1. INTRODUCTION

If an observer gazes at a curved reflector, such as a mirrored sphere, then the scene appears to be distorted, as in Fig. 1. We ask: are there any surfaces that do not appear distorted in a given reflector? If so, then such surfaces pass unchanged through the optical system, and so we refer to them as eigensurfaces, and the corresponding reflectors as eigenmirrors.

 

Fig. 1. Distortion of a checkerboard pattern in a spherical mirror. This is a panoramic view of a test scene, with the camera appearing in the center of the photo. The checkers are 0.2159 m square, and the mirror, of radius 0.031 m, is 0.34 m above the floor. Two squares are labeled: square (1) is 0.10795 m from the camera and square (2) is 0.4572 m away, yet square (1) consists of disproportionately many more pixels. For the robot soccer application, we sought greater equity between the two.

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The motivation for this work is an aspherical reflector designed for vision-based control of soccer robots, as depicted in Fig. 2. In Ref. [1], the authors investigated the use of convex mirrors for endowing robots with panoramic vision, thus simplifying the control component of the problem. In fact, Fig. 1 depicts an image taken from such experiments. Here the spherical mirror is mounted above the sensor (no robot appears in this test case), which is pointed skyward. The checkerboard pattern surrounds the camera on the ground plane. The lens of the camera is clearly visible in the center of the image. The spherical mirrors were low cost and worked moderately well for this application, giving a panoramic view of the horizon, and even capturing some fluorescent lights on the ceiling, but the distortion was somewhat problematic. Each white square appearing in Fig. 1 was an ANSII letter-sized sheet of paper cut square, and so was $0.2159\times 0.2159$ m. Hence, it is easy to see that close objects, such as the square labeled (1), which is 0.10795 m from the lens, are allocated many more pixels then a square such as (2), which is a mere 0.6096 m from the lens. Additionally, valuable pixels are wasted on the camera lens itself. The response to these problems was to design a convex, rotationally symmetric mirror so that when mounted above the robot and viewed along the axis of rotation, the ground plane appeared undistorted. Thus, the squares should look like squares and each should be allocated an equal number of pixels. The cross section of such a reflector, which we refer to as a rectifying mirror, appears in Fig. 3. An image of a checkerboard plane reflected in a rectifying mirror appears in Fig. 4, where the field of view is $360°\times 145°$. Thus, we have an example of a curved reflector that does not distort everything viewed in it. The flat plane, at an appropriate distance, is an eigensurface of the rectifying mirror. To design this mirror, the authors derive a first-order differential equation for the cross section, using the fact that the slope of the mirror at each point is prescribed.

 

Fig. 2. Soccer robots at the UPenn GRASP Lab, circa 1997. Each robot was equipped with a spherical mirror to allow for panoramic views.

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Fig. 3. Cross section of a rotationally symmetric mirror designed to give a soccer robot a $360°\times 145°$ view of the playing field. This reflector has plane as its eigensurface. At the lowest point, the radius of curvature is 1.7 m.

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Fig. 4. Case of the rectifying mirror. This curved reflector was designed to be mounted on a soccer playing robot, 34 cm above the playing field, which here contains a checkerboard pattern.

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Given the above, one may ask the dual question: starting with a reflector, are there textured shapes that appear undistorted in the reflector? To address this question we must be careful about what our assumptions and definitions are.

We should point out that we view the problem under consideration as a foundational question. To keep things tractable, we work within the framework of geometric optics and assume a monocular observer that is ideal pinhole camera. Thus, for us, images are formed ultimately by a system of perfect perspective projection, and so the only aberration under consideration is distortion.

In the original robot soccer application, the control of the robots was based on the locations of objects of interest on the field, which, from the vision standpoint, were all color blobs. Focus was thus not a major concern, and indeed was not part of our model of the rectifying mirror. Hence, the blur that is evident in Fig. 4 was not a concern. Distortion was our concern mainly because of the pixel count problem mentioned above. Therefore, throughout, the reader should imagine holding up a surface such as a checkerboard, coffee mug, etc., to a chromed water faucet and observe the reflection.

