What’s the best lens for stereo? An exploration of lens models and theirstereoscopic depth performance
Leonid Keselman
Stanford University
leonidk@stanford.edu

Abstract
We introduce the use of various projection functions in
the analysis of stereoscopic depth sensors. Through thesemethods, we are able to design stereo systems which are notbound by traditional quadratic depth error. Additionally, wedemonstrate how existing correspondence algorithms canbe modified to handle these lens designs. In addition, we
can construct lens projection models which are more suited
to natural lens designs, and build stereo systems with minimal re-sampling errors. This also allows us to constructwide-angler stereoscopic systems than previously possible,
without significant re-sampling or sacrificing accuracy.

1. Introduction
Stereoscopic correspondence methods are at the heart ofmany practical depth sensors. This is because they’re practical and simple to build, requiring only a rigid baseline andtwo sensors. Their use is widespread, from dental toothscanners, to consumer electronics and advanced driver assist systems [7].
All of these systems, to the best of our knowledge, arebuilt on an assumption that the lens used is a rectilinear
lens. Rectilinear lenses create a projection model like thatof a pinhole camera; characteristic of such a lens is thatstraight lines are always straight. This maps to humans own
physical perception of the world. This is also known as perspective projection, and we will discuss some of its shortcomings, as applied to stereo, in this paper. The goal of thispaper is to ask and analyze a simple question, if one could
design any lens system possible, how could you fully utilize
it and what would the impact be?

1.1. Stereoscopic depth systems
To generate valid stereo correspondence, we must first
setup epipolar geometry. In this paper, we will use the convention of mapping the epipole at infinity, with the epipolarlines parallel to the horizontal axis of the imagers. While

Figure 1. Two dimensional overview of our coordinate system.
The field-of-view of each camera is identical and equal to θ. The
spacing between the cameras, or baseline, is B. The projection
function of the system is that which maps a 3D point (X,Y,Z) to a
position on the sensor (using U and V as coordinate values)

exist methods exist for searching along other configurationsof epipolar geometry, this paper will not explore them. Additionally, we assume that both camera use the same exactly
lens model.
Additionally, the creation of valid depth correspondence
requires that a lens projection equation be fully captured.
this is because real lenses typically have some non-designed
distortions. This is true for traditional rectilinear lenses,
fisheye lenses, and likely any of the other lens models presented here. This distortion from a non-ideal lens is first
captured and corrected for in a lens distortion step, before
the lens intrinsics are estimated. While this is sometimes

a two-step process, many modern implementations [4] estimate both distortion parameters and the camera matrix simultaneously through the use of a Levenberg–Marquardt
solver. This process, of capturing the full camera projection model, is required to create a consistent and known
mapping from correspondences on the image plane to correspondences in the real world. And holds for all lens models presented here, although the camera model solver would
have to be modified accordingly.
The model of projection most often used is that of arectilinear lens, with a Brown-Conrady distortion model.
While our paper discusses other lenses, the existing methods [24, 4] of using nonlinear optimization to capture a reallens distortion model can be trivially augmented to handle
our proposed mechanisms and methods. For a general introduction to the basic geometries and methods, we recommend Szeliki’s book [21].

1.2. Lens Projections
This work proposes that lenses are analyzed strictly with
their projection models. This allows us to perform straightforward analysis of the stereoscopic depth errors, along withbuilding algorithms that perform directly depth matchingdirectly on the projected images. Typical optical literature
characterizes lenses via an angular specification, while thisanalysis is easier in Cartesian coordinates. A summary table of some common projections and their characteristics
is available in table 1. A straightforward discussion of thebenefits and types of common lens projections is readily
available [3], from where the angular forms of these projections is obtained. The Cartesian forms were derived forthe table on page 3.
1.2.1

Rectilinear lenses

A traditional lens is based on the pinhole model of light.
It’s the projection that would be given with a pinhole lens.
It is also know as rectilinear or perspective projection. It’s
use is common because it preserves straight lines remaining straight. Additionally, it’s sampling is constant across aplane positioned in front of the camera, allowing a constantsized window to across the entire frame, as
x=

