Baseline and Triangulation Geometry in a Standard Plenoptic Camera

The SPC can be seen as a complex derivative of a stereo vision system. The stereo triangulation concept is presented hereafter to serve as a groundwork.

Figure 2 illustrates a stereoscopic camera setup where sensors are coplanar. The depicted setup may be parameterised by the spacing of the cameras’ axes, denoted as B for baseline, the cameras’ image distance b and the optical centres \(O_L\), \(O_R\) for each camera, respectively. As seen in the diagram, an object point is projected onto both camera sensors indicated by orange dots. With regard to corresponding image centres, the position of the image point in the left camera clearly differs from that in the right. This phenomenon is known as parallax and results in a relative displacement of respective image points from different viewpoints. To measure this displacement, the horizontal disparity \(\Delta x\) is introduced given by \(\Delta x = x_R - x_L\), where \(x_R\) and \(x_L\) denote horizontal distances from each projected image point to the optical image centre. Nowadays, image detectors are composed of discrete photosensitive cells making it possible to locate and measure \(\Delta x\). The disparity computation is a well studied task (Marr and Poggio 1976; Yang et al. 1993; Bobick and Intille 1999) and is often referred to as solving the correspondence problem. Algorithmic solutions to this are applied to a set of points in the image rather than a single one and thus yield a map of \(\Delta x\) values, which indicate the depth of a captured scene.

An object point’s depth distance Z can be directly fetched from parameters in Fig. 2. As highlighted with a dark tone of grey, \(\Delta x\) may represent the base of any acute scalene triangle with b as its height. Another triangle spanned by the base B and height Z is a scaled version of it and shown in light grey. This relationship relies on the method of similar triangles and can be written as an equality of ratiosTo infer the depth distance Z, Eq. (1) may be rearranged toAs seen by these equations, it is feasible to retrieve information about the depth location Z. Likewise, if \(\Delta x\) is constant, it may be obvious that by decreasing the baseline B, the object distance Z shrinks. Given a case where the depth range is located at a far distance, it is thus recommended to aim for a large baseline. Note that this relationship and corresponding mathematical statements only hold for cases where optical axes of \(O_L, O_R\) are aligned in parallel.

Reasonable scenarios exist in which a camera’s optical axis is tilted with respect to the other. In such a case, the principle of similar triangles does not apply in the same manner as in Eq. (1).

Taking the left camera as the orientation reference, the right lens \(O_R\) is seen to be tilted as shown in Fig. 3. In this case, perspective image rectification is commonly employed to correct for non-coplanar stereo vision setups (Burger and Burge 2009). Iocchi (1998) concludes that optical axes intersect in a point \(Z_0\) as both axes lie on the x, z plane if angle rotation occurs around the y-axis, whereas image planes of both cameras are still seen to be parallel. In traditional stereo vision, this yields deviations such that Iocchi’s (1998) method serves as a first-order approximation for small angle rotations in the absence of image processing. As demonstrated in Sect. 3.2, this approach, however, is suitable for our plenoptic triangulation model where imaginary sensor planes of virtual cameras are coplanar, whilst their optical axes may be non-parallel. Let \(\varPhi \) be the rotation angle, then laws of trigonometry allow to putandwhich may be shortened toafter substituting for \(Z_0\). This approximation suffices to estimate the depth Z for small rotation angles \(\varPhi \) in stereoscopic systems without the need of an image rectification.

It has been shown in Adelson and Wang (1992), Ng (2006), Dansereau (2014), Bok et al. (2014) that extracting viewpoints from an SPC can be attained by collecting all pixels sharing the same respective micro image position. To comply with provided notations, a 1-D sub-aperture image \(E_{i}\left[ s_j\right] \) with viewpoint index i is computed withwhere u and c have been omitted in the subscript of \(E_{i}\) since i is a sufficient index for sub-aperture images in the 1-D row. Equation (6) implies that the effective viewpoint resolution equals the number of micro lenses. Figure 6 depicts the reordering process producing 2-D sub-aperture images \(E_{(i,g)}\) by means of index variables \(\left[ s_j, \, t_h\right] \) and \(\left[ u_{c+i} \, , \, v_{c+g}\right] \) for spatial and directional domains, respectively. As can be seen from colour-highlighted pixels, samples at a specific micro image position correspond to the respective viewpoint location in a camera array.

