830 ieee journal of selected topics in …jocelyn.chanussot/publis/ieee_jstsp...830 ieee journal of...

830 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 7, NOVEMBER 2012

Efficient Robust d-Dimensional Path OperatorsFrançois Cokelaer, Hugues Talbot, Member, IEEE, and Jocelyn Chanussot, Fellow, IEEE

Abstract—Path openings and closings are efficient morpholog-ical operators that use flexible oriented paths as structuring ele-ments. They are employed in a similar way to operators with ro-tated line segments as structuring elements, but are more effectiveat detecting linear structures that are not necessarily locally per-fectly straight. While their theory has always allowed paths in ar-bitrary dimensions, de facto implementations were only proposedin 2D. Recently, a new implementation was proposed enabling thecomputation of efficient -dimensional path operators. Howeverthis implementation is limited in the sense that it is not robust tonoise. Indeed, in practical applications, for path operators to be ef-fective, structuring elements must be sufficiently long so that theycorrespond to the length of the desired features to be detected. Yet,path operators are increasingly sensitive to noise as their lengthparameter increases. To cope with this limitation, we proposean efficient -dimensional algorithm, the Robust Path Operator,which uses a larger and more flexible family of flexible structuringelements. Given an arbitrary length parameter G, path propaga-tion is allowed if disconnections between two pixels belonging to apath is less or equal to G and so, render it independent of . Thissimple assumption leads to constant memory bookkeeping and re-sults in a low complexity.

Index Terms—Morphological Algebraic operators, flexiblestructuring elements, noise robustness, -dimensional processing.

I. INTRODUCTION

I N this paper we focus on the description of efficient op-erators for the detection of thin, elongated, oriented, but not

necessarily straight features in -dimensional images, in a noisycontext. Due to their thin nature, such features are easily cor-rupted by noise typically resulting in disconnections. This as-pect of image filtering has lead to number of applications in anumber of research fields such as vessels detection and segmen-tation in biomedical imaging [1]–[4], road network and riversdetection in remote sensing images [5]–[7] wood fiber lengthmeasurements in 3D images [8] or fiber extraction from com-posite material [9].While thin and oriented object filtering or segmentation may

not be the most popular subject in image processing and anal-ysis, there exist however a large variety of existing approaches.Aswedealwithmorphologicaloperators in thispaperwewill justbriefly detail some of the more common solutions to these prob-lems. For a recent and rather complete line extraction review the

Manuscript received February 09, 2012; revised June 06, 2012; accepted July13, 2012. Date of publication August 16, 2012; date of current version October12, 2012. This work was supported by the DGCIS (General Directorate forCompetitiveness, Industry and Services), the FUI (Interministerial Fund) withinDELPIX project, and the KIDICO ANR-2010-BLAN-0205-03 grant. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Dan Schonfeld.F. Cokelaer and J. Chanussot are with the GIPSA-Lab/DIS, Grenoble INP,

38402 Saint Martin D’Heres Cedex, France.H. Talbot is with the Université Paris-Est, Laboratoire d’Informatique Gas-

pard-Monge, Equipe A3SI, ESIEE Paris, France.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JSTSP.2012.2213578

reader should refer to [10]. Making the assumption that we areseeking tubular objects in 3D, [11]–[13] use the convolution ofthe image with multiple scale derivatives of Gaussian kernels.At a given scale, the tensor of second order derivatives (Hessianoperator) convolved by Gaussian kernels can be considered asprobes, enabling the evaluation of contrast differences betweenthe regions inside and outside the range given by the scale and ina given direction. One can then extract local structure orientationinformation by an eigenvalue analysis and apply a so-called ves-selness function, which yields the strongest response at a scalethat best matches the size of the tubular object to detect. In suchan approach the difficulties are the appropriate choice of parame-ters,moreprecisely the size andnumber of scales ensuring agoodfeatures detection without enhancing the noise. This, of course,while retaining a reasonable computation time.Here, we consider the subclass of image filters that are de-

