3D Shape Analysis: Fundamentals, Theory, and Applications

Preface

The primary goal of this book is to provide an in‐depth review of 3D shape analysis, which is an important problem and a building block to many applications. This book covers a wide range of basic, intermediate, and advanced topics relating to both the theoretical and practical aspects of 3D shape analysis. It provides a comprehensive overview of the key developments that have occurred for the past two decades in this exciting and continuously expanding field.

This book is organized into 14 chapters, which include an introductory chapter (Chapter 1) and a Conclusions and Perspectives chapter (Chapter 14). The remaining chapters (Chapters 2-13) are structured into three parts. The first part, which is composed of two chapters, introduces the reader to the background concepts of geometry and topology (Chapter 2) that are relevant to most of the 3D shape analysis aspects. It also provides a comprehensive overview of the techniques that are used to capture, create, and preprocess 3D models (Chapter 3). Understanding these techniques will help the reader, not only to understand the various challenges faced in 3D shape analysis but will also motivate the use of 3D shape analysis techniques in improving the algorithms for 3D reconstruction, which is a long‐standing problem in computer vision and computer graphics.

The second part, which is composed of two chapters, presents a wide range of mathematical and algorithmic tools that are used for shape description and comparison. In particular, Chapter 4 presents various global descriptors that have been proposed in the literature to characterize the overall shape of a 3D object using its geometry and/or topology. Chapter 5 covers the key algorithms and techniques that are used to detect local features and to characterize the shape of local regions using local descriptors. Both local and global descriptors can be used for shape‐based retrieval, recognition, and classification of 3D models. Local descriptors can be also used to compute correspondences between, and to register, 3D objects. This is the focus of the third part of the book, which covers the three commonly studied aspects of the registration and correspondence problem, mainly: rigid registration (Chapter 6), nonrigid registration (Chapter 7), and semantic correspondence (Chapter 8).

The last part and its five chapters are more focused on the application aspects. Specifically, Chapter 9 reviews some of the semantic applications of 3D shape analysis. Chapter 10 focuses on a specific type of 3D object, human faces, and reviews some techniques which are used for 3D face recognition and classification. Chapter 11 focuses on the problem of recognizing objects in 3D scenes. Nowadays, cars, robots, and drones are equipped with 3D sensors, which capture their environments. Tasks such as navigation, target detection and identification, and object tracking require the analysis of the 3D information that is captured by these sensors. Chapter 12 focuses on a classical problem of 3D shape analysis, i.e. how to retrieve 3D objects of interest from a collection of 3D models. It provides a comparative analysis and discusses the pros and cons of various descriptors and similarity measures. Chapter 13, on the other hand, treats the same problem of shape retrieval but this time by using multimodal queries. This is one of the emerging fields of 3D shape analysis and it aims to narrow the gap between the different visual representations of the 3D world (e.g. images, 2D sketches, 3D models, and videos). Finally, Chapter 14 summarizes the book and discusses some of the open problems and future challenges of 3D shape analysis.

The purpose of this book is not to provide a complete and detailed survey of the 3D shape analysis field. Rather, it succinctly covers the key developments of the field in the past two decades and shows their applications in various 3D vision and graphics problems. It is intended to advanced graduate students, postgraduate students, and researchers working in the field. It can also serve as a reference to practitioners and engineers working on the various applications of 3D shape analysis.

May 2018

Hamid Laga

Yulan Guo

Hedi Tabia

Robert B. Fisher

Mohammed Bennamoun

Acknowledgments

The completion of this book would not have been possible without the help, advice, and support of many people. This book is written based on the scientific contributions of a number of colleagues and collaborators. Without their ground breaking contributions to the field, this book would have never matured into this form.