Note that all of our examples are rotationally symmetric, but for our general problem statement no such assumption is needed. If a surface is not a portion of a surface of revolution, then we say that the surface is freeform. Working without any symmetry assumption typically makes a problem considerably more complex, but also adds degrees of freedom. For example, one expects that a freeform surface be modeled by a partial differential equation, rather than a more friendly differential equation. Our work is heavily informed by freeform optics, a topic that has recently seen explosive growth, and so some discussion of the development of freeform optics is warranted.

An early example of a freeform surface in optical design (possibly the first) is the 1959 progressive spectacle design by Kanolt [2]. A well-known example that was remarkable not only for its novelty of design but also for its commercial success, was the Polaroid SX-70 collapsible camera [3–5]. Stone and Forbes’ prescient 1992 work considered the Hamiltonian optics of freeform surfaces [6,7]. Thompson soon after began a systematic analysis of aberrations in freeform systems with a mind towards astronomical instrumentation [8]. More recent work on aberrations includes the work by Bauer and Rolland [9].

Illumination systems are a natural application area for freeform systems. Early work using aspheres in laser beam shaping in the 1970s was certainly in the freeform spirit [10–12]. Classic references on illumination are Elmer [13] (which exhibits elements of freeform thinking), Welford and Winston [14], and Winston, Miñano, and Benítez [15]. The simultaneous multiple surface design method in 3D is described in Ref. [16]; see also [17] for applications.

The prescribed intensity problem for a single source, which is to illuminate a surface according to a prescribed intensity, was solved in the 1990s by Oliker et al. in Refs. [18,19]. The solution makes use of optimal transport and Monge–Ampère equations. These rich ideas have led to a considerable amount of work, such as [20–23]. Source-target maps are also an approach to the problem, such as those described by Fournier, Cassarly, and Rolland [24].

The prescribed projection problem is to redirect a two-parameter bundle of rays to a target surface using a single reflector, is formulated by Hicks and Perline in Ref. [25]. With a single reflector, this problem is ill-posed in the sense of Hadamard. Croke and Hicks introduced the tool of exterior differential systems for the design of two or four freeform reflector systems [26,27]. Related work by Plakhov, Tabachnikov, and Treschev investigates the number of reflections needed to achieve an arbitrary transformation [28]. Other work on bundles and perfect imaging includes [29–31].

The reader should bear in mind that the above only scratches the surface of the subject of freeform optics, and reflects the authors’ interest in the mathematical foundations of the subject. The total number of papers on applications of freeform optics far exceeds the above.

Returning to the problem at hand, we need to give a more precise meaning to the term “distortion.” Here we use it for whenever an image is not geometrically similar to one formed by a perspective (or orthographic) projection. In particular, in the case of undistorted image formation, straight lines in the scene map to straight lines in an image plane. If that is not the case, then one might imagine that some nonlinear transformation is at work, although the details of such a transformation (domain, range, etc.) are not a priori clear. Nevertheless, the association of nonlinear transformations with distortions is reasonably natural, given our everyday experience with the curved reflectors that surround us.

As we saw in the case of the rectifying mirror, even though a reflector is curved, one cannot immediately rule out the possibility that some surfaces (with recognizable texture) could appear unchanged when viewed in the reflector. (Weaker examples are abundant. For example, all rotationally symmetric systems image lines through the axis of rotation to lines in the image plane, be they curved or not.) Are there rotationally symmetric mirrors that perfectly image other surfaces?

The answer is “yes.” Our first result is that almost every rotationally symmetric reflector has an eigensurface when viewed along its axis of rotation. (At this point we should point out that in this paper we will ignore the issue of obstruction.) That is, there are surfaces that remain invariant, in a manner to be made precise, under the above-mentioned nonlinear transformation. This means that the observer gazing at the mirror sees a scaled, flipped, rotated, etc., version of what an observer in another fixed position sees.

Of course, the idea of having an optical system produce a rectified image is fundamental in imaging optics. The idea of having a single lens or mirror perform a prescribed transformation on some ray bundle could be said to have originated in the study of conics in ancient times, with more modern examples including the theory of perfect optical instruments, such as Cartesian ovals and more exotic examples such as the Luneberg lens, Maxwell’s fisheye, etc. [32–34].

