Relaxation labelling algorithms — a review

Relaxation labelling a lgorithms - a review J Kittter and J ~l~i~~wort~*

An ;m~orta/~t research topic in image ~ro~essjng and image ~~ter~ret~tio~ methodology is the d~ve~~~rne~t of methods to ~ncur~or~te context#a~ ;~format;o~ into the ~nter~retatj~~ of objects. Over the last decade, relaxation /abelling has been a useful and much studied approach to this problem. It is an attractive technique because it is highly parallel, involving the propagation of locai information via iterative processing. The paper . surveys the literature pertaining to reiaxation la~ei~~~g and highlights the ~rn~~rta~t theoretical advances and the interesting a~p~i~ati~~s for which it has proven useful. keywords: image processkg, reja~at~o~ label~~~g

image

i~t~r~retat~o~,

Many sensory data understanding problems involve the, task of detecting and identifying objects the data represents. For instance, in continuous speech recognition the sensory data is an acoustic signal, and- objects to be detected and identified may be words or phonemes. The richest form of sensory data is the image. Objects at a high level of description will be the elements of an imaged scene while at lower levels objects will be edges, image segments, simple geometric shapes etc. The ident~fiGation of each object is usually based on a set of measurements which can be extracted from the sensory data and which characterize and specifically relate only to one object. For instance, In the case of edges the measurements may be local directional derivatives of the image intensity function. In shape recognition, the set of measurements may comprise the Fourier descriptors of the object boundary. Such object-specific information is seldom sufficient to allow unambiguous interpretation of the object. ~eve~hel~ss, as collections of objects in the real world are invariably ordered in some sense, the constraints imposed by the mutuaf relationships between individual objects in ordered collections may be used to Department of Electronic and ElectrIcal Englneerlng, University of Surrey, GuMford, SurreyGU2 5XH. UK “SERC Rutherford Appleton Laboratory, Chllton. Dldcot. Oxon OXl’l OQX. UK

206

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reduce or even eliminate the ambigui~. The idea can be best exemplified by the properties of collections of objects in natural speech where words in a sentence are subject to grammatical rules of the language. These rules provide a powerful source of information which may be adequate to eradicate any errors that might otherwise be made if word recognition were based simply on the properties of the acoustic signal representing each word. Depending on the nature of the rules which describe the relationships between objects, the presence of a particular object in a collection may impose constraints on certain classes of objects in its neighbourhood and by the same token may actively provide support for others. In general, therefore, irrespective of whether it has an inhibiting or supportive role. each object can be considered as a source of contextual information about other objects and may therefore aid their interpretation. The idea of using contextual information is far from new. Ullmannl exploited constraints imposed by triplets of pattern primitives to reduce substantially the errors that occur with a pattern recognition system after a learning sequence of fixed length. Clowes2 and Huffman~ used constraints between straight line segments to eliminate nonsensical interpretations of an ideal tine drawing representing a set of polyhedra. Many other methods of using contextual information have been suggested in the literature (eg by Zuckei”, Kittler and F6glein5 and Haralick”). They broadly fall into the following categories: group classification methods, contextual decision rules, rewriting rules (grammatical or structural methods) and relaxation methods. This paper will concentrate on the last category. namely on relaxation labelfing methods in general and on probabilistic relaxation labefling in particular. The discussion will be restricted to imagery data but generally any method suggested for two-dimensional image data is equally applicable to one-dimens;onal sensory signals. The pioneering work in relaxation labe~ling is normally credited to Waltz7 who considered the problem of line drawing interpretation studied earlier by Clowes2 and Huffman3. His formulation of the consistent labelling problem allowed only unambiguous interpretation of line segments. This was achieved by

& Co. ~Publishers) Ltd

image and vis~un computing

sequentially filtering out inconsistent label pairs on connected segments. This approach was popularized by Rosenfeld et a/’ who showed that Waltz’s filtering can be carried out in parallel and could therefore be implemented as a network of processors, each associated with one object in the image. An extensive theoretical underpinning of the consistent labelling problem has been given by Montanarig, Mackworth”. Haralick et al” 14, Henderson15, Ullmann et aP6, and Ikeda” and Kasif Shapiro”. NudeI” 20, Nishihara and Rosenfeld”. However, the problem considered by Waltz’ is somewhat unrealistic since no information as to the identity of each line segment is assumed to be available. In practice. when analysing real imagery, it is reasonable to assume that some useful information could be extracted from the raw Image data. On the other hand, the edge representation of a scene is unlikely to look like an ideal lrne drawing. Rosenfeld et aP argued that, perhaps more appropriate than discrete relaxatron which enforces unambiguous labelling, it is better to formulate the problem in a continuous domain. Fuzzy set and probabilrstic frameworks were considered in this respect but the latter seems to have attracted most attention to date. This paper reviews the literature on probabrlistic relaxation now comprising more than 100 contributions The last major reviews of this subject area, which were carried out by Davis and Rosenfeldz3, Faugeras 24, Berthod and Ballard and Brown26. date from 1981 and by now the number of papers written on the topic has more than doubled. In the next section, the classical probabilistic relaxation labelling algorithm of Rosenfeld et a/* will be overviewed. The three sections after that summarrze theoretical and methodological developments of probabilistic relaxation. The material is discussed under three major headings: probability updating schemes, compatibrlity and support functions, and general issues which do not fall neatly under any of the previous two headings. A drscussion of applications of relaxation is followed by some brief conclusions. It should be noted that relaxation processes are widely used for solving problems of the Image ‘filtering’ type, where the task consrsts in recovering a function defined on the Image domain from the observed image such a task could be intensity value?’ 3’ Although formulated as a labelling problem, eg each grey level could be associated with a different label, its solution using probabrlistic labelling methodology would be unnatural and inefficient An excellent survey of many of these approaches has been given by Glazer3*.