Some of the research presented in this book was developed in collaboration with our PhD students, collaborators, and supervisors. We were very fortunate to work with them and are very grateful for their collaboration. Particularly, we would like to thank (in alphabetical order) Ian H. Jermyn, Sebastian Kurtek, Jonathan Li, Stan Miklavcic, Jacob Montiel, Michela Mortara, Masayuki Nakajima (Hamid Laga's PhD advisor), David Picard, Michela Spagnuolo, Ferdous Sohel, Anuj Srivastava, Antonio Verdone Sanchez, Jianwei Wan, Guan Wang, Hazem Wannous, and Ning Xie. We would also like to thank all our current and previous colleagues at Murdoch University, Tokyo Institute of Technology, National University of Defense Technology, Institute of Computing Technology, Chinese Academy of Sciences, the University of South Australia, the ETIS laboratory, the Graduate School in Electrical Engineering Computer Science and Communications Networks (ENSEA), the University of Western Australia, and The University of Edinburgh.

We are also very grateful to the John Wiley team for helping us create this book.

This work was supported in part by funding from the Australian Research Council (ARC), particularly ARC DP150100294 and ARC DP150104251, National Natural Science Foundation of China (Nos. 61602499 and 61471371), and the National Postdoctoral Program for Innovative Talents of China (No. BX201600172).

Lastly, this book would not have been possible without the incredible support and encouragement of our families.

Hamid Laga dedicates this book to his parents, sisters, and brothers whose love and generosity have always inspired him; to his wife Lan and son Ilyan whose daily encouragement and support in all matters make it all worthwhile.

Yulan Guo dedicates this book to his parents, wife, and son. His son shares the same time for pregnancy, birth, and growth as this book. This would be the first and best gift for the birth of his son.

Hedi Tabia dedicates this book to his family.

Robert B. Fisher dedicates this book to his wife, Miesbeth, who helped make the home a happy place to do the writing. It is also dedicated to his PhD supervisor, Jim Howe, who got him started, and to Bob Beattie, who introduced him to computer vision.

Mohammed Bennamoun dedicates this book to his parents: Mostefa and Rabia, to his children: Miriam, Basheer, and Rayaane and to his seven siblings.

1

Introduction

1.1 Motivation

Shape analysis is an old topic that has been studied, for many centuries, by scientists from different boards, including philosophers, psychologists, mathematicians, biologists, and artists. However, in the past two decades, we have seen a renewed interest in the field motivated by the recent advances in 3D acquisition, modeling, and visualization technologies, and the substantial increase in the computation and storage power. Nowadays, 3D scanning devices are accessible not only to domain‐specific experts but also to the general public. Users can scan the real world at high resolution, using devices that are as cheap as video cameras, edit the 3D data using 3D modeling software, share them across the web, and host them in online repositories that are growing in size and in number. Such repositories can include millions of every day objects, cultural heritage artifacts, buildings, as well as medical, scientific, and engineering models.

The increase in the availability of 3D data comes with new challenges in terms of storage, classification, and retrieval of such data. It also brings unprecedented opportunities for solving long‐standing problems; First, the rich variability of 3D content in existing shape repositories makes it possible to directly reuse existing 3D models, in whole or in part, to construct new 3D models with rich variations. In many situations, 3D designers and content creators will no more need to scan or model a 3D object or scene from scratch. They can query existing repositories, retrieve the desired models, and fine‐tune their geometry and appearance to suit their needs. This concept of context reuse is not specific to 3D models but has been naturally borrowed from other types of media. For instance, one can translate sentences to different languages by performing cross‐language search. Similarly, one can create an image composite or a visual art piece by querying images, copying parts of them and pasting them into their own work.

Second, these large amounts of 3D data can be used to learn computational models that effectively reason about properties and relationships of shapes without relying on hard‐coded rules or explicitly programmed instructions. For instance, they can be used to learn 3D shape variation in medical data in order to model physiological abnormalities in anatomical organs, model their natural growth, and learn how shape is affected by disease progression. They can be also used to model 3D shape variability using statistical models, which, in turn, can be used to facilitate 3D model creation with minimum user interaction.

Finally, data‐driven methods facilitate high‐level shape understanding by discovering geometric and structural patterns among collections of shapes. These patterns can serve as strong priors not only in various geometry processing applications but also in solving long‐standing computer vision problems, ranging from low‐level 3D reconstruction to high‐level scene understanding.