fined as neighborhood operators, meaning that the output valueof a given pixel is obtained by combining the values of sur-rounding input pixels in a specified window or structuring el-ement [14]. Among such operators, structuring element basedmorphological filters are well known for their capacity to extractor filter out features that correspond to a particular given shape.In the case of the detection of line-like structures in an image,straight lines segments covering a large range of orientationscan achieve good results [15]. Unfortunately, thin and orientedstructures in images are rarely perfectly straight, even locally.Consequently, segment-based operators can fail in detecting lo-cally non-straight features. One solution may be to use shapesthat can adapt to these structures. As an extension of straightlines operators, path operators were proposed in [16], [17] and[18]. They have the ability to adapt to local image structures byincreasing the structuring element (SE) flexibility while keepinga sufficient level of anisotropy. They generate a narrow, flexibleand oriented family of SEs from a given length and an adja-cency relation which sets the global orientation of the path. Forboth filtering efficiency and computation time, path operatorshave consistently been proven to yield better results comparedto straight lines operators [8].A path is a connected one-dimensional subgraph of an image,

composed of nodes (or vertices) linked by edges. In images,nodes are composed of pixels, and vertices are defined by anadjacency relation. Typically adjacency relation for paths are asubset of the connectivity relation between neighboring pixels.In 2D, four adjacency relations (see Fig. 1) are commonly used.For a given orientation, and for a path opening, the result ofthe operator image is equivalent to computing the union of theopenings with all the paths of length compatible with the ad-jacency as SEs. For a 3-neighbour adjacency relation this cor-responds to considering different paths and so as manyopenings, which is clearly not practical. In addition, path opera-tors using unbroken paths of length are called “Complete PathOperators.” It is possible also to consider paths which allow a

1932-4553/$31.00 © 2012 IEEE

COKELAER et al.: EFFICIENT ROBUST D-DIMENSIONAL PATH OPERATORS 831

Fig. 1. (a) Two vertical (N-S) paths; (b) (E-W) horizontal and (SE-NW) diag-onal path.

certain amount of missing nodes. Such paths are called “Incom-plete Paths,” which are in even greater number.However, Talbot and Appleton have proposed an efficient,

ordered implementation for both 2D complete and incom-plete path operators [19]. Their algorithm complexity is in

in the complete case and withmissing nodes in the incomplete case. Like the idea of rank-maxoperators [20] in 2D, incomplete path operators provide thebest results for low SNR features detection compared to otherclassical approaches [10]. However, there are some remainingproblems with incomplete path operators as proposed in [19].First they require a significant amount of extra memory andcomplexity for path bookkeeping, which increases with thedimensionality. So their implementation represents a significantchallenge in 3D and up. To the best knowledge of the authors,such an implementation has not been proposed yet. Also, theparameter is typically an increasing function of . Indeed,as increases, so does the probability of encountering a noisepixel, so must increase also to retain effectiveness. Howeverthis impacts negatively the algorithm’s computational andmemory complexity.To solve these problems, in the remainder of the paper, we

propose a new framework for handling noise pixels.We design a-dimensional algorithm, the robust path operator (RPO), basedon the dimensionality-independent path openings implementa-tion of [8]. For each pair of parameters (L,G), a larger family offlexible and incomplete SEs than the incomplete path operatoris considered. Given an arbitrary length G, path propagation isallowed if the disconnection length between two elements of anincomplete path is less or equal to G. This renders G indepen-dent of . This simple assumption leads to the generation of amore flexible and more efficient family of SEs, due to constantmemory bookkeeping requirements and a lower-complexity im-plementation.

II. THEORETICAL ASPECTS

In this section we briefly recall the theoretical foundations forthe complete path operators established in [18].We also proposea new extension for the robust path operators.

A. Complete Path Operators

Starting from a set of points E (i.e., image domain), we definea spatial connectivity graph (e.g., Fig. 2) between these pointsfrom an adjacency relation . The set of all these relationsspecify the edges going from to . In this context, is calledthe successor and the predecessor.

Fig. 2. Connectivity graph for a 3-neighbors N-S adjacency relation.

We define the dilation of subset considering all thepoints in that have a predecessor in by:

(1)

A -path of length L is a L-tuple:

(2)

It is equivalent to saying that:

(3)

If is a -path of length , we define the corresponding reversepath:

(4)

called -path of length . We define the set of -path of lengthin by and conversely as the set of all -path of lengthin . Then from a path , we define the set correspondingto path elements. We denote the set of all -path in a subsetof by:

(5)

and conversely the set of all -path of length in by .Path openings are defined as the union of all -path of lengthcontained in :

(6)

Path closings are defined similarly by complementation, as isusual in mathematical morphology.

