adaptive mathematical morphology – a survey of the field

Pattern Recognition Letters xxx (2014) xxx–xxx

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Pattern Recognition Letters

journal homepage: www.elsevier .com/locate /patrec

Adaptive mathematical morphology – A survey of the field q

http://dx.doi.org/10.1016/j.patrec.2014.02.0220167-8655/� 2014 Elsevier B.V. All rights reserved.

q This paper has been recommended for acceptance by Gabriella Sanniti di Baja.⇑ Corresponding authors. Tel.: +46 737252830 (V. Curic).

E-mail addresses: [email protected] (V. Curic), [email protected](A. Landström), [email protected] (M.J. Thurley), [email protected] (C.L. LuengoHendriks).

1 These authors contributed equally to this paper.

Please cite this article in press as: V. Curic et al., Adaptive mathematical morphology – A survey of the field, Pattern Recognition Lett. (2014),dx.doi.org/10.1016/j.patrec.2014.02.022

Vladimir Curic a,⇑,1, Anders Landström b,*,1, Matthew J. Thurley b, Cris L. Luengo Hendriks a

a Centre for Image Analysis, Uppsala University and Swedish University of Agricultural Sciences, Uppsala, Swedenb Department of Computer Science, Electrical and Space Engineering, Luleå University of Technology, Luleå, Sweden

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:OverviewMathematical morphologyAdaptive morphologyAdaptive structuring elementsAdjunction property

a b s t r a c t

We present an up-to-date survey on the topic of adaptive mathematical morphology. A broad review ofresearch performed within the field is provided, as well as an in-depth summary of the theoreticaladvances within the field. Adaptivity can come in many different ways, based on different attributes,measures, and parameters. Similarities and differences between a few selected methods for adaptivestructuring elements are considered, providing perspective on the consequences of different types ofadaptivity. We also provide a brief analysis of perspectives and trends within the field, discussing possi-ble directions for future studies.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Background

Mathematical morphology, introduced by Matheron [48] andSerra [64], is a powerful framework for nonlinear image process-ing, thereby providing a useful tool for a number of tasks such asimage filtering, image segmentation, shape comparison, etc. Themost common way to define morphological operators is by usingthe concept of structuring elements. Structuring elements are usu-ally small shapes used to probe the image, defining the output ofthe filter operation from the interaction between the two. Mathe-matical morphology can also be defined on a lattice structure usingalgebraic tools [39,61], in the continuous framework using partialdifferential equations [2,18,45], and on graph-like structures[23,38,78]. For a comprehensive study on the theory and differentapplications of mathematical morphology, the interested reader isreferred to the excellent books on the topic by Soille [69] andNajman and Talbot [53].

In classical mathematical morphology, structuring elements re-main the same for all points in the image domain, i.e. one singlestructuring element is used to process the whole image by trans-lating it to every point in the image. We will refer to such non-adaptive structuring elements as rigid. One rigid structuring

element is often not suitable for the whole image however, sincethe variation of image structures regarding e.g. shape, size, and ori-entation often provides a challenge when processing all pointsidentically. For instance, morphological operators that stretch overedges as a result of unsuitable structuring elements can quickly de-stroy or corrupt important information in the image. In some casesseveral different structuring elements are considered by repeatedlyprocessing the image and selecting one single result for each point[9,44], but this means that the whole image is processed once foreach considered structuring element, while only a small fractionof the computed values may actually be needed to produce thefinal result. The usefulness and necessity of using adaptive struc-turing elements is thereby evident.

Consequently, adaptive morphology has attracted a lot of atten-tion in recent years and constitutes a topic of ongoing research.Adaptive structuring elements should adapt to the imagestructures, considering different image attributes such as gray levelvalues (luminance, contrast), spatial distances between pixels,edges in the image, image gradient, and noise. There are manyways in which this task can be achieved, and hence, due tovariation of different image attributes, there is a number of recentpapers that deal with adaptive mathematical morphology. Adap-tivity in mathematical morphology could also be considered fromdifferent perspectives depending on the mathematical structureconsidered, such as group invariance of morphological operators,or partial differential equations. Nevertheless, recent papersaddress two important aspects of adaptive mathematical morphol-ogy: (1) how to construct adaptive structuring elements that aresuitable for the image analysis task at hand, and (2) how to prop-erly define morphological operators with adaptive structuringelements.

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2 V. Curic et al. / Pattern Recognition Letters xxx (2014) xxx–xxx

1.2. Contribution

To our knowledge only one survey paper on the different ap-proaches in adaptive mathematical morphology has appeared inthe literature so far [47], and, although of excellent quality, it doesnot include the latest developments. In the light of these remarks,the primary contribution of this paper is to:

1. provide an up-to-date survey of existing approaches for adap-tive mathematical morphology (Section 2),

2. at one place summarize and clarify how adaptive morphologicaloperators should be properly computed (Section 3),

3. present four methods for the construction of adaptive structur-ing elements (Section 4),

4. present application-oriented examples of approaches in adap-tive mathematical morphology (Sections 5 and 6), and

5. briefly discuss possible future directions in which this fieldmight further develop (Section 7).

2. Overview of adaptive mathematical morphology

We here present a brief history of adaptive mathematical mor-phology and include (to our knowledge) all important work donein this field.

2.1. History

Mathematical morphology constitutes a well defined nonlineartheoretical framework based on set relations, using shapes or func-tions known as structuring elements to probe the processed data.Classical morphological operators are non-adaptive, i.e. the imageis probed by a single rigid structuring element. This is often notideal however, as discussed in Section 1, which has motivatedthe development of adaptive mathematical morphology.