CLASSICAL

PROBABILISTIC

RELAXATION

Before reviewing the literature on probabilistic relaxation. it is best to give a brief overview of the classical probabrlistic relaxation labelling method introduced in the seminal paper of Rosenfeld el aP. This widely cited and used scheme WIII provide a reference point to which subsequent developments of the methodology can be conveniently related. Consider a network of objects a,. i=l,2,. , N. Unit

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a, can belong to one of m, possible classes from the set A,. For the sake of simplicity we shall assume that the label sets A, are identical for all the nodes of the network, re

A, = A =

{h,ik

= 1,

, m;

VI

(1)

Each unit a, will usually be characterized by a set of measurements X,. In addition, all the objects in the network will provide contextual information which may be relevant to the interpretation of objects a,. The task addressed by relaxation labelling is to assign labels 8, to objects a,. i=l , , N takrng Into account the total evidence provided by measurements X, and the contextual information conveyed by the network. Ideally we would like to choose for each node i just one label from the set A, thus giving an unambiguous labelling to the whole network However, in practice the available information will be insufficient to make such unambiguous Interpretation possible. Instead we shall have to be content with an ambiguous interpretation obtained by allowing all the possible labels h, Vk at every node a, and by assigning probability value P(9) = h,lX,, contextual information) to each of these labels. In the framework of this generalized formulation the unambiguous labelling of object a, becomes a special case of a probabilistrc labelling corresponding to the situation when the probability for one of these labels equals unity and that of the other labels assumes the value zero. To describe the relaxation labellrng scheme8, it is first necessary to introduce essential notation. The contextual informatron conveyed by label h, at object a, about label h, at object a, will be represented by a compatibility coefficient r(8,=h,, 9,=x,,). Let the current (sth) estimate of the probability that node a, has label h, be P(O,,=h,). Then the support for label h, at a, given by node ican be defined as cJ;(e,=h,)

=

&e,=;l,,e,=k,) k=i

P”(8,=h,)

(2)

and the total support for this label lent by all the objects in the network is defined as Q:(e,=h,) where

f

,=1

=

&$r(R,=i,. ,=I

C,, are weights

c,, =

9,=X,)

P(Q,=h,)

(3)

satisfying

1

The coefficients C,, can be used to express the relative importance assigned to individual nodes in the network. Withthe basic notation introduced it IS now possible to write down the expression suggested by Rosenfeld et aI8 for computing updated probabilities of object labels. Given the current value of probability P(f3,=h,), the (s+ 1 )th estimate Ps+‘(Ql=h,) is defined as

where

the

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term

In

the

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207

guarantees that the updated quantities satisfy the axiomatic properties of probabilities. The probability updating formula in equation (5) is largely intuitive and its basis is that the current label probabilities at each node should be adjusted according to the relative support afforded to the different labels by the complete network. It is often referred to as the nonlinear probabilistic relaxation scheme. its linear counterpart, also suggested by Rosenfeld et a18, aimed to find label probabilities satisfying P($,=h,)

= Q(e,=?L,)

Vi,/

(6)

However, the linear relaxation has been shown to yield a unique solution which is independent of any relevant information that may be contained in measurements X, unless the formulation is modified as suggested by E3lake39and Elfving and Eklundh4*. In general, this information is incorporated in the relaxation process by means of initial probabilities PO(C),=&) Vi,/ which are determined as

P*(e,=h,)

= P(ej=h,lXi)

(7)

where the right-hand side in equation (7) denotes the a posteriori probability of label 8, assuming the value h, given the measurement set X,. It is, in fact, the ‘nocontext’ probability of the ith node having label hi. Note that this information in X, V’i is used directly only once at the algorithm initialization step. The algorithm in equation (5) generated considerable interest which is partly reflected in the number of its applications reported in the literature. These will be reviewed in some detail below. First the following section summarizes the basic properties of the algorithm which have been observed by various researchers. This is followed by a survey of the theoretical and methodological developments which have taken place to date.