B. Incomplete Path Operators

Incomplete paths operators are designed to increase flexi-bility and noise robustness by allowing an arbitrary number ofmissing pixels along a -path of length . They are partic-

ularly suitable for practical applications as they utilize a largerfamily of SEs better able to cope with noisy image features.An incomplete path openings of parameter and of lengthconsiders a collection of path with at least vertices

inside . Considering as the complementary set of inwe have:

(7)

Where represents cardinality, i.e., number of nodes. Fromthis definition follows immediately the incomplete path openingwhich is the union of all path contained in .For

(8)


C. Robust Path Operators

1) Principle: Given an arbitrary length G, path propagationis allowed if the disconnection length between two elements ofthe path is less or equal to G . Pixels belonging tothis disconnections will be termed noise pixels.2) Searching for Noise Pixels: As in [18], we define the op-

erator such that contains every -length -path’s firstpoints in ( is defined analogously):

(9)

Nowwe can define a noise path condition meaning that a-path of with must have at least one of itsfirst element’s predecessors and at least one of its last element’ssuccessors in . For we have:

(10)

So we define the set as:

(11)

3) -Robust Paths: Here we define the set of -robust-path of lengthFor

(12)

with

(13)

So we define the robust path opening as:

(14)

It is easy to show that robust path openings (resp. closings)have all the algebraic properties of an opening (resp. a closing):increasingness, idempotence and anti-extensivity (resp. exten-sivity). Specifically, we rely on previous results showing thatcomplete path operators are openings (resp. closings) [17]. Al-lowing paths to admit noise pixels simply means that we admita larger family of structuring elements over which we perform asupremum (resp. an infimum), which does not change the basicproperties of these operators.

III. PROPOSED ALGORITHM

This section describes the main changes to robustify the di-mensionality independent path operators.

A. Dimensionality-Independent Path Openings

Dimensionality-independent path operators are based on theprinciples behind complete ordered path operators given byTalbot and Appleton in [19] but provide a more convenientimplementation suitable to process n- images. One couldcorrectly implement a grayscale path openings (resp. closings)by stacking the results of all binary path operators from thelowest to the highest threshold (resp. highest to lowest). At

the end of the process, the value of the stacked operator at aparticular point corresponds to the highest threshold for whichthe binary operator remains true at that point (meaning thatat this grey level, this pixel fits into the structuring elementdefined by the length and by the adjacency relation).Here we provide some implementation details about this al-

gorithm. For an opening (resp. a closing) the algorithm startsby creating a linear array of pixels memory addresses orderedby grey level value (low to high for opening and high to low forclosing). Starting from the lowest to the highest grey level value,each of the pixels that is still active will be treated independentlyand will be considered as a “seed” to update downstream andupstream length change in the image. Each of the downstreamand upstream length values (respectively and ) is storedin an image and is used to accumulate path length for each ofthe pixels for the upstream and the downstream direction. Up-stream and downstream direction are given by the adjacencyrelation: for example in the case of a 2D North-South path, theupstream neighbors are .The downstream direction is defined by the opposite direction

.Initially, for all pixels, and are set to the target path

length value L and all the pixels are set to active, meaning thatthey are all part of a path of length L at the lowest grey level.Starting from an active “seed” pixel, two propagation passes aremade sequentially to update and in the image by en-queuing upstream and downstream active neighbors iteratively.To enqueue temporarily pixels during the propagation passes,a FIFO queue is used. For example, in the upstream pass,starting from an active “seed” , for each of the upstream pixelin the queue, the maximum length of its downstream neigh-

bors is found and increased by one:

(15)

If is smaller than is assigned to and itsupstream active neighbors are enqueued. As the target lengthwas set to L, the propagation would stop after a maximum of Literations or when there would not be any active neighbors to en-queue anymore. During a propagation pass, all the pixels whosedownstream value changed are enqueued in a FIFO queue .After the upstream pass, the downstream pass is done in thesame way as for the upstream pass but considering the oppo-site neighborhood relationship. When the two passes are over,the maximum length of a path which is going through a pixelstored in is computed as:

(16)

If falls under the target value , this means that this pixel isnot part of a path of length . Its output value is set to the currentselected “seed” pixel value. Moreover, as it will no be a part ofa path of length further during the process, its status is set toinactive and its downstream and upstream length are set to 0.Note that to render this algorithm suitable for processing n-

images, the implementation starts by creating a linear arrayof pixel memory addresses. From a given pixel, accessingto the desired neighbors is simply performed by adding thecorresponding memory address offset. To simply constrain the


Fig. 3. Noise pixels research for (a) The candidate is flagged; (b)The candidate is not flagged.

path to the image domain, a dark border around the image isadded in the opening (resp. light border for closing).This algorithm is slightly more time consuming than the

Talbot and Appleton version, however, it can be readily ex-tended to higher dimensions, and also can be extended toprovide more robust operators, as we now present.