1
·u·z
f

x
u=f·
z
f
∂u
=
∂x
z
This is easy to see, as changes in real world position (x),have no dependence on position in the frame (u), only on
the distance of the object and the focal length of the lens.
Hence, for fronto-parallel objects, a constant sized window

Figure 2. A figure taken from [3], showing the behavior of various
lens projection types when given a fixed focal length.

would provide consistent matching over the entire frame,
provided the focal lengths and distances of the images arematched. However, as described below, it has some very
natural shortcomings.
1.2.2

Shortcomings of traditional lenses

Building very wide angle lenses is impractical under pinhole projection. This is for two reasons: first, the numberof pixels required for any object grows as tan(θ), whereθ is the off-axis angle of the optical ray. This trends
to infinity as pinhole imagers try to cover a hemisphere(180 degrees)1 . Second, due to the physical properties ofsuch a sampling function, there is strong natural vignetting.
Specifically, perspective projection imagers exhibit cos4 (θ)[2] light loss. In fact, this rule of thumb is often an underestimate due to the transmission properties of the optics involved; for details, see the cited paper.
1.2.3

Fish-eye Lenses

The common alternative lens model to perspective projection is a fish-eye projection. These have recently taken off
for use in security, automotive, and consumer use. Unfortunately, this is a catch-all term for a variety of different
lens projection models. We pick two to analyze here, the1 Rectilinear lenses project as u = tan(θ) which naturally has x → ∞as θ → π2

Projection
Rectilinearf-θAngular Formula
u = f · tan(θ)
u=f ·θ

Cartesian Formula
u = f · xz
u = f · tan( xz )Orthographic

u = f · sin(θ)

u = f · √x2x+z2

Cubic

u = f · tan(θ)3

u = f · ( xz )3

Comments
Also known as perspective or pinhole projection
A common type of fish-eye projection, also
known as equidistant projection
A somewhat rarer type of fish-eye projection, but
one that exhibits no natural vignetting.
A lens model that has no practical examples that
we know of, but has interesting theoretical properties described below.

Table 1. A table of the lens projections used in paper and their various forms

Although her work didn’t focus on stereo systems, it discussed the concept that rectilinear projections have notable
shortcomings for certain computer vision applications.

Figure 3. A picture of a real anamorphic lens with an oval aperture. On the right there is a demonstration of the stretching effectexhibited by the lens. These images and further information isavailable [22]f-θ model and the orthographic model, which are both incommon use. Additional information about the variationsof fish-eye projection is available in [3], and their diagram
is repeated in figure 2.

1.3. Anamorphic Lenses
Another interesting property of lens projections is thatlenses often use spherical lens components, which create
circular projections, and can be characterized by the off-axis
angle of interest. While this simplifies lens design, there exist alternative lenses called anamorphic lenses [11]. These
lenses have asymmetric behavior, typically stretching theimage down only a single axis, while leaving the other axis
undisturbed. This is used in the film industry to fit wide aspect ratio images onto a more square 35mm film. They’recommonly used in cinema for these reasons and are readily
available for purchase. This can be seen in figure 3.

2. Related WorkIn some ways, this work is motivated by Fleck’s article
on the use of various lens projections in computer vision [5].