PROBABILITY

UPDATING

SCHEMES

Chile the probabilistic relaxation scheme in equation (5) has been demonstrated in several examples to have a tendency to increase the consistency of labelling of objects in a given relational network, its formal convergence properties were not established by Rosenfeld et ~1~. Moreover, it has been observed that the convergence of the algorithm can be s/ow~~. The first attempt to fill this gap in the understanding of these properties was reported by Zucker et a/4’. However, the condition ,,C __.-.<1_1,.._V.__ .-.,,f <,V+h :_ +L,;, ___,.” ,..-nrl “UL ,., ,+ +n “I L”Il”C~yr;I ILt: r.JJuL IVIIII III 111c?‘11 1JdrJG:’tlLUllICCl L” be necessary but not sufficient. In any case the paper contained no suggestion as to how the convergence condition could be translated into conditions on the actual compatibility and support functions to ensure convergence of the updating scheme. The problem of slow convergence was dealt with in a heuristic fashion by effectively taking the right-hand side of equation (5) to the power of w. This remedy appeared to have QfiC.nlnr3t;~n ~“,l”~iy~liL.” ‘y the r0n\,nmanro tha affact of I;C._?*i\* /// rCa,ly UC_,~c21~1”111 process by a factor of approximately W. Hummet and Rosenfeld4’ defined a geometric framework in which relaxation processes can be conveni-

ently studied. They postulated conditions which any relaxation labelling algorithm must satisfy and showed how a variety of updating rules can be formulated to meet these conditions. In particular the scheme in equation (5) and its accelerated version can be regarded as special cases of an approximate log-linear updating process. As relaxation processes are essentially parallel, it is easy to envisage that the computation can be done by an array of identical processors, each associated with a single object-label pair in the network. Ullman43 and Faugeras 44 have investigated the types of problem that EStcht distriht #a-i ~‘“Y”Y”“.Y rwcw~s~~nrc rnn .,.“.. pffsvti\mlv vI’UY.*.-., s&y L1#UC VU”/ “.“.I OUYCYY and showed that these include relaxation labelling viewed as a constrained optimization problem. They also obtained the conditions under which a relaxation process will be local and discussed the implication of these conditions on object interconnections and network symmetry. The difficulties of establishing convergence properties of the heuristic updating schemes motivated c_.,_,-.,,-.r. ,...A o,r+h,~45.46 +n 1, rm,,, _+n +kr\ r\.nh,nm -4 I” I”I,II”IcILc: IIIC? p,“ulollI “I I dU!_Jt$lclb cl, I” “CI LI I”U consistent labelling as an optimization problem. They introduced an objective function which comprises two components: a criterion of consistency, and a measure of ambiguity. As in linear relaxation, labelling may be considered consistent when the current probability estimate P(E),=&) is equal to the support @(0,=X,) V’i,/, but other definitions of consistency have been suggested47. Ambiguity of labelling. on the other hand, is measured in terms of a quadratic entropy function. The overall objective function is then optimized with respect to label probabilities subject to the constraint that the optimal solution satisfies ,z

p,,,(e,=h,j

60,=x,)

= > 0

I

vi

(8)

Vi.1

(9)

This new outlook stimulated attempts to view the updating scheme in equation (5) as an algorithm otimizing some objective function which quantifies labelling consistency. This function has eventually been identifjed by Berthod and Faugeras4* as having the form F = f

f

I=1 i=l

p(e,= h,) cqe,=

h,)

(10)

The form of this function has been adopted by Hummel criterion and Zucker49--5’ as their basic consistency which can be generalized to cater for higher-order compatibilities (triplets,. , n-tuples of objects). With the identity of the consistency function established, efforts were made to improve on the optimization procedure embodied in equation (5). Lloyd5* has rewritten the probability updating formula as

PS+‘(e,=h,) = PS(6,=h,) + ap,,

(11)

where PS(B,=h,) [@(@,=A,) -

p(e,=&,

as(s;=h,)]

Pii = ~,P‘(R,=h,WV,=h,)

(12)

