Equitable mirrors Casey Douglas,1,* Emek Köse,1,2 Nora Stack,1,3 and Caroline VanBlargan1,4 1
St. Mary’s College of Maryland, 18952 E. Fisher Rd, St. Mary’s City, Maryland 20686, USA 2 3 4
e-mail: [email protected]
e-mail: [email protected]
e-mail: [email protected]
*Corresponding author: [email protected] Received 12 June 2014; revised 3 November 2014; accepted 10 November 2014; posted 14 November 2014 (Doc. ID 214027); published 17 December 2014
Mirror surfaces used in catadioptric sensors are often designed so as to minimize one particular kind of image distortion. In this article we explore some finer properties of equi-areal mirrors, those that feature no area distortion, and we propose novel ways to measure compound forms of distortion. Specifically, we develop new mirror surfaces with large fields of view that simultaneously minimize angular and areal distortion with respect to different cost functions. © 2014 Optical Society of America OCIS codes: (110.0110) Imaging systems; (230.4040) Mirrors; (060.3735) Fiber Bragg gratings. http://dx.doi.org/10.1364/AO.53.008471
1. Introduction
The main problem in catadioptric sensor design is to realize a given projection with a camera-mirror pair; that is, one seeks to determine one or multiple mirror surfaces that, when coupled with a camera, reflect the world in a desired way. A camera is a device that maps R3 to R2 via a specified projection. Our designs use simplistic but common models for cameras, namely perspective and orthographic models. A pinhole camera realizes perspective projection since all light rays pass through a common point to form an image. The distance between this pinhole and the image plane is the focal distance. Orthographic projection can then be viewed as a limiting case of perspective projection wherein the focal distance becomes infinite and light rays travel parallel to the optical axis. For the purposes of this paper, cameras are modeled via orthographic projection. Mirror surfaces can be designed so that the catadioptric sensor preserves certain geometrical 1559-128X/14/368471-10$15.00/0 © 2014 Optical Society of America
properties. For example, image-to-world mappings can be conformal, equi-areal, equidistant, equiresolution, etc., as in [1–4]. In efforts to achieve uniform resolution, for instance, Conroy and Moore designed catadioptric sensors that have solid angle-pixel density invariance [2], while Chahl and Srinivasan studied mirror surfaces that preserve a linear relationship between the angle of incidence of light onto the surface and the angle of reflection [1]. However, neither of these sensors feature a uniformly distributed correspondence between points in the image plane and points in the object sphere. Hicks and Perline address this in their work on the family of equi-areal mirrors, so named because they induce projections that uniformly rescale area [3]. In summary, then, catadioptric sensors are designed based on the needs of their users, and, as indicated by the preceding examples, when these needs involve minimizing a single kind of distortion much can be said. The equi-areal mirrors studied in [3] minimize areal distortion, for instance, while parabolic mirrors have long been known to minimize angle distortion—indeed, this result is re-derived and generalized in Theorem 2.1. Minimizing multiple types of distortion, however, has not been as popular. 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8471
In this paper we propose and analyze precise ways in which to measure and simultaneously minimize multiple kinds of distortion. The so-called equitable mirrors that we seek allow for both area and angle distortion but attempt to minimize their combined, overall impact. This problem overlaps with the “map maker’s problem” from cartography since a flat map of the earth cannot simultaneously preserve angles and rescale area, a fact that also follows from Gauss’ Theorem Egregium; indeed, a catadioptric generalization of this result appears in Theorem 2.2. As such, we first review useful and common notions of angle and area distortion. A.
