed. Polar repre-sentation has three advantages. First, it limits the interpola-tion interval to a finite domain of [0, 2π], second, it exploitsthe correlation between the radial and angular component,and third, it reduces the number of dimensions compared toGP based on parametric Cartesian representation.

In GP with polar form (GP-PF), the contour evidencepoints are represented as a function of the radial and angu-lar components. Given the contour evidence points x and y,their polar transform is defined as follows:

θ = arctan(yx

),

r =√x2 + y2.

(23)

Here, the angular component θ is considered to be the inputto the GP and the radial r as the corresponding observationsdescribed by the latent function r = f(θ). To enforce theinterpolation to a closed periodic curve estimation such thatf(θ) = f(θ + 2π) = r, the input set is defined over thefinite domain θ∗ ∈ [0, 2π] as D = (θi, ri)ni=1 ∪ (θi +2π, ri), ...ni=1. The best estimate for r∗ is obtained by themean of the distribution

µθ∗ = kTθ∗K−1θθ r. (24)

The GP-PF method is summarized in Algorithm 1.

4 Experiments

This section presents the data, the performance measures,and the results for the concave points detection, contour es-timation, and segmentation.

Algorithm 1: Contour Estimation by GP-PFInput: Contour evidence coordinates x, y.Output: Estimated objects’ full contour points x∗, y∗.Parameters: Covariance function k1. Perform mean centering for x, y by subtracting their mean from

all their values.2. Retrieve polar coordinates θ, r from x, y using Eq. [23].3. Define the input set and the corresponding output for the

observed contour points asD = (θi, ri)ni=1 ∪ (θi + 2π, ri), ...ni=1.

4. Define the new input set for unobserved contour points byθ∗ ∈ [0, 2π].

5. Estimate the covariance function, Kθ∗θ∗ , kθ∗ ,Kθθ usingEq. [18].

6. Predict the output, r∗, on the new input set, θ∗, using thepredictive mean using Eq. [24].

7. Transform θ∗, r∗ from polar to Cartesian coordinates x∗, y∗.

4.1 Data

The experiments were carried out using one syntheticallygenerated dataset and one dataset from a real-world applica-tion.

The synthetic dataset (see Fig. 4(a)) consists of imageswith overlapping ellipse-shape objects that are uniformlyrandomly scaled, rotated, and translated. Three subsets ofimages were generated to represent different degrees ofoverlap between objects. The dataset consists of 150 sam-ple images divided into three classes of overlap degree. Themaximum rates of the overlapping area allowed betweentwo objects are 40%, 50%, and 60%, respectively, for thefirst, second, and third subset. Each subset of images in thedataset contains 50 images of 40 objects. The minimum andmaximum width and length of the ellipses are 30, and 45pixels. The image size is 300× 400 pixels.

The real dataset (nanoparticles dataset) containsnanoparticles images captured using transmission electronmicroscopy (see Fig. 4(b)). In total, the dataset contains 11images of 4008 × 2672 pixels. Around 200 particles weremarked manually in each image by an expert. In total thedataset contains 2200 particles.The annotations consist ofmanually drawn contours of the particles. This informa-tion was also used to determine the ground truth for con-cave points. Since not all the particles were marked, a pre-processing step was applied to eliminate the unmarked par-ticles from the images. It should be noted that the imagesconsist of dark objects on a white background and, there-fore, pixels outside the marked objects can be colored whitewithout making the images considerably easier to analyze.

4.2 Performance Measures

To evaluate the method performance and to compare themethods, the following performance measures were used:


(a) (b)

Fig. 4: Example images from the datasets studied: (a) Synthetic dataset with the maximum overlap of 40%; (b) Nanoparticlesdataset.

