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This book presents recent advances in the field of shape analysis. Written by experts in the fields of continuous-scale shape analysis, discrete shape analysis and sparsity, and numerical computing who hail from different communities, it provides a unique view of the topic from a broad range of perspectives.

Over the last decade, it has become increasingly affordable to digitize shape information at high resolution. Yet analyzing and processing this data remains challenging because of the large amount of data involved, and because modern applications such as human-computer interaction require real-time processing. Meeting these challenges requires interdisciplinary approaches that combine concepts from a variety of research areas, including numerical computing, differential geometry, deformable shape modeling, sparse data representation, and machine learning. On the algorithmic side, many shape analysis tasks are modeled using partial differential equations, which can be solved using tools from the field of numerical computing. The fields of differential geometry and deformable shape modeling have recently begun to influence shape analysis methods. Furthermore, tools from the field of sparse representations, which aim to describe input data using a compressible representation with respect to a set of carefully selected basic elements, have the potential to significantly reduce the amount of data that needs to be processed in shape analysis tasks. The related field of machine learning offers similar potential.

The goal of the Dagstuhl Seminar on New Perspectives in Shape Analysis held in February 2014 was to address these challenges with the help of the latest tools related to geometric, algorithmic and numerical concepts and to bring together researchers at the forefront of shape analysis who can work together to identify open problems and novel solutions. The book resulting from this seminar will appeal to researchers in the field of shape analysis, image and vision, from those who want to become more familiar with the field, to experts interested in learning about the latest advances.​



Numerical Computing for Shape Analysis


Chapter 1. Ornament Analysis with the Help of Screened Poisson Shape Fields

In this chapter, some thought-provoking application problems in Ornament Analysis are examined. Fields constructed via Screened Poisson Equation are used as intermediate level representations towards developing solutions. In the considered problems, the fields serve to a variety of purposes – i.e., to embed critical point detection process into a suitable morphological scale space, to regularise an ill-posed search problem, and finally to integrate features in a context – extending the visual functions of the Screened Poisson Equation based shape fields.
Sibel Tari

Chapter 2. A Comparison of Non-Lambertian Models for the Shape-from-Shading Problem

In this paper we present in a unified approach Shape-from-Shading models under orthographic projection for non-Lambertian surfaces and compare them with the classical Lambertian model. Those non-Lambertian models have been proposed in the literature by various authors in order to take into account more realistic surfaces such as rough and specular surfaces. The advantage of our unified mathematical model is the possibility to easily modify a single differential model to various situations just changing some control parameters. Moreover, the numerical approximation we propose is valid for that general model and can be easily adapted to the real situation. Finally, we compare the models on some benchmarks including real and synthetic images.
Silvia Tozza, Maurizio Falcone

Chapter 3. Direct Variational Perspective Shape from Shading with Cartesian Depth Parametrisation

Most of today’s state-of-the-art methods for perspective shape from shading are modelled in terms of partial differential equations (PDEs) of Hamilton-Jacobi type. To improve the robustness of such methods w.r.t. noise and missing data, first approaches have recently been proposed that seek to embed the underlying PDE into a variational framework with data and smoothness term. So far, however, such methods either make use of a radial depth parametrisation that makes the regularisation hard to interpret from a geometrical viewpoint or they consider indirect smoothness terms that require additional consistency constraints to provide valid solutions. Moreover the minimisation of such frameworks is an intricate task, since the underlying energy is typically non-convex. In this chapter we address all three of the aforementioned issues. First, we propose a novel variational model that operates directly on the Cartesian depth. In contrast to existing variational methods for perspective shape from shading this refers to both the data and the smoothness term. Moreover, we employ a direct second-order regulariser with edge-preservation property. This direct regulariser yields by construction valid solutions without requiring additional consistency constraints. Finally, we also propose a novel coarse-to-fine minimisation framework based on an alternating explicit scheme. This framework allows us to avoid local minima during the minimisation and thus to improve the accuracy of the reconstruction. Experiments show the good quality of our model as well as the usefulness of the proposed numerical scheme.
Yong Chul Ju, Daniel Maurer, Michael Breuß, Andrés Bruhn

Chapter 4. Amoeba Techniques for Shape and Texture Analysis

Morphological amoebas are image-adaptive structuring elements for morphological and other local image filters introduced by Lerallut et al. Their construction is based on combining spatial distance with contrast information into an image-dependent metric. Amoeba filters show interesting parallels to image filtering methods based on partial differential equations (PDEs), which can be confirmed by asymptotic equivalence results. In computing amoebas, graph structures are generated that hold information about local image texture. This chapter reviews and summarises the work of the author and his coauthors on morphological amoebas, particularly their relations to PDE filters and texture analysis. It presents some extensions and points out directions for future investigation on the subject.
Martin Welk

Chapter 5. Increasing the Power of Shape Descriptor Based Object Analysis Techniques