Below, we will compute the cross sections of rotationally symmetric eigenmirrors and eigensurfaces using differential equations, and in the general 3D case, we will model the problem using a first-order nonlinear partial differential equation. Other rotationally symmetric work of Hicks and Perline include [35], which describes reflectors for panoramic imaging that provide uniform resolution in the sense that equal solid angles are mapped to equal numbers of sensor pixels. These surfaces, in the far-field case, are shown to be rotationally symmetric surfaces of constant Gaussian curvature. Later, Roitman showed that all constant Gauss curvature surfaces have this property [36]. Hicks and Coletta derive and classify rotationally symmetric equicylindrical reflectors in Ref. [37], i.e., reflectors that give uniform resolution with respect to a cylindrical unwarping transformation in software.

2. PURELY PLANAR CASE

The problem statement alone is tricky, so we will begin with the purely planar case. Assume that we have a curve given by (some portion of) the graph of $f(u)$, which is reflective, and two points ${\mathbf{p}}_1$ and ${\mathbf{p}}_{\mathbf{2}}$ in the plane, which can be thought of as observers or sources or centers of projection; see Fig. 5. We treat ${\mathbf{p}}_1$ as the observer looking directly at the reflector (eigenmirror) $f$, and ${\mathbf{p}}_2$ as an observer looking at the eigensurface. Take $B$ to be the bundle of rays that emanate from ${\mathbf{p}}_1$ and then strike the graph of $f$. Part of the data of the problem will be that we are given a transformation $T$ between $B$ and the rays that emanate from ${\mathbf{p}}_2$. It is important to get clear here some terminology and notation regarding $T$ and these two bundles.

When working with bundles, it is of course necessary to have some sort of coordinates on them. To describe a ray sometimes we will give its footpoint (source) and the direction of the ray. Another method is to choose a reference plane that intersects the bundle in such a way that each ray crosses the reference plane at a unique point.

Therefore, given $T$ and a ray $r$ in $B$, while it is technically correct to write $T(r)$, which we will sometimes denote as $\tilde{r}$, we will abuse notation sometimes and write $T(\mathbf{V})$, where $\mathbf{V}$ is the unit direction of $r$. Note that $T$ does not give a correspondence between point on $r$ and $T(r)$ for a given $r$.

It is useful to write $S^1(\mathbf{p})$, the unit circle at $\mathbf{p}$. In the 3D case, we write the unit sphere at $\mathbf{p}$ as $S^2(\mathbf{p})$. Let $r$ be a ray in $B$ from ${\mathbf{p}}_1$ through $[u,f(u)]$, i.e., it has direction

This allows us to identify $B$ with a subset of the unit sphere $S^1({\mathbf{p}}_1)$. One may think of $T$ as prescribing a certain view. The rays at ${\mathbf{p}}_2$ that correspond to those in $B$ we will call $B^{\prime }$. In other words, $B^{\prime }=T(B)$.

For the reflector $f$ to have an “eigencurve,” the following must hold: if $r$ is traced to $f$, and reflects in the direction

In 2D, the fundamental equation generically does have a solution for a given $u$. That is, if we come to the problem thinking that the reflector $f$ is given, then the system Eq. (3) will most likely have a solution for a given $u$. When $s$ and $t$ are solved for, they will depend on $u$, and so the eigencurve, $E_c$, can be described parametrically using $t(u)$ as

 

Fig. 5. Problem in 2D, with no assumed symmetry.

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Fig. 6. Sample 2D problem. The eigenmirror is $y=u^3$, with $u\in [-3/5,1]$, ${\mathbf{p}}_1=[0,1/3]$, ${\mathbf{p}}_2=[2,3/2]$, and $T([X_1,X_2])=R([X_1,-X_2])$, where $R$ is the rotation matrix given in Eq. (8).

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3. ROTATIONALLY SYMMETRIC EXAMPLE

Suppose one has a rotationally symmetric mirror that is to be viewed along its axis of symmetry. In that case, our schematic now has the form depicted in Fig. 7, which depicts a cross section of the mirror. As earlier, we have viewpoints ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$, and a transformation $T$.

 

Fig. 7. 2D schematic of the cross section of a rotationally symmetric eigenmirror with eigensurface. One observer at ${\mathbf{p}}_1$ gazes at the mirror and the other, ${\mathbf{p}}_2$, looks directly at the eigensurface. Observe that what we have here is not a two reflector problem—there is no reflective relation between the ray $\overline{r}$ and $T(r)$ at their point of intersection on the eigensurface.