B. Robust Path Openings

An overview of our proposed algorithm for robust path open-ings is outlined in Fig. 4. A more complete form of this algo-rithm can be found in the Appendix section.1) Principle: At each threshold change, given a length G,

the set of active pixels is enlarged with the pixels satisfying aspecific condition: they belong to a path of maximum length Gformed by inactive pixels. These pixels are flagged as noise andtheir upstream and downstream lengths are not reset. Duringthe propagation of the path, noise pixels act like active pixelsand will prevent the propagation from being interrupted bysmall disconnections. Obviously, in the grayscale case, at eachthreshold change we have to update the set of noise pixels fromprevious thresholds considering the newly deactivated pixels.After the updating procedure, a noise pixel which is no longer

a part of a path of maximum length G formed by deactivedpixels is enqueued is to update length change inthe image, otherwise its status and lengths are not changed. Thiswill ensure that robust path openings will respect the stackingproperty.2) Flags Description: As seen above, we need to add a

“status” flag to handle noise pixels . For efficiency reasons,two other flags are added to prevent active pixels that aredeactivated during the propagation of the path to, enter at nextthreshold into the noise pixels research procedure (as

this pixel cannot be flagged as a noise pixel at the nextthreshold) and, to prevent from being considered as a “seed”again . This is due to the fact that this pixel will not be a partof a path of length .3) Determine Noise Pixels: The search procedure of noise

pixels is performed at each threshold change when:• we determine noise pixels from the newly deactivatedpixels set.

• we update the set of previous noise pixels.Given an arbitrary length , from a candidate noise pixel, twopropagation passes are made from (in upstream and down-stream direction) seeking active pixels in the range of a max-

Fig. 4. Overview of -dimensional robust path openings.

imum of G iterations. We define and as respectivelythe minimum of upstream and downstream iterations necessaryto find an active pixel from . If , weflag as noise pixel and enqueue it in , otherwise this pixelcannot be a noise pixel and have to be used as a “seed” to prop-agate the change in lengths (its corresponding and areset to 0).Obviously, as long as its status is set to noise, a pixel cannot

be considered as a seed. Fig. 3 illustrates two possible cases for. In a), we can find an active pixel (in black) making G

iterations considering the upstream and downstream directions.We have and , this pixel is flagged as noise.In b), we have and , this pixel cannot beflagged as a noise pixel.4) Robust Propagation: Now that we have extended the set

of propagation pixels, we enter into the path propagation func-tion following the same principle as in dimensionality inde-pendent path openings [8]. Noise pixels are enqueued inin the same way as active pixels are enqueued. Fig. 11 in theAppendix section gives the robust propagation algorithm. Con-sidering both active and noise pixels, at the end of the twopropagation passes, each of the pixels whose or valuechanges is enqueued in . For each of the pixels in thelength of the longest path passing through it is computed as wehave seen previously in the explanation of dimensionality inde-pendent path openings algorithm. For an active pixel , if this


Fig. 5. 3D rendering of (a) ground truth image; (b) noisy input image ; (c) result of path openings ; (d) result of robust path openings (); (e) path openings applied on (d); the same color mapping was applied for (b), (c), (d) and (e); image is -bit.

value falls under , its corresponding upstream and downstreamlengths are set to 0. Its status is set to inactive. Obviously, thispixel will neither be considered as a “seed” nor as a noise pixel(flags ). For a noise pixel , if thisvalue falls under , its corresponding upstream and downstreamlengths are set to 0 and its noise flag is reset. To prevent this pixelto be considered as a seed in the current threshold its flagis set to true. As in Luengo’s algorithm it is possible to writedirectly in the input linear buffer, an additional output buffer isnot needed.5) Border Issue: As with dimensionality independent path