These technological developments and the opportunities they bring have motivated researchers to take a fresh look at the 3D shape analysis problem. Although most of the recent developments are application‐driven, many of them aim to answer fundamental, sometimes philosophical, questions such as: What is shape? Can we mathematically formulate the concept of shape? How to compare the shape of objects? How to quantify and localize shape similarities and differences? This book synthesizes the critical mass of 3D shape analysis research that has accumulated over the past 15 years. This rapidly developing field is both profound and broad, with a wide range of applications and many open research questions that are yet to be answered.

1.2 The 3D Shape Analysis Problem

Shape is the external form, outline or surface, of someone or something as opposed to other properties such as color, texture, or material composition.

Source: Wikipedia and Oxford dictionaries.

Humans can easily abstract the form of an object, describe it with a few geometrical attributes or even with words, relate it to the form of another object, and group together, in multiple ways and using various criteria, different objects to form clusters that share some common shape properties. Shape analysis is the general term used to refer to the process of automating these tasks, which are trivial to humans but very challenging to computers. It has been investigated under the umbrella of many applications and has multiple facets. Below, we briefly summarize a few of them.

3D shape retrieval, clustering, and classification. Similar to other types of multimedia information, e.g. text documents, images, and videos, the demand for efficient clustering and classification tools that can organize, automatically or semi‐automatically, the continuously expanding collections of 3D models is growing. Likewise, users, whether they are experts, e.g. graphics designers who are increasingly relying on the reuse of existing 3D contents, or novice, will benefit from a search engine that will enable them to search for 3D data of interest in the same way they search for text documents or images.

Correspondence and registration. This problem, which can be summarized as the ability to say which part of an object matches which part on another object, and the ability to align one object onto another, arises in many domains of computer vision, computer graphics, and medical imaging. Probably, one of the most popular examples is the 3D reconstruction problem where usually a 3D object is scanned by multiple sensors positioned at different locations around the object. To build the complete 3D model of the object, one needs to merge the partial scans produced by each sensor. This operation requires a correct alignment, i.e. registration, step that brings all the acquired 3D data into a common coordinate frame. Note also that, in many cases, 3D objects move and deform, in a nonrigid way, during the scanning process. This makes the alignment process even more complex. Another example is in computer graphics where a 3D designer creates a triangulated 3D mesh model, hereinafter referred to as the reference, and assigns to each of its triangular faces some attributes, e.g. color and material properties. The designer then can create additional models with the same attributes but instead of manually setting them, they can be automatically transferred from the reference model if there is a mechanism which finds for each point on the reference model its corresponding points on the other models.

Detection and recognition. This includes the detection of low level features such as corners or regions of high curvatures, as well as the localization and recognition of parts in 3D objects, or objects in 3D scenes. The latter became very popular in the past few years with the availability of cheap 3D scanning devices. In fact, instead of trying to localize and recognize objects in a scene from 2D images, one can develop algorithms that operate on the 3D scans of the scene, eventually acquired using commodity devices. This has the advantage that 3D data are less affected than 2D images by the occlusions and ambiguities, which are inherent to the loss of dimensionality when projecting the 3D world onto 2D images. 3D face and 3D action recognition are, among others, examples of applications that have benefited from the recent advances in 3D technologies.

Measurement and characterization of the geometrical and topological properties of objects on one hand and of the spatial relations between objects on the other hand. This includes the identification of similar regions and finding recurrent patterns within and across 3D objects.

Summarization and exploration of collections of 3D models. Given a set of objects, one would like to compute a representative 3D model, e.g. the average or median shape, as well as other summary statistics such as covariances and modes of variation of their shapes. One would like also to characterize the collection using probability distributions and sample from these distributions new instances of shapes to enrich the collection. In other words, one needs to manipulate 3D models in the same way one manipulates numbers.

Implementing these representative analysis tasks requires solving a set of challenges, and each has been the subject of important research and contributions. The first challenge is the mathematical representation of the shape of objects. 3D models, acquired with laser scanners or created using some modeling software, can be represented with point clouds, polygonal soup models, or as volumetric images. Such representations are suitable for storage and visualization but not for high‐level analysis tasks. For instance, scanning the same object from two different viewpoints or using different devices will often result in two different point clouds but the shape remains the same. The challenge is in designing mathematical representations that capture the essence of shape. A good representation should be independent of (or invariant to) the pose of the 3D object, the way it is scanned or modeled, and the way it is stored. It is also important to ensure that two different shapes cannot have the same representation.