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Obviously, a parabolic mirror would be natural to analyze, so consider the case where the mirror cross section is given by the graph of

Our association $T$ will be between the ray $r$ through [0, 0] of slope $m$ and the ray $\tilde{r}$ through [0, 20] of slope $-m$, i.e., $T([{\mathbf{V}}_1,{\mathbf{V}}_2])=[{\mathbf{V}}_1,-{\mathbf{V}}_2]$. Taking $\mathbf{In}={\mathbf{p}}_1-[u,f(u)]=[-u,-1/2u^2-3]$, gives

In Fig. 8, we see a cross section the (black) parabolic eigenmirror and (red) eigensurface.

 

Fig. 8. System for which the reflective surface is a portion of the parabola $f(u)=\frac{1}{2}{u}^2+3$. The first viewpoint, ${\mathbf{p}}_1=[0,0]$, observes the (black) parabola from below, via the dashed ray. The second viewpoint, ${\mathbf{p}}_2=[0,20]$ views that (red) eigensurface from above, via the dotted ray.

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Using the POV-Ray ray-tracer, we may simulate how an observer would see this pair, and compare the eigensurface with a similarly textured surface. A side view of the configuration appears in Fig. 9(A). Here the pair of surfaces lies in the center of a cubical test room, with checkerboard patterned walls. In particular, the floor is green and white, and the ceiling is green and black. A flat disk with a sandy texture has been placed on the back on the parabolic mirror. From above, at $[0,0,20]={\mathbf{p}}_2$, an observer looks down and sees the sandy back of the parabolic mirror and the rings in the eigensurface. (It just so happens that they appear fairly evenly spaced.)

 

Fig. 9. (A) Side view of a parabolic mirror (upper) and the corresponding eigensurface (below). The entire scene lies within a room with a cubical room with checkered walls. (B) From above, at [0,0,20], an observer looking down sees the sandy back of the parabolic mirror and the rings in the eigensurface. (C) The view from ${\mathbf{p}}_1$ of the parabola. The proof of concept is that the rings look the same as in (B). (D) Here we see a flat disk reflected in the parabola. It is not, of course, an eigensurface of the parabola, and so the rings do not appear evenly spaced, as in (B) and (C).

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If we remove the eigensurface and replace it with a flat disk with a ringed pattern, we can view it reflected in the parabolic mirror and see that it is distorted in comparison to the above view in Fig. 9(B), i.e., it is not an eigensurface. The crucial comparison is between Fig. 9(C) and this view, where the observer’s eye ${\mathbf{p}}_1$ is at the origin, beneath the parabolic mirror. The result is that the rings are undistorted and look identical to those in Fig. 9(B). That is, we have found the eigensurface. To underscore the specialness of the eigensurface, in Fig. 9(D) we view from ${\mathbf{p}}_1$ a flat disk with evenly spaced rings, reflected in the parabola. Here we see that the rings are distorted, i.e., a flat disk is not, of course, an eigensurface of the parabola.

4. NONEXISTENCE OF AN EIGENSURFACE IN THE OFF-AXIS CASE

We have shown that an eigensurface can be constructed for any rotationally symmetric mirror with viewpoints both along the axis of symmetry. Let us now consider the case of a rotationally symmetric mirror with only ${\mathbf{p}}_1$ on the axis of symmetry. Note that in our above rotationally symmetric discussion, with the exception of a few degenerate choices for viewpoints on the axis, an eigenmirror always exists. Here we show that if one takes the viewpoint ${\mathbf{p}}_2$ to be off the axis of rotation, then the eigensurface can fail to exist.

Again, it is natural to consider a quadratic mirror, so we consider a rotationally symmetric mirror given by the parabolic surface $z=u^2+v^2-c$, with $c>0$. Let ${\mathbf{p}}_1=[0,0,0]$ and take the reference plane $z=-1$ to label rays that emanate from ${\mathbf{p}}_1$. Let ${\mathbf{p}}_2=[0,-a,b]$, with $a,b>0$, and use the reference plane $y=-a+1$ for labeling rays.