openings algorithm, we add a dark border (resp. bright borderfor closing) around the image to constrain the path to the imagedomain. At the initialization step, to prevent these pixels to enterinto the noise pixels research function we set their flags totrue. Their corresponding lengths and are set to 0.6) Path Reconstruction: With no extra cost in execution

time, we can modify in a simple way the robust path operator toreconstruct disconnections that affect thin features. The idea is

to enhance noise pixels to the highest threshold value for whichthe binary robust path operator remains true in order to fill thegaps between two active elements of the path (see Figs. 10,11 in Appendix section). In the case of a robust path opening(resp. closing), with respect to the result of the opening, theresult of this reconstruction operator forms a closing (resp.opening). Indeed it is clearly extensive (resp. anti-extensive),it is also increasing and idempotent. However the study of theprecise properties of this operator fall outside of the scope ofthe present article.

IV. EXPERIMENTAL STUDY

In this section, we evaluate the benefits of using robust pathopenings for the detection of thin human vessels in MagneticResonance Angiography (MRA) using the simulated data setpresented in [2]. The vessel length and radius are respectively90 and 2 pixels.From the evaluation method of [22] and [23], we set the back-

ground at a grey level of 100, the vessel at grey level of 160


Fig. 6. 2D vessels detection (a) input image: top-hat filter applied on the green channel of a retinal image (contrast enhanced), the red rectangle represents theregion of interest used for comparisons of the different algorithms’ results; (b) Thresholded version of the input image; (b) Result of complete ordered path openings

; (c) Result of incomplete ordered path openings ; (d) Result of robust path openings ; the same threshold wasapplied for (b), (c), (d), and (e) for a better visualization.

which are typical values of MRA acquisition. We then addedwhite Gaussian noise with standard deviation equal to 35 totest the performance of our method (typical MRA noise levelsare more around ).We filtered this image with both Com-plete and Robust path openings of length equal to 90in the z-axis direction. Fig. 5 shows three dimensional render-ings of ground truth, input and result images using the max-imum intensity projection method. We can note that disconnec-tions due to noise causes the complete path opening to fail todetect the vessel (orange color for the vessel (c)) whereas therobust path opening provides a higher detection rate (red colorfor the vessel (d)). As could be expected, the noise is greatlyreduced by the complete path opening (green background) butcontrast in the feature is also greatly reduced. The robust pathopening preserves the feature much better but also some of thenoise compared with the complete path opening. This phenom-enon results in a higher false alarm rate (yellow color for thebackground (d)).To cope with this limitation in a simple fashion, we can apply

a shorter complete path openings ( in the same orienta-tion) on the result of the robust path openings to eliminate thesmaller residuals. Fig. 5(e) shows that we are able to detect thefeature of interest and to simultaneously reduce significantly theamount of false positives (on this image, the feature is in red andbackground appears in green). Note that the same colormap isused for images (b) to (e).

V. APPLICATIONS AND TIMINGS

Here we present results regarding to the regular path opening,the incomplete version and the robust version.

Fig. 7. Running times of Complete Ordered Path Openings (COPO), Incom-plete Ordered Path Openings (IOPO) and Robust Path Openings (RPO) withdifferent tolerance parameters ( and ). Timings are in seconds. Image is565 585 8-bit.

A. Retinal Image for Vessel Extraction

Improving the detection of narrow and low-contrast featureswhile reducing the false positive detections is the goal of vesselextraction in a retinal image. This problem has been widely con-sidered in the past few years [24]–[26].Firstly the green channel of the color retinal image is ex-

tracted to obtain a grayscale image, as the green channel pro-vides the best signal to noise ratio. Secondly a square top-hatfilter is applied on the image to extract the local minima fromthe image. The top-hat filter extracts the vessels but generatesa large amount of noise. We compare the results of complete,incomplete and robust path openings on the top-hat image, per-forming the union of the four orientations in each case. Fig. 6shows the thresholded version of all the results on a region ofinterest (the region of interest was chosen where the vessels arethin and low contrasted). Using complete path openings, the to-tality of the noise is eliminated, however, we notice that thisoperator is not able to recover the vessels presenting some dis-connections. Incomplete path openings give better results, thevessels are better recovered but some vessels which are veryaffected by the disconnections are not (if the number of gapsin a path of length is greater than , path propagation isstopped). We can note also the generation of some artefacts due


Fig. 8. Surface rendering of (a) input noisy angiogram; (b) complete path openings; (c) robust path openings .