Most likely the first ideas on morphological operators based onnon-rigid structuring elements appeared in the work of Serra [65],providing a foundation for adaptive morphology. One of the firstapplications of adaptive structuring elements was presented byBeucher et al. [10] who used structuring elements that adapt withrespect to their position in the image, following the law of perspec-tive for traffic cameras. Similarly, Verly and Delanoy [77] designedadaptive structuring elements for application to range images.Charif-Chefchaouni and Schonfeld [20] considered general theoryfor adaptive morphology in the binary case. Other early work onadaptive structuring elements was undertaken by Morales [50],Chen et al. [21] and Cheng and Venetsanopoulos [22].

Attention to this area of mathematical morphology has in-creased within the last ten years, including the introduction ofmethods such as general adaptive neighborhoods [28] and mor-phological amoebas [42]. These papers present two different meth-ods for constructing adaptive structuring elements based on localsimilarity measures for neighboring pixels, and seem to have in-cited momentum for adaptive morphology within the mathemati-cal morphology community as well as with the wider audience. Atabout the same time, gray valued input-adaptive morphology wasdefined by Bouaynaya and Schonfeld [14], based on the earlierwork for the binary case by Charif-Chefchaouni and Schonfeld[20] and the umbra transform. Theoretical advances as well as lim-itations of adaptive mathematical morphology have been exploredby Bouaynaya et al. [12] and Bouaynaya and Schonfeld [15]. Fur-thermore, Roerdink [59] addressed theoretical issues importantfor the development of the field, pointing out often overlooked is-sues regarding the properties of adaptive morphological operators.This study provides a good ground for further development of thefield, as well as a discussion on the terminology used for adaptivestructuring elements.

Please cite this article in press as: V. Curic et al., Adaptive mathematical modx.doi.org/10.1016/j.patrec.2014.02.022

The latest studies of adaptive mathematical morphology in-clude work on adaptive structuring functions, mostly inspired byrecent image filtering techniques such as bilateral filtering [4]and nonlocal means [63,75]. Furthermore, an interesting applica-tion of adaptive mathematical morphology to image regularizationfor inverse problems has also recently been presented by Purkaitand Chanda [57].

It should be noted that adaptive structuring elements play asimilar role as kernels used in other well-known methods for adap-tive filtering [49,52,54,73] where a desirable property of the filter-ing kernel is that it does not operate across edges in the image. Thegoals are different, however, as morphological operators are usednot only to remove the noise but also to preserve the shapes inthe image. Morphological operators are based on the maximumor minimum value, rather than the median or weighted mean overpixel values within the kernel.

2.2. Overview of the field

There exists a variety of different methods for constructingadaptive morphological operators that differ in how they adaptto different image attributes. As a result, there are different waysin which approaches to adaptive mathematical morphology canbe grouped. Maragos and Vachier [47] considered the followingthree groups:

� adaptivity with respect to the spatial neighborhood position,� adaptivity with respect to gray level image values, and� algebraic principles such as group and representation theory.

At the same time, Roerdink [59] defined two categories of adaptive-ness, referring to them as:

� location-adaptive mathematical morphology (adaptability withrespect only to the position in the image domain) and� input-adaptive mathematical morphology (adaptability with

respect to the image content, i.e. the position in the imagedomain as well as values in the image range),

which obviously coincide with the two first groups considered byMaragos and Vachier [47].

We follow the aforementioned categories, but identify a largerdiversity of adaptivity in mathematical morphology. We identifyseveral aspects on adaptivity to which the various methods canbe associated. Note that neither the aforementioned grouping ofdifferent methods nor the aspects we have identified providestrictly disjoint groups, since one method can belong to several dif-ferent groups or aspects. Our primary goal is not to present a newcategorization of different methods in adaptive mathematical mor-phology, but rather to complement the work by other authors.

2.2.1. SimilarityMost methods for defining adaptive structuring elements rely

on local similarity of neighboring pixels. Structuring elements in-clude points that are similar to the considered central point (originof the structuring element) according to some measure of localsimilarity such as spatial similarity, similarity between gray levelvalues, etc.

Some methods allow for complete adaptivity of the shape of thestructuring elements. For instance, Debayle and Pinoli [28] pre-sented a method for adaptive morphology based on homogeneousregions, obtaining spatially adaptive structuring elements deter-mined by the connected component that contains the origin ofthe structuring element. This approach is closely related to earlierwork by Braga-Neto [16]. Cuisenaire [24] presented locally adap-tive mathematical morphology based on the distance transform.

rphology – A survey of the field, Pattern Recognition Lett. (2014), http://



V. Curic et al. / Pattern Recognition Letters xxx (2014) xxx–xxx 3

Possibly the best known method for adaptive structuring elements,morphological amoebas, is based on a geodesic distance takingboth spatial distance and gray level difference into account [43].Similarly, Grazzini and Soille [35] considered spatially variableneighborhoods by utilizing another cost function for geodesic dis-tances, while Curic et al. [25] defined adaptive structuring ele-ments that are computed from the salience distance transform ofthe edge image. In the same line, Morard et al. [51] presented amethod that uses a region growing technique based on the similar-ity between neighboring points.

Other methods use a predefined shape where the size of theshape is set according to a measure of local similarity. For example,Dokládal and Dokládalova [33] used rectangles, while Curic andLuengo Hendriks [26] used circles.

Recent studies on adaptive structuring functions have also con-sidered measures of local similarity between neighborhood points[4,27] as well as on similarity between image patches [63,75]. Re-cently, an interesting work that introduces adaptive morphologicaloperators into a stochastic framework has been presented by Ang-ulo and Velasco-Forero [7]. This method is based on random walksimulations as an estimation for adaptive structuring functions.

Apart from including pixels which are similar to the consideredcentre of the structuring element, as defined by spatial or geodesicdistance, adaptive morphological operators can also be adapted tolevel set decomposition of the image [74,46]. These studies are clo-sely related to viscous mathematical morphology and viscous lat-tices [66].

2.2.2. StructureRelated to similarity (or lack thereof), but still different by def-

inition, is the concept of structure. Structure-based methods alignthe structuring elements to edges and contours, rather thanrestricting them by a measure of similarity. This can be done byconsidering orientation only, or by considering rate of anisotropyor distances to edges as well.