Our computations here will label the ray bundles, not with unit directions, but with points in chosen reference planes. That is, a plane will be associated with each viewpoint (essentially an image plane), and if a ray through the viewpoint strikes that plane at a point, then we will consider that point to be the label (or coordinates) of the ray. This makes writing $T$ down in those coordinates easy.

Let the transformation from the bundle $B$ at ${\mathbf{p}}_1$ to the bundle at ${\mathbf{p}}_2$ be given by $T(u,v,-1)=[-u,-a+1,b-v]$. Given an arbitrary point $q_1=[u,v,u^2+v^2-c]$ on the mirror, we can calculate the line to it from ${\mathbf{p}}_1$. This line intersects the reference plane of ${\mathbf{p}}_1$ at

5. ROTATIONALLY SYMMETRIC OFF-AXIS EXAMPLE

Next, we consider the case where the eigenmirror is unknown, but may be parametrized in terms of $z$ and $\theta$ as

Thus, our axis of symmetry is the $z$ axis. We will index rays in $B$ with the reference plane $z=-1$, and index rays in $B^{\prime }$ with the plane $y=-3$. Thus, rays in $B$ are labeled by points of the form $[\alpha ,\beta ,-1]$, and rays in $B^{\prime }$ are labeled by points of the form $[\alpha ,-1,\beta ]$. For $T\text{:}B\to B^{\prime }$, we take $T([\alpha ,\beta ,-1])=[-\alpha ,-3,4-\beta ]$. Remember–this means that a ray from ${\mathbf{p}}_1$ to $[\alpha ,\beta ,-1]$ is transformed to the ray from ${\mathbf{p}}_2$ to $[\alpha ,-1,\beta ]$.

Our only unknown is $\rho (z)$, and we will compute the ordinary differential equation that $\rho$ must satisfy in order for an eigensurface to exist.

Let $\mathbf{n}(z,\theta )$ denote the normal to the surface at $\mathbf{x}(z,\theta )$ and define $\mathbf{Out}$ to be the reflection about $\mathbf{n}$ of $\mathbf{In}={\mathbf{p}}_1-\mathbf{x}(z,\theta )$. Thus, the reflected ray at $\mathbf{x}(z,\theta )$, $\overline{r}$, parametrized by $t>0$, has the form, $\mathbf{x}(z,\theta )+t\mathbf{Out}(z,\theta )$. Writing $\dot{\rho }$ for $\frac{d\rho }{dz}$, we have

The second factor also gives a spurious solution but warrants close inspection. Writing $r\dot{\rho }+z=0$ as $rdr=-zdz$, we see that the solutions are $r=\pm \sqrt{-z^2+C}$, so the eigenmirror is

Thus, one may conclude that for the data given in this case that no interesting eigensurfaces exist. This is fairly general, of course, since there is nothing special about ${\mathbf{p}}_1$ other than it should lie on the axis of rotation, and that ${\mathbf{p}}_2$ not lie on the axis of rotation. What is special to this example is the choice of $T$. It is not at all clear if another reasonable choice of $T$ would lead to true solutions.

6. GENERAL 3D CASE

For the general case in 3D, much of our above notation can be recycled. The main difference is now that the mirror will be given parametrically in the form $\mathbf{x}(u,v)=[X(u,v),Y(u,v),Z(u,v)]$, although of course one can often use the graph form of $\mathbf{x}(u,v)=[u,v,f(u,v)]$.

Suppose that $r$ is the ray from ${\mathbf{p}}_1$ to $\mathbf{x}(u,v)$, a point on a reflective surface $\mathbf{x}$. Let $\mathbf{V}\in S^2({\mathbf{p}}_1)$ denote the direction of $r$. The ray $r$ reflects at $\mathbf{x}(u,v)$, then proceeds along the ray $\overline{r}$, which is parametrized by $t\mapsto \mathbf{x}(u,v)+t\mathbf{Out}$, where $\mathbf{Out}$ is the unit direction after reflection. To have an eigenmirror means that for every $u$, $v$ in the domain of $\mathbf{x}$, we $\overline{r}$ intersects with $T(r)=\tilde{r}$ (which is given by $s\mapsto {\mathbf{p}}_2+sT(\mathbf{V})$); see Fig. 10. Then the fundamental equation describing eigensurfaces and eigenmirrors is $\overline{r}=\tilde{r}$, i.e.,

 

Fig. 10. Derivation for the partial differential equation in the freeform case.

