Daniela Giorgi

Describing shapes by geometrical-topological properties of real functions

Differential topology, and specifically Morse theory, provide a suitable setting for formalizing and solving several problems related to shape analysis. The fundamental idea behind Morse theory is that of combining the topological exploration of a shape with quantitative measurement of geometrical properties provided by a real function defined on the shape. The added value of approaches based on Morse theory is in the possibility of adopting different functions as shape descriptors according to the properties and invariants that one wishes to analyze. In this sense, Morse theory allows one to construct a general framework for shape characterization, parametrized with respect to the mapping function used, and possibly the space associated with the shape. The mapping function plays the role of a lens through which we look at the properties of the shape, and different functions provide different insights. In the last decade, an increasing number of methods that are rooted in Morse theory and make use of properties of real-valued functions for describing shapes have been proposed in the literature. The methods proposed range from approaches which use the configuration of contours for encoding topographic surfaces to more recent work on size theory and persistent homology. All these have been developed over the years with a specific target domain and it is not trivial to systematize this work and understand the links, similarities, and differences among the different methods. Moreover, different terms have been used to denote the same mathematical constructs, which often overwhelm the understanding of the underlying common framework. The aim of this survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner. The term geometrical-topological used in the title is meant to underline that both levels of information content are relevant for the applications of shape descriptions: geometrical, or metrical, properties and attributes are crucial for characterizing specific instances of features, while topological properties are necessary to abstract and classify shapes according to invariant aspects of their geometry. The approaches surveyed will be discussed in detail, with respect to theory, computation, and application. Several properties of the shape descriptors will be analyzed and compared. We believe this is a crucial step to exploit fully the potential of such approaches in many applications, as well as to identify important areas of future research.

Reeb graphs for shape analysis and applications

Reeb graphs are compact shape descriptors which convey topological information related to the level sets of a function defined on the shape. Their definition dates back to 1946, and finds its root in Morse theory. Reeb graphs as shape descriptors have been proposed to solve different problems arising in Computer Graphics, and nowadays they play a fundamental role in the field of computational topology for shape analysis. This paper provides an overview of the mathematical properties of Reeb graphs and reconstructs its history in the Computer Graphics context, with an eye towards directions of future research.

Multidimensional Size Functions for Shape Comparison

Size Theory has proven to be a useful framework for shape analysis in the context of pattern recognition. Its main tool is a shape descriptor called size function. Size Theory has been mostly developed in the 1-dimensional setting, meaning that shapes are studied with respect to functions, defined on the studied objects, with values in ℝ. The potentialities of the k-dimensional setting, that is using functions with values in ℝk, were not explored until now for lack of an efficient computational approach. In this paper we provide the theoretical results leading to a concise and complete shape descriptor also in the multidimensional case. This is possible because we prove that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance. Experiments are carried out to show the feasibility of the method.

Retrieval of trademark images by means of size functions

We propose a new, effective system for content-based retrieval of figurative images, which is based on size functions, a geometrical-topological tool for shape description and matching. Three different classes of shape descriptors are introduced and integrated, for a total amount of 25 measuring functions. The evaluation of our fully automatic retrieval system has been performed on a benchmark database of 10,745 real trademark images, supplied by the United Kingdom Patent Office. Comparative results show that our method actually outperforms other existing whole-image matching techniques, comprising features incorporated in the MPEG-7 standard.

Size functions for comparing 3D models

This paper proposes an original framework to use size functions in the 3D context. Size functions are a mathematical tool, that have already shown its effectiveness for image retrieval and classification. They are introduced here for the first time to discriminate among 3D objects represented by triangle meshes, through the proposal of a method for defining size graphs independently of the underlying triangulation. We first derive a skeletal signature, which guarantees the topological coding and the geometric description of an object surface, then this signature is used as a size graph to compute discrete size functions. The attractive feature of size functions is that it readily gives a similarity measure between shapes. The result is the introduction of a new technique for 3D model retrieval, devised to capture both local and global properties of a shape. Finally, we demonstrate the potential of our approach in a set of experiments, and discuss the results with respect to existing techniques.