Figure 1.1Complexity of the shape similarity problem. (a) Nonrigid deformations. (b) Partial similarity. (c) Semantic similarity.

Second, almost every shape analysis task requires a measure that quantifies shape similarities and differences. This measure, called dissimilarity, distance, or metric, is essential to many tasks. It can be used to compare the 3D shape of different objects and localize similar parts in and across 3D models. It can also be used to detect and recognize objects in 3D scenes. Shape similarity is, however, one of the most ambiguous concepts in shape analysis since it depends not only on the geometry of the objects being analyzed but also on their semantics, their context, the application, and on the human perception. Figure 1.1 shows a few examples that illustrate the complexity of the shape similarity problem. In Figure 1.1a, we consider human body shapes of the same person but in different poses. One can consider these models as similar since they are of the same person. One may also treat them as different since they differ in pose. On the other hand, the 3D objects of Figure 1.1b are only partially similar. For instance, one part of the centaur model can be treated as similar to the upper body of the human body shape, while the other part is similar to the 3D shape of a horse. Also, one can consider that the candles of Figure 1.1c are similar despite the significant differences in their geometry and topology. A two‐year‐old child can easily match together the parts of the candles that have the same functionality despite the fact that they have different geometry, structure, and topology.

Finally, these problems, i.e. representation and dissimilarity, which are interrelated (although many state‐of‐the‐art papers treat them separately), are the core components of and the building blocks for almost every 3D shape analysis system.

1.3 About This Book

The field of 3D shape analysis is being actively studied by researchers originating from at least four different domains: mathematics and statistics, image processing and computer vision, computer graphics, and medical imaging. As a result, a critical mass of research has accumulated over the past 15 years, where almost every major conference in these fields included tracks dedicated to 3D shape analysis. This book provides an in‐depth description of the major developments in this continuously expanding field of research. It can serve as a complete reference to graduate students, researchers, and professionals in different fields of mathematics, computer science, and engineering. It could be used for courses of intermediate level in computer vision and computer graphics or for self‐study. It is organized into four main parts:

The first part, which is composed of two chapters, provides an in‐depth review of the background concepts that are relevant to most of the 3D shape analysis aspects. It begins in Chapter 2 with the basic elements of geometry and topology, which are needed in almost every 3D shape analysis task. We will look in this chapter into elements of differential geometry and into how 3D models are represented. While most of this material is covered in many courses and textbooks, putting them in the broader context of shape analysis will help the reader appreciate the benefits and power of these fundamental mathematical tools.

Chapter 3 reviews the techniques that are used to capture, create, and preprocess 3D models. Understanding these techniques will help the reader, not only to understand the various challenges faced in 3D shape analysis but will also motivate the use of 3D shape analysis techniques in improving the algorithms for 3D reconstruction, which is a long‐standing problem in computer vision and computer graphics.

The second part, which is composed of two chapters, presents a range of mathematical and algorithmic tools that are used for shape description and comparison. In particular, Chapter 4 presents the different descriptors that have been proposed in the literature to characterize the global shape of a 3D object using its geometry and/or topology. Early works on 3D shape analysis, in particular classification and retrieval, were based on global descriptors. Although they lack the discrimination power, they are the foundations of modern and powerful 3D shape descriptors.

Chapter 5, on the other hand, covers the algorithms and techniques used for the detection of local features and the characterization of the shape of local regions using local descriptors. Many of the current 3D reconstruction, recognition, and analysis techniques are built on the extraction and matching of feature points. Thus, these are fundamental techniques required in most of the subsequent chapters of the book.

The third part of the book, which is composed of three chapters, focuses on the important problem of computing correspondences and registrations between 3D objects. In fact, almost every task, from 3D reconstruction to animation, and from morphing to attribute transfer, requires accurate correspondence and registration. We will consider the three commonly studied aspects of the problem, which are rigid registration (Chapter 6), nonrigid registration (Chapter 7), and semantic correspondence (Chapter 8). In the first case, we are given two pieces of geometry (which can be partial scans or full 3D models), and we seek to find the rigid transformations (translations, scaling and rotations) that align one piece onto the other. This problem appears mainly in 3D scanning where often a 3D object is scanned by multiple scanners. Each scan produces a set of incomplete point clouds that should be aligned and fused together to form a complete 3D model.