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It is at this point that our thinking diverges from the above derivations and examples. If the eigensurface and eigenmirror are both treated as unknowns, then we have a first-order nonlinear partial differential equation for $f$.

As an example, take ${\mathbf{p}}_1=[0,0,0]$, ${\mathbf{p}}_2=[1,0,1]$, and $T(\mathbf{V})=\mathbf{V}$. Then the fundamental partial differential equation [Eq. (18)] takes the form

 

Fig. 11. Eigenmirror $z=\frac{1-v^2}{2u}$ and its corresponding eigensurface.

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Fig. 12. Simulation of the view from ${\mathbf{p}}_1$ of an eigenmirror (below) and an eigensurface (above), which has a checkerboard pattern on it. The eigenmirror reflects this pattern, and the view is identical to the view of the upper surface from viewpoint ${\mathbf{p}}_2$, as in Fig. 13. The numbers are actual objects in the simulation, placed in the back of the scene to emphasize the use of a different viewpoint in Fig. 13.

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Fig. 13. View of the eigensurface from viewpoint ${\mathbf{p}}_2$.

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To strengthen this claim, we performed the following experiment. Imagine that we had printed the two images in Figs. 12 and 13 and then cut out all of the white from the paper version of Fig. 12, creating a lattice of sorts. Then one could overlay that onto the printed version of Fig. 13 and any mismatch between the patterns would be visible. A variant of this procedure was performed using the fact that png format supports transparency. The (positive) result appears in Fig. 14.

 

Fig. 14. View of the eigensurface from viewpoint ${\mathbf{p}}_2$, overlaid with the view from ${\mathbf{p}}_1$. To do this, all pure white in Fig. 12 was first converted to transparency and then pasted onto the image in Fig. 13. As a result, a ghost image of the surface appears floating above the mirror. If there were a significant discrepancy between the image of the eigensurface and eigenmirror, it would appear in the lower composite, i.e., there is no “interference” from the overlaying.

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Remarks:

7. CONCLUSION

A conventional flat mirror has a remarkable but underappreciated property: one observer can experience the view of another observer at a different location, without physically moving to that location. This application is remarkable and is used to great dramatic effect in an early scene from the film Saving Private Ryan. Pinned down by enemy fire, Tom Hanks’ character cannot safely see the enemy position, and so cannot make a plan of action. But using chewing gum, a small mirror and stick, he is able to obtain the viewpoint he wants—but without putting himself in danger. Applications of our work for the human eye are the most natural. Security mirrors, dental tools, etc., all are designed to work in conjunction with the human eye, but flat or spherical mirrors are not always optimal for these applications. (The problem we consider is not solved by flat reflectors because we ask for the ability to have the stolen viewpoint include rotations, flips, scales, etc.)

When can one “steal” another viewpoint using a curved reflector? Certainly, this is a fundamental-sounding question of geometric optics, and we argue that writing down and analyzing the ordinary and partial differential equations for this problem is sensible if we seek to understand the complex transformations induced on ray bundles by optical elements.

A classic struggle for photographers is having to photograph a wall or building, but being unable to stand at the ideal location to do so because of some physical obstruction. The result is keystone distortion, or the “convergence of verticals”—see Kingslake [38]. Classically, this problem was solved by using a camera that incorporated bellows and a “rising front/swing back.” In principle though, this problem could be framed in terms of eigenmirrors and eigensurfaces, with the building face being the eigensurface. The convergence of verticals is similar to the problem of anamorphosis that required the SX-70 mentioned above, to incorporate freeform elements. As an element of a photographic system, such a mirror might be able to image a surface from an awkward position in a precise fashion. (Of course, incorporating such a freeform element into an photographic system might require other complex optical elements to deal with the aberrations that we have ignored in our approximation.)

A summary of the technical development of our work is as follows:

Note that to say that one viewer sees an object in the same way as another could reasonably allow a change in of field of view, i.e., a contraction or expansion of the bundle $B$. We did not consider such examples, but the equation that one obtains when taking $T$ to be practically anything nontrivial is a quite complicated first-order nonlinear partial differential equation. The study of such equations appears to be a rich and unexplored intersection among optical design, numerical analysis, and partial differential equations.