Since raw SPC captures do not naturally feature the \(E_{f_s}\left[ s_j, \, u_{c+i}\right] \) index notation, it is convenient to define an index translation formula, considering the light field photograph to be of two regular sensor dimensions \(\left[ x_k, \, y_l\right] \) as taken with a conventional sensor. In the horizontal dimension indices are converted bywhich means that \(\left[ x_k\right] \) is formed bybearing in mind that M represents the 1-D micro image resolution. Similarly, the vertical index translation may beand thereforeThese definitions comply with Fig. 6 and enable to apply our 4-D light field notation \(\left[ s_j, \, u_{c+i}, \, t_h, \, v_{c+g}\right] \) to conventionally 2-D sampled representations \(\left[ x_k, \, y_l\right] \), where k and l start to count from index 0. To apply the proposed ray model and image process, the captured light field has to be calibrated and rectified such that the centroid of each micro image coincides with the centre of a central pixel. This requires an image interpolation with sub-pixel precision, which was first pointed out by Cho et al. (2013) and confirmed by Dansereau et al. (2013).

In the previous section, it was shown how to render multi-views from SPC photographs by means of the proposed ray model. Because a 4-D plenoptic camera image can be reorganised to a set of multi-view images as if taken with an array of cameras, it is supposed that each of these images has an optical centre of a so-called virtual camera with a distinct location. The localisation of such is, however, not obvious. This problem was first recognised and addressed in publications by our research group (Hahne et al. 2014a, b), but, however, lacked of experimental verification. As a starting point, we deploy ray functions that proved to be viable to pinpoint refocused SPC image planes (Hahne et al. 2016) and further refine the model by finding intersections along the entrance pupil. Once theoretical positions of virtual cameras are derived, we examine in which way the well established concept of stereo triangulation (see Sect. 2) applies to the proposed SPC ray model.

In order to geometrically describe rays in the light field, we first define the height of optical centres \(s_j\) in the MLA bywith \(o = (J-1) /2\) as the index of the central micro lens where J is the overall number of micro lenses in the horizontal direction. Geometrical MIC positions are denoted as \(u_{c,j}\) and can be found by tracing main lens chief rays travelling through the optical centre of each micro lens. This is calculated bywhere \(f_s\) is the micro lens focal length and \(d_{A'}\) is the distance from MLA to exit pupil of the main lens, which is illustrated in Fig. 4b. Micro image sampling positions that lie next to MICs can be acquired by a corresponding multiple i of the pixel pitch \(p_p\) as given byChief ray slopes \(m_{c+i,j}\) that impinge at micro image positions \(u_{c+i,j}\) can be acquired byLet \(b_U\) be the objective’s image distance, then a chief ray’s intersection at the refractive main lens plane \(U_{i,j}\) is given bywhere c has been left out in the subscript of \(U_{i,j}\) as it is a constant and will be omitted in following ray functions for simplicity. The spacing between principal planes of an objective lens will be taken into account at a later stage.

Since the main lens works as a refracting element, chief rays possess different slopes in object space, which can be calculated as followswith a chief ray passing through a point \(F_{i,j}\) along the main lens focal plane F by means of its image side slope \(m_{c+i,j}\) and the main lens focal length \(f_U\). Consequentially, a chief ray slope \(q_{i,j}\) of that beam in object space is given byas it depends on the intersections at refractive main lens plane U, focal plane \(F_U\) and the chief ray’s travelling distance, which is \(f_U\) in this particular case. With reference to preliminary remarks, an object ray’s path may be provided as a linear function \(\widehat{f}_{i,j}\) of the depth z, which is written asAs the name suggests, sub-aperture images are created at the main lens’ aperture. To investigate ray positions at the aperture, it is worth introducing the aperture’s geometrical equivalents to the proposed model, which have not been considered in our previous publications (Hahne et al. 2014a). An obvious attempt would be to locate a baseline \(B_{A'}\) at the exit pupil, which is found bywhere \(m_{c+i,j}\) is obtained from Eq. (14). Practical applications of an image-side baseline \(B_{A'}\) are unclear at this stage.

However, the baseline at the entrance pupil \(A''\) is a much more valuable parameter when determining an object distance via triangulation in an SPC. Figure 7 offers a closer look at our light field ray model by also showing principal planes \(H_{1U}\) and \(H_{2U}\). There, it can be seen that all rays having i in common (e.g. blue rays) geometrically converge to the entrance pupil \(A''\) and diverge from the exit pupil \(A'\). Intersecting chief rays at the entrance pupil can be seen as indicating object-side-related positions of virtual cameras \(A''_{i}\).