The detection performance for both concave points and ob-jects were measured using True Positive Rate (TPR), Pos-itive Predictive Value (PPV), and Accuracy (ACC) definedas follows:

TPR =TP

TP + FN, (25)

PPV =TP

TP + FP, (26)

ACC =TP

TP + FP + FN, (27)

where True Positive (TP) is the number of correctly detectedconcave points or segmented objects, False Positive (FP) isthe number of incorrectly detected concave points or seg-mentation results, and False Negative (FN) is the numberof missed concave points or objects. To determine whethera concave point was correctly detected (TP), the distanceto the ground truth concave points was computed and thedecision was made using a predefined threshold value. Thethreshold value was set to 10 pixels. Moreover, the averagedistance (AD) from detected concave points to the groundtruth concave points was measured for the concave pointsdetection. To decide whether the segmentation result wascorrect or incorrect, Jaccard Similarity coefficient (JSC) [11]was used. Given a binary map of the segmented object Osand the ground truth particle Og , JSC is computed as

JSC =Os ∩OgOs ∪Og

. (28)

The threshold value for the ratio of overlap (JSC threshold)was set to 0.6. The average JSC (AJSC) value was also usedas a performance measure to evaluate the segmentation per-formance.

4.3 Parameter selection

The proposed method requires the following major parame-ters.

Weighting parameters α, and β define to what extentthe ellipticity and symmetry features must be weighted inthe computation of the grouping cost J . A lower value ofα and β emphasizes non-ellipticity and non-symmetry fea-tures where higher values ensure that objects have perfect el-liptical symmetry shapes. This parameter is useful when theobjects are not perfectly elliptical. In this work, the weight-ing parameters α, and β were set to 0.1 and 0.9 to obtain thehighest performance of segment grouping.

Type of the covariance function determines how the co-variance function measure the similarity of any two points ofthe data, and specifies the desired form of the output func-tion and affects the final contour estimation result. In thiswork, the contour estimation is performed using the GP-PFmethod with the Matern 5/2 covariance function as a moreflexible and general one.

The parameters of the concave points detection methodswere set experimentally to obtain the best possible result foreach method.

4.4 Results

4.4.1 Concave points detection

The results of the concave points detection methods appliedto real nanoparticles and synthetic datasets are presented inTables 1 and 2, respectively. The results have been reportedwith the optimal parameters setup for each listed methods.The chosen nanoparticle dataset is challenging for CPD ex-periments since it contains objects with noisy boundaries.


Here, the noise includes the small or large scale fluctuationson the object boundaries that can affect the performance ofthe concave points detection methods. The results on thenanoparticles dataset (Table 1) show that the Prasad+Zhangmethod outperforms the others with the highest TPR andACC values and the competitive PPV and AD values. Fromthe results with the synthetic dataset (Table 2) it can beseen that Prasad+Zhang achieved the highest value of ACC.Zhang [59] scored the highest values of TPR and AD, butit suffers from the low values of PPV and ADD. This is be-cause the method produces a large number of false concavepoints. Comparing the Zhang [59] and Prasad+Zhang [55]CPD methods shows the advantage of the proposed parame-ter selecting strategy in concave points detection that makesthe Prasad+Zhang method a more robust in presence of ob-jects with noisy boundaries. In terms of PPV, Zafari [54] andKumar [50] ranked the highest. However, these approachestend to underestimate the number of concave points as theyoften fail to detect the concave points at a low depth of con-cave regions. Considering all the scores together, the re-sults obtained from the synthetic dataset showed that thePrasad+Zhang method achieved the best performance.

The results presented in Tables 1 and 2 were obtainedby a fixed distance threshold value of 10 pixels. To checkthe stability of the results with respect to distance threshold(ρ1), the distance threshold was examined by the nanopar-ticles datasets. The effect of ρ1 on the TPR, PPV and ACCscores captured with the concave point detection methodsare presented in Fig. 5. As it can be seen, the Prasad+Zhangmethod achieved the highest score of TPR and ACC regard-less of the selected threshold. It is worth mentioning thatat low values of ρ1, the Prasad+Zhang method outperformsall methods with respect to PPV. This is clearly since thePrasad+Zhang method can detect the concave points moreprecisely.

Table 1: Comparison of the performance of concave pointsdetection methods on the nanoparticles dataset.