There have been many existing methods on expanding
stereoscopic correspondence algorithms to handle non ideal
lenses. Many authors build on wide angle lenses, distorting
the imagers back to a linear projection for stereo matching[8] [6]. Other authors [18] [12] have pursued using fish
eye lenses, which are a form of nonlinear lens, but did somostly through the use of features which were invariant under their chosen lens projection. The shortcomings of using such invariant features are that these methods produced
sparse depth maps.
Abraham and Förstner developed a method setting up
valid epipolar geometry for using panoramic and wide angle lenses [1]. They analyze various model properties andgenerate a constraint under which epipoles at infinity can bepreserved. However, while their work would correctly discover nonlinear lens distortion, they do not explore densematching on their images. Without the restriction of rectilinear sampling down the matching axis, their model does
not allow for use of standard correlation algorithms, instead
requiring resampling at every disparity candidate. However,Roy et al. demonstrate the value of a cylindrical model likethat proposed here [16]. However, they do not analyze its
properties, use dense matching algorithms on such models,
nor do they apply their model to anything but synthetic data.
Li [14] [15] developed methods of using nonlinear lensfor generating depth, including calibration and search, but
the method only applied to only spherical projections. Additionally, their use of nonlinearities was limited to only theaxis perpendicular to the epipolar line.
However, since almost all of these methods undistortback to a rectilinear projection, there is no analysis of how
to utilize and design lens projection geometries that provide
favorable properties for stereoscopic imagers — including
improved depth invariance at the expense non-uniform accuracy across the image.

3. Model for Lens Projections
3.1. Stereoscopic System
To setup the principles of a matching stereo system, we
define the coordinate system defined in 1. There is a baseline, a focal length, and a projection function from the realworld to the image.
When discussing various projection geometries, there is
first a choice of parametrization. For an overview of various parametrization and models, please see [19]. While optics papers general prefer an angular parametrization (where
fish-eye lenses are modeled as as f-θ lenses, with a linear
mapping between angle and sensor position), computer vision literature typically prefers a parametrization of Cartesian coordinates. While we’ve explored both parametrization, we’ve found that the Cartesian (on image plane) sampling of of stereoscopic correspondence makes the Cartesian parametrization significantly simpler. For an example
of various projection functions in both forms, see figure 1.

3.2. Stereo geometry with projection functions
To parametrize the equations of accuracy and correspondence, we can setup a general equation to define stereoscopic correspondence between two imagers as follows∂z, and substitute
Additionally, we can take the derivative, ∂din our relationship between z and d to generate an error relationship. Traditionally we take the error relative to a single
pixel disparity error (∂d = 1).

z=

f ·B
d

f ·B
∂z
= − 2 · ∂d
∂d
d
∂z
f ·B
= − f ·B · ∂d∂d( z )2
∂z
z2
=−
· ∂d
∂d
f ·B

(4)

3.4. Fish-eye Lens Derivation
The derivations in the rectilinear section can be repeatedto generate results for stereoscopic depth systems with fisheye lenses. The result for typical f-θ lenses is straightforward, although it’s very complicated for orthographic
lenses. However, both types of fish-eyes exhibit much lessvignetting than rectilinear lenses and support imaging anentire hemisphere, so they’re both adequate to address ourissues with rectilinear lenses. Results are in table 2.

3.5. Cubic Lens Derivationx(u) = x(u − d) + B(1)

where x(u) is the projection function from the imager coordinate (u) to the real world (x), z is the distance from theplane of the imagers, B is the baseline, and d is the disparity. This is a two-dimensional parametrization, where theaxis perpendicular to the epipolar lines is ignored. With thisequation and the projection forms in table 1, it’s easy to derive various properties of the stereo system, such as depth
error, and mapping from disparity to depth.

3.3. Rectilinear Lens Derivation

∂z
3z 2
=−
∂d
Bf 3

For a traditional rectilinear projection, we’d see
x(u) =

1
·u·z
f

(2)

where f is the effective focal length. This equation can thenbe solved to generate the relationship between disparity and
depth
1
1
· u · z = · (u − d) · z + Bffu = (u − d) +
z=

f ·B
d

In the scope of this project, we also explored a variety
of other lens projection functions, to see if we can solve aninteresting issue in stereoscopic depth systems, their depth
error function. Namely, in a traditional rectilinear system,
depth error is quadratic with distance, as seen in equation
4. We looked for optics which used extreme distortion tocounteract these properties. In our exploration, we foundone, namely cubic projections. These create extremely barrel distorted images in the center of the image, but theirdepth error is

Bf
z
(3)



Bf 3
− u3
z

 23

∂d

If we simplify the expression, by removing the dependence
on the focal length and baseline, which work in tandem to
set a coefficient of proportionality, we can better see that theerrors are
( 4
3
if Bf
z3,
∂z
z ≫u
(5)
∝
Bf
2
2
3
∂d
z · u , if u ≫ zWhat this means is that for the center of the image, when u
is small, we have sub quadratic depth error. Outside that, wehave traditional quadratic depth error, which grows quadratically worse towards the edges. To the best of our knowledge, this is the first known result of a stereo system getting
sub-quadratic depth error, even in theory.