She showed that the probability update direction defined by p!,, V;,/leads to an extremal point of function F and that the nonlinear probabilistic relaxation algorithm given by equation (5) corresponds to setting the stepsize a equal to unity. The algorithm convergence can be considerably accelerated by choosing an optimal value of a. Since the objective function f is quadratic, the optimal step can easily be determined Following the foundations laid by Hummel and Rosenfeld4’, Mohammed et a/53 have developed an optimization procedure along more conventional lines. In general the candidate direction of update is defined in terms of the gradient direction of function Fat the current label probabrlity values. The actual direction of update is obtained by projecting the gradient vector onto the planes defined by equation (8). In contrast with Lloyd’s approach, this optimization process does not terminate when any one of the probabilities assumes the value zero. Even zero probabilities are updated, provrded the new values become positive as a result of the update. If the emphasis is on minimizing ambiguity and the __~...__I. -X _I^‘-^rI___ -- ^-^_ _L_:- ~^-^I^-.I At.__

about each other, the initial label probabilities should not be changed by the updating formula. Pavlrdis”’ showed that the algorithm of Rosenfeld et a/* did not posess this basic property. Similarly, when all labels at every node have equal initial probabilities, ie the measurements X, contain no useful information, an unbiased process should leave probabilities these unchanged. Zucker and Mohammed”‘and Haralrck eta/“’ derived the conditrons which must be satisfied by the compatrbility and support functions to eliminate bias both In the general case and in the homogeneous case where all nodes have the same label sets, reciprocal compatrbilrty relationships and equal node weighting. They also showed that all fixed points of the updating scheme In equation (5) can be determined and their stability investigated by means of the Frechet derivative. A more general characterrzatron of solutions yrelded by probabilrstrc relaxation algorithms is given by Elfving and Eklundh4’. The limited justification of the arithmetic averaging for the support function in equation (5) motivated Zucker and Mohammed”” to consider more general forms for combining the support from the objects ---. ^. :-^ ___I_ .L _I _I_ ~~__ _ TI_I- _ -I ~I~

the probabilistrc labelling problem can be solved by means of an iterative application of dynamic programming54,55. This methodology can be applied to suitably structured two-dimensional networks such as lattices by optimizing at each iteration, first along the rows and then along the columns of the network nodes. Generally, relaxation labelling procedures will find a local extremum of the consistency/ambiguity criterion function empioyed. As discussed by Faugeras”“, for example, this should not necessarily be considered a drawback. A local optimum is more likely to reflect the information content introduced by measurements X, than a global optimum which will invariably be independent of the input data. However, for some image matching applrcations which are formulated as consistent labelling problems it may be important to find the global optrmum corresponding to the best possible match. In such situations, stochastic optimization as discussed by Geman and Geman56, Cohen et d5’, Hinton5* and Trivedi5’, will have to be deployed instead of deterministic procedures.

the arithmetic average is a special case of the geometric average and advocated the use of the latter. In the same paper Zucker and Mohammed” also introduced a simple transformation which permitted the compatibility coefficients to be viewed as conditional probabilities. The aim of this transformation was to get a better handle on the relaxation process properties and to facilitate its interpretatron. Although this simplistic change in itseif did not contribute much to the body of knowledge relating to relaxation labellrng. It may have inspired others to approach the problem of determining compatibrlrty and support functions In a probabrlistic framework. By setting out to combine the evidence furnished by individual nodes sequentrallv. KrrbvR3 showed that the support function should be of a product form. This result provided a theoretical justification for the heuristic geometric-average rule of Zucker et aP4 Other authors were more concerned with compatibility functions. Peleg and RosenfeldGS Investigated the effectiveness of mutual information as a compatibility function In relaxation labellrng The use of mutual information was also advocated by Yamamoto66 (referred to by Narayanan”‘). In later work. the use of the somewhat unnatural concept of probabrlrty distributions of label probabrlitres enabled Peleg6* to apply formal rules for manrpulatrng conditional probabrlrties and to derive. under fairly restrictrve assumptrons. an expression for the probabrlrty of the jornt occurrence of label 1: at unit a, and label h, at unit a, grven the current values of the respective label probabilitres. Fortunately some of the most suspect probabilistrc entitles cancel out when they appear in both the numerator and denomrnator of the updatrng formula used. However, the main conclusion of this analysrs, namely that the compatibility coefficients should be computed as the ratio of the joint label probabrlities and the product of marginal probabrlrtres. has met with little controversy. Most importantly, the relaxation process with ^---,.+:&:l:r.. -,._~I:-:__r^1_X.-_^1 L”IIIc)dLI”IIILy L”elllLlt!ll,s uelllleu In this fashion can be shown to be unbiased both In the no-InformatIon

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COMPATIBILITY

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SUPPORT

FUNCTIONS As pointed out by several authors (eg Hummel and Rosenfeld4*. Zucker and Mohammed” and Haralick et aP), one of the main problems of designing a relaxation labellrng process lies in selecting suitable compatibility and support functions. Because of their heuristic basis, the compatibility and support functions in equations (2) and (3) have been found to suffer from a number of drawbacks which include bias and lack of interpretability of the computed probabilities. Any bias of a relaxation process is best manifested $n two simple situations: the independent-node case, and the no-information case. The properties of equation (5) under the assumption of independent __J_^ _..^ X’__r _r..-._^I I^.. n_..,-_I:_67 \n,,__- rl__ ___I__ iluueb were IIIS~ stuu~eu uy rdvliuis--. vvfiefi trit: IIUU~S In a network do not convey any contextual information