Shih and Cheng [67] defined elliptical structuring elementsfrom local path curvature obtained by edge linking in binary data.Tankyevych et al. [72] used adaptive morphology for vesselenhancement: directions for line structuring elements are obtainedfrom a so called ‘‘vesselness function’’ depending on the eigen-values of the Hessian for each neighborhood. Adaptive line andrectangular structuring elements, where the latter are constrainedin width by nearby distinct edges, were considered by Verdú-Monedero et al. [76], who obtained directions from diffusedsquared gradient fields. Landström and Thurley [40] used the LocalStructure Tensor (LST) to define elliptical structuring elementswhich vary from lines to disks depending on the rate of anisotropy.This method has also been extended to enable processing of partlymissing data [41].

A slightly different approach [5] is based on multiscale imagedecomposition, adapting the size of the structuring elements tothe local scale of structures in the image. This approach processestwo points of similar scales in the same way, rather than using alevel set decomposition of the image.

2.2.3. Partial Differential EquationsA quite different approach to mathematical morphology is

achieved by considering Partial Differential Equations (PDEs)where the main morphological operators can be defined using dif-fusion equations. This strategy defines morphological filters with-out explicit use of structuring elements. The implicit structuringelement is a unit ball that can deform over time. PDE-based meth-ods may very well take similarity and structure into account,depending on the formulation of the problem, and operate in acontinuous framework. Breuß et al. [17] proposed continuous mor-phology based on tensors, solving PDEs. Maragos and Vachier [46]


introduced adaptivity of morphological operators for PDEs, defin-ing viscous dilation and erosion. A generalization of the PDEs fornonlocal erosion and dilation has also been defined on a graphstructure [71].

Welk et al. [80] proposed differential equations for morpholog-ical amoebas and presented an interesting connection betweenmorphological amoebas and self-snakes. In the same line, Welk[79] also considered a connection of morphological amoebas witha curvature-based PDE. It should be stressed that these connectionsof morphological amoebas with PDEs are specific for the amoebamedian filter, and not general for morphological operators.

2.2.4. GraphsAs already mentioned, there exists work on nonlocal morpho-

logical operators on graphs by Ta et al. [71]. Adaptive morpholog-ical operators, in particular ones based on morphological amoebas,are defined using the concept of minimal spanning trees [70].Cousty et al. [23] presented a unified framework for graphs thatincludes discrete spatially variant structuring elements.

Path openings and closings are morphological operations withflexible line segments as structuring elements [37]. Unfortunately,path openings as well as area openings and other attribute open-ings are often not considered as a part of adaptive mathematicalmorphology.

2.2.5. Group adaptivityMorphological operators are mostly defined as translation

invariant operators. Nevertheless, for certain applications, the use-fulness of translation-invariant morphological operators can belimited [60]. Therefore, group morphology has been introduced[58]. In this framework, morphological operators are invariant un-der different types of transformations such as non-commutativesymmetry groups, rotation groups, or similar.

Morphological operators with fixed structuring elements aretranslation-invariant operators. However, as pointed out by Roerd-ink [59], most adaptive morphological operators are also transla-tion-invariant operators. If an operator adapts its structuringelement to local features in the image, in a translation invariantmanner, the operator is translation-invariant, even if the structur-ing element is not the same at all locations in the image. Therefore,adaptive morphological operators are not necessarily translationinvariant operators.

2.2.6. Efficient implementationsEfficient algorithms have been proposed for morphological

operators with rigid structuring elements (see e.g. [1,34]). This islikely a strong reason for the wide use of morphological operatorswithin the image processing community. Nevertheless, the num-ber of presented algorithms for adaptive morphological operatorswith focus on efficiency is still quite limited. An efficient algorithmfor adaptive morphological operators on binary images has beenproposed [36]. Stawiaski and Meyer [70] proposed fast implemen-tation of morphological amoebas using the graph-based approachand the concept of minimal spanning trees, while Velasco-Foreroand Angulo [75] presented an efficient implementation of nonlocalmorphological operators using sparse matrices.

3. Theory

The definition of adaptive morphological operators has oftenbeen overlooked in the literature, as pointed out by Roerdink[59]. In this section we summarize general theoretical work aimedat properly and efficiently performing morphological operationsusing adaptive structuring elements, i.e. adaptive morphologicaloperators. Approaches for generalizing classical non-adaptive





morphology into the adaptive case are covered. Our aim is to: (1)provide a short summary of how to properly compute adaptivemorphological operators; and (2) put the considered approachesin perspective by using a unified notation. For more details onthe approaches, we refer the interested reader to the originalpublications.

We start by addressing an important property known asadjunction, which is required to define mathematically correctopenings and closings from the combination of an erosion and adilation. Morphological operations are then formulated for therigid case, providing a basis for a coherent notation, beforeaddressing the theoretical work on adaptive morphology. Morespecifically, we consider three suggested approaches for definingadaptive morphology, based on structuring elements, structuringfunctions, and using the notion of impulse functions.

3.1. Adjunction

Let ðL;6Þ be a complete lattice of gray valued functions withthe domain D and range T . Two morphological operators e and ddefined on a lattice L form an adjunction ðe; dÞ when

dðf Þ 6 g () f 6 eðgÞ ð1Þ

for any pair of f ; g 2 L. One of the main issues within adaptivemathematical morphology concerns proper definition of adjunctmorphological operators, which is addressed in a number of papers[15,59,75]. Adjunction is important because if the morphologicalerosion e and dilation d fulfill (1), i.e. form an adjunction ðe; dÞ,the morphological opening c and closing u can be defined as d � eand e � d, respectively [61].