The calculation of virtual camera positions \(A''_{i}\) is provided in the following. By taking object space ray functions \(\widehat{f}_{i,j}(z)\) from Eq. (18) for two rays with different j but same i and setting them equal as given bywe can solve for the equation system which yields a distance \(\overline{A''H_{1U}}\) from entrance pupil \(A''\) to object-side principal plane \(H_{1U}\) (see Fig. 7). Recall that the index for the central micro lens \(s_j\) is found by with o as the image centre offset.

The object-side-related position of \(A''_i\) can be acquired byWith this, a baseline \(B_{G}\) that spans from one \(A''_i\) to another by gap G can be obtained as follows

For example, a baseline \(B_1\) ranging from \(A''_{0}\) to \(A''_{1}\) is identical to that from \(A''_{-1}\) to \(A''_{0}\). This relies on the principle that virtual cameras are separated by a consistent width. To apply the triangulation concept, rays are virtually extended towards the image space bywhere \(b_N\) is an arbitrary scalar which can be thought of as a virtual image distance and \(N_{i,j}\) as a spatial position at the virtual image plane of a corresponding sub-aperture. The scalable variable \(b_N\) linearly affects a virtual pixel pitch \(p_N\), which is found bySetting \(b_U=f_U\) aligns optical axes \(z'_i\) of virtual cameras to be parallel to the main optical axis \(z_U\) (see Fig. 7). For all other cases where \(b_U\ne f_U\) (e.g. Fig. 8), the rotation angle \(\varPhi _i\) of a virtual optical axis \(z'_i\) is obtained byThe relative tilt angle \(\varPhi _G\) from one camera to another can be calculated withwhich completes the characterisation of virtual cameras.

Figure 8 visualises chief rays’ paths in the light field when focusing the objective lens such that \(b_U>f_U\). In this case, \(z'_i\) intersects with \(z_U\) at the plane at which the objective lens is focusing. Objects placed at this plane possess a disparity \(\Delta x = 0\) and thus are expected to be located at the same relative 2-D position in each sub-aperture image. As a consequence, objects placed behind the \(\Delta x = 0\) plane expose negative disparity.

Establishing the triangulation in an SPC allows object distances to be retrieved just as in a stereoscopic camera system. On the basis of Eq. (5), a depth distance \(Z_{G,\Delta x}\) of an object with certain disparity \(\Delta x\) is obtained byand can be shortened towhich is only the case where \(b_U=f_U\). One may notice that Eq. (28) is an adapted version of the well-known triangulation equation given in Eq. (2).

Experimentations are conducted with two different micro lens designs, denoted as MLA (I.) and (II.), which can be found in Table 1. Input parameters relevant to the triangulation are \(f_s\) and \(p_m\). Besides this, Table 1 provides the lens thickness \(t_s\), refractive index n, radii of curvature \(R_{s1}\), \(R_{s2}\) and principal plane distance \(\overline{H_{1s}H_{2s}}\). The number of micro lenses in our MLA amounts to 281 \(\times \) 188 for horizontal and vertical dimensions, respectively. These values allow for modelling the micro lenses in an optical design software.

It is well known that the focus ring of today’s objective lenses moves a few lens groups whilst others remain static, which, in consequence, changes the lens system’s cardinal points. To prevent this and simplify the experimental setup, we only shift the plenoptic sensor away from the main lens to vary its image distance \(b_U\) by keeping the focus ring at infinity. In doing so, we assure cardinal points remain at the same relative position. However, the available space in our customised camera constrains the sensor’s shift range to an overall focus distance of \(d_f \approx \) 4 m where \(d_f\) is the distance from the MLA’s front vertex to the plane that the main lens is focused on. For this reason, we examine two focus settings \((d_f \rightarrow \infty ~\text {and}~d_f \approx ~4~\text {m})\) in the experiment. To acquire the main lens image distance \(b_U\), we employ the thin lens equation and solve for \(b_U\) as given bywith \(a_U=d_f - b_U - \overline{H_{1U}H_{2U}}\) as the object distance. After substituting for \(a_U\), however, it can be seen that \(b_U\) is an input and output parameter at the same time, which turns out to be a typical chicken-and-egg case. To treat this problem, we define the initial image distance to be the focal length (\(b_U:=f_U\)) and substitute the resulting \(b_U\) for the input variable afterwards. This procedure is iterated until both values are the same. Objective lenses are denoted as \(f_{193}\), \(f_{90}\) and \(f_{197}\) with index numbers representing focal lengths in millimetres. The lens designs for \(f_{193}\) and \(f_{90}\) were found in Caldwell (2000), Yanagisawa (1990) whilst \(f_{197}\) is obtained experimentally using the technique provided by TRIOPTICS (2015). Table 2 lists calculated image, exit pupil and principal plane distances for the main lenses. It is noteworthy that all parameters are provided with respect to 550 nm wavelength. Precise focal lengths \(f_U\) are found in the image distance column at the infinity focus row.