3D models can not only undergo rigid transformations but also nonrigid deformations. Think, for instance, of the problem of scanning a human body. During the scanning process, the body can not only move but also bend. Once it is fully captured, we would like to transfer its properties (e.g. color, texture, and motion) onto another 3D human body of a different shape. This requires finding correspondences and registration between these two 3D objects, which bend and stretch. This is a complex problem since the space of solutions is large and requires efficient techniques to explore it. Solutions to this problem will be discussed in Chapter 7.

Semantic correspondence is even more challenging; think of the problem of finding correspondences between an office chair and a dining chair. While humans can easily match parts across these two models, the problem is very challenging for computers since these two models differ both in geometry and topology. We will review in Chapter 8 the methods that solve this problem using supervised learning, and the methods that used structure and context to infer high‐level semantic concepts.

The last part of the book demonstrates the use of the fundamental techniques described in the earlier chapters in a selection of 3D shape analysis applications. In particular, Chapter 9 reviews some of the semantic applications of 3D shape analysis. It also illustrates the range of applications involving 3D data that have been annotated with some sort of meaning (i.e. semantics or labels).

Chapter 10 focuses on a specific type of 3D objects, which are human faces. With the widespread of commodity 3D scanning devices, several recent works use the 3D geometry of the face for various purposes including recognition, gender classification, age recognition, and disease and abnormalities detection. This chapter will review the most relevant works in this area.

Chapter 11 focuses on the problem of recognizing objects in 3D scenes. Nowadays, cars, robots, and drones are all equipped with 3D sensors that capture their environments. Tasks such as navigation, target detection and identification, object tracking, and so on require the analysis of the 3D information that is captured by these sensors.

Chapter 12 focuses on a classical problem of 3D shape analysis, which is how to retrieve 3D objects of interest from a collection of 3D models. Chapter 13, on the other hand, treats the same problem of shape retrieval but this time by using multimodal queries. This is a very recent problem that has received a lot of interest with the emergence of deep‐learning techniques that enable embedding different modalities into a common space.

The book concludes in Chapter 14 with a summary of the main ideas and a discussion of the future trends in this very active and continuously expanding field of research.

Figure 1.2Structure of the book and dependencies between the chapters.

Readers can proceed sequentially through each chapter. Some readers may want to go straight to topics of their interest. In that case, we recommend to follow the reading chart of Figure 1.2, which illustrates the inter‐dependencies between the different chapters.

1.4 Notation

Table 1.1 summarizes the different notations used throughout the book.

Table 1.1 List of notations used throughout the book.

Part I

Foundations

2

Basic Elements of 3D Geometry and Topology

This chapter introduces some of the fundamental concepts of 3D geometry and 3D geometry processing. Since this is a very large topic, we only focus in this chapter on the concepts that are relevant to the 3D shape analysis tasks covered in the subsequent chapters. We structure this chapter into two main parts. The first part (Section 2.1) covers the elements of differential geometry that are used to describe the local properties of 3D shapes. The second part (Section 2.2) defines the concept of shape, the transformations that preserve the shape of a 3D object, and the deformations that affect the shape of 3D objects. It also describes some preprocessing algorithms, e.g. alignment, which are used to prepare 3D models for shape analysis tasks.

2.1 Elements of Differential Geometry

2.1.1 Parametric Curves

A one‐dimensional curve in c02-i0001 can be represented in a parametric form by a vector‐valued function of the form:

(2.1)

The functions c02-i0002 , and c02-i0003 are also called the coordinate functions. If we assume that they are differentiable functions of c02-i0004 , then one can compute the tangent vector, the normal vector, and the curvature at each point on the curve. For instance, the tangent vector, c02-i0005 , to the curve at a point c02-i0006 , see Figure 2.1, is the first derivative of the coordinate functions:

(2.2)

Note that the tangent vector c02-i0007 also corresponds to the velocity vector at time c02-i0008 .