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and the independent-node cases. An attempt to establish a relationship between Peleg’s approach and the compatibility coefficient of Rosenfeld et a/’ has been reported by Wang and Zhuang6g. In general all the approaches to determining compatibility functions discussed so far can handle higher-order dependences. However, the actual number of studies in which higher-order dependences were experimented with are very few. Some results have been reported by Peleg7’. Eklundh and Rosenfeld” and Fekete et a/72. The work on contextual statistical classification alnnrkhmc u’y”1,L’““d

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close links between the recursive implementation of these algorithms and probabilistic relaxation. This suggested that both compatibility and support functions should be derived using the mathematical apparatus of statistical decision theory. The approach advocated has been illustrated in a number of specific examples by Kittler and Foglein73. The main implication of this work is that compatibility and support functions rl^,^_rl ueperru on the 2SSiiiTipTiGiiS That Gii be iTlade Sbi3Lii the problem in hand. The approach also offers a mechanism for incorporating contextual information in the case of nonmemoryless noise. An interesting alternative to the above methods of determining compatibility and support functions is advocated by Faugeras74. He casts the problem of gathering evidence for relaxation labelling in the framework of Schafer’s theory75 of mathematical evidence. The approach provides a direct mechanism for handling ignorance, ie the situation where none of the possible labels can be assigned with reasonable confidence. However, a reject or no-label option can also be introduced in the usual probabilistic framework as discussed by Berthod”.

labelling performance*‘. Haralicka2 set out to explain this characteristic by attempting to provide interpretation for the probabilities yielded by the updating process. He achieved some degree of success with the product updating rule where the compatibility coefficients were defined as by Peleg6* and also argued that the conceptual framework in which Peleg’s relaxation algorithm was developed precluded any meaningful interpretation of the computed probabilities. This argument, however, is largely academic as Peleg’s relaxation labelling scheme involving the product rule is identical to that interpreted by Haralick. While the relaxation nrnrn~c in\,nl\,inn z,rithmntir_~\,nrona *I IIP ;c uIILIIIII~,LI~~uY~Iuy~ IUIti I.3 IPCC IticI.2 pl”bu.B.J II lY”lYll my the obvious, nevertheless the interpretation for the probabilities computed by the classical scheme put forward by Kittler and Foglein 73 is directly applicable also to Peleg’s updating formula and renders it less objectionable. The need for interpretation is less important when compatlbrlrty and support functions are derived as advocated by Kittler and Foglein73 as the assumptions ..-^l_V1..:_- ..-?.I^~:--I ^^^^^^^ ^I^ _&-r-2 _ -“:A_: _--I UflU~ll~lll~

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The complexity of Image understanding processes necessitates that image description be computed at multiple levels of abstraction. Although the methodology of relaxation processes has been developed to deal primarily with the object-labelling problem at any single level of image representation, it can be extended to handle interlevel interaction as demonstrated by Zucker and Mohammed”. This potential may prove to be very important, as the ability to draw on information from higher levels of description which is of more global character is considered essential for maximum rerll lrtinn “I nf ““‘“‘yu\Ly amhinl lit\/ USI” and “yp_,“L1’“~“u h\/nnthncic \/arifirntinn IYUU”LIVII ““IIIIL,ULI”II. _A

do not have to be deduced retrospectively. In general, however, even here the interpretation is valid only for the first iteration. The meaning of the probabilities yielded by subsequent iterations is increasingly speculative. To obviate the lack of adequate theory which would allow the selection of well behaved compatibility and support functions, Peleg and Rosenfelda3 adopt a pragmatic view and devise evaluation measures which can be used as criteria for stopping the relaxation process. They suggest that an evaluation measure should reflect the best compromise between the information conveyed by measurements and the contextual information conveyed by the model of the relationships between objects. The problem has also been addressed by Elfving and Eklundh4’. Richards et a/*’ 84 on the other hand suggest heuristic rules that must be satisfied by the compatibility and support function so that the performance deterioration is overcome. These rules can be used as the basis for a possible probabilistic relaxation procedure design strategy. As an alternative approach Richards et a/85 advocate the use of a modified updating scheme where the label probabilities computed according to equation (5) are adjusted by a factor which is dependent on the initial label probabilities. With the increasing proliferation and variety of relaxation labelling techniques, a need has been identified to carry out comparative studies to assess the relative advantages and drawbacks of individual methods. However , u C, rnmnrohnnci\/o c\/ctom~t;~ rnmn~r;~nn nf bulIlplLIIGIIal”G JyJIcIIIclLIL LvlIlyallJul! VI

similar approach has been pursued by Kuschel’*, Kuschel and Page7g and by Davis and Rosenfeld*‘. Glazer38 and Narayanan et a133 recommend a multilevel relaxation process involving either the raw image or one of its lowest levels of abstraction at different resolutions. The main aim of the multiresolution representatlon is to increase the rate of convergence of the relaxation process.