To verify claims made about SPC triangulation, experiments are conducted as follows. Baselines and tilt angles are estimated based on Eqs. (22) and (26) using parameters given in Tables 1 and 2. Thereof, we compute object distances from Eq. (27) for each disparity and place real objects at the calculated distances. Experimental validation is achieved by comparing predicted baselines with those obtained from disparity measurements. The extraction of a disparity map from an SPC requires at least two sub-aperture images that are obtained using Eq. (6). Disparity maps are calculated by block matching with the Sum of Absolute Differences (SAD) method using an available implementation (Abbeloos 2010, 2012). To measure baselines, Eq. (27) has to be rearranged such thatThis formula can also be written aswhich yields a relative tilt angle \(\varPhi _G\) in radians that can be converted to degrees by multiplication by \(180/\pi \).

Stereo triangulation experiments are conducted such that \(B_4\) and \(B_8\), just as \(\varPhi _4\) and \(\varPhi _8\), are predicted based on main lens \(f_{197}\) and MLA (II.) with \(d_f \rightarrow \infty \) and \(d_f \approx 4\) m focus setting. Real objects were placed at selected depth distances \(Z_{G,\Delta x}\) calculated from this setup.

An exemplary sub-aperture image \(E_{(i,g)}\) with infinity focus setting and related disparity maps is shown in Fig. 10. A sub-pixel precise disparity measurement has been applied to Fig. 10b, d as the action figure lies between integer disparities. It may be obvious that disparities in Fig. 10b, d are nearly identical since both viewpoint pairs are separated by \(G=4\), however placed at different horizontal positions. This justifies the claim that the spacing between adjacent virtual cameras is consistent. Besides, it is also apparent that objects at far distances expose lower disparity values and vice versa. Comparing Fig. 10b, c shows that a successive increase in the baseline \(B_G\) implies a growth in the object’s disparity values, an observation also found in traditional computer stereo vision.

Table 3 lists baseline measurements and corresponding deviations with respect to the predicted baseline. This table is quite revealing in several ways. First, the most striking result is that there is no significant difference between baseline predictions and measurements using the model proposed in this paper. The reason for a 0% deviation is that objects are placed at the centre of predicted depth planes \(Z_{G,\Delta x}\). An experiment conducted with random object positions would yield non-zero errors that do not reflect the model’s accuracy, but rather our SPC’s capability to resolve depth, which depends on MLA and sensor specification. Hence, such an experiment is only meaningful when evaluating the camera’s depth resolution. A more revealing percentage error is obtained by a larger number of disparities, which in turn requires the baseline to be extended. These parameters have been maximised in our experimental setup making it difficult to further refine depth. To obtain quantitative error results, Sect. 4.3 aims to benchmark proposed SPC triangulation with the aid of a simulation tool (Zemax 2011).

A second observation is that our previous methods (Hahne et al. 2014a, b) yield identical baseline estimates, but fail experimental validation exhibiting significantly large errors in the triangulation. This is due to the fact that our previous model ignored pupil positions of the main lens such that virtual cameras were seen to be lined up on its front focal plane instead of its entrance pupil. Baseline estimates calculated according to a definition provided by Jeon et al. (2015) further deviate from our results with \(B_4 = 290.7293\) mm and \(B_8 = 581.4586\) mm. As the authors disregard optical centre positions of the sub-aperture images, it is impossible to obtain distances via triangulation and assess results using percentage errors.

Whenever \(d_f \rightarrow \infty \), virtual camera tilt angles in our model are assumed to be \(\varPhi _G=0^{\circ }\). Accurate baseline measurements inevitably confirm predicted tilt angles as measured baselines would deviate otherwise. To ensure this is the case, a second SPC triangulation experiment is carried out with \(d_f \approx 4\) m, yielding images shown in Fig. 11.