Equation

World-to-Image

Image-to-World

Depth

Depth Error

Rectilinear

u = f · tan(θ)

z uf

Bf
d

z
− Bf

B
1
tan ( uf )+tan ( f (d−u))

1
− Bf

fθ

u=f ·θ

z tan
1

Cubic
u = (f · tan(θ)) 3
Orthographic u = f · sin(θ)

3

 
u
f

z uf 3z √ 2uf −u2

Bf 3
u3 +(d−u)3Omitted

2



2



z + B − z tan

2
Bf 3
3z 2
3 3
− Bf
−
u
3
z
OmittedTable 2. Comparison of various projection functions. Due to the fact that spatial accuracy and depth depend on the derivatives of the
projection function, the rectilinear projection is the only one with a constant derivative, and hence the only one with no dependence on
where on the imager projection is taking place. These results were all verified using a computer algebra system [20]. The orthographic
results are very challenging to derive as they’re not defined everywhere, and the results are omitted for clarity, although the included script
can provide the results.

Figure 4. A simulation of the depth errors at given distances with arectilinear lens, a fish-eye lens, and a cubic lens. The sub-quadratic
performance of the cubic imager can be seen, along with the position invariance of the rectilinear lens.

3.6. Conclusion: Projections of Interest
Of these various lens projections, there are two promising ones. First, the cubic lens projection is interesting because of it’s guarantee of superior depth accuracy in the
middle of the image. However, as results in the next section show, these lenses require extreme magnification in the
middle of the frame, and may not be feasible to build Second, it seems that there are minimal accuracy advantages tousing other lens projection. This is demonstrated in figure 4.
This suggests that the reasons to use a different lens shouldcome only from the angle-of-view and noise benefits, andnot for any accuracy reason.

4. Simulation Results
In general, we will demonstrate the results of this workon anamorphic frames, with non-rectilinear distortion down
only one axes. The methods for dealing with each case separately can then be combined for results that work for symmetric lenses. Constraining distortion to one axis allows fora simplification of the analysis. We will briefly discuss what
it means to put the anamorphic axis in either direction.
When the axis of nonlinear distortion is perpendicular to
the epipolar line, there is no change in the projection function as defined earlier. This allows for predictable depth
accuracy. However, it requires that we change traditional

matching algorithms to search along a curved line insteadof down a scan-line. We also need to take care to use thecorrect projection function when projecting a stereo disparity into a real-world coordinate (using the inverse of the
projection equation chosen). While this causes variation in
spatial resolution of image lens across the image, it is one
simply way to create wide-angle stereo imagers. By mounting anamorphic lenses to both cameras, perpendicular to theepipolar line, we can create wide-angle stereoscopic systems which don’t suffer from heavy vignetting or perspective distortion. The limitation of this method is that the resulting images have very asymmetric aspect ratios (being
much ”taller” than ”wide”).
When the axis of nonlinear distortion is parallel to theepipolar line, this affects the accuracy and matching algorithms required to generate stereoscopic depth. Matchingalgorithms then need to be modified to use variable sized
windows. Additionally, algorithms must use the correction
functions in table 2 to map back to consistent disparities.