Angle Distortion
As detailed in [5], there is a natural way in which to measure angle distortion for mappings between planar domains. A bijective mapping w:U⊆C → C from a subset of the complex plane into the complex plane is said to be K quasi-conformal if ∂w ∂w sup ∂z ∂¯z ≤ K: z∈U ∂w ∂w − ∂z ∂¯z
(1)
In Eq. (1) the quantity K is a fixed real number that satisfies K ≥ 1, and we have used the complex notation ∂w 1 ∂w ∂w −i ∂z 2 ∂x ∂y
and
∂w 1 ∂w ∂w i ; ∂¯z 2 ∂x ∂y
where i2 −1 and z¯ x − iy denotes the complex conjugate of z x iy. The theory of quasi-conformal mappings is typically developed for orientation-preserving functions but can be extended to orientation-reversing functions in a simple way: we say that an orientationreversing mapping, wz, is K quasi-conformal if ¯ its orientation-preserving conjugate, wz, is K quasi-conformal. The prototypical example of a K-quasi-conformal mapping is obtained by “shearing” the complex plane in a fixed direction. For instance, the function wz 2x iy stretches the complex plane by a factor of 2 along lines parallel to the x axis. In this example, the fraction appearing in Eq. (1) equals the constant 2 at every input z, making the mapping 2-quasiconformal, and intuition for this fact can be gained in several ways. First, one can note that at every point z the function wz stretches infinitesimal squares into infinitesimal rectangles with a lengthto-height ratio of 2. Alternatively, one can also note that at every point z the function wz stretches infinitesimal circles into infinitesimal ellipses whose major-axis is twice as long as its minor-axis. 8472
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
In general, angle-preserving or conformal mappings take infinitesimal circles to infinitesimal circles, but quasi-conformal mappings take infinitesimal circles to infinitesimal ellipses. In fact, the fraction appearing in definition (1) is directly related to the eccentricity of an infinitesimal ellipse centered at the point wz. If this quantity is one, then w preserves angles at z. If it is infinite, then w effectively destroys angles at z, and therefore fails to be quasi-conformal. Definition (1) can be rephrased using the complex dilatation of a mapping, μ, which is defined by
∂w ∂¯z μz : ∂w ∂z A mapping w is K quasi-conformal, then, if its complex dilatation satisfies ‖μ‖∞ k, and 1k ≤ K; 1−k where ‖μ‖∞ denotes the suprema of jμzj. Furthermore, angles are preserved when μ 0 and destroyed when jμj 1. The complex dilatation μ need only be a measurable, essentially bounded function, which allows the notion of quasi-conformality to apply to a much wider class of functions. For the purposes of this paper, though, we can restrict the definition to orientation-reversing and orientation-preserving diffeomorphisms—differentiable functions with differentiable inverses. One readily checks that the dilatation for an orientation-reversing map, w, is the ¯ reciprocal of the dilatation for w. A rotationally symmetric mirror surface with profile curve f r induces a planar mapping between the image plane and stereographically projected object spheres; this is accomplished by tracing reversed, orthographically projected light rays from the image plane onto an object sphere. As in [3], we use the sphere at infinity to model distant object spheres, referring to it as a “reflection sphere” since it keeps track of the directions of reflected light rays. Moreover, because our mirrors are rotationally symmetric, we can replace this sphere with the circle at infinity, parameterizing it as a unit circle. The dilatation associated to such a mapping under orthographic projection is then given by the formula μr
rf 00 r − f 0 r : rf 00 r f 0 r
(2)
More generally, if we produce a mirror surface by revolving the planar curve γt rt; zt about the vertical z axis, then the dilatation of its induced mapping is given by
μt
z00 r0 − z0 r00 r0 r − r0 3 z0 ; z00 r0 − z0 r00 r0 r r0 3 z0
(3)
where differentiation in Eq. (2) is taken with respect to r but in Eq. (3) it is taken with respect to t. For instance, surfaces of the form f r arp have constant μ p − 2∕p and therefore have K p − 1 (for p ≥ 2). These uniformly quasi-parabolic mirrors distort angles by a factor of p − 1 at every point. B.
