Refractive Back-Projection

The goal of refractive back-projection is the computation of the ray w in water from a given image point I, as illustrated on the left side of Figure 5.4. The proposed approach is an alternative to the explicit 3D ray tracing commonly used in the literature (See Section 3.3.1). In the depicted situation, ray tracing would comprise the computation of the direction vectors of the rays a, g and w, as well as the computation of the points of refraction R₁ and R₂. On the contrary, the proposed approach comprises only the computation of two ray directions and one point, as will be described in the following. Requirements. The required quantities are the refractive indices n_a, n_w, n_g, the glass layer thickness t_g= d(S₁, S2), the air layer thickness ta= d(C, S1) and the system axis

s₀ with a normalized direction vector ¯s.

Definitions. The ray w in question can be expressed by:

5 Calibration of Shared Flat Refractive Systems

Figure 5.4: Refractive back-projection. Left: Ray path (a, g and w) from full ray tracing. Right:

Avoid full ray tracing by application of the VOP model to compute the ray w

expressed by Equation5.2.

This comprises a position vector c and a direction vector w. With an according scalar

λ, every real object point O on this ray can be expressed by its position vector o. The

position vector c belongs to the point of intersection of the system axis s₀ and the virtual part of ray w. This point of intersection is a VOP. On the right side of Figure5.4 it is labeled by ˇC. Note that the illustration of the camera and its coordinate system is

omitted for the sake of clarity.

Computation of the Position Vector. As can be seen on the right side of Figure5.4, the setup resembles the one of Section4.3, which has been used to formulate the VOP model for two flat refractive interfaces. The only difference is that the direction of consideration of the ray path is inverted. Here, the ray w in water is considered as the incident ray and its virtual part (dashed) intersects the system axis s₀ at the VOP ˇC.

This VOP ˇC belongs to the real object point C and both can be related by the previously

proposed VOP model.

The first preparatory step is the computation of the normalized direction vector ¯a of the

ray a in air, with respect to the camera coordinate system. With the aid of Equation 2.17, it can be computed from image point I and the known intrinsic camera parameters. Subsequently, the angle of incidence α, which is the angle between the two normalized direction vectors ¯a and ¯s, can be computed straightforward. Under these conditions, the

position vector c can be computed as follows:

• Compute the distance ˇx = d(S1, ˇC) from distance x = ta, with tg, α, na, ng and n_w by application of Equation4.8 under consideration that the direction of the ray path is inverted: replace angle α by angle β and switch n_a and n_w.

• Compute the angle of refraction β according to Snell’s law from the angle of incidence α by Equation 2.20.

5.2 Virtual Object Point Model and Monocular Vision

Figure 5.5: Refractive back-projection: Computation of the direction vector w of ray w.

• Since the normalized direction vector ¯s of the system axis s₀ is known, the position vector c can be determined straightforward by: c = ˇx · ¯s.

Computation of the Direction Vector. The computation of the direction vector w of the ray w in water can be realized by the basic VOP model for a single flat refractive interface from Section 2.3.3. The actual presence of two flat refractive interfaces does not influence this computation.

The ray a, with the normalized direction vector ¯a, and the ray w, with direction vector w, share the same plane of refraction Π. Only the known, normalized direction vector

¯

a is necessary for the computation of w. Based on the basic VOP model for a single

flat refractive interface from Section 2.3.3, the exploited relations can be constructed from the setup on the left side of Figure 5.5. This construction can be seen on the right side of Figure 5.5. Note that the glass layer can be ignored and that the ray w is represented by its direction vector w. Let vector w point to an arbitrary real object point A₃, which is chosen in a way that the normalized direction vector ¯a points exactly

to a VOP ˇA₃. With this, all the prerequisites for the application of the VOP model are fulfilled. Therefore, the computation of w consists of the following steps:

• Compute the distance ˇx = d(A₁, ˇA₃) = d(R, A₂) by orthogonal projection of the normalized vector ¯a onto the normalized vector ¯s, with the aid of Equation 2.7. • Compute the distance x = d(A₁, A₃) from the known angle of incidence α, distance

ˇ

x and the refractive indices n_a and n_w with the aid of an appropriately rearranged

Equation 2.26.

• Since the normalized direction vector ¯s of the system axis s₀is known, the direction vector w can be determined straightforward by: w = ¯a + (x − ˇx)¯s

5 Calibration of Shared Flat Refractive Systems