many methods on a range of data represents a large research effort and as yet few studies have been reported. O’Leary and Peleg’” consider the effect of several relaxation methods in the simple case of labelling two nodes with a set of two labels, while Nagin et a/*’ have performed experiments on segmenting test images which contain a variety of spatial structures using a range of compatibility coefficients and/or

GENERAL

COMMENTS

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processes is an initial reduction in ambiguity of interpretation which peaks after the first few Iterations and is then followed by gradual degradation of the

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interpretation, Price’; has give”l~m~i”;~~~lt?v~r oi number of methods applied to the relaxation matching of semantic network descriptions of images but his

work is only briefly reported and a more detailed description of methods and results should be compiled. Some comparative results have also been reported by Fekete et alT2. Although these studies throw new light on a number of aspects of various methods, without exception they have proved inconclusive as they tend to be unsystematic and compare different updating schemes that have different compatibility and support functions.

interrelationships of labels are derived empirically from ‘ground truth’ data by measuring and globally averaging transitional probabilities, correlations or the mutual information@‘j’ of neighbouring pixels. Upon applying these methods, most investigators report a sharp initial decrease in classification error rates of several per cent followed by a smaller increase as the process converges to a stable solution. Kalayeh and Landgrebeg7 suggest that many problems can be overcome by deriving estimates of compatibility coefficients over a local, say 5 x 5 pixel, window.

APPLICATIONS Edge enhancement

Toy problems Several authors have considered relaxation on particularly simple graph structures with only a few labels. The most famous example is the three-node-four-label toy triangle example introduced in the original paper by Rosenfeld eta18. However, as mentioned above, O’Leary and Peleg*” have studied a two-node-two-label graph and Haddow8’ has analysed the relaxation scheme of Rosenfeld et a/* for the particularly symmetric case of a tetrahedral graph with a two-label set at each node. All these are nontrivial examples which produce a range of solutions dependent on both the initial probabilities and the assumed compatibilities between labels on pairs of connected nodes. They are particularly useful in helping to understand the nature of fixed points ?f the .,I LI 8” relauatinn IVIUI\ULI”I

nrnr~c< ylvv”““.

Classification to binarize Bhanu and Faugeras go have used relaxation a grey-level image. Initial probabilities p,, and pb of :he pixel being object or background are assigned on :he basis of individual pixel grey level and these are then iteratively updated using the supporting evidence q0 and qb for the labels from the immediate-neighbour pixels. They use a criterion function approach which IS based on the dot product of the pixel probability vector with its nerghbourhood support vector. By maximizing the criterion for the sum of all the pixels, they hope to obtain the most consistent and least ambigIJOUS label assignments. Optimization is achieved by a projected gradient method which finds the nearest local maxrmum to the initial labelling. Good results, which can be controlled by variation of a few parameters pertaining to the initial assignment scheme and the optimizatron procedure, can be obtained. The use of relaxation for the more general case of improving many-label classification of multispectral data has been attempted by many authors, especially In relation to remotely sensed data5,8’ w’~” g4 and colour images8’ g5gb. Most of these applications have been approached in the same way. Model clusters are defined in the measurement hyperspace either by automatic clustering or by hand segmentation of ‘ground truth’ data. The initial label probabilities are calculated Act~nro as a C;MF\I0 l”lUl l”l”l l”Vl.3 UlJL”l IL_ Jll I ,p’L fiFC?iGn =f the h/lc,h3lonnh;r between the models and the measured pixel data. The

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Labelling points as edge pixels IS another important application where relaxation has been triedg8. Schacter et algg posed this problem as an edge/no-edge classification problem and used edge direction Information to provide measures of compatibilrty between neighbouring pixels. Edge label probabilities were adjusted using a heuristic function, and at the end of each :+,.“?.t;--. +k, ,,rl,, rl:,,,.+;,?,r I> >“vere IetZsLIIIlaLc;” ,,,c+,-n+,.rl as ,leldll”II LIIt: euyt; UIIt?LLI”I weighted averages of edge directions of all pixels in a small neighbourhood. The weighting factors were the probabilities of the pixels being labelled as edges. Schacter et alclarmed the method proved highly effec tive as a noise-cleaning technique but il requires the adjustment of several arbitrary parameters. Hanson and Risemang5 suggested an edge relaxation scheme which is based on applying collinearity constraints to binary arrays of edge segmefnts. The assignment of the strength of an edge segment was adjusted on the basis of the number of edge segments meeting at four-way vertices after thresholding the edge strength. Between zero and three other edges can meet an edge at a vertex. If an edge segment occurs which has no neighbours at either end, then it IS completely Isolated and IS very unlikely to be part of a true edge The edge strength of such a segment IS severely reduced. However, If exactly one other edge occurs at each end of an edge, then there is good continuation and the strength assignment of such edges is strongly supported. Cases where more than two edges meet at a vertex are treated as ambiguous evrdence and edge label strengths are left unadjusted Thresholding and edge strength reassignment are Iterated many times. Results were reported as very favourable Faugeras and Berthod4h rllustrate their optrmrzatron technique for relaxation by consrdenng labellrng pixels with a set of four edge labels {north. east, south, west; and a no-edge label Initial edge probabrlrttes were derived using the normalized output of a standard differencing 3 x 3 edge operator Only edge magnitude was used in the calculation. The crrterron whtch was optimized was a combinatton of a consrstency measure (the agreement of label assignment based on evidence of the pixel and the evidence of Its nerghbourhood) and an ambiguity measur-e (the quadratic entropy of the labellrng). The compatrbrlrty coefficients were derived empirrcally from hand-labelled data A few rteratrons of the process produced good results \/Vamhnrn . ““#, ‘yu”y