Methods TPR PPV ACC AD Time[%] [%] [%] [pixel] (s)

Prasad+Zhang [55] 96 90 87 1.47 0.50Zhang [59] 84 38 35 1.42 2.28Bai [5] 79 40 36 1.55 2.52Zafari [54] 65 95 64 2.30 2.48Wen [40] 94 51 50 1.97 2.13Dai [12] 54 87 50 2.84 2.50Samma [34] 80 84 70 2.28 2.95Wang [39] 65 58 44 2.37 4.95Kumar [21] 33 94 31 3.09 8.17Farhan [16] 77 15 14 2.09 8.82

Fig. 6 shows example results of the concave points de-tection methods applied to a patch of a nanoparticles image.It can be seen that while Zafari [54], Dai [12], Kumar [50]

Table 2: Comparison of the performance of concave pointsdetection methods on the synthetic dataset.

Methods Overlapping TPR PPV ACC AD[%] [%] [%] [%] [pixel]

Prasad+Zhang [55] 40 97 97 94 1.47Zhang [59] 40 98 37 37 1.25Bai [5] 40 92 46 44 1.34Zafari [54] 40 81 98 80 2.04Wen [40] 40 97 61 60 1.75Dai [12] 40 71 97 70 2.58Samma [34] 40 75 83 65 2.20Wang [39] 40 87 86 76 1.87Kumar [21] 40 64 98 62 2.64Farhan [16] 40 91 18 18 1.77



and Samma [34] suffer from false negatives and Zhang [59],Bai [5], Wen [40] and Farhan [16] suffer from false posi-tives, Prasad+Zhang [55] suffers from neither false positivesnor false negatives.

4.4.2 Contour estimation

Tables 3 and 4 show the results of contour estimation withthe proposed GP-PF using various types of commonly usedcovariance functions on the nanoparticles dataset using boththe ground truth contour evidences and the contour evi-dences obtained using the proposed contour evidence ex-traction method. As it can be seen the Matern family outper-forms the other covariance functions with both the groundtruth and obtained contour evidences.

To demonstrate the effect of the more accurate contourestimation on the segmentation of partially overlapping ob-jects, the performance of the GP-PF method was comparedto the BS and the LESF methods on the nanoparticle dataset.Table 5 shows the segmentation results produced by the con-tour estimation methods applied to the ground truth contour


0 2 4 6 8 10 12 14 16 18 20

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(a) (b) (c)

Fig. 5: Performance of concave detection methods with different values of the distance threshold (ρ1) in the nanoparticlesdataset: (a) TPR; (b) PPV; (c) ACC.

(a) (b) (c)

(d) (e) (f) (g)

(h) (i) (j) (k)

Fig. 6: Comparison of the performance of the concave points detection methods on the nanoparticles dataset: (a) GroundTruth; (b) Prasad+Zhang [55]; (c) Zhang [59]; (d) Bai [5]; (e) Wen [40]; (f) Zafari [54]; (g) Dai [12]; (h) Kumar [50];(i) Farhan [16]; (j) Wang [39]; (k) Samma [34].

evidences and Table 6 presents their performance with theobtained contour evidences.

The results show the advantage of the proposed GP-PFmethod over the other methods when applied to the groundtruth contour evidences. In all terms the proposed contourestimation method achieved the highest performance. Theresults on the obtained contour evidences (Table 6) confirmthe out-performance of the GP-PF method over the otherswith the highest values.

In general, the LESF contour estimation takes advantageof objects with an elliptical shape which can lead to desir-able contour estimation if the objects are in the form of el-lipses. However, in situations where objects are not close toelliptical shapes, the method fails to estimate the exact con-tour of the objects. The BS contour estimation is not limitedto specific object shapes, but requires to learn many con-tour functions with prior shape information. The proposedGaussian process contour estimation can resolve the actual


Table 3: The performance of the proposed GP-PF methodwith different covariance functions on the ground truth con-tour evidences.

Methods TPR PPV ACC AJSC[%] [%] [%] [%]

Matern 3/2 96 96 92 89Matern 5/2 96 97 93 89Squared exponential 91 92 84 85Rational quadratic 95 96 92 88

Table 4: The performance of the proposed GP-PF methodwith different covariance functions on the obtained contourevidences.


Matern 3/2 87 83 74 87Matern 5/2 87 83 74 87Squared exponential 83 78 68 83Rational quadratic 87 82 73 85

Table 5: The performance of the contour estimation methodson the ground truth contour evidences.


GP-PF 96 96 92 90BS 92 92 86 89LESF 94 95 89 88

Table 6: The performance of the contour estimation methodson the obtained contour evidences.