Fig. 9. Surface rendering of (a) robust rath openings with no reconstruction; (b) robust path openings with reconstruction ;

to the linkage of some random noise pixels together. Robustpath openings visibly provide the best results for the detection,as some vessels which are not present in the incomplete pathopenings result were recovered. However, as we are increasingthe tolerance of the path opening operators, we generate a largeramount of artifacts.Fig. 7 compares the running times of the different algorithms

using the top-hat retinal image as input. As expected, the com-plexity of the robust path openings is lower compared to in-complete path openings in all cases. For example, forand , robust path openings is 7 times faster than incom-plete path openings. Moreover, robust path openings has con-stant memory requirements irrespective of the parameter . Weobserve that, given a fixed , the running time of robust pathopenings is linear with G (about 0.4 G). We can also notethat the parameter does not affect computation time much forgiven a fixed , indeed, we believe that it is regulated betweennoise pixels research and path propagation.

B. 3D Data

Herewegiveanexample for theuseof robustpaths forbiomed-ical imaging on 3D data. We propose to process a 3D cerebralangiogram image in order to detect the network of vessels whileremoving the noise. To do this, we apply both complete androbust path openings of length , performing the unionof three orientations (orientations are in the x, y and z axis witha 9-neighbors adjacency relation). Fig. 8 shows that the robustoperators have the ability to recover a larger part of the vessels(e.g., the part of the vessel in the red circle) compared to thecomplete ones while eliminating most of the noise. The runningtimes of complete and robust grayscale path openingsare respectively 10.5 and 18.6 seconds (45 and 124 secondsfor and respectively). The image dimensionswere 256 256 256 8-bit. In this case, our implementation

achieves quite good performance since the background is flat.For anon-flat background (in theworst case, for awhiteGaussianbackground) our implementation could result in higher runningtimes. Fig. 9 shows that the robust path openings is able to bothdetect and then reconstruct the thin and corrupted structures. Asnoise pixels are handled like active pixels during the propagationof thepath,wecanmakeuseof the status changeof noise pixels togive them the highest value for which they belong to a path. Thisis very efficient in practice. This could be useful to recover lostinformation, as is sometime the case after a thinning operation,for instance in vascular network analysis.

VI. CONCLUSION

In this paper, we have defined the robust path operators andwe have proposed an efficient algorithm for their computation.We have specified the theoretical foundations of robust path op-erators. Our algorithm uses a constant amount of memory andits complexity is proportional to the jump length (0.4 G asshown experimentally in 2D). We have shown the effectivenessof our algorithm for detecting thin, oriented features in noisyimages on both simulated and real 2D and 3D data, and wehave compared our results to complete and incomplete path op-erators. Timings comparisons with path based algorithms werealso achieved and have proven the robust path operators to bequite efficient in practice. Although there is no equivalent im-plementation of incomplete path operators in 3D for robust pathoperators to compare with, the authors believe that the 2D re-sults would carry over to 3D as the complexity scales with thenumber of voxels involved. We have also proposed an exten-sion to robust path openings which can be used to reconstructthe noisy gaps between feature elements. As the robust path op-erators increase flexibility significantly, in the case of a WhiteGaussian background, they may generate a higher false alarmrate because isolated noise pixels may be lumped together withaligned pixels. In this paper we have proposed to combine bothcomplete and robust path openings to cope with this limitation.In this case, we reach a high detection rate for a lower falsepositive detections rate. While the implementation of our al-gorithm is inspired from the dimensionality-independent pathoperators, we provide a new framework to cope with discon-nections corrupting thin, oriented and flexible noisy features in-dimensional images. Future work will focus on the quantita-tive evaluation of the efficiency of robust path operators, con-sidering other classical pre-processing methods aiming at thedetection of thin, elongated and noisy features in 2D and 3Dimages.


APPENDIX

Fig. 10. -dimensional robust path openings.

Fig. 11. Robust path propagation.


ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir comments, which greatly improved the paper. Volume ren-derings were performed with VolView 3.4. Surface renderingswere performed using Visilog 6.9.

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[17] H. J. A. M. Heijmans, M. Buckley, and H. Talbot, “Path-based mor-phological openings,” in Proc. ICIP, 2004, pp. 3085–3088.

[18] H. J. A. M. Heijmans, M. Buckley, and H. Talbot, “Path openings andclosings,” J. Math. Imag. Vis., vol. 22, no. 2–3, pp. 107–119, 2005.