process

with

of -, n/loo ,,_“_ ha\/0 yvvy IISP~ _.I ;~n PTI~P yyJI “.

a label

set srmrlar

to that

enhancement

of

Faugeras

211

but using the standard updating scheme of Rosenfeld et a15. The results of edge enhancement enable decisions relating to the choice of a suitable post processing image filter to be made more confidently; howstudy also stresses that a suitable ever. this methodology for the assignment of initial probabilities, compatibilities and the choice of relaxation updating scheme is lacking. Danker and Rosenfeld”’ have reported combining edge and region information in a joint relaxation scheme to extract ‘blobs’ from images, They found that the two processes complemented one another, leading to the suppression of spurious edges within the background image area and prevention of the expansion of blob assignments into the nonblob areas. They also reported that the speed of convergence was improved.

Features are extracted in two images. The features of one image can be regarded as nodes of a graph while the features in the second image are possible labels for the nodes. The initial probabilities of labels can be taken as a simple function of the distance between node and label points in the two images. Node and label assignments are then compatible if similar neighbourhood states exist in both images. Although these relaxation schemes might not have great computational advantages relative to other standard matching methods, they are more tolerant of image distortion. In addition, the reinforcing processes are local and therefore missing matches can be tolerated and hence occluded objects can be recognized.

High-level Feature

enhancement

Detection of lines, strips and curves has been attempted by several authors”,“’ lo4. Zucker et a/lo4 use a thin rectangular neighbourhood mask oriented in one of eight directions defined by a line operator whose output forms a measurement vector for initial probability assignments. By weighting pixels in the contextconveying neighbourhood. the curvature of the lines which are to be enhanced can be controlled. The main difficulty reported is a tendency to produce curves which are too thick. However, a thinning constraint can be incorporated into the compatibility coefficients to overcome this tendency. Peleg and Rosenfeld65 have considered how compatibility coefficients can be computed from suitable test data. They found that curves can be found equally satisfactorily using either correlation coefficients or mutual information measures and the coefficients derived from one image were suitable for use ;n CIUlYtj C.,,r\,n IGlOhaLI”II rnln”qt;nn “II _rl nthnr ),I “II1X71;ln*Cloc. llIlcl$jXC%. Danker and Rosenfeld’** suggested that strips, which are formed from two antiparallel edges, are important features in many images. They may represent long straight roads, runways or structures such as tree trunks. Relaxation processes can be used to reinforce collinear but antiparallel edge responses and hence to identify strips. Danker and Rosenfeld use a simple iterated convolution method to enhance strips of an rr”.“..r^“.r:..+r. ..:-.7. n,,l “IIt5I’LcILIVII. ,.“:,...+,,+;,?dppI”PI,cILe >ILt: al’”

Shape and stereo

matching

Relaxation labelling has been much used in matching problems such as two-dimensional shape matchproblem”2~‘15 ing lo5 “’ or the stereo correspondence Representative feature points such as corners are extracted from the template shape and the real-world image. Initial probabilities can be assigned on the basis of the degree of match of these local features and these probabilities can then be iteratively reinforced on the basis of the occurrence of matches with other features which are found in approximately correct relative nncitinnc ~“Y’&‘“‘I.,. The stereo correspondence problem is very similar.