4.1. Data source and algorithms
For this set of simulation, we will simply distort highresolution stereo images from a a standard dataset [17].
This allows us to have a ground truth and use the extra resolution from the image to compute arbitrary sampling functions. In addition, we implemented basic stereoscopic correspondence algorithms with sub-pixel resolution
from scratch. Most of the results are with sum-of-absolutedifferences block matching, but we also tried Sobel + BM
[13] and Census [23]. These latter algorithms are important as they’re robust to noise and can be used to validate
the best-case impact of photon noise. Additionally, we implement a standard left-right check [10] to remove noisy
matches. All algorithms were implemented in C++, withthe use of no external libraries (except for image loading),
and tested on both Windows and Linux.
The modification of the algorithms is fairly straightforward. The easiest method is simply to adjust the location of 2 ufRectilinear

Horizontal Anamorphic

Vertical Anamorphic

Left Image

Computed Depth

Left
Image

Rectilinear

Depth
Estimate

GroundTruth

Figure 5. Example results for cubic anamorphic lenses from theMiddlebury stereo dataset [17]. These images were distorted at thesame time they were sub-sampled, to produce comparable results.
The first row are left images from left/right pairs, while the middle row is results from an implemented depth matching algorithm.
The last row are ground truth results.

searches based on where they’d be if the image were rectilinear. This can be done with equations from table 1. With
the anamorphic assumption, this means that images from X(or horizontal) (or parallel to epipolar lines) distorted images require widening and shrinking the window. In ourcase, we simply use a fixed sized window on the smaller of
the two windows and sampled the larger window according
to the difference in projected patch sizes between the two u
coordinates. For the vertical anamorphic distortion, one canchange the window to keep consistent sampling, but keeping a fixed sized window for the whole image is acceptablein practice. We always used a fixed disparity search size forall the images.

4.2. Cubic Lens Performance
In our simulation of cubic lenses, we tried both anamorphic distortions in the horizontal and vertical axis. An example input image, the depth output, and the ground truth
depth are seen in figure 5. As described earlier, the resulting algorithm causes a loss of data in he middle of the X
anamorphic image. This is due both to the limited algorithm
search window, and the fact that cubic distortion requires ahuge magnification in the middle of the image (200 pixel
wide image requires 30x more magnification in the center
than the corresponding rectilinear) means that the dataset
doesn’t have any texture and the stereo algorithm fails. This
math comes from the projection function gradient, wherea rectilinear camera is constant, while a cubic camera is∇u ∝ 3 ∗ x2 .
The vertical anamorphic results work fine, but don’t givethe interesting depth properties that were discovered in the

Orthographic
horizontal
anamophic

Orthographic
verticalanamophic

Figure 6. Example results for orthographic anamorphic lenses
from the Middlebury stereo dataset [17]. These images were distorted at the same time they were sub-sampled, to produce comparable results.

analysis.

4.3. Orthographic Lens Performance
Since the cubic projection results were theoretically interesting, but the images were so extreme that it became
obvious such lenses would be challenging to build, we decided to shift our focus to a more traditional projection function, the orthographic projection. Such lenses are interesting, practical to build [3] (e.g. Nikor 10mm f/5.6 OP), andexhibit no vignetting. That means, if such lenses can beused for stereo, they’d have better SNR performance due
to less photon noise in the corners of the image. Our algorithms were validated on such images. The results areshown in figure 6. One caveat to this method is that

4.4. Proposal for a hardware system
The basic next steps for this component of the project
would be to actually build a stereo system with either a rectilinear anamorphic lens, which are commonly available on
the internet. Or build a stereo matching system with an orthographic fish-eye camera. Both would demonstrate thehigh fidelity results of a wide angle stereo depth system
without any vignetting loss or a requirement for an infinite
number of pixels (as would be needed in the traditional rectilinear lens case).

5. Synthetic stereoscopic matching
In order to further validate the contributions of these various lens models, we implemented a customized version of
the tungsten C++ path-tracer which could handle arbitrary
anamorphic projection functions. We used the well-known
Sponza scene as our example data.

5.1. Synthetic GoPro Data
To select a camera design for these synthetic results, we
decided to try and match the well-known cameras from GoPro. Specifically, we used a fish-eye lens, with an f-θ projection, and a horizontal field of view of 120 degrees. Our
images were 2:1 aspect ratio (instead of the typical 1.7:1),
and can be seen as slightly cropped versions of the images
one would get from a GoPro.