GP-PF 87 83 74 87BS 82 76 66 87LESF 85 83 71 87

contour of the objects without any strict assumption on theobject’s shape.

Fig. 7 shows example results of contour estimationmethods applied to a slice of nanoparticles dataset.

It can be seen that BS tends to underestimate the shapeof the objects. With small contour evidences, the BS methodresults in a small shape while LESF can result in largershapes. The proposed GP-PF method estimates the objectsboundaries more precisely.

Figs. 8 and 9 show the effect of the Jaccard similaritythreshold on the TPR, PPV, and ACC scores with the pro-posed and competing contour estimation methods. As ex-pected, the segmentation performance of all methods de-grades when the JSC threshold is increased. However, thethreshold value has only the minor effect on the rankingorder of the methods and the proposed contour estimationmethod outperforms the other methods with higher JSCthreshold values.

(a) (b)

(c) (d)

Fig. 7: Comparison of the performance of contour estima-tion methods on the ground truth contour evidences: (a)Ground truth full contour; (b) GP-PF; (c) BS; (d) LESF.

4.4.3 Segmentation of overlapping convex objects

The proposed method is organized in a sequential mannerwhere the performance of each step directly impact the per-formance of the next step. In each step, all the potentialmethods are compared and the best performer is selected.The performance of the full proposed segmentation methodusing the Prasad+Zhang concave points detection and theproposed GP-PF contour estimation methods was comparedto three existing methods: seed point based contour evidenceextraction and contour estimation (SCC) [53], nanoparticlessegmentation (NPA) [29], and concave-point extraction andcontour segmentation (CECS) [59]. These methods are par-ticularly chosen as previously applied for segmentation ofoverlapping convex objects. Fig. 10 shows an example ofsegmentation result for the proposed method.

The corresponding performance statistics of the compet-ing methods applied to the nanoparticles dataset are shownin Table 7. As it can be seen, the proposed method outper-formed the other four methods with respect to TPR, PPV,ACC, and AJSC. The highest TPR, PPV, and ACC indicatesits superiority with respect to the resolved overlap ratio.

Fig 11 shows the effect of the Jaccard similarity thresh-old on the TPR, PPV, and ACC scores with the proposedsegmentation and competing methods. As it can be seen,the performance of all segmentation methods degrades whenthe JSC threshold is increased, and the proposed segmenta-tion method outperforms the other methods especially withhigher JSC threshold values.


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Fig. 8: Evaluation results for the contour estimation methods with the different JSC thresholds value applied to the groundtruth contour evidences: (a) TPR; (b) PPV; (c) ACC.

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Fig. 9: Evaluation results for the contour estimation methods with the different JSC threshold values applied to the obtainedcontour evidences: (a) TPR; (b) PPV; (c) ACC.

(a) (b)

Fig. 10: An example of the proposed method segmentation result on the nanoparticles dataset: (a) Ground truth; (b) Proposedmethod.


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Fig. 11: Evaluation results for the segmentation methods with the different JSC threshold values: (a) TPR; (b) PPV; (c) ACC.

Table 7: Comparison of the performance of the segmentationmethods on the nanoparticles dataset.


Proposed 87 83 74 87SCC 74 79 62 82NPA 57 80 50 81CECS 58 66 45 83

5 Conclusions

This paper presented a method to resolve overlapping con-vex objects in order to segment the objects in silhouette im-ages. The proposed method consisted of three main steps:pre-processing to produce binarized images, contour evi-dence extraction to detect the visible part of each object,and contour estimation to estimate the final objects contours.The contour evidence extraction is performed by detectingconcave points and solving the contour segment groupingusing the branch and bound algorithm. Finally, the contourestimation is performed using Gaussian process regression.To find the best method for concave points detection, com-prehensive evaluation of different approaches was made.The experiments showed that the proposed method achievedhigh detection and segmentation accuracy and outperformedfour competing methods on the dataset of nanoparticles im-ages. The proposed method relies only on edge informa-tion and can be applied also to other segmentation problemswhere the objects are partially overlapping and have an ap-proximately convex shape.

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resolving overlapping convex objects in silhouette images ...arxiv:1906.01049v1 [cs.cv] 3 jun 2019....

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