Moreover, it has been shown [59] that morphological operatorse and d satisfy the adjunction property (1) only if adaptive structur-ing elements are derived only once from the input image. Thatmeans that we should first compute adaptive structuring elementsfor every point in the image, i.e., have a set of structuring elementsfSx : x 2 Dg, where D denotes the domain of the image. These sameshapes, sometimes referred to as the structuring element map,should be used when computing both the erosion and dilation,since only then they will form an adjunction. These structuring ele-ments could be computed from the input image or from asmoothed version of the input image, called the pilot image [43].

3.2. Non-adaptive morphology

We first consider classical non-adaptive morphology, formulat-ing it in a notation that simplifies subsequent generalization to theadaptive case. We here distinguish between (flat) structuring ele-ments, which are defined solely as sets of pixels, and (non-flat)structuring functions, which have the range ½�1; 0�. Most com-monly, morphological operations are performed using (flat) struc-turing elements, i.e. set-based structuring elements. Given a rigidstructuring element S, its translation Sx to a point x 2 D can be ex-pressed as [64]

Sx ¼ y 2 D : y � x 2 Sf g; ð2Þ

which simply yields the structuring element S corresponding to thelocal variable ðy � xÞ.

Each structuring element Sx has another corresponding elementS�x. For non-adaptive structuring elements this is simply the trans-lated reflection through the origin, given by

S�x ¼ y 2 D : x� y 2 Sf g: ð3Þ

Note that S�x can also be implicitly, but equivalently, defined by therelation

y 2 S�x () x 2 Sy; x; y 2 D: ð4Þ


The erosion eS : L ! L and dilation dS : L ! L of a function f 2 L bythe structuring element S are then defined as [64]

eSðf ÞðxÞ ¼^

y2Sx

f ðyÞ; x 2 D; ð5Þ

dSðf ÞðxÞ ¼_

y2S�x

f ðyÞ; x 2 D; ð6Þ

whereV

andW

denote the infimum and supremum operators,respectively.

The entity S�x is known under various different names, such asthe reflected, transposed, or reciprocal structuring element, andthere is currently no consensus regarding proper terminology.While the aforementioned names have all been used for boththe rigid and the adaptive case, they risk leading to ambiguitiesin a general terminology for adaptive morphology as there is rea-sonable room for misinterpretation (for instance, in the adaptivecase S�x cannot simply be described as the reflection of Sx throughthe origin). However, for the general case we observe thefollowing:

1. Given the set of structuring elements fSx : x 2 Dg there is a one-to-one relation between Sx and S�x for each point x.

2. We can alternate between Sx and S�x in a cyclic manner using theð�Þ� notation based on Eqs. (3) and (4) above, i.e.

rpholog

S��x ¼ ðS�xÞ� ¼ Sx: ð7Þ

These properties form a duality with respect to the set of structur-ing elements fSx : x 2 Dg, i.e. the whole set is needed to retrieve S�xfor a given point x, and we will consequently refer to S�x as the dualstructuring element.

We then turn our focus to the non-flat case, i.e. structuringfunctions. Let s : D ! ½�1;0� be an arbitrary rigid structuring func-tion and let sx represent its translation to a point x 2 D. For anypoint y we then have

sxðyÞ ¼ sðy � xÞ; ð8Þ

while the corresponding dual structuring function s� is defined as[15]

s�xðyÞ ¼ syðxÞ ¼ sðx� yÞ: ð9Þ

It should here be noted that for structuring functions Eq. (7)becomes

s��x ¼ ðs�xÞ� ¼ sx; ð10Þ

in analogy with the flat case. Erosion and dilation by the structuringfunction s are then defined for each point x 2 D by [64]

esðf ÞðxÞ ¼^

y2DðsxÞf ðyÞ � sxðyÞð Þ; x 2 D; ð11Þ

dsðf ÞðxÞ ¼_

y2Dðs�xÞf ðyÞ þ s�xðyÞ� �

; x 2 D; ð12Þ

where DðsxÞ is the domain of the structuring function s in point x. Itshould be noted that the flat case is given by expressing a structur-ing element S using a corresponding structuring function s definedas

sðyÞ ¼0; y 2 S;

�1; y R S;

�ð13Þ

which inserted into (11) and (12) yields (5) and (6), respectively.

y – A survey of the field, Pattern Recognition Lett. (2014), http://



Fig. 1. Adaptive structuring elements are computed in points x; y and z. Since x 2 Sy

then y 2 S�x , and x R Sz then z R S�x .


3.3. Adaptive structuring elements

By far the most commonly used strategy within adaptive mor-phology relies on the use of (flat) structuring elements (see e.g.[25,40,43]). Given such a pixel-dependent structuring elementS½x�, which may vary with the considered pixel x, the translatedstructuring element Sx is defined as

Sx ¼ y : y � x 2 S½x�f g: ð14Þ

Its dual structuring element S�x (see Fig. 1) is then implicitly definedby (4) or, equivalently, defined explicitly as [15]

S�x ¼ y : x� y 2 S½y�f g: ð15Þ

Note that this approach simply generalizes the rigid structuring ele-ments, i.e. (14) and (15) simplify to (2) and (3) if S½x� ¼ S; 8x 2 D.Once structuring elements are constructed, the morphological ero-sion and dilation are again defined by (5) and (6). These morpholog-ical operators satisfy the adjunction property (1) only if fSx : x 2 Dgare computed once for the input image and used to compute bothoperations.

Velasco-Forero and Angulo [75] defined a structuring elementsystem that overcomes the issue of adjunction. These adaptivestructuring elements satisfy the following properties: (1) x 2 Sx,and (2) y 2 Sx ) x 2 Sy. Note that this is a sufficient but not neces-sary condition for e and d to form an adjunction ðe; dÞ.

