Andrea Cerri

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.

Recent Trends, Applications, and Perspectives in 3D Shape Similarity Assessment

The recent introduction of 3D shape analysis frameworks able to quantify the deformation of a shape into another in terms of the variation of real functions yields a new interpretation of the 3D shape similarity assessment and opens new perspectives. Indeed, while the classical approaches to similarity mainly quantify it as a numerical score, map‐based methods also define (dense) shape correspondences. After presenting in detail the theoretical foundations underlying these approaches, we classify them by looking at their most salient features, including the kind of structure and invariance properties they capture, as well as the distances and the output modalities according to which the similarity between shapes is assessed and returned. We also review the usage of these methods in a number of 3D shape application domains, ranging from matching and retrieval to annotation and segmentation. Finally, the most promising directions for future research developments are discussed.

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.

SHREC'13 track: retrieval on textured 3D models

This contribution reports the results of the SHREC 2013 track: Retrieval on Textured 3D Models, whose goal is to evaluate the performance of retrieval algorithms when models vary either by geometric shape or texture, or both. The collection to search in is made of 240 textured mesh models, divided into 10 classes. Each model has been used in turn as a query against the remaining part of the database. For a given query, the goal was to retrieve the most similar objects. The track saw six participants and the submission of eleven runs.

k-dimensional Size Functions for Shape Description and Comparison

This paper advises the use of k-dimensional size functions for comparison and retrieval in the context of multidimensional shapes, where by shape we mean something in two or higher dimensions having a visual appearance. The attractive feature of k-dimensional size functions is that they allow to readily establish a similarity measure between shapes of arbitrary dimension, taking into account different properties expressed by a multivalued real function defined on the shape. In particular, we outline the potential of this approach in a series of experiments.

The persistence space in multidimensional persistent homology

Multidimensional persistent modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit one. Therefore, it is reasonable to look for a generalization of persistence diagrams concerning those properties that are related only to persistent Betti numbers. In this paper, the persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense. Furthermore, it is presented a method to visualize topological features of a shape via persistence spaces. Finally, it is shown that this method is resistant to perturbations of the input data.

SHREC'12 track: stability on abstract shapes

This contribution reports the results of the SHREC 2012 track: Stability on Abstract Shapes. This track saw six registrations of which only three participants effectively sent the results of their runs.

3D Artifacts Similarity Based on the Concurrent Evaluation of Heterogeneous Properties

Archaeological artifacts are often classified in homogeneous groups, according to either intangible properties (e.g., origin, use, age) or physical features (e.g., color, material, geometric shape, size, style). In particular, a single property is usually not enough to characterize artifacts’ peculiar traits, as most of the objects are affected by degradation or only partially preserved. In this article, we propose a shape analysis and comparison pipeline specifically targeted to the similarity assessment of real-world 3D artifacts. The proposed methodology is able to concurrently evaluate heterogeneous properties, such as geometric (e.g., curvature, size, roundness, or mass distribution) and photometric (e.g., texture, color distribution, or reflectance) aspects. The geometric description is based on a statistical technique to select properties that are mutually independent; the photometric information is handled according to a topological perspective and complemented by the analysis of color distribution. The outcome is a mixed description of each 3D artifact, which is used to derive a similarity measure between objects. The potential of our approach is high because any property representable as real- or vector- valued functions can be easily added in our framework. Experimental results carried on an existing collection of textured triangle meshes are exhibited to show the potentiality of the method in retrieval and classification tasks.