Proposed refractive magnification function for the analysis of refraction

pc water air paw

Figure 4.4.: Restriction of the validity of the SVP model: The image shows that the single viewpoint model is only applicable to small angles so that according rays meet in the common point pc. Larger angles result in a shift of the refrax paw with a different virtual focal point p.

4.4. Proposed refractive magnification function for the

analysis of refraction

In this section, we present a novel fundamental mathematical function that completely characterises the refractive magnification in the image of a camera with two refractive interfaces being parallel to the camera’s image plane. The equation also incorporates the single interface case and can be easily extended for arbitrary multi interface systems. In contrast to previous literature in which the magnification effects by arbitrary thick ports were only partially described, this proposed function allows the comprehensive understanding of refractive magnification, its dependency to all parameters in the system and its exact calculation. The function also allows a simplified qualitative evaluation of refractive effects, and is an alternative to an exact quantitative calculation requiring a more complicated refractive forward projection, which is not available in a closed form. The proposed function corresponds to an extension for arbitrary thick ports of the radial distortion correction function for single interfaces presented in [68, 71]. In con- trast to a representation of the relation between the undistorted and distorted radial coordinates in the form of a radial distortion function, we prefer the representation of the occurring magnification.

The importance of our novel fundamental magnification function is the ability to describe and calculate numerous properties of an underwater camera with a thick, flat port being parallel to the camera’s image plane (see section 4.4.1, 4.4.3, and 4.4.2).

4.4.1. Newly discovered properties and methods

In addition to known properties, the function also served the authors as a basis to present newly discovered properties and to obtain more accurate calculations of the optical system, such as:

1. Closed-form calculation of the magnifications occurring in the image perceived through a thick, flat and parallel port. The extension for multiple thick layers is straight forward.

2. Increase/decrease in magnification with increasing object distance depending on the port thickness and the port distance

3. The magnification of close objects at the image centre does not only depend on the refractive indices of air and water, but also on the object’s distance and the port’s distance, thickness and index of refraction.

4. Calculation of an optimal port distance for reduced object distance dependency 5. Increased object distance dependency with increasing distance to the characteristic

radial distance

6. Closer objects are less distorted than more distant objects. 7. The magnification increases with growing port thickness.

8. Chromatic aberration is object distance dependent at small object distances and loses its distance dependency at large object distances.

The derivation of these properties and methods is described in more detail in section 4.5.

4.4.2. Proposed extensions of the radial distortion correction

function for arbitrary thick ports

Treibitz et al. [68, 71] described the distance dependent radial distortion correction function for thin ports. Using our refractive magnification function, we extended their function for arbitrary thick ports and obtained the following results:

1. Precise radial distortion correction function for arbitrary thick ports, which are parallel to the image plane (see (4.7))

2. At large object distances the image magnification is identical to the thin flat in- terface case, which allows a simple correction of image distortion and a straight forward integration into the perspective camera model (see (4.8)).

4.4. Proposed refractive magnification function for the analysis of refraction

4.4.3. Existing properties and methods

To date, different effects and properties of flat refractive geometry have been described. The following list summarises the effects and properties, which can be derived from our proposed fundamental refractive magnification function. The detailed derivations are omitted here.

1. In Gaussian optics, objects appear larger as if they were taken with a larger focal length, which is about nw/na times larger than the actual focal length [63]. 2. Distortion/magnification increases with radial distance (pincushion distortion) [58] 3. Increase in magnification with increasing object distance or decreasing port dis-

tance [68, 71]

4. Port distance independence if water is replaced by air [73]. The calibration of the port distance is not possible in air.

5. Object distance independent magnification with a thin port located on the camera’s centre of projection [68, 71]

6. The magnification of objects at large distances is independent of the port’s thick- ness [58], distance [68, 71] and index of refraction.

7. The change in the magnitude of magnification with respect to the port thickness is relatively small compared to the change caused by the radial distance [68, 71] 8. The magnification increases if the index of refraction of water or the index of

refraction of the port increase. It decreases with increasing index of refraction of air.

9. As a result of the wavelength specific image magnification (chromatic aberration), red coloured fringes appear relative to the image centre on the inside and blue coloured fringes on the outside of edges in the image [63, 58].

4.4.4. Derivation of the refractive magnification function

In this section, the proposed refractive magnification function is derived. The magni- fication function describes the ratio between the radial distances r0 and r of the image points of an object point at the distance zp perceived by a perspective camera with and without a parallel port (see Fig. 4.5)

θp d _f t θw θa r r z r na np nw p= (rp, zp) rpw rap pc = (0,0)

Figure 4.5.: Illustration to calculate the refractive magnification function

fm is composed of a refractive back projection of the image pointr0 on an object plane, which is parallel to the camera’s image plane and located at the object distance zp, and a subsequent perspective forward projection. As the complicated refractive forward projection is not required, fm can be solved in a less complex, closed form.

The refractive back-projection is calculated as follows. It starts with a perspective back projection of the image point r0 onto the air-port interface. For the refrax on the air-port interface, we have

rap = r0d

f , (4.2)

where f denotes the focal length of the camera and d the port distance (see Fig. 4.5). Secondly, the relation between the object pointp= (rp, zp) and the refrax rap is defined by

rp =rap+ttanθp+ (zp−t−d) tanθw, (4.3) witht denoting the thickness of the port, andθp andθw representing the refracted angle in the port and the incident angle in water. Both angles are obtained from the refracted angle θa in air using Snells law

nasinθa =npsinθp =nwsinθw, (4.4) withna,np,nw denoting the indices of refraction of air, the port and water. Here, sinθa