Retrieving the dual structuring elements in the adaptive caserequires computing and storing the structuring elements for allpoints in the image. It may therefore be simpler to compute ad-junct operators based on (4): computations can be done withoutexplicitly calculating any dual structuring elements S�y , by observ-ing that a processed pixel x constitutes a part of the dual structur-ing element S�y for any pixel y within its structuring element Sx. Theadjunct dilation can be calculated by the following algorithm usedby Lerallut et al. [43]:

for each point x 2 D docompute Sx

for each y 2 Sx dodðyÞ ¼ maxðf ðxÞ; dðyÞÞ

end forend for

This way of computing the adjunct adaptive dilation does notrequire storage (or frequent recomputation) of all structuringelements.

3.4. Adaptive structuring functions

Since the adaptive structuring function now varies for eachpoint x 2 D, it is here denoted with s½x�. In the theory providedby Bouaynaya and Schonfeld [14,15], non-adaptive morphological


operators based on rigid structuring functions (Section 3.2) are di-rectly generalized to the adaptive case. Hence, for adaptive struc-turing functions we have

sxðyÞ ¼ s½x�ðy � xÞ: ð16Þ

The corresponding dual structuring function is defined as

s�xðyÞ ¼ syðxÞ ¼ s½y�ðx� yÞ: ð17Þ

Note that, as for rigid structuring elements, (13) can be used to con-vert adaptive structuring elements into structuring functions,which gives the flat case covered in Section 3.3. For rigid structuringfunctions, i.e. if s½x� ¼ s; 8x 2 D, the expressions (16) and (17) aresimplified into (8) and (9), respectively.

Similarly to a structuring element system, Velasco-Forero andAngulo [75] defined a morphological weight system where thestructuring functions satisfy the following properties: (1) sxðxÞ ¼ 0,and (2) sxðyÞ ¼ syðxÞ. This is a sufficient, but not necessary, conditionfor ðe; dÞ to form an adjunction.

3.5. Impulse functions

Adaptive morphological operators can also be defined using thenotion of impulse functions, without explicit use of the adjunctionproperty [7,76]. In this case, the morphological opening and closingare computed directly without resorting to compositions of theerosion and dilation. For h 2 D and t 2 T , the impulse function iy

is defined by

iyðxÞ ¼1; if x ¼ y;0; if x – y

�ð18Þ

for all x 2 D, where 0 is the smallest element within the range T .Every function f 2 LðD; T Þ can then be written as

f ¼_x2D

ix � f ðxÞ ð19Þ

and the erosion e, dilation d, and opening c are defined, respectively,as [11]

eðf Þ ¼_t2Tfix � t : CDðsxÞ;t 6 f ;x 2 Dg; ð20Þ

dðf Þ ¼_fCDðsxÞ;f ðxÞ : x 2 Dg; ð21Þ

cðf Þ ¼_t2TfCDðsxÞ;t 6 f : x 2 Dg; ð22Þ

where DðsxÞ is the support of the structuring function in point x andCDðsxÞ;t is the cone of base DðsxÞ and height t.

4. Selected methods

In this section we consider two of the most influential (and cur-rently most cited) works of adaptive mathematical morphology, aswell as our own recent methods on this topic. We present a morein-depth study of the following approaches:

� General Adaptive Neighborhoods (GANs) [28],� Morphological Amoebas (MAs) [43],� Salience Adaptive Structuring Elements (SASEs) [25], and� Elliptical Adaptive Structuring Elements (EASEs) [40].

These methods are presented in order of increasing constraints onthe structuring element shapes, ranging from the first method(GANs), which provides complete adaptivity, to the last one (EASEs),where shapes are predefined.





4.1. General adaptive neighborhoods

General adaptive neighborhoods (also called intrinsicstructuring elements) were proposed by Debayle and Pinoli[28–30]. A connected neighborhood is considered for each pointx 2 D, which includes points based on a measure hðxÞ 2 R (suchas gray level values, contrast, or similar). This adaptiveneighborhood, called a weak General Adaptive Neighborhood, isdefined as

VmðxÞ ¼ y 2 D : hðyÞ � hðxÞj j < m and y 2 CCðxÞf g; ð23Þ

where CCðxÞ denotes the connected component of the point x andm > 0 is a tolerance that determines the size of the neighborhood.In order to have a proper structuring element that can directly con-struct adjunct morphological erosion and dilation, the concept ofstrong GANs is introduced. They are used as adaptive structuringelements for each point x 2 D, and are defined as

Smx ¼

[z2DfVmðzÞ : x 2 VmðzÞg: ð24Þ

The GAN framework has been used in a connection with theChoquet filtering [31] and for the logarithmic image processing[55]. Also, the GAN approach has been recently extended to spa-tially and intensity adaptive morphology [56] where level setsare processed at different scales.

The computational complexity of the direct implementation ofthe GANs is OðN2Þ where N is the number of pixels in the image.For each point in the image it is necessary to check if all pointsin the image are in the predefined tolerance m and in the sameconnected component as the considered point.

4.2. Morphological amoebas

Morphological amoebas likely constitute the most well-knownmethod for adaptive mathematical morphology. They rely on geo-desic distances. When computing geodesic distances, a 2D image isembedded into a 3D surface with two spatial coordinates and onethat represents the gray level value. A geodesic distance betweentwo points ðx; f ðxÞÞ and ðy; f ðyÞÞ is the shortest distance over thesurface between the two points [68]. Let Pðx; yÞ ¼ fx ¼ x1; . . . ;

xi;xiþ1; . . . ;xn ¼ yg be a path connecting points x and y. The costof the path is computed as the sum of the cost of all adjacent pointsin the path, i.e., cðxi;xiþ1Þ; i ¼ 1; . . . ;n� 1. The cost of the pathPðx; yÞ is then equal to the distance

dðx; yÞ ¼ minPðx;yÞ

Xn�1

i¼1

cðxi;xiþ1Þ: ð25Þ

A structuring element Srx centered in a point x can then be com-

puted as

Srx ¼ y 2 D : dðx; yÞ < rf g; ð26Þ

where r > 0 is the parameter that determines its size.For morphological amoebas cðxi;xiþ1Þ ¼ 1þ kjf ðxiÞ � f ðxiþ1Þj,

where k > 0 is a weight parameter. The difference jf ðxiÞ � f ðxiþ1Þjpenalizes the changes in gray level values, i.e., restricts structur-ing elements from growing across high gradients. The number 1stands for the spatial distance between adjacent pixels andbetter properties of morphological amoebas can be achievedif Euclidean or weighted distances [13] are used insteadof 1.