[19] H. Talbot and B. Appleton, “Efficient complete and incomplete pathopenings and closings,” Image Vis. Comput., vol. 25, no. 4, pp.416–425, 2007.

[20] P. Soille, “On morphological operators based on rank filters,” Patt.Recog., vol. 35, no. 2, 2002.

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[23] A. Pacureanu, J. Rollet, C. Revol-Muller, V. Buzuloiu, M. Langer, andF. Peyrin, “Segmentation of 3D cellular networks from sr-microct im-ages,” in Proc. IEEE ISBI, 2011, pp. 1970–1973.

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François Cokelaer received the M.Sc. degree inelectrical engineering from the Grenoble Instituteof Technology (Grenoble INP), Grenoble, France,in 2009. He is currently working toward the Ph.D.degree in image processing at the University ofGrenoble, France. His research interests includemathematical morphology, linear features filteringand their applications in the fields of biomedical andnon-destructive testing imaging.

Hugues Talbot (M’05) received the Ph.D. degree inmathematical morphology from Ecole Nationale Su-perieure des Mines de Paris (ENSMP) with highesthonors in 1993, under the guidance of professorsLinn W. Hobbs (MIT), Jean Serra, and DominiqueJeulin (ENSMP). He was affiliated with CSIRO,Mathematical and Information Sciences, Sydney,Australia, between 1994 and 2004. He is currentlya computer science professor at ESIEE, affiliatedwith Universite Paris- Est, France. He has workedon a number of applied projects in image analysis

with various companies, earning the Australian Institute of Engineers Awardin 2004 and the DuPont Australian and New-Zealand Innovation Award in2005 for work related to melanoma diagnosis. He has contributed more than 80publications to international journals and conferences. His research interestsinclude image segmentation, optimization and algorithms with application toimage analysis and computer vision. He is a member of the IEEE and the IEEEComputer Society.

Jocelyn Chanussot (M’04–SM’04–F’12) receivedthe M.Sc. degree in electrical engineering fromthe Grenoble Institute of Technology (GrenobleINP), Grenoble, France, in 1995, and the Ph.D.degree from Savoie University, Annecy, France, in1998. In 1999, he was with the Geography ImageryPerception Laboratory for the Delegation Generalede l’Armement (DGA—French National DefenseDepartment). Since 1999, he has been with GrenobleINP, where he was an Assistant Professor from1999 to 2005, an Associate Professor from 2005

to 2007, and is currently a Professor of signal and image processing. He iscurrently conducting his research at the Grenoble Images Speech Signals andAutomatics Laboratory (GIPSA-Lab). His research interests include imageanalysis, multicomponent image processing, nonlinear filtering, and data fusionin remote sensing. Dr. Chanussot is the founding President of IEEE Geoscienceand Remote Sensing French chapter (2007–2010) which received the 2010IEEE GRS-S Chapter Excellence Award “for excellence as a Geoscienceand Remote Sensing Society chapter demonstrated by exemplary activitiesduring 2009.” He was the recipient of the NORSIG 2006 Best Student PaperAward, the IEEE GRSS 2011 Symposium Best Paper Award and of the IEEEGRSS 2012 Transactions Prize Paper Award. He was a member of the IEEEGeoscience and Remote Sensing Society AdCom (2009–2010), in charge ofmembership development. He was the General Chair of the first IEEE GRSSWorkshop on Hyperspectral Image and Signal Processing, Evolution in Remote


sensing (WHISPERS). He is the Chair (2009–2011) and was the Cochair ofthe GRS Data Fusion Technical Committee (2005–2008). He was a memberof the Machine Learning for Signal Processing Technical Committee of theIEEE Signal Processing Society (2006–2008) and the Program Chair of theIEEE International Workshop on Machine Learning for Signal Processing,(2009). He was an Associate Editor for the IEEE GEOSCIENCE AND REMOTE

SENSING LETTERS (2005–2007) and for Pattern Recognition (2006–2008).Since 2007, he is an Associate Editor for the IEEE TRANSACTIONS ON

GEOSCIENCE AND REMOTE SENSING. Since 2011, he is the Editor-in-Chief ofthe IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONSAND REMOTE SENSING.

830 ieee journal of selected topics in …jocelyn.chanussot/publis/ieee_jstsp...830 ieee journal of...

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