A number of papers in the Computer Vision and Pattern Recognition
literature have demonstrated that invariants, or equivalently
structure modulo a 3D linear transformation, are sufficient for object
recognition. The final stage in the recognition process is
verification, where an outline is transferred from an acquisition
image of the object to the target image.
For the most part recognition based on invariants has concentrated on
planar objects, though some 3D invariants have been measured from
single and multiple images for polyhedra, point sets, surfaces of
revolution and algebraic surfaces. The work so far on surfaces of
revolution has only exploited isolated points on the outline (such as
bitangents), and has not addressed transfer or verification.
This thesis, for the first time, extends the transfer and extraction
of invariants to surfaces of revolution using the entire outline.
Given a single view of the surface, it is possible to obtain the
projection in any other given view, given a minimal number of points
in the target image. In particular it is is possible to reconstruct
the generating curve, and thereby a rich set of invariants.
Table of Contents
0.1 Introduction
0.2 Specification
0.3 Relevant Literature
1 Introduction
1.1 The Object Class of Interest
1.2 The Task
1.3 The Chosen Imaging Geometry
1.4 Contributions of this Thesis
1.5 Outline of this Thesis
2 Distinguished features
2.1 Tangents
2.1.1 The Tangent Cone
2.1.2 The Outline
2.2 The Affine Basis
3 The Weak Perspective Camera
3.1 The underlying Geometry
3.1.1 The Surface of Revolution
3.1.2 The Weak Perspective Camera
3.1.3 Recovering the Generating Function
3.1.4 How to calculate the viewing direction
3.1.5 Transfer using two arbitrary views
3.2 Method 1. Using the Generating Curve
3.2.1 Summary
3.2.2 The Implementation
3.2.3 Results
3.3 Method 2. Using the Outline’s Envelope
3.3.1 The underlying Geometry
3.3.2 The Implementation
3.3.3 Results
3.4 Comparing the two Methods
3.5 Affine Extensions
3.5.1 Unknown Aspect Ratio
3.5.2 Full Affine Distortions
4 The Affine Camera
4.1 Theoretical Background
4.1.1 The Affine Camera
4.1.2 The Surface’s 3D Geometry and its Image
4.1.3 Acquisition — Calculating the Conics
4.1.4 Transfer
4.1.5 Summary
4.2 Implementation
4.2.1 The Common Frame
4.2.2 The Acquisition
4.2.3 Transfer
4.3 Results
4.4 Possible Enhancements and Open Questions
4.4.1 Better Features than Intersections
4.4.2 Unused Constraints
5 The Projective Camera
5.1 The underlying geometry
5.1.1 The projective Camera
5.1.2 The Surface’s 3D Geometry
5.1.3 Summary
5.2 A possible Implementation
5.2.1 Acquisition
5.2.2 Transfer
5.2.3 Transfer into the Canonical Frame
5.3 Results
6 Conclusions
6.1 A Recognition System
6.1.1 Transfer between two Views
6.1.2 Transfer into a Canonical Frame
6.1.3 How to build a Recognition System
6.2 Future Work
Research Objectives and Key Topics
This thesis explores novel methods for transferring the entire outline of a surface of revolution across different imaging geometries, including weak perspective, affine, and projective cameras. The primary objective is to enable robust object verification and the extraction of viewpoint-independent invariants from a canonical frame, without the strict necessity of a calibrated camera system.
- Application of bitangent points and conics as viewpoint-independent features.
- Development of transfer algorithms for weak perspective and affine projection models.
- Introduction of canonical frames for reconstructing generating functions of surfaces.
- Analysis of perspective distortions and their impact on outline alignment.
- Robustness evaluation of recognition systems for partially occluded objects.
Excerpt from the Book
The Object Class of Interest
A surface of revolution results from rotating a function f(z) — the so called generating curve — around the z-axis (see figure 1.1) where the z-axis is the axis of symmetry.
In practice, a surface of revolution is the surface of a solid, or object, of revolution. Objects of revolution have been known to man for thousands of years, since the invention of the potter’s wheel, and later the lathe, allowed man to produce highly symmetric objects with ease, and today we are surrounded by rotationally symmetric objects such as bottles, pens, vases, glasses, lamp-shades and light-bulbs. Some of the objects of revolution found in every household are shown in figure 1.2.
Summary of Chapters
1 Introduction: Provides an overview of surfaces of revolution as the object class of interest, defines the task of outline-based transfer, and outlines the thesis structure.
2 Distinguished features: Explains the geometric properties of surfaces of revolution, focusing on viewpoint-independent features such as conics and bitangent points.
3 The Weak Perspective Camera: Details two methods for transferring outlines under weak perspective geometry using generating curves and outlines' envelopes.
4 The Affine Camera: Generalizes the transfer methods to affine projections and introduces a canonical frame for generating function reconstruction.
5 The Projective Camera: Presents transfer methods for projective camera models, utilizing cross-ratios for mapping points on the axis of symmetry.
6 Conclusions: Discusses how the developed transfer methods can be integrated into an automated recognition system and suggests directions for future work.
Key Terms
Surface of revolution, Generating curve, Outline, Bitangent, Conic, Weak perspective camera, Affine camera, Projective camera, Transfer, Verification, Canonical frame, Cross-ratio, Tangent cone, Symmetry, Invariance.
Frequently Asked Questions
What is the primary focus of this research?
The research focuses on developing techniques to transfer the entire outline of a surface of revolution from one image to another, enabling object recognition and verification.
Which camera geometries are addressed?
The thesis addresses three specific imaging geometries: the weak perspective camera, the affine camera, and the projective camera.
What is the core scientific methodology?
The work employs geometric computer vision techniques, using viewpoint-independent features such as bitangent-pairs and conics to map outlines between different viewpoints.
What are the main research goals?
The goals are to enable transfer of surface projections between views and to extract invariants by transforming surfaces into a canonical frame.
Why is a canonical frame useful?
A canonical frame allows the retrieval of the surface's generating function, which simplifies the process of calculating invariants for object recognition.
Which features are used to calculate the transfer?
The methods primarily use distinguished features on the outline, specifically conics and bitangent points, which remain invariant across viewpoints.
How does perspective distortion affect the transfer results?
Perspective distortions can cause a noticeable displacement of the outline along the axis of symmetry, especially when the camera distance is relatively close to the object.
What role do swallowtails and cusps play in the analysis?
Cusps and swallowtails are identified as typical geometric events during the transfer of outlines, often marking the start of self-occlusion on the surface.
- Quote paper
- Sven Utcke (Author), 1993, Transfer and Invariants of Surfaces of Revolution, Munich, GRIN Verlag, https://www.grin.com/document/185569