Similarly to morphological amoebas, Grazzini and Soille [35]used the following costs between two adjacent pixels xi and xiþ1

to define adaptive structuring elements:

cðxi;xiþ1Þ ¼12jrf ðxiÞj þ jrf ðxiþ1Þjð Þ � kxi � xiþ1k ð27Þ


and

cðxi; xiþ1Þ ¼12jf ðxiÞ � f ðxiþ1Þj � kxi � xiþ1k: ð28Þ

Region growing structuring elements [51] lie halfway betweenGANs and morphological amoebas. The points that belong to astructuring element are included with a region growing procedure,and the number of points in the structuring element is predefined.The growing procedure is based on the difference between gray le-vel values of the adjacent points, which makes it similar to mor-phological amoebas if cðxi;xiþ1Þ ¼ f ðxiÞ � f ðxiþ1Þj j. This methodproduces adaptive structuring elements that do not necessarily oc-cupy a whole homogeneous region as is the case with GANs.

The computational complexity for morphological amoebas isOðN � r2 log r2Þ where r determines the size of the distance propa-gation, i.e., is the radius of the morphological amoeba. Notice thatthe computational cost could reach OðN2 log NÞ if the size of thestructuring elements is the whole image.

4.3. Salience adaptive structuring elements

Salience adaptive structuring elements [25] are adaptive struc-turing elements computed with path-based distances on the sal-ience map of the input image. The salience map is obtained fromthe salience distance transform [62] of the weighted edges or someother attributes in the input image, containing information aboutthe important structure in the image. For this particular method,the salience map SM is computed as

SMðzÞ ¼ Offsetþ_

w2DNMSðf ÞðwÞ � kw� zkð Þ; z 2 D; ð29Þ

where NMSðf Þ is the result of non-maximal suppression applied tothe Gaussian gradient magnitude, as used in the Canny edge detec-tor, and

Offset ¼^z2D

_w2D

NMSðf ÞðwÞ � kw� zkð Þ: ð30Þ

The smoothness used in the Gaussian derivatives is comparable tothat of the pilot image in morphological amoebas. Since the saliencemap SM already incorporates spatial and tonal information, the costof the path between two adjacent points xi and xiþ1 is computed as

cðxi; xiþ1Þ ¼ SMðxiÞ þ SMðxiþ1Þ ð31Þ

and a salience adaptive structuring element Srx is defined by Eq. (26)

using the cost given by Eq. (31).To allow a larger flexibility in size, the radii of the salience adap-

tive structuring elements also depend on the salience map SM, sothat structuring elements located close to edges in the input imageare smaller in size while structuring elements in homogeneousareas are larger. The salience map SM can be further controlledby scaling the initial edge map or using a different distance prop-agation for its construction.

Salience adaptive structuring elements have the same computa-tional cost as morphological amoebas since are based on the geo-desic distance propagation on the salience map.

4.4. Elliptical adaptive structuring elements

A structure-adaptive morphological filtering method was pre-sented by Landström and Thurley [40]: elliptical adaptive struc-turing elements are based on the well-known Local StructureTensor (LST), which is a 2 � 2 matrix for each pixel (in 2D) whoseeigenvectors and eigenvalues, respectively, contain informationabout the orientation of structures in the image (i.e. edges) andthe rate of anisotropy [19]. More specifically, the LST T is givenby




Table 1Parameters used in the performed experiments.

Experiment Imagesize

GAN MA SASE EASE

Fig. 2 186� 186 m ¼ 5 k ¼ 0:25; r ¼ 40 k ¼ 25 M ¼ rw ¼ 20Fig. 3, left 91� 91 m ¼ 30 k ¼ 0:25; r ¼ 10 k ¼ 7 M ¼ rw ¼ 10Fig. 3, right 170� 170 m ¼ 10 k ¼ 0:25; r ¼ 4 k ¼ 4 M ¼ rw ¼ 4


TðxÞ ¼ Gr � $f ðxÞ$Tf ðxÞ� �

; ð32Þ

where $ ¼ @@x1

@@x2

� �Tis the gradient (nabla) operator and Gr is a

Gaussian kernel with standard deviation r, which acts to regularizethe matrix as well as setting the scale at which structures should beconsidered.

The orientation of the elliptical structuring elements is thengiven by the orientation of the eigenvalues of T, while the semi-major and -minor axes a and b are set from

aðxÞ ¼ k1ðxÞ þ �k1ðxÞ þ k2ðxÞ þ 2�

M; ð33Þ

bðxÞ ¼ k2ðxÞ þ �k1ðxÞ þ k2ðxÞ þ 2�

M; ð34Þ

where M is the user-defined semi-major axis, � > 0 is a small con-stant (i.e. machine epsilon), and k1 and k2 are the eigenvalues of T.

The resulting elliptical structuring elements vary dynamicallybetween lines, where the image is highly anisotropic, and disks,in isotropic regions. The LST is calculated on a neighborhood de-fined by a user-supplied radial bandwidth rw defining r, whereaf-ter structuring elements are defined from the eigenvectors andeigenvalues of the LST based on a user-set maximum semi-majoraxis M.

Given a set of pre-defined structuring elements, which can bestored in a look-up table since a certain number of specific shapesare considered, the computational cost for the method is OðN � RÞ.R here represents the maximum number of neighborhood pixelsused in the operations, resulting from the choice of M and rw.