Area Distortion
A catadioptric sensor is called equi-areal if its magnification factor, denoted by mf , is constant. This quantity compares the area of a region on the reflection sphere to the area of its projection on the image plane. Hicks and Perline compute the magnification factor of a sensor under orthographic projection by considering two concentric circles of radii r and r Δr around the optical axis [3]. The infinitesimal annulus with area πr Δr2 − r2 is then mapped to the topological annulus on the sphere, bounded by ϕ and ϕ Δϕ, whose area is π1 − cosϕ Δϕ− 1 − cosϕ. The limiting ratio of these two areas, taken as Δr → 0, is the magnification factor mf lim
Δr→0
πcosϕ − cosϕ Δϕ sinϕ dϕ : 2r dr πr Δr2 − r2
Using the fact that ϕ 2 arctanf 0 r, we obtain the following expression for the magnification factor: mf
2f 0 rf 00 r ; r1 f 0 r2 2
(4)
which, as noted in [3], is a multiple of the revolved surfaces’ Gaussian curvature. 2. Preserving Area or Angles
For completeness sake, we begin with a brief but thorough treatment of conformal mirrors and then discuss equi-areal mirrors. Although it is well known that the only rotationally symmetric, conformal mirror is the paraboloid, Theorem 2.1 classifies all conformal mirror surfaces. Theorem 2.1. The only angle-preserving mirrors (with respect to orthographic projection) are graphs of harmonic functions z f x; y and graphs of the quadric surfaces z ax − x0 2 y − y0 2 d; where a, d ∈ R and x0 ; y0 ∈ R2 are fixed. Proof. After parameterizing a portion of the mirror surface as a graph, z f x; y, one finds that the angle-preserving properties are determined by the mapping gx; y ∇f x; y. This follows since stereographic projection is a conformal mapping between the sphere at infinity and the plane. In particular, our mirror preserves angles if and only if
∂g 0 ∂¯z
∂g 0; ∂z
or
where the first equation implies g is orientationpreserving, and the second one implies it is orientation-reversing. For the orientation-preserving case, we find ∂g 0⇔f xx f yy ∂¯z
and
f xy −f yx :
From these equations it easily follows that f x; y ax − x0 2 ay − y0 2 d; where x0 ; y0 ; d ∈ R are fixed. Analogous computations confirm that when g is an orientation-reversing, angle-preserving map the function f x; y must satisfy f xx −f yy ; f xy f yx : In other words, f x; y is harmonic. □ Of course, large portions of harmonic graphs are not viable as mirror surfaces since they can cause self-reflections. In addition, the only rotationally symmetric harmonic functions are unbounded near zero as they are given by f r c ln r, where c ∈ 0; ∞ determines the mirror’s field of view at r 1. As a result, we will use paraboloids as the premiere example of angle-preserving mirrors, and after vertically and horizontally shifting a parabolic mirror, it can be assumed to have the more familiar form f r ar2 . For the remainder of the paper, then, we use r ∈ 0; ∞ as our independent variable and use y to denote the height coordinate of the reflected light ray on the circle at infinity. Before developing equitable mirrors that efficiently interpolate between parabolic mirrors and equi-areal mirrors, we first collect some facts about the “extreme cases.” That is, we analyze the area distortion caused by a large field-of-view paraboloid and then summarize the angle distortions caused by large field-of-view equi-areal mirrors. The magnification factor for a parabolic mirror, f r ar2 , is given by mf r
8a2 : 1 4a2 r2 2
If our parabolic mirror is to have a large field of view over a finite interval r ∈ 0; r1 , then it will suffer significant area distortion. This follows from the fact that mf 0 8a2 → ∞
as a → ∞:
In order for f r to achieve a full field of view at r r1 the parameter a necessarily tends to infinity. Moreover, as mf 0 becomes unbounded, mf r → 0 for all other values of r. In summary, then, conformal 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8473