An advantage of shape based techniques, for object analysis tasks, is that shape allows a large number of numerical characterizations. Some of these have an intuitively clear meaning, while others do not, but they are still very useful because they satisfy some desirable properties (e.g. invariance with respect to a set of certain transformations). In this chapter we focus on numerical shape characteristics that have a clear intuitive interpretation – i.e. based on such numerical values, we can predict, to some extent, what the considered object looks like. This is beneficial, since it enables a priori appraisal of whether certain shape characteristics have suitable discriminative potential that make them appropriate for the intended task. By their nature, the number of such methods cannot be as large as the number of methods to allocate shape/object characteristics based on some formalism (algebraic, geometric, probabilistic, etc.). Because of that, some other possibilities to increase the discriminative capacity of the methods based on numerical shape characteristics, with an intuitively predictable meaning, are considered. Herein, we observe two such possibilities: the use of tuning parameters to obtain a family of shape characteristics, and the use of multiple shapes derived from the objects analyzed.
Joviša Žunić, Paul L. Rosin, Mehmet Ali Aktaş

Chapter 6. Shape Distances for Binary Image Segmentation

Shape distances are an important measure to guide the task of shape classification. In this chapter we show that the right choice of shape similarity is also important for the task of image segmentation, even at the absence of any shape prior. To this end, we will study three different shape distances and explore how well they can be used in a trust region framework. In particular, we explore which distance can be easily incorporated into trust region optimization and how well these distances work for theoretical and practical examples.
Frank R. Schmidt, Lena Gorelick, Ismail Ben Ayed, Yuri Boykov, Thomas Brox

Chapter 7. Segmentation in Point Clouds from RGB-D Using Spectral Graph Reduction

In this chapter, we tackle the problem of segmentation in point clouds from RGB-D data. In contrast to full point clouds, RGB-D data only provides a part of the volumetric information, the depth information of the one view given in the corresponding RGB image. Still, this additional information is valuable for the segmentation task as it helps disambiguating texture gradients from structure gradients. In order to create hierarchical segmentations, we combine a state-of-the-art method for natural RGB image segmentation based on spectral graph analysis with an RGB-D boundary detector. We show that spectral graph reduction can be employed in this case, facilitating the computation of RGB-D segmentations in large datasets.
Margret Keuper, Thomas Brox

Sparse Data Representation and Machine Learning for Shape Analysis


Chapter 8. Shape Compaction

We cover and discuss techniques that are designed for compaction of shape representations or shape configurations. The goal of compaction is to reduce storage space, a fundamental problem in many application domains. We consider compaction both at the representation level (i.e., digital storage) and in physical domains (i.e., physical storage). Shape representation compaction focuses on reducing the memory space allocated for storing the shape geometry data, whilst shape compaction techniques in the physical domain reduce the physical space occupied by shape configuration. We use the term shape configuration to refer to how a shape, real or conceptual, is physically modeled (e.g., design and composition of its parts) and spatially arranged (e.g., shape parts positioning and possibly in relation to other shapes). In this paper we briefly cover the representation compaction techniques whilst placing our focus on the less explored realm of shape compaction approaches on physical configurations.
Honghua Li, Hao Zhang

Chapter 9. Homological Shape Analysis Through Discrete Morse Theory

Homology and persistent homology are fundamental tools for shape analysis and understanding that recently gathered a lot of interest, in particular for analyzing multidimensional data. In this context, discrete Morse theory, a combinatorial counterpart of smooth Morse theory, provides an excellent basis for reducing computational complexity in homology detection. A discrete Morse complex, computed over a given complex discretizing a shape, drastically reduces the number of cells of the latter while maintaining the same homology. Here, we consider the problem of shape analysis through discrete Morse theory, and we review and analyze algorithms for computing homology and persistent homology based on such theory.
Leila De Floriani, Ulderico Fugacci, Federico Iuricich

Chapter 10. Sparse Models for Intrinsic Shape Correspondence

We present a novel sparse modeling approach to non-rigid shape matching using only the ability to detect repeatable regions. As the input to our algorithm, we are given only two sets of regions in two shapes; no descriptors are provided so the correspondence between the regions is not know, nor do we know how many regions correspond in the two shapes. We show that even with such scarce information, it is possible to establish very accurate correspondence between the shapes by using methods from the field of sparse modeling, being this, the first non-trivial use of sparse models in shape correspondence. We formulate the problem of permuted sparse coding, in which we solve simultaneously for an unknown permutation ordering the regions on two shapes and for an unknown correspondence in functional representation. We also propose a robust variant capable of handling incomplete matches. Numerically, the problem is solved efficiently by alternating the solution of a linear assignment and a sparse coding problem. The proposed methods are evaluated qualitatively and quantitatively on standard benchmarks containing both synthetic and scanned objects.
Jonathan Pokrass, Alexander M. Bronstein, Michael M. Bronstein, Pablo Sprechmann, Guillermo Sapiro

Chapter 11. Applying Random Forests to the Problem of Dense Non-rigid Shape Correspondence

We introduce a novel dense shape matching method for deformable, three-dimensional shapes. Differently from most existing techniques, our approach is general in that it allows the shapes to undergo deformations that are far from being isometric. We do this in a supervised learning framework which makes use of training data as represented by a small set of example shapes. From this set, we learn an implicit representation of a shape descriptor capturing the variability of the deformations in the given class. The learning paradigm we choose for this task is a random forest classifier. With the additional help of a spatial regularizer, the proposed method achieves significant improvements over the baseline approach and obtains state-of-the-art results while keeping a low computational cost.
Matthias Vestner, Emanuele Rodolà, Thomas Windheuser, Samuel Rota Bulò, Daniel Cremers

Chapter 12. Accelerating Deformable Part Models with Branch-and-Bound

Deformable Part Models (DPMs) play a prominent role in current object recognition research, as they rigorously model the shape variability of an object category by breaking an object into parts and modelling the relative locations of the parts. Still, inference with such models requires solving a combinatorial optimization task. In this chapter, we will see how Branch-and-Bound can be used to efficiently perform inference with such models. Instead of evaluating the classifier score exhaustively for all part locations and scales, such techniques allow us to quickly focus on promising image locations. The core problem that we will address is how to compute bounds that accommodate part deformations; this allows us to apply Branch-and-Bound to our problem. When comparing to a baseline DPM implementation, we obtain exactly the same results but can perform the part combination substantially faster, yielding up to tenfold speedups for single object detection, or even higher speedups for multiple objects.
Iasonas Kokkinos

Deformable Shape Modeling


Chapter 13. Non-rigid Shape Correspondence Using Surface Descriptors and Metric Structures in the Spectral Domain

Finding correspondence between non-rigid shapes is at the heart of three-dimensional shape processing. It has been extensively addressed over the last decade, but efficient and accurate correspondence detection still remains a challenging task. Generalized Multidimensional Scaling (GMDS) is an approach that finds correspondence by mapping one shape into another, while attempting to preserve distances between pairs of corresponding points on the two shapes. A different approach consists in detecting correspondence between shapes by matching their pointwise surface descriptors. Recently, the Spectral GMDS (SGMDS) approach was introduced, according to which the GMDS was re-formulated in the natural spectral domain of the shapes. Here, we propose a method that combines matching based on geodesic distances and pointwise surface descriptors . Following SGMDS, in the proposed solution the entire problem is translated into the spectral domain, resulting in efficient correspondence computation. Efficiency and accuracy of the proposed method are demonstrated by comparing it to state-of-the-art approaches, using a standard correspondence benchmark.
Anastasia Dubrovina, Yonathan Aflalo, Ron Kimmel

Chapter 14. The Perspective Face Shape Ambiguity

When a face is viewed under perspective projection, its shape (i.e. the 2D position of features) changes dramatically as the distance between face and camera varies. This causes substantial variation in appearance which is significant enough to disrupt human recognition of unfamiliar faces. However, a face viewed at any distance is still perceived as natural and humans are bad at interpreting the subject-camera distance given only a face image. We show that perspective viewing of faces leads to an ambiguity. Namely, that observed configurational information (position of projected vertices) and shading can be explained by a continuous class of possible faces. To demonstrate the ambiguity, we propose a novel method for efficiently fitting a 3D morphable model to 2D vertex positions when the subject-camera distance is known. By varying this distance, we obtain a subspace of faces, all of which are consistent with the observed data. We additionally show that faces within this subspace can all produce approximately the same shading pattern via a spherical harmonic lighting model.
William A. P. Smith

Chapter 15. On Shape Recognition and Language

Shapes convey meaning. Language is efficient in expressing and structuring meaning. The main thesis of this chapter is that by integrating shape with linguistic information shape recognition can be improved in performance. It broadens the concept of shape to visual shapes that include both geometric and optical information and explores ways that additional linguistic information may help with shape recognition. Towards this goal, it briefly describes some shape categories which have the potential of better recognition via language, with emphasis on gestures and moving shapes of sign language, as well as on cross-modal relations between vision and language in videos. It also draws inspiration from psychological studies that explore connections between gestures and human languages. Afterwards, it focuses on the broad class of multimodal gestures that combine spatio-temporal visual shapes with audio information. In this area, an approach is reviewed that significantly improves multimodal gesture recognition by fusing 3D shape information from motion-position of gesturing hands/arms and spatio-temporal handshapes in color and depth visual channels with audio information in the form of acoustically recognized sequences of gesture words.
Petros Maragos, Vassilis Pitsikalis, Athanasios Katsamanis, George Pavlakos, Stavros Theodorakis

Chapter 16. Tongue Mesh Extraction from 3D MRI Data of the Human Vocal Tract

In speech science, analyzing the shape of the tongue during human speech production is of great importance. In this field, magnetic resonance imaging (MRI) is currently regarded as the preferred modality for acquiring dense 3D information about the human vocal tract . However, the desired shape information is not directly available from the acquired MRI data. In this chapter, we present a minimally supervised framework for extracting the tongue shape from a 3D MRI scan. It combines an image segmentation approach with a template fitting technique and produces a polygon mesh representation of the identified tongue shape. In our evaluation, we focus on two aspects: First, we investigate whether the approach can be regarded as independent of changes in tongue shape caused by different speakers and phones. Moreover, we check whether an average user who is not necessarily an anatomical expert may obtain acceptable results. In both cases, our framework shows promising results.
Alexander Hewer, Stefanie Wuhrer, Ingmar Steiner, Korin Richmond


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