PHOG: photometric and geometric functions for textured shape retrieval

In this paper we target the problem of textured 3D object retrieval. As a first contribution, we show how to include photometric information in the persistence homology setting, also proposing a novel theoretical result about multidimensional persistence spaces. As a second contribution, we introduce a generalization of the integral geodesic distance which fuses shape and color properties. Finally, we adopt a purely geometric description based on the selection of geometric functions that are mutually independent. The photometric, hybrid and geometric descriptions are combined into a signature, whose performance is tested on a benchmark dataset.

A new algorithm for computing the 2-dimensional matching distance between size functions

Size Theory has proven to be a useful geometrical/topological approach to shape comparison. Originally introduced by considering 1-dimensional properties of shapes, described by means of real-valued functions, it has recently been generalized to taking into account multi-dimensional properties coded by functions valued in R^k. This has led to the introduction of a shape descriptor called k-dimensional size function, and the k-dimensional matching distance to compare size functions. This paper presents new theoretical results about the 2-dimensional matching distance, leading to the formulation of an algorithm for its approximation up to an arbitrary error threshold. Experiments on 3D object comparison are shown to discuss the efficacy and effectiveness of the algorithm.

3D shape description and matching based on properties of real functions

This tutorial covers a variety of methods for 3D shape matching and retrieval that are characterized by the use of a real-valued function defined on the shape (mapping function) to derive its signature. The methods are discussed following an abstract conceptual framework that distinguishes among the three main components of these class of shape matching methods: shape analysis, via the application of the mapping function, shape description, via the construction of a signature, and comparison, via the definition of a distance measure. Goal of the tutorial is to facilitate the understanding of the performance of the various methods by a methodical analysis of the properties of various methods at the three different stages.

Information-theoretic selection of high-dimensional spectral features for structural recognition

Pattern recognition methods often deal with samples consisting of thousands of features. Therefore, the reduction of their dimensionality becomes crucial to make the data sets tractable. Feature selection techniques remove the irrelevant and noisy features and select a subset of features which describe better the samples and produce a better classification performance. In this paper, we propose a novel feature selection method for supervised classification within an information-theoretic framework. Mutual information is exploited for measuring the statistical relation between a subset of features and the class labels of the samples. Traditionally it has been measured for ranking single features; however, in most data sets the features are not independent and their combination provides much more information about the class than the sum of their individual prediction power. We analyze the use of different estimation methods which bypass the density estimation and estimate entropy and mutual information directly from the set of samples. These methods allow us to efficiently evaluate multivariate sets of thousands of features. Within this framework we experiment with spectral graph features extracted from 3D shapes. Most of the existing graph classification techniques rely on the graph attributes. We use unattributed graphs to show what is the contribution of each spectral feature to graph classification. Apart from succeeding to classify graphs from shapes relying only on their structure, we test to what extent the set of selected spectral features are robust to perturbations of the dataset.

3D relevance feedback via multilevel relevance judgements

Relevance feedback techniques are expected to play an important role in 3D search engines, as they help to bridge the semantic gap between the user and the system. Indeed, similarity is a cognitive process that depends on the observer. We propose a novel relevance feedback technique, which relies on the assumption that similarity may emerge from the inhibition of differences, i.e., from the lack of diversity with respect to the shape properties taken into account. To this end, a user is provided with a variety of shape descriptors, each analyzing different shape properties. Then the user expresses his/her multilevel relevance judgements, which correspond to his/her concept of similarity among the retrieved objects. Finally, the system inhibits the role of the shape properties that do not reflect the user’s idea of similarity. The feedback technique is based on a simple scaling procedure, which does not require neither a priori learning nor parameter optimization. We show examples and experiments on a benchmark dataset of 3D models.