mirrors with large fields of view have highly nonuniform magnification. At the other end of the spectrum lie the constantGaussian curvature surfaces of revolution, mirror surfaces that Hicks and Perline showed are equiareal (see [3]). As categorized in [6], these surfaces fall into one of six families: spheres, bulges, spindles, pseudospheres, hyperboloid types, and conic types. The first three have positive Gaussian curvature and are generated by the curve
plane or a singularity. For instance, μB and μS have unit length when y 1, while μSp has unit length when y 1 − 2d. By restricting the field of view to avoid those points, where Eqs. (5)–(7) have unit length, one can limit the amount of angle distortion caused by an equi-areal mirror. These considerations lead us to the following Theorem 2.3. Let m ∈ 0; 1 be fixed. Of all (portions of) equi-areal mirrors with
s Z t∕a t s 2 2 γt b cos ; ds ; a − b sin a a 0
‖μ‖∞ ≤ m;
where a; b ∈ 0; ∞ are fixed. The latter three are orientation reversing mirrors generated by hyperbolic versions of γt; these surfaces are similarly determined by two, positive parameters a and b. The size of d b2 ∕a2 determines a particularly oriented surface’s type, with d 1 corresponding to the sphere and pseudosphere, d > 1 corresponding to the bulge and hyperboloid type, and d ∈ 0; 1 corresponding to the spindle and conic type. Using Eq. (3) and the explicit profile curves, γt, the dilatation μ is easily expressed in terms of the parameter d and the height coordinate, y, on the circle at infinity. These expressions are related in elementary ways. For the sphere and the pseudosphere one has μS y
1y μPs −y; 3−y
(5)
with y ∈ −1; 1 for the sphere and y ∈ −1; 1 for the pseudosphere. The dilatation for the bulge and hyperboloid type are given by μB y; d
4d − 3 2y y2 μH −y; d 1 4d − 1 2y − y2
(6)
with d > 1 and y ∈ −1; 1. Lastly, the spindle and the conic type have μSp y; d
4d − 3 2y y2 −μC −y; 1 − d 4d − 1 2y − y2
(7)
with d ∈ 0; 1 and y ∈ 1 − 2d; 1 for the spindle and y ∈ −1; 1 − 2d for the conic type. Note that neither the spindle nor the conic type is capable of achieving a full field of view, i.e., image a full hemisphere. These expressions allow us to easily prove a surprising generalization of the map maker’s problem that we call Theorem 2.2. (The mirror maker’s problem). There is no quasi-conformal, equi-areal mirror surface (of revolution). Proof. In each of the expressions above one can determine values of y where jμj 1. These occur at the specified end points for y and correspond to points where our mirrors either have a vertical tangent 8474
the sphere and pseudosphere offer the largest field of view. Proof. An elementary but careful analysis of the dilatations in Eqs. (5)–(7) lies at the heart of our proof. As indicated in Fig. 1, one first observes that to achieve ‖μ‖∞ m, values of y near the end points should be discarded. The key observation is that as d → 1, the surfaces of constant positive (resp. negative) Gaussian curvature tend to a sphere (resp. pseudosphere), and their fields of view improve as they do so. This phenomenon is easy to establish when comparing spheres and bulges (resp. pseudospheres and hyperboloid types). No matter what value we set for ‖μ‖∞ m, the sphere will always offer a larger field of view than the bulge. This is depicted in Fig. 1 where we have set ‖μ‖∞ 0.9. As indicated in Fig. 2 though, comparing the sphere and the spindle (resp. the pseudosphere and the conic type) is more subtle. The spindle can offer a larger field of view when y is sufficiently close to yf where jμyf j ‖μ‖∞ m. Solving Eq. (7) for y yields
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
y yμSp
q μSp − 1 2 1 μSp 2 d − d μSp 1
;
and from this we can conclude that the largest total field of view for the spindle will occur when we set Μ
0.9
1.0
Μ
0.8 0.6
Bulge
0.4 Sphere 0.2
y 1.0
0.5
0.5
1.0
Fig. 1. Dilatation jμj for sphere and bulge with d 1.5.
0.9
1.0 0.8 0.6
Spindle
0.4 Sphere 0.2
y 1.0
0.5
0.0
0.5
distortion, μ, and area distortion, mf , or comparing mirrors to those with desirable properties, like spheres and paraboloids. In much of what follows, our functionals are defined in terms of yr ∈ −1; 1, the height coordinate on the circle at infