3.3 Stripe Scanners

Fig. 3.4 Cross-section of a pinhole camera. The 3D points Q1 and Q2 are projected into the image plane as points q1 and q2 respectively. Finally, d is the distance between the image plane and the projection center. Figure courtesy of NRC Canada

3.3.1 Camera Model

The simplest mathematical model that can be used to represent a camera is the pinhole model. A pinhole camera can be assembled using a box in which a small hole (i.e. the aperture) is made on one side and a sheet of photosensitive paper is placed on the opposite side. Figure 3.4 is an illustration of a pinhole camera. The pinhole camera has a very small aperture so it requires long integration times; thus, machine vision applications use cameras with lenses which collect more light and hence require shorter integration times. Nevertheless, for many applications, the pinhole model is a valid approximation of a camera. In this mathematical model, the aperture and the photo-sensitive surface of the pinhole are represented respectively by the center of projection and the image plane. The center of projection is the origin of the camera coordinate system and the optical axis coincides with the Z-axis of the camera. Moreover, the optical axis is perpendicular to the image plane and the intersection of the optical axis and the image plane is the principal point (image center). Note that when approximating a camera with the pinhole model, the geometric distortions of the image introduced by the optical components of an actual camera are not taken into account. Geometric distortions will be discussed in Sect. 3.5. Other limitations of this model are described in Sect. 3.8.

A 3D point [Xc , Yc , Zc ]T in the camera reference frame can be transformed into pixel coordinates [x, y]T by first projecting [Xc , Yc , Zc ]T onto the normalized camera frame using [x , y ]T = [Xc /Zc , Yc /Zc ]T . This normalized frame corresponds to the 3D point being projected onto a conceptual imaging plane at a distance of one unit from the camera center. The pixel coordinates can be obtained from the normalized coordinates as

sy

where sx and sy are the dimensions of the sensor in millimeters divided by the number of pixels along the X and Y -axis respectively. Moreover, d is the distance in millimeters between the aperture and the sensor chip and [ox , oy ]T is the position

104 M.-A. Drouin and J.-A. Beraldin

in pixels of the principal point (image center) in the image. The parameters sx , sy , d, ox and oy are the intrinsic parameters of the camera. Note that the camera model used in this chapter is similar to the one used in the previous chapter, with one minor difference: f mx and f my are replaced by d/sx and d/sy respectively. This change

focus relative to the camera center (in general, d and f are not equal, with d often being slightly larger). This will be discussed further in Sect. 3.8. Note that intrinsic camera parameters can be determined using the method described in Chap. 2.

The extrinsic parameters of the camera must be defined in order to locate the position and orientation of the camera in the world coordinate system. This requires three parameters for the rotation and three parameters for the translation. (Note that in many design situations, we can define the world coordinate system such that it coincides with the camera coordinate system. However, we still would need to estimate the pose of the light projection system, which can be viewed as an inverse camera, within this frame. Also, many active 3D imaging systems use multiple cameras, which can reduce the area of the ‘missing parts’ caused by self occlusion.)

(3.9)

where θx , θy and θz are the rotation angles around the X, Y , and Z axis and translation is represented by a vector Tc = [Tx , Ty , Tz]T . Note that the rotation matrix Rc

Then, the point Qc is projected onto the normalized camera frame [x , y ]T =

Moreover, one may verify using Eq. (3.10) that a point in the camera reference frame can be transformed to a point in the world reference frame by using

3.3.2 Sheet-of-Light Projector Model

In the projective-geometry framework of the pinhole camera, a digital projector can be viewed as an inverse camera and both share the same parameterization. Similarly, a sheet-of-light projection system can be geometrically modeled as a pinhole camera with a single infinitely thin column of pixels. Although this parameterization is a major simplification of the physical system, it allows the presentation of the basic concepts of a sheet-of-light scanner using the same two-view geometry that was presented in Chap. 2. This column of pixels can be back-projected as a plane in 3D and the projection center of this simplified model acts as the laser source (see Sect. 3.8.5).

In order to acquire a complete range image, the laser source is rotated around a rotation axis by an angle α. The rotation center is located at Tα and Rα is the corresponding rotation matrix.2 For a given α, a point Qα in the laser coordinate frame can be transformed into a point Qw in the world coordinate frame using

In a practical implementation, a cylindrical lens can be used to generate a laser plane and optical components such as a mirror are used to change the laser-plane orientation.

3.3.3 Triangulation for Stripe Scanners

Triangulation for stripe scanners essentially involves intersecting the back-projected ray associated with a camera pixel with the sheet of light, projected at some angle α. Consider a stripe scanner, where projector coordinates are subscripted with 1 and camera coordinates are subscripted with 2. Suppose that an unknown 3D scene point, Qw , is illuminated by the laser for a given value of α and this point is imaged in the scanner camera at known pixel coordinates [x2, y2]T . The normalized camera point [x2, y2]T can be computed from [x2, y2]T and the intrinsic camera parameters

2The rotation matrix representing a rotation of θ around an axis [a, b, c]T of unit magnitude is