Fig. 2. First row: calculated structuring elements for (a) GANs, (b) morphological amoe


5. Experimental results

Studying the behaviour of different methods is important forunderstanding how different types of adaptivity affect the opera-tions. We therefore present examples where a selected set of meth-ods have been applied to the same images, considering thedifferent behaviors caused by the different definitions as well asmain differences and similarities. We do not intend to present anextensive comparison between different methods for adaptivemorphological operators since they rely on: (1) different imageattributes, (2) different measures of these attributes, and (3) differ-ent user-set parameters for these measures. As such they are notdirectly comparable, and a quantitative comparison can easily be-come biased due to the choice of images and parameters.

The choice of parameters will of course play an important rolein determining the size and/or shape of the structuring elements,and a number of different parameter setups were investigated foreach of the considered methods. However, as the parameters arenot directly comparable we focus on analyzing the behaviour ofthe structuring elements based on their definition and the under-lying theory, and the presented results are based on parametersfor which each of the methods will have similar size of structuringelements on average. The parameter values used in the experi-ments are summarized in Table 1. Note that for SASE the adaptiveradius rðxÞ ¼ k �meanðSMÞ � SMðxÞ;x 2 D is used.

All methods considered here depend on the computation ofadaptive structuring elements and the subsequent finding of theminimum or maximum of gray level values in the input imagewithin the resulting neighborhood. In this study we perform threeexperiments.

5.1. Experiment 1: structuring element shapes

We first compare the shapes of adaptive structuring elementsfor a noiseless synthetic image, as well as for the same image withadded independent identically distributed white Gaussian noisewith standard deviation r ¼ 5. Fig. 2 shows the shape of adaptivestructuring elements for three points in the image. The circles de-note the considered points in the image. Note how the two GANsfor the green and the blue points blend together in Fig. 2(a), whichdemonstrates that the structuring elements overlap. In the noisy

bas, (c) SASEs, (d) EASEs. Second row: same as the first row but for a noisy image.




Fig. 3. Left column: dilation of the diamond image; Right column: opening of thehistorical text document. (a) Input image, (b) GANs, (c) Morphological amoebas, (d)SASEs, (e) EASEs.


case (Fig. 2e), the GAN for the red point contains many holes, whilethe other GANs shrink to a single point. We see that GANs indeeddisregard any spatial distances, and are completely based on zonesof homogeneous intensity. With increasing noise level, originallywide structuring elements become more and more sparse (i.e. haveholes) while thin structuring elements risk being reduced in size toa few (or even a single) pixel. Morphological amoebas align strictlyto strong edges, while forming more disk-like shapes in regions ofmore isotropic structure (Fig. 2(b), red structuring element). Thatis, edges cut off the structuring elements but do not otherwiseaffect the size of the amoebas. For noisy data (Fig. 2(f)), structuringelements shrink as the intensity variation within the amoebas in-creases, causing amoeba distances to increase more rapidly withrespect to spatial distances. Note a few holes that appear in thestructuring element denoted with red color. SASE cross edges morethan amoebas. Comparing the sizes of the larger blue and the smal-ler green structuring elements (Fig. 2(c)), we also see that the sizeof these structuring elements are affected more near strong edges(in contrast to the case with amoebas). The size of SASE is affectedin the noisy image due to large difference between the strong andweak edges in the input image (Fig. 2(g)). In most cases they formshapes with fewer holes than morphological amoebas. Adaptiveelliptical structuring elements have convex and symmetric shapeswithout holes in them and they are least affected by the noise(Fig. 2(g) and 2(h)). Note that the line structuring element, denotedwith red color, is the same for the noiseless and the noisy image.The elliptical structuring elements become line-shaped wherethere is a clear dominant directional structure in the image, whilechanging towards disks where directional structure is more ambig-uous: the red structuring element is a vertical line while the bluestructuring element is wider than its green counterpart. Note thatthe salience adaptive and the elliptical structuring element in theupper part of the image breach the border between the shapesmore than the other two methods (although this is dependent onthe choice of k for morphological amoebas).

5.2. Experiment 2: gray level contrast

The second experiment concerns morphological dilation of asynthetic image with diamonds of different size and various con-trast with smooth intensity transition between the objects andbackground. The results are depicted in Fig. 3, left column. Dilationwith GANs changes the complete background, assigning it the va-lue of the tolerance m, while leaving the inner parts of diamondsalmost intact (Fig. 3(b)). Note the artefacts that appear around dia-monds for dilation with morphological amoebas, where the shapesof diamonds are all changed (Fig. 3(c)). These effects result fromsmall differences in contrast, in this case caused by the smoothingprefiltering which decreases the gray level values at the corners ofthe diamonds in the pilot image. Although this could be changedby using another type of prefiltering to create the pilot image (herea gaussian filter with standard deviation r ¼ 1 was used), the re-sults indicate a sensitivity towards small contrast changes thatcan give rise to clear artefacts in the filtered image. SASEs do nothandle the corners of the shapes well, but cause fewer artefactsthan amoebas. This is the only method where dilation grows theshapes with low contrast more than high-contrast ones(Fig. 3(d)). Elliptical adaptive structuring elements cause the leastartefacts, and the resulting shapes of the dilated diamonds arenot affected by their different levels of contrast (Fig. 3(e)).

5.3. Experiment 3: structures in real world data

Our third experiment demonstrates morphological openings ofa gray-level image containing parts of a historical text document(see Fig. 3(a), right). The contours of the letters constitute structure


that are occasionally discontinuous, i.e. have small gaps. Note thatthe text printed on the back side is partly visible in the background.The text is not significantly changed by GANs (Fig. 3(b)), but noisein the background is reduced. Morphological amoebas (Fig. 3(c)) fillin the smallest gaps in the letters and reduce noise. The salienceadaptive structuring elements close gaps in the contour, but maycause blurred letters in the result (Fig. 3(d)). The elliptical structur-ing elements, however, efficiently fill in the gaps without muchchange to the outer contours of the letters (Fig. 3(e)).





