Persons used to the Euclidean plane generally find it difficult to envisage the projective plane with its ideal points at infinity, but without the usual invariance of angles and length (or at least ratio of length). So it is only understandable that other models have been proposed.
2.9.1 The Ray-Space Model
Perhaps the most widespread model is that of a ray space, a space of coincident rays embedded into a three-dimensional spaceIR3, as described in e. g. [103]. In this space each ray — all rays emanate from a common origin — represents a projective point. Only the direction of the ray matters in this model. In projective space the crossproduct of two points defines a line, see Section 2.4. Consequently in ray space a line is represented by the crossproduct of two rays — a plane through the
54 The Gaussian Sphere
Figure 2.18:Ray-space model. Each ray corresponds to a point in the image plane. Two rays span a plane, two planes define a ray. Ideal points correspond to a ray parallel to the image plane.
Figure 2.19: Gaussian sphere model.
2.9.2 The Gaussian Sphere Model 55
origin spanned by the two rays. Conversely, the crossproduct of two planes is the ray common to both planes, representing a projective point. This is illustrated in Figure 2.18.
The process of image formation is modelled as the intersection of all theses rays and planes with a planenot through the origin (note the similarity between this model and the viewer centred camera model in Section 2.3). Consequently, ideal points with respect to this image plane are represented by rays parallel to the image plane, while a plane through the origin and parallel to the image plane represents the ideal line. It is easy to see from this model that the distinction between ideal points and other points is really quite arbitrary, since the image plane can be chosen randomly. The same model can be used to describe the mapping from one plane onto a second plane, where the origin of the rays is the centre of projection, and it is possible to model arbitrary relationships between two planes by a composition of rotations and anisotropic scaling inIR3, compare [103].
2.9.2 The Gaussian Sphere Model
A slightly different model, but based on the one above, is the model of a Gaussian or unit sphere. A projective point corresponds to the point where a line from the sphere’s origin to the projective point intersects the sphere, and a projective line corresponds to the great circle that is the intersection between the sphere and a plane through the sphere’s origin and the projective line (compare Figure 2.19).
Note that any line through the origin will intersect the sphere at 2 points on opposite sides of the sphere. It is therefore customary to avoid this ambiguity by considering only a semi-sphere.
It is obvious that the ray-space model can easily be converted into the Gaussian sphere model by calculating the intersections between rays and planes on the one side and the sphere on the other. Easier still, if a ray is expressed ask(x, y, z)T, the corresponding point on the Gaussian sphere is simply its normalisation into a unit-vector, √ 1
x2+y2+z2(x, y, z)T.
2.9.3 Calibrated Cameras and Gaussian Sphere
The Gaussian sphere model has some particularly convenient features when deal-ing with calibrated cameras. It is then possible to calculate coordinates on the
56 The Gaussian Sphere
Gaussian sphere (x, y, z)Tfrom image coordinates (X, Y)Tas
x y z
= 1 pX2+Y2+f2
X Y f
(2.62)
wheref is the distance between the centre of projection and the image plane. It is often called the camera’s focal length, although this is strictly only true for a camera focused at a point at infinity, compare the discussion on Page 30. The beauty of this construction is that directions which are perpendicular in reality will also be perpendicular on the Gaussian sphere (compare the rays pointing to the vanishing points in Figure 2.19), although they arenot perpendicular in the image5. Conversely, assuming that a sufficient number of directions in the image are known to be perpendicular in reality, this can then be used to calibrate an unknown camera, compare Section 6.3.2. In addition, Kanatani [69] showed that using the model in Equation (2.62), which he calledN-vectors6, has several advantages with respect to numerical computations as well as error distribution, compare also Section 4.
5The same is of course also true for the ray-space model, into which the Gaussian sphere model can be transformed.
6Nstanding for normalised.
Chapter 3
Probability and Statistics
The most may err as grossly as the few.
John Dryden, Absalom and Achitophel, 1631–1700
58 Introduction
3.1 Introduction
Measurements in any discipline are generally encumbered with measurement errors.
This is particularly true for image measurements, where a less than ideal imaging-process is followed by a discretisation of the image. It is, on the other hand, a reasonable assumption that knowledge about the accuracy of our measurements is essential when decisions based on these measurements are required. Systems which ignore this are at best cumbersome, requiring the user to fine-tune a generally high number of sometimes obscure parameters; at worst they will often simply fail.
Although this chapter covers the general aspects of statistical properties and error propagation, I will use examples from computer vision throughout this chapter.
Virtually all image measurements boil down to measuring edgel positions, possi-bly with subpixel accuracy. Section 3.2 gives a short introduction into the kinds of errors customarily encountered when dealing with any measurements, as well as some basic concepts used in statistics. The edgel positions are then used to con-struct higher order con-structures — contours, line segments, conics, and ever more complex configurations. Section 3.3 describes how the measurement error in im-age coordinates — or any random variable — is propagated into derived quantities.
One of the standard tasks in computer vision is to decide whether some structure derived from image measurements conforms to a given model. Section 3.4 explains how confidence tests, and in particular theχ2-test can be used as a decision mak-ing tool. Section 3.5 finally describes some common probability distributions on the sphere; this is applicable to angles and other measurements with only finite support.
Much of what is said in this chapter can be found e. g. in [29, 100], or [43, pp. 151–
164]. Books on photogrammetry [144]1 or [49] can be another rich source of in-formation and inspiration for someone working in computer vision, in particular where error propagation is concerned. An introduction into confidence testing can be found in any textbook on statistics, examples are [81, 145]. Books concerned with statistics on directional data in contrast are much harder to find, the reader is referred to [95, 156].
1Note, however, that at least the 4th edition of the Manual of Photogrammetry contains several gross errors.
Basic Concepts in Statistics 59