212

interpretation

At the highest image interpretation levels, the matching of relational networks has been studied by Kitchen and Rosenfeld”“~‘“. Discrete relaxation is applied to relational networks consisting of areas from maps, atoms of P-codeine and components in a complex switching network’16. The algorithm converged very quickly in all these cases to a fairly unambiguous result. Kitchen”? also considered relational networks with numeric as well as symbolic attributes using a fuzzy relaxation matching scheme. Although these methods were sensitive to the exact form of the updating rule, it was found they could work very well and tolerate quite large measurement errors. A similar fuzzy relaxation scheme has been used by Barrow and Tenenbaum”’ in their reasoning system for image analysis, MSYS. Faugeras and Price’lg used probabilistic relaxation to match high-level semantic network descriptions of images. Images were segmented using an automatic system based on region growing and line extraction r%lT\fiCic.P0P tlInq+s,rnr nf thn rnnmnntar-4 oz-a~c .-,snh ~JI”LGJJGjJ. I ~cl~UIc7.2 “I Li 3-z zq.jl,Kz1 llcIt.4 cilGU3 3ULl r as average colour. average texture, size. shape, position and orientation were derived and compatibility relationships were defined in terms of adjacencies and relative distances and orientations of segments. In a typical image, between 100 and 200 segments might have had to be matched. A world model was defined with similar attributes and a global criterion for the relaxation matching of segments was optimized. The scheme ;~~~r~,,,r~+~,4 _ nvnheh;l:t\, thvnphfilrl 07,c-d ;4 PI rnnmnnt IIIL”ly”lolG” a pl”“aulllLy Llll~311”IU CalILl, II a ~r;ylIIt:IIL label probability exceeded it, the segment was given this label with certainty. This meant that good matches were quickly made. Faugeras and Price have reported good results, most of the disagreements being attributable to poor initial segmentations.

Other applications implementations

and hardware

In addition to applications of direct relevance to images, the relaxation labelling technique has been applied to problems in character recognition’~‘20, handwriting interpretation’*‘, the breaking of substitution rinhorc122 rcarnnctn lrtinn nf mrrtrirnc frnm smmd Y’y’8V.U , thn IIlY ,“““IIYLIUV~I”II UI II,UII,Y”U ,#“1,, CI U”I,.III”U projections lz3, the segmentation of curves’o3 and wave-

image and vision computing

forms*0’24 and the solution of layout problems’? Little literature exists concerning the practical hardware implementation of probabilistic relaxation even though they have long been recognized as highly parallel schemes. McLean and Dyer’26 discuss how line and curve enhancement in the updating scheme of Rosenfeld et al’ can be achieved using analogue electronics. Logarithmic amplifiers can be used to perform the necessary multiplications fast enough to achieve multi-iteration relaxation in real time. Duncan and Frei”’ have introduced modular VLSI operators based on a relaxation scheme which repre~nn+~ thrr nvnhs,h;l;+T, A;r+r;h,,t;nn Th,,,,t JCI 1~3 LIIC ptuuau~~~~y UI~LIIUULIUII ~UVUL 3 jZiXd 2s F3i7 ellipse. Using a family of 5 x 5 pixel basis operators, they hope to implement a relaxation scheme using a set of three chips. It has been reported that their approach gives results which are comparable with those of standard line and curve relaxation schemes.

CONCLUSIONS This survey shows that many significant advances have been made since Rosenfeld et a/’ first introduced probabilistic relaxation labelling. It has shown success or great promise in many application areas. However, many issues stall remain to be resolved, especially in respect of defining suitable compatibility relationships, Interpreting In detail the effects of some of the suggested updating schemes and proving convergence properties. Relaxation labelling should remain a fertile research field for several years to come

relations’ Artif Intel/. Vol 8 (1977) pp 99-l 18 11 Haralick, R M ‘Structural pattern recognition, homomorphisms, and arrangements’ Pattern Recogn;tjoff Vol 10 (1978) pp 223-236 R M and Kartus, J S ‘Arrangements. 12 Haralick. homomorphisms and discrete relaxation’ /EEE Trans. SMC Vol 8 No 8 (1978) pp 600-612 13 Haralick, R M and Shapiro, L G ‘The consistent labeling problem: part I’ /EEE Trans. Pattern Anal. Mach. Intel/. Vol 1 No2 (1979) pp 173-l 84 14 Haralick, R M and Shapiro, L G ‘The consistent labeling problem- part II’ lEEE Trans. Pattern Anal. iI”,rh ,,+,Y,, \/,I hl- 3 T3\/Inon\ _“, 103 1n13 /“1cK,,,. ,,,1t?,(. ““1 L7 ,\I” I JO”, 1J&.J I z7.YL”3 G ‘A note on discrete relaxation’ 15 Henderson, Comput. Vision, Graphics, Image Process. Vol 28 (I 984) pp 3844388 J R, Haralick, R M and Shapiro, 16 Ullmann, L G ‘Computer architectures for solving consistent labelling problems; Comput. J. Vol 28 No 2 (1985) 17 Shapiro, L G ‘Solvrng consrstent labelrng problems havrng the separation property’ Proc. 7th Int. ~” r--x,,,, on Pattern n”ecog,Titio,y, nA_.-.__-l r-_-J_ iVIUf/ll~dl,

18

19

20

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image and vision computing