Now, let c02-i0009 . The length c02-i0010 of the curve segment defined between the two points c02-i0011 and c02-i0012 is given by the integral of the tangent vector:

(2.3)

Here, c02-i0013 denotes the inner dot product. The length c02-i0014 of the curve is then given by c02-i0015 .

Figure 2.1An example of a parameterized curve and its differential properties.

Let c02-i0016 be a smooth and monotonically increasing function, which maps the domain c02-i0017 onto c02-i0018 such that c02-i0019 . Reparameterizing c02-i0020 with c02-i0021 results in another curve c02-i0022 such that:

(2.4)

The length of the curve from c02-i0023 to c02-i0024 , is exactly c02-i0025 . The mapping function c02-i0026 , as defined above, is called arc‐length parameterization. In general, c02-i0027 can be any arbitrary smooth and monotonically increasing function, which maps the domain c02-i0028 into another domain c02-i0029 . If c02-i0030 is a bijection and its inverse c02-i0031 is differentiable as well, then c02-i0032 is called a diffeomorphism. Diffeomorphisms are very important in shape analysis. For instance, as shown in Figure 2.2, reparameterizing a curve c02-i0033 with a diffeomorphism c02-i0034 results in another curve c02-i0035 , which has the same shape as c02-i0036 . These two curves are, therefore, equivalent from the shape analysis perspective.

Figure 2.2Reparameterizing a curve c02-i0037 with a diffeomorphism c02-i0038 results in another curve c02-i0039 of the same shape as c02-i0040 . (a) A parametric curve c02-i0041 . (b) A reparameterization function c02-i0042 . (c) The reparametrized curve c02-i0043 .

Now, for simplicity, we assume that c02-i0044 is a curve parameterized with respect to its arc length. We can define the curvature at a point c02-i0045 as the deviation of the curve from the straight line. It is given by the norm of the second derivative of the curve:

(2.5)

Curvatures carry a lot of information about the shape of a curve. For instance, a curve with zero curvature everywhere is a straight line segment, and planar curves of constant curvature are circular arcs.

2.1.2 Continuous Surfaces

We will look in this section at the same concepts as those defined in the previous section for curves but this time for smooth surfaces embedded in c02-i0046 .

A parametric surface c02-i0047 (see Figure 2.3) can be defined as a vector‐valued function c02-i0048 which maps a two‐dimensional domain c02-i0049 onto c02-i0050 . That is:

(2.6) equation

where c02-i0051 , and c02-i0052 are differentiable functions with respect to c02-i0053 and c02-i0054 . The domain c02-i0055 is called the parameter space, or parameterization domain, and the scalars c02-i0056 are the coordinates in the parameter space.

Figure 2.3Example of an open surface parameterized by a 2D domain c02-i0057 .

As an example, consider c02-i0060 to be the surface of a sphere of radius c02-i0061 and centered at the origin. This surface can be defined using a parametric function of the form

In practice, the parameter space c02-i0062 is chosen depending on the types of shapes at hand. In the case of open surfaces such as 3D human faces, c02-i0063 can be a subset of c02-i0064 , e.g. the quadrilateral domain c02-i0065 or the disk domain c02-i0066 . In the case of closed surfaces, such as those shown in Figure 2.4a,b, then c02-i0067 can be chosen to be the unit sphere c02-i0068 . In this case, c02-i0069 and c02-i0070 are the spherical coordinates such that c02-i0071 is the polar angle and c02-i0072 is the azimuthal angle.

