A Systematic Approach to Continuous Graph Labeling With Application to Computer Vision

The discrete and continuous graph labeling problem are discussed. A basis for the continuous graph labeling problem is presented, in which an explicit connection between the discrete and continuous problems is made. The need for this basis is argued by noting conditions which must be satisfied before solutions can be pursued in a formal manner. Several cooperative solution algorithms based on the proposed formulation and results of the application of these algorithms to the problem of extracting line drawings are presented. I XHECIONTINlrnrlrGRAPHLAREXWGJzQRLEM A graph labeling problem is one in which a unique label, A from a set A of possible labels must be assigned to each vertex of a graph G = (V,E). The assignment must be performed given information about the relationship between labels on adjacent vertices and incomplete local information about the correct label at each vertex. In a discrete graph labeling problem [ 1,2,3], the local information consists of a subset, & s A, of the label set associated with vertex vi E V, from which the correct label for each vertex must be chosen. The contextual information consists of binary relations Ru s Axh, referred to as constraint relations, assigned to each edge vivj E E. The function of the constraint relations is to make explicit which labels can co-occur on adjacent vertices. The graph, label set, and constraint relations together form a constraint network [2,5]. An (unambiguous) labeling is a mapping which assigns a unique label h E A to each vertex of the graph. A labeling is consistent if none of the constraint relations is violated, that is, if label h is assigned to vertex Vi and label h’ is assigned to vertex vj then the pair (h,X’) is in the constraint relation Rii for the edge ViVj E E. Given initial labeling information, several search techniques have been developed which can be used to derive consistent labelings. The original backtracking search described by Waltz [ 11 was later implemented in parallel by Rosenfeld et al. [6], resulting in the discrete relaxation operator. At the same time a continuous analogue, the continuous graph labeling problem was * This work was supported in part by the Robotics Research Laboratory, and in part by the Ultrasonics Imaging Laboratory both in the Department of Electrical and Computer Engineering, University of Michigan. proposed, as well as a continuous relaxation algorithm for its solution, and since then several other relaxation algorithms have been proposed [7,8]. In a continuous graph labeling problem, the initial information consists of strength measures or figures of merit, pi (Aj), given for each label Aj E A on each vertex Vi E I! The strength measures are assumed generated by feature detectors which are making observations in the presence of noise. They usually take onvalues in the range [O,l], a 0 indicating no response, and a 1 indicating a strong response. The contextual information, which is represented in terms of constraint relations for the discrete graph labeling problem, are replace by measures of compatibility, usually taking values in the range [-1,1] or [O,l], which serve to indicate how likely the pairs of labels are to co-occur on adjacent vertices. Several problems have resulted in the extension of the graph labeling problem from the discrete to continuous case. In the discrete case the presence or absence of a pair in a constraint relation can be determined with certainty depending on what labelings are to be considered consistent. In the continuous case, however, there is apparently no formal means to assign specific numeric values to the compatibility coefficients, particularly for shades of compatibility between “impossible” and “very likely”, although several heuristic techniques have been proposed [7,9,10]. Furthermore, with respect to a constraint network, the concept of consistency is well defined, The objective the continuous relaxation labeling processes has often been stated to be that of improving consistency, however, the definition for consistency has not been given explicitly. This latter issue is circumvented in several of the optimization approaches which have been proposed [11,12,13], where the an objective function, defined in terms of the compatibility coefficients and the initial strength measures is given. However, because of the dependence of the objective functions on the compatibility coefficients, and because no real understanding of the role which these coefficients play yet exists, it is often difficult to describe the significance of these approaches in terms of what is being achieved in solving the problem. In an alternate approach to the continuous graph labeling problem [14] an attempt has been made to maintain the characteristics of the original problem while allowing for more systematic approaches toward a 50 From: AAAI-82 Proceedings. Copyright ©1982, AAAI (www.aaai.org). All rights reserved. solution. It is felt that solutions to the reformulated problem will be more useful because it will be easier to relate the results of the solution algorithm to what is being achieved in the problem domain. In order to develop this approach, we review the characteristics of the solutions to the graph labeling problem which have appeared so far (refer to Fig. 1). The inputs to the process are the initial strength measures ipi'( i=l,s..,W 0 j=l ,...,mj which can be represented by an 7~ x’I)?. dimensional vector: F = (p 10 (AI), p : @2), . . . , p:(&)) E Pm. Since the selection of a particular label at a given vertex is related to the label selections made at other (not necessarily adjacent) vertices, information about the label selection at that vertex is contained in the initial labeling values distributed over the extent of the network. The function of the global enhancement process, g, is to accumulate this evidence into the labeling values at the given vertex. The output vector: is used by a process of local maxima selection [ 151, s , to choose a labeling: r; = (A,, 43 . . . I &J, where & is the label assigned to vertex ui. Thus g is a function, g:Rnm + Rnm and s is a function is s:Rnm + C,,(A), where C,(A) is the set of possible labelings. The hope is that labeling resulting from the process s (e @)) is an improvement over the labeling resulting from direct local maxima selections @). If a numerical solution is to be sought for this problem, then a formal definition must be given to the concept of an improved labeling. In previous work, particularly with respect to computer vision, improvements were rated subjectively, or in the case of an experiment