6. Discussion

From classical non-adaptive morphology we are used to well-defined and often quite small structuring elements that representsome type of intuitive shape, e.g. disks, diamonds, lines, andcrosses. In the adaptive case however, the diversity of shapes usedas structuring elements is more or less unlimited and varies sub-stantially between the methods. Nevertheless, it should be notedthat all structuring elements in the more closely investigatedmethods share two fundamental properties: (1) they are connectedcomponents and (2) they include a point x that they belong to (i.e.their origin).

Methods based solely on connected homogeneous areas in theimage, such as General Adaptive Neighborhoods, are completelyunaware of spatial relations as long as connectivity with the originof the structuring element is preserved. As a consequence, suchmethods are unaffected by scale (i.e. large vs. small objects). Also,for GANs, all points in the same connected homogeneous regionwill share the same adaptive structuring element independentlyof the neighboring structures in the image. Moreover, when apply-ing this type of structuring elements to a close-to-uniform region,such as a black background with contrast variations, local varia-tions in the input image may affect large regions in the output.On the other hand, if an image is highly corrupted by noise, meth-ods based solely on gray-level values will still disregard highlynoisy values and only include points for which the intensity levelsare more similar to the considered origin. Hence, more extremeoutliers (in terms of intensity levels) are not allowed to affect theresult, which makes methods such as GANs suitable for noiseremoval.

Methods based on weighted combinations of gray-level andspatial information provide a tool for spatial constraints, therebytaking the geometry of shapes into account. This is more similarto what we are used to from classical morphological processing.In the case of morphological amoebas, for instance, the parameterk regulates addition between two incommensurate image domainsand has a strong influence on the size and shape of the morpholog-ical amoebas. The resulting structuring elements adapt well toedges when gray-level values have larger impact, but increasingthe weight of the spatial distance enables crossing of even strongedges. The weighting of the two thereby provides a tradeoff be-tween flexibility, i.e. avoid crossing edges, and constraints thatforces them to cross edges. In the case of morphological amoebas,the two parameters k and r should be set simultaneously.

Setting an explicit weight parameter, such as the k in morpho-logical amoebas, can be challenging. To overcome this issue, adap-tive structuring elements can be computed from a mapping thatholds information about both spatial distances and gray-level val-ues. This is done for salience adaptive structuring elements, whichcross edges but are smaller in size near strong edges.

In general, the structuring elements used in similarity-basedmethods adapt very well to image regions. Such methods shouldtherefore be very useful for noise removal but are hard to use forlinking segments together into larger connected regions, as is oftenthe purpose of morphological operations such as closings. Suchlinking of structures can be obtained by enforcing further restric-tions on the structuring elements with regard to shape, as is donefor the adaptive elliptical structuring elements. With the constraintof adaptivity to a predefined (but variable) shape, it is possible toprolong or link existing dominant structure in the image while pre-serving the rest of the structure.

An important practical aspect is the computational cost for eachof the methods considered in this study. Generally speaking, meth-ods based on the propagation of geodesic distance (e.g. Morpholog-ical amoebas and Salience adaptive structuring elements) have


relatively high computational complexity due to the distance prop-agation. Nonetheless, the complexity may be reduced by usingoptimized algorithms for distance propagation. Methods based so-lely on differences between gray level values, such as Generaladaptive neighborhoods, require less computational time. If onlya limited number of predefined shapes is required, such as forthe Elliptical adaptive structuring elements, precalculated librariesof structuring elements can be used for fast processing (especiallyfor batch processing).

7. Perspectives and trends

As shown in Section 5 and discussed in Section 6, the methodspresented in this survey have both benefits and drawbacks,depending on the specific task. Then, one could ask if a single‘‘ideal’’ method for adaptive mathematical morphology exists?The answer to this question is probably no, since everybody (andevery application) has a different idea of what ‘‘ideal’’ means in thiscontext. None of the methods for adaptive morphological operatorscan be considered best for all purposes, and their usefulness isapplication dependent. Nonetheless, adaptive mathematical mor-phology seems to have room for several improvements and possi-ble future studies. We note here some possible directions for futureresearch.

Few, if any, methods for adaptive morphological operators arespread to the wider image analysis community. This might bedue to relatively high computational cost required to computeadaptive structuring elements. Efficient algorithms for computingadaptive structuring elements need to be developed, and thismight be necessary in order to allow adaptive morphology to be-come more extensively used.

Furthermore, most adaptive morphological operators that havebeen defined for the Euclidean space are, in fact, special cases ofthe recently introduced mathematical morphology on Riemannianmanifolds [6]. This seems to be a prominent direction to generalizeadaptive mathematical morphology and make a unified frameworkfor all methods. Hence, we believe that this direction will be fur-ther explored.

Despite that basic morphological operators have been definedproperly (as presented in Section 3), it is still an open problemhow to obtain useful granulometries based on adaptive structuringelements. As the absorption property is not necessarily satisfied,care must be taken. The granulometry is an historically importanttool in mathematical morphology, and its relationship to adaptivemorphological operators needs to be defined.

As could be easily noted, all methods presented in Section 4 aredefined for gray valued images and not for multivalued ones. Evenan extension of classical mathematical morphology to multivalueddata is a challenging task, since it depends on ordering vectors, andthere is no unambiguous way to order vectors [3,8]. This extensionbecomes even more complicated when structuring elements adaptto the image content, since structuring elements could adaptdifferently for different image channels. So far only a few studieson adaptive morphological operators for color images have beenpresented [32,43], and it is an interesting topic for furtherinvestigations.

Acknowledgements

We would like to thank to Dr. Johan Debayle for providing uswith his code on General Adaptive Neighborhoods. Fig. 3(a, right)is used courtesy of Per Cullhed, Uppsala University Library andFredrik Wahlberg, the Handwritten Text Recognition project atUppsala University.





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adaptive mathematical morphology – a survey of the field

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