The flow of this thesis goes from the theoretical foundations (projective geometry, error propagation, and their combination) to practical applications showcasing one or more of the previously described theoretical principles; within the application chapters I go from the 2D case of a single planar homography to the case of sev-eral homographies all within one image and from there to the case of an even less restricted class of objects, surfaces of revolution.
In more detail, I’m starting this thesis with an overview of the state of the art in projective geometry (Chapter 2) and error propagation (Chapter 3) respectively.
These chapters do not contain anything new and are for a huge part lifted straight out of [103] and a couple of other books, in spirit if not in words. If you know your way around projective geometry or error propagation I would recommend to simply skip the respective chapter, they are here for completeness, and as a handy reference for later work.
The actual thesis starts with Chapter 4, which combines projective geometry and error propagation. The underlying idea is not new, and as far as the application to homogeneous coordinates is concerned can be found in [75]; however, in this chapter I also consider the application of these principles to other parameterisations than homogeneous coordinates and, starting from first principles, derive a number of new results such as an excellent approximation to the covariance of a line segment fitted to edgels, a new stopping-criterion for incremental fits based on a χ2-test, and a new algorithm for the calculation of the cross-ratio of 4 lines which due to the use of error propagation in fact performs faster than current algorithms. I will also give an intuitive explanation why the spherical normalisation used by many authors is indeed superior to an Euclidean normalisation; and finally I will give an overview on how to compare a number of common stochastic entities. Just this last section alone could already put away with many of the numerous, finely tuned parameters so common to computer vision algorithms.
The next three chapters describe different application scenarios. In Chapter 5 I describe the application of error-propagation principles to the grouping and recog-nition of zebra crossings and other repeated structure. This application was first described by me in [6], and is a nice example of an implementation which I be-lieve would have been impossible without the use of error propagation due to the high variations of a zebra-crossing’s size and quality even within a single image;
of particular interest here is how only a few confidence-tests can replace a host of manually chosen parameters, resulting in a uniquely stable algorithm. It describes the groundbreaking work on which later publications such as [135] build.
In Chapter 6 I outline an algorithm for the grouping of houses (or, indeed, any struc-ture consisting of orthogonal and parallel elements). Over the years we have seen a few algorithms for the reconstruction of buildings from monocular images [36, 87, 97], however, in contrast to multi-view approaches these nearly always require manual
16 The Outline of this Thesis segmentation of image regions. The algorithm outlined in this chapter could be seen as an attempt to remedy this situation. It is, however, included in this thesis for a different reason: buildings show a number of diverse features at different scales, and I will in particular have a closer look at collinear line segments of only a few pixels to several hundreds of pixels in length and distance as well as vanishing points, the image of intersection of parallel lines at infinity, which can be anywhere from lit-erally in the image to litlit-erally at infinity. What is more, these features come with differing accuracies, and even one and the same feature can have different accuracies attached to it depending on context. This application is therefore well suited as a showcase for several different ideas and approaches such as a new algorithm for the iterative improvement of vanishing-point position and one for the automatic group-ing of vanishgroup-ing points; a new objective function for the (partial) calibration of a camera from vanishing-points which takes the different uncertainties in the positions of the vanishing points into account and extends the usual Legoland assumption to more general setups; an extension on previous work which takes the vanishing-point information into account when merging line-segments; and finally a comparison of the performance of several different error-measures, both new ones first introduced in this thesis as well as established ones from the literature, for the identification of collinear line segments.
Chapter 7 finally describes part of the grouping algorithm underlying some of my older publications on the recognition of surfaces of revolution such as [3–5, 9], but also newer publications on their reconstruction, such as [8]. An important feature for both recognition as well as reconstruction of SORs is the object’s axis. The axis can be calculated, e. g., based on the intersections of bitangents, which can vary considerably in their accuracy; it is therefore an excellent example to compare the performance of a number of established algorithms on a number of different features and to demonstrate how even a well-known and often-used algorithm like total least squares will fail if the underlying assumptions (iiid-data) are violated; much better alternatives are introduced and an extensive comparison and discussion shows the merit of error propagation for a problem which, in similar form, one can see tackled with unsuitable tools at nearly any computer-vision conference, even today. The comparisons are done on real contour-data derived from real images which previously appeared in publications about the grouping and recognition of SORs.
This thesis ends, as all theses do, with a conclusion and outlook in Chapter 8.
Due to the diverse nature of the underlying problems, ranging from projective ge-ometry to error propagation, from intrinsically two-dimensional problems like the recognition of repeated structure to intrinsically three-dimensional problems like the grouping of box-like and even (partly) free-form objects (surfaces of revolution), there is no separate chapter entitled “literature survey”. Instead you can find a small overview over the then relevant literature in each chapter’s introduction, and then again whenever a direct reference can help to set the work described in context.
The bibliography itself comes in two parts, starting with a list of my own relevant work on page 195 and the bibliography proper on page 197.
Chapter 2
Projective Geometry
. . . experience proves that anyone who has studied geometry is in-finitely quicker to grasp difficult subjects than one who has not.
Plato, The Republic, Book 7, 375 B. C.
18 Projective Transformations
2.1 Introduction
When working in computer vision and image understanding, one of the first things one often seeks to describe is the image formation process, i. e. how are the real world and any specific image of this world related to each other. This connection can be made elegantly by projective geometry.
Projective geometry is much older than computer vision. According to [138] the first systematic treatise on projective geometry was published 1822 by Poncelet in his Trait´e des propri´et´es projectives des figures. Prompted by Felix Klein’s Er-langen programme of 1872 [79] as well as a general interest in invariant theories, projective geometry became rather fashionable among the mathematicians of the late 19th and early 20th century (e. g. [39]). The book that by many in the vision community is considered the standard reference on projective geometry, Algebraic Projective Geometry by J. G. Semple and G. T. Kneebone [138], dates back to 1952.
Only comparatively recent trends in computer vision require a somewhat more in-volved algebra; mostly tensor algebra as it is used in shape from multiple view approaches [59]. However, since this thesis concentrates on single view geometry, only standard projective geometry is used here.
This chapter describes the theory and principles of projective geometry as they apply to this thesis. Starting from 2D projective transformations, the notion of ho-mogeneous coordinates is introduced and several subgroups of the projective group are presented (Section 2.2). This leads naturally to the discussion of different cam-era models in Section 2.3. Points, lines and conics are introduced (Sections 2.4 and 2.5) as well as the crossratio of four collinear points or four coincident lines respectively (Section 2.6). Finally some special transformations (canonical frames in Section 2.7 and “projective symmetry” in Section 2.8) are presented, and an al-ternative representation of the projective plane is introduced: the Gaussian sphere (Section 2.9), which has proven useful for error-propagation purposes or algorithms like the grouping by vanishing points discussed in Section 6. This introduction is naturally a rather brief and incomplete one, the interested reader can find additional information in, e. g., [43, 69, 103, 138, 146].