Left Image

Depth estimate

Rectilinear

Rectilinear+ vignettingfθ horizontal
anamorphicfθ verticalanamorphic

5.2. Rectilinear Depth
As a baseline for these synthetic results, we also included
a rectilinear projection with the same field of view and position. To better simulate the impact of a real f-theta lens,
we also added synthetic vignetting to the rectilinear lens inorder to account for its light loss properties. The model forvignetting was cos(θ)4 [2], while our model for noise coming from vignetting was simply assuming photon noise [9].
We assumed 5,000 electron capacity wells and sampled theappropriate Poisson distribution at each location to emulate
the noise.

5.3. Anamorphic Depth
Figure 7 includes an example of these results. It is visible that our proposed matching algorithms perform stereo
matching correctly on these synthetic anamorphic picture,with little to no loss of performance. Although there’s somevariation in performance across the projections (due to having different image content), it’s clear that the proposed
methods provide comparable or better performance than arectilinear projection, if real vignetting (+ photon noise) istaken into account, even using a robust matching function
like Census [23] isn’t enough to overcome the noise issues
introduced. Our quantitative table of performance, normalized to the results of a rectilinear lens (without vignetting),
is shown in figure 8.
We had originally planned to report the depth errors resulting from using a rectilinear un-distortion model insteadof an anamorphic one. However, we found our generated
datasets (which take 4 hours to render all example images)
had scaled depth values which made it impossible to compute mean squared error or any other depth-based error metric. There was simply not enough time to fix the error in thetungsten renderer, regenerate these datasets, and computedepth-based error metrics. However, it is planned future
work.

Figure 7. Example results for f-θ anamorphic lenses. The example
scene is the commonly used Dubrovnik Sponza scene used in thegraphics community. The renders were generated from a custom
version of the Tungsten C++ ray-tracer.

Despite that, these results suggest that anamorphic software undistortion models would minimize resampling andmake it easier to build stereo depth systems with fish-eye
lenses.

6. Conclusions
We have demonstrated and derived a generalized formof stereoscopic correspondence, along the requisite generalized sampling algorithm for modifying existing algorithms.
Additionally, we have shown synthetic renders supporting
our mathematical conclusions, and demonstrated the intuition behind desiring non-linear lenses. This should open
the path to making wide angle stereoscopic systems that naturally use the depth accuracy properties of the wide angle
lenses they’re designed with, instead of being distorted into
rectilinear projections.
Of equal interest, we’ve shown that stereoscopic systems
with properly designed optics are not limited to quadratic
depth error, and instead have be built with a desired tradeoff between spatial and depth invariant accuracy. This interesting mathematical and synthetic result paves the way
for many interesting new stereoscopic systems. We’d liketo pursue implementing a practical version of this concepts
on top of consumer wide-angle cameras, to the aim of better
computational efficiency and accuracy.

Density of Depth Result (higher is better)

Block Matching

Census

1.2

1

0.8

0.6
RectilinearRectilinear fThetaX+ vignetting

fThetaY

Figure 8. Accuracy results. Quantitative results from figure 7.
The fThetaX and fThetaY examples are anamorphic distortions(fθ) along a single axis. The metric is simply density of data after
passing a standard left-right check [10]. Values are normalized tothe scores from a rectilinear projection. There are two stereo costcorrelation functions, a standard sum of absolute differences andCensus [23]. The results from the proposed anamorphic methods
are better than those from Rectilinear stereo, when realistic noise
is taken into account.

7. Next Steps
We plan to validate these theoretical and synthetic resultswith real optics. We’d like to build stereo systems with actual anamorphic lenses to show the benefit of a new optical
distortion model. In additional, we can use these resultswith traditional spherical lenses, in the form of an anamorphic computational model to minimize re-sampling, while
continuing to use traditional correlation algorithms. Thisshould make stereo systems with cameras like GoPros more
efficient and practical.

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