Apparent Depth

When an observer is looking at an object submerged in water, it seems to be closer to him as it actually is and hence seems to be bigger. This effect occurs due to refraction and implies, that a virtual image of the real object is seen. The question that arises is

2 Basics

Figure 2.7: Apparent depth in the case of a perpendicular incident ray.

where this virtual image is located. This location is called apparent depth [TMB12]. The following considerations are concerning point objects, single flat refractive interfaces and monocular viewing.

Perpendicularity of Incident Rays

The case that is most commonly explained in physics textbooks is an observer at a viewpoint V with a viewing direction perpendicular to a flat refractive interface Φ

between two media like air and water, as illustrated in Figure 2.7. The light ray is emitted from a point object O. The incident ray in this case is the ray w in water and the refracted ray is the ray a in air.

According to Tipler et al. [TMB12], an equation for the refraction at a single interface can be set up by utilizing Snell’s law and the small-angle approximation (sin α = α) by:

n_w x + n_a ˇ x = n_a− n_w r , (2.24)

with x = d(A, O) being the object distance, ˇx = d(A, ˇO) being the image distance and r being the curvature radius of the interface. Since the interface is flat, the curvature

radius amounts to r = ∞. As the incident ray coincides with the interface normal, the incidence angle is zero degrees and there is no refraction. After reorganization we end up with:

ˇ

x = −na

nw

· x. (2.25)

Since ˇx is negative, the image is virtual and on the same side of the refractive interface

Π as the object. This is what most physics textbooks teach for the case of an incidence angle of zero degrees. As can be seen, Equation2.25is independent of the distance to the viewpoint V of the observer. Hence, V could be situated anywhere on the ray a in air. In the remainder of this thesis, the point object O will be called real object point and ˇO will be called VOP.

2.3 Physics

Non-Perpendicularity of Incident Rays

Equation 2.25can also be used as an approximation for small deviations of the angle of incidence from zero degrees. Larger angles of incidence are most commonly not considered explicitly in physics textbooks. Since these perpendicular incident rays do not represent the more general case that this thesis is confronted with, some further research results for non-perpendicular rays need to be considered in the following. According to Tipler et al. [TMB12], Equation 2.24will not result in the correct location for non-perpendicular rays, since approximate perpendicularity is a prerequisite for its application. In the case of non-perpendicular rays, the apparent depth of an object is a controversial topic in physics.

Bartlett et al. [BLJ84] propose that the image of a point object, submerged in a medium with refractive index n₂ (here: water with n_w), from a viewpoint in a medium with refractive index n₁ (here: air with n_a), is in general astigmatic. This means that two virtual images can be formed, which do not coincide except for an angle of incidence of zero degrees. The schematic derivation of both locations of the virtual image can be seen in Figure 2.8. In the observations of Bartlett et al. [BLJ84], the ray w in water is the central ray of a cone representing a bundle of rays, which are emitted from a real object point O. The derivation of the locations is done by a limit value observation

in two planes. According to the definition of astigmatism [TMB12], these planes are perpendicular to each other and are called the meridional plane Ψ_m and the sagittal plane Ψ_s, respectively. For the application of the limit value observation, a second ray in each of these planes (green ray in Ψ_s and red ray in Ψ_m) is chosen from the cone in Figure2.8a (the cone’s size is exaggerated for visualization). The limit value observation

is done for an infinitesimal radius of the cone, as is illustrated in Figure 2.8 b − d. The

final results are the two possible locations of the virtual image at the VOPs ˇO_m and ˇO_s

in Figure 2.8d. These VOPs are the points of intersection of the virtual extensions of

the rays into water (dashed). Due to occlusions, the two rays in the sagittal plane can only be seen in the top views of Figure 2.8 b − d and the two rays in the meridional

plane in the side views. In the sagittal case, the planes of refraction of both ray paths obviously differ. However, both of them have to intersect in the interface normal s, which is passing through the real object point O. Hence, the virtual extensions of the rays have to intersect in the interface normal s and the virtual image is located at the sagittal

VOP ˇOs. The limit value observation is just performed formally. The meridional plane

Ψ_m and the plane of refraction Π, which belongs to the ray w, are identical. Therefore, the location of the meridional VOP ˇO_m has to be situated in this plane as well. It is located higher than the real object point O and closer to the observer. The viewpoint of the observer can be located arbitrarily on the ray a in air. As can be seen, both locations are situated on the backwards extended ray a.

The apparent depth of an object is still a controversial topic in physics. This is shown, for example, by the recent work of Quick et al. [QGP15]. The authors revise the findings of Bartlett et al. [BLJ84] and conclude in favor of the sagittal VOP ˇO_s.

2 Basics

Figure 2.8: Derivation process of the VOP locations in the case of a non-perpendicular incident

ray.

Model for a Single Flat Refractive Interface

As will be seen in Chapter 4, the physical foundation of the VOPs is not of primary importance for the developments in this thesis. Therefore both locations represent valid possibilities. Since the location of the VOP ˇOm will not be utilized for the approaches

to be developed in this thesis, as will be justified in Section 4, only the geometrical derivation from the previous section is presented. The interested reader may be referred

to [BLJ84] or [QGP15] for a more detailed derivation. In contrast, the location of the

VOP ˇO_s will be used in the following model for a single flat refractive interface.

The relation between the location of the VOP ˇO = ˇOs and the real object point O in

Figure2.9 can be derived from Snell’s law and trigonometry in right-angled triangles

[BLJ84]. The path of the light is inverted once again and with that, the relation can be

expressed by: ˇ x x = cos α r _n w na 2 − sin2α , (2.26)

with ˇx = d(A, ˇO) being the distance to the VOP, x = d(A, O) being the distance to the

real object point and α being the angle of incidence. Note that this is also valid for media that are different from water and air. Therefore, the refractive indices n_a and

n_w have to be replaced accordingly. The substitution of an angle of incidence that is equal to zero degrees into this equation and some rearranging leads to Equation2.25, which represents the apparent depth in the case of a ray that hits the refractive interface perpendicularly. Note that the negative algebraic sign, which is used from the physics perspective to make clear that a virtual image is produced, is omitted. Equation2.26is independent of the distance to the viewpoint of the observer. Hence, this viewpoint can be situated anywhere on the ray a.