Let c02-i0073 be a diffeomorphism, i.e. a smooth invertible function that maps the domain c02-i0074 to itself such that both c02-i0075 and its inverse are smooth. Let also c02-i0076 denote the space of all such diffeomorphisms. c02-i0077 transforms a surface c02-i0078 into c02-i0079 . This is a reparameterization process and often c02-i0080 is referred to as a reparameterization function. Reparameterization is important in 3D shape analysis because it produces registration. For instance, similar to the curve case, both surfaces represented by c02-i0081 and c02-i0082 have the same shape and thus are equivalent. Also, consider the two surfaces c02-i0083 and c02-i0084 of Figure 2.4b where c02-i0085 represents the surface of one long hand and c02-i0086 represents the surface of the hand of another subject. Let c02-i0087 be such that c02-i0088 corresponds to the thumb tip of c02-i0089 . If c02-i0090 and c02-i0091 are arbitrarily parameterized, which is often the case since they have been parameterized independently from each other, c02-i0092 may refer to any other location on c02-i0093 , the fingertip of the index finger, for example. Thus, c02-i0094 and c02-i0095 are not in correct correspondence. Putting c02-i0096 and c02-i0097 in correspondence is equivalent to reparameterizing c02-i0098 with a diffeomorphism c02-i0099 such that for every c02-i0100 , c02-i0101 and c02-i0102 point to the same feature, e.g. thumb tip to thumb tip. We will use these properties in Chapter 7 to find correspondences and register surfaces which undergo complex nonrigid deformations.

Figure 2.4Illustration of (a) a spherical parameterization of a closed genus‐0 surface, and (b) how parameterization provides correspondence. Here, the two surfaces, which bend and stretch, are in correct correspondence. In general, however, the analysis process should find the optimal reparameterization which brings c02-i0058 into a full correspondence with c02-i0059 .

2.1.2.1 Differential Properties of Surfaces

Similar to the curve case, one can define various types of differential properties of a given surface. For instance, the two partial derivatives

(2.7)

define two tangent vectors to the surface at a point of coordinates c02-i0103 . The plane spanned by these two orthogonal vectors is tangent to the surface. The surface unit normal vector at c02-i0104 , denoted by c02-i0105 , can be computed as:

(2.8) equation

Here, c02-i0106 denotes the vector cross product.

First Fundamental Form

A tangent vector c02-i0107 to the surface at a point c02-i0108 can be defined in terms of the partial derivatives of c02-i0109 as:

(2.9) equation

The two real‐valued scalars c02-i0110 and c02-i0111 can be seen as the coordinates of c02-i0112 in the local coordinate system formed by c02-i0113 and c02-i0114 . Let c02-i0115 and c02-i0116 be two tangent vectors of coordinates c02-i0117 and c02-i0118 , respectively. Then, the inner product c02-i0119 , where c02-i0120 is the angle between the two vectors, is defined as:

(2.10)

where c02-i0121 is a 2D matrix called the first fundamental form of the surface at a point c02-i0122 of coordinates c02-i0123 . Let us write

The first fundamental form, also called the metric or metric tensor, defines inner products on the tangent space to the surface. It allows to measure:

Angles. The angle between the two tangent vectors c02-i0124 and c02-i0125 can be computed as:

Length. Consider a curve on the parametric surface. The parametric representation of the curve is c02-i0126 . The tangent vector to the curve at any point is given as:

Since the curve is on the surface defined by the parametric function c02-i0127 , then c02-i0128 and c02-i0129 are two orthogonal vectors that are tangent to the surface. Using Eq. (2.3), the length c02-i0130 of the curve is given by

Area. Similarly, we can measure the surface area c02-i0131 of a certain region c02-i0132 :

where c02-i0133 refers to the determinant of the square matrix c02-i0134 .

Since the first fundamental form allows measuring angles, distances, and surface areas, it is a strong geometric tool for 3D shape analysis.

Second Fundamental Form and Shape Operator

The second fundamental form c02-i0135 of a surface, defined by its parametric function c02-i0136 , at a point c02-i0137 is a c02-i0138 matrix defined in terms of the second derivatives of c02-i0139 as follows:

(2.11) equation

where

(2.12)

Here, c02-i0140 refers to the standard inner product in c02-i0141 , i.e. the Euclidean space. Similarly, the shape operator c02-i0142 of the surface is defined at a point c02-i0143 using the first and second fundamental forms as follows:

(2.13) equation

The shape operator is a linear operator. Along with the second fundamental form, they are used for computing surface curvatures.

2.1.2.2 Curvatures

Let c02-i0144 be the unit normal vector to the surface at a point c02-i0145 , and c02-i0146 a tangent vector to the surface at c02-i0147 . The