Qualitative similarity measures—The case of two-dimensional outlines

https://doi.org/10.1016/j.cviu.2007.05.002Get rights and content

Abstract

In this paper qualitative similarity measures are introduced. Depending on the underlying representation such similarity measures are based on specific qualitative distinctions which are frequently motivated by perceptual clear distinctions. Here, we discuss one such representation and show how it applies to different domains. In particular, qualitative methods are useful as soon as specific qualitative features can be defined for the purpose of characterising specific objects. Accordingly, we set two examples, namely for a domain of historical objects and for the geographic domain. Afterwards, however, we also demonstrate that our qualitative representation performs quite well when applied to a well-known test data set, without specifying any specific features. Instead, frequencies of qualitative relations are taken into account. The results indicate that qualitative measures not only relate to distinctions which can be easily comprehended by vision but that they are especially efficient in terms of runtime complexity, both issues being of particular importance in the case of image databases.

Introduction

In computer vision, similarity matching is frequently viewed as a matter of taking a number of measurements using quantitative reference systems: given two vectors (each one describing an image or a single object); we suppose that each vector consists of n attributes. Then, the question is how large their difference is. Similarity measures which are based on real valued attributes reflect how similar objects are along n quantitatively partitioned spectra, each of which comprises a metric on some particular granularity level. The similarity between objects, then, can be determined with corresponding precision. From what follows, we are concerned with the usual n-dimensional feature space.

One might ask whether precision is always that important; indeed, whether quantitative reference systems are to be taken at all; or, is there an alternative approach for the purpose of determining similarities in computer vision? The class of methods we advocate in this article, do in fact also pertain to methods at the feature space level—though, what distinguishes them is that they rely on differences in kind, rather than of measurement. According to Freksa [5], this means taking into account qualitative distinctions obtained by comparing features within the object domain rather than by measuring them in terms of some artificial external scale. In this sense, qualitative features are of a relative kind where the reference entity is a single value rather than a whole set of categories. For instance, in a scene we distinguish whether two points lie on the same side with respect to a line (the reference entity) or whether they lie on different sides: we focus on how features are altogether arranged, and we determine their ordering in the two-dimensional plane but omit any quantitative distinctions. As a consequence, a new category of appropriate similarity measures is required. From now on, we shall speak of measures only in the sense of those qualitative distinctions.

It is not our purpose to propose just any alternative class of similarity measures. Our concept of qualitative similarity measures distinguishes itself to be robust and efficient. However, we have to pay for it with imprecision. Interestingly, imprecise descriptions are frequently sufficient, and in particular, frequently related to the human’s visual system, who can comprehend such qualitative, imprecise features better than precise quantitative measurements (we are able to state whether two points lie on the same side of a line or not, but we are not able to state how far they are apart). This is where our method will be useful: either coarse descriptions are sufficient for our purposes, or features have to be provided which can be comprehended by the user.

In order to relate our method to the state of the art we shall finally compare it with a number of quantitative approaches. Since many methods have been devised in computer vision we have to select some of them; we do this by focusing on those methods which fall into the same complexity class (concerning the costs for the comparison of two-dimensional outlines). Also, there exist some qualitative methods which have been proposed over the last decade. These include Cohn [1], who describes a region-orientated technique that distinguishes different concave shapes by considering the notion of the connection of regions and their convex hulls; Schlieder [21], who introduced a point-oriented approach by describing how triples of vertices of a polygon are related relative to each other; Galton and Meathrel [6] proposed a representation of outlines by means of strings over an alphabet of seven qualitative curvature types. Eventually, the slope projection approach of Jungert [17] maps the vertices of polygons onto both the x-axis and the y-axis; depending on the ordering of vertices on these axes several features can be derived, for instance, whether a vertex forms a convex or concave part of a closed polygon. All these approaches characterise outlines qualitatively, as we will do below. They are detailed and discussed in the context of our method in [15]. It shows that opposed to quantitative approaches qualitative approaches can be easily compared on a conceptual level. This allows their representational expressiveness easily to be determined, and as a consequence, what features they are able to distinguish.

In Section 2 we shall motivate the employment of qualitative features by showing shape properties we are going to look at. Section 3 introduces our qualitative feature scheme. Sections 4.1 Scenario I, 4.2 Scenario II provide example applications which show the usefulness of qualitative similarity measures in different domains; Section 4.3 shows how the qualitative representation performs quite well in comparison to other well-known approaches. Section 5 analyses how stable the approach is, in particular when being faced with different degrees of precision, distortions, changes in viewpoint, and segmentation errors. Then, Section 6 defines the concept of qualitative similarity measures. We conclude with a discussion about granularity and complexity issues in Section 7.

Section snippets

Accessing qualitative features

The motivation for describing qualitative features of objects is as follows. Collections of objects d’art, historical tools, or natural objects such as in the geographic domain are some examples of collections to which experts want to access efficiently. Most notably, such collections show a number of objects from the same category, each exemplar showing specific details being typical for that exemplar. Frequently, the expert has the visual appearance of those specifics in mind. Therefore, a

A qualitative feature scheme

In this section we will summarise previous work on a qualitative feature scheme. It serves as an example approach to which we shall apply the more general notion of qualitative similarity measures. For the purpose of providing a manageable overview on how qualitative representations are defined and on how they are employed, we focus on the description of linear objects (open as well as closed objects) which can be approximated by polygons. Accordingly, our feature scheme consists of a number of

Applications

The following applications demonstrate:

  • How to use the qualitative representation in a query-by-sketch system (using the basis relations and scopes and their extent, i.e. BA relations).

  • How it shows to be useful in the geographic domain (employing the circulation direction and reversals, i.e. Gestalt features).

  • How it performs regarding a well known test data set (applying scope histograms, i.e. frequency distributions).

All example applications especially show how well the qualitative description

Robustness

Having shown the application of the qualitative approach, the question arises as to how stable it is. This is important inasmuch deviations between polygons might exist although these polygons approximate the same object. This is due to noisy data or differently approximated outlines. In order to analyse how qualitative shape concepts behave under such circumstances we have to address a number of issues:

  • The shapes are to be approximated with different degrees of precision for the purpose of

Qualitative similarity measures

What is the distinction between our concept of qualitative similarity measures (as applied in the previous sections) and the classical approach? Many methods have been devised most of which rely on similarity measures which are defined in some quantity space, as on the Cartesian coordinate system. Such metric-driven approaches use external reference systems which define an artificial scale relative to which objects are described [5]. By contrast, a qualitative approach allows objects to get

Discussion

In the first scenario it was sufficient and not difficult to find a number of appropriate properties to make the distinctions required, but this may well be more difficult in other domains. On the other hand, having modelled a domain by a number of such qualitative properties, according to our results, they form a robust set of features appropriate in a query-by-sketch system. This is similar in the geographic domain. However, the MPEG-test additionally shows that even the renunciation of

Acknowledgments

I thank Arne Schuldt for his contributions to the qualitative shape framework and for carrying out the MPEG-test. I should also like to thank the Bamberger Naturkundemuseum for making available the historical fruit collection and Mona Hess, Bamberg University, for taking the pictures of the fruits. Prof. Otthein Herzog gave me the opportunity to carry out the research. Eventually, I am grateful to the anonymous referees for a number of useful suggestions for improving this article.

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