Disparity maps in Fig. 11b, d give further indication that the spacing between adjacent virtual cameras is consistent. Results in Table 4 demonstrate that tilt angle predictions match measurements. It is further shown that virtual cameras are rotated by small angles of less than a degree. Nevertheless, these tilt angles are non-negligible as they are large enough to shift the \(\Delta x=0\) disparity plane from infinity to \(d_f \approx 4\) m, which can be seen in Fig. 11.

Generally, Tables 3 and 4 suggest that the adapted stereo triangulation concept proves to be viable in an SPC without measurable deviations if objects are placed at predicted distances. A maximum baseline is achieved with a short MLA focal length \(f_s\), large micro lens pitch \(p_M\), long main lens focal length \(f_U\) and a sufficiently large entrance pupil diameter.

A baseline approximation of the first-generation Lytro camera may be achieved with the aid of the metadata (*.json file) attached to each light field photograph as it contains information about the micro lens focal length \(f_s=0.025\) mm, pixel pitch \(p_p~\approx ~0.0014\) mm and micro lens pitch \(p_M~\approx ~0.0139\) mm, yielding \(M=9.9286\) samples per micro image. The accommodated zoom lens provides a variable focal length in the range of \(f_U = 6.45\)–51.4 mm (43–341 mm as 35 mm-equivalent) (Ellison 2014). It is unclear whether the source refers to the main lens only or to the entire optical system including the MLA. From this, hypothetical baseline estimates for the first-generation Lytro camera are calculated via Eqs. (20)–(22) and given in Table 5.

Disparity analysis of perspective Lytro images should lead to baseline measures \(B_G\) similar to those of the prediction. However, verification is impossible as the camera’s automatic zoom lens, settings (current principal planes and pupil locations) are undisclosed. Reliable measurements of such require disassembly of the main lens, which is impractical in the case of present-day Lytro cameras as main lenses are unmountable.

To obtain quantitative measures, this section investigates the positioning of a virtual camera array by modelling a plenoptic camera in an optics simulation software (Zemax 2011). Table 6 reveals a comparison of predicted and simulated virtual camera positions just as their baseline \(B_{G}\) and relative tilt angle \(\varPhi _G\). Thereby, the distance from an objective’s front vertex \(V_{1U}\) to entrance pupil \(A''\) is given bybearing in mind that \(\overline{A''H_{1U}}\) is the distance from entrance pupil \(A''\) to object-side principal plane \(H_{1U}\) and \(\overline{V_{1U}H_{1U}}\) separates the front vertex \(V_{1U}\) from its object side principal plane \(H_{1U}\). Simulated \(\overline{V_{1U}A''}\) are obtained by extending ray slopes \(q_{i,j}\) towards the sensor, whilst these virtually elongated rays are seen to ignore lenses and finding the intersection of \(q_{i,j}\) and \(q_{i,j+1}\).

Observations in Table 6 indicate that the baseline grows withgiven that the entrance pupil diameter is large enough to accommodate the baseline. Besides, it has been proven that tilt angle rotations become larger with decreasing \(d_f\). Baselines have been estimated accurately with errors below 0.1% on average, except for one example. The key problem causing the largest error is that MLA (I.) features a shorter focal length \(f_s\) than MLA (II.) which produces steeper light ray slopes \(m_{c+i,j}\) and hence severe aberration effects. Tilt angle errors remain below 0.3% although results deviate by only \(0.001^{\circ }\) for \(f_{90}\) and are even non-existent for \(f_{193}\). However, entrance pupil location errors of about \(\le \)1% are larger than in any other simulated validation. One reason for these inaccuracies is that the entrance pupil \(A''\) is an imaginary vertical plane, which in reality may exhibit a non-linear shape around the optical axis.

An experiment assessing the relationship between disparity \(\Delta x\) and distance \(Z_{G,\Delta x}\) using different objective lenses is presented in Table 7. From this, it can be concluded that denser depth sampling is achieved with larger main lens focal length \(f_U\). Moreover, it is seen that a tilt in virtual cameras yields a negative disparity \(\Delta x\) for objects further away than \(d_f\), which is a phenomenon that also applies to tilted cameras in stereoscopy. The reason why \(d_f \approx Z_{G,\Delta x}\) when \(\Delta x=0\) is that \(Z_{G,\Delta x}\) reflects the separation between ray intersection and entrance pupil \(A''\), which lies nearby the sensor and \(d_f\) is the spacing between ray intersection and MLA’s front vertex. Overall, it can be stated that distance estimates based on the stereo triangulation behave similar to those in geometrical optics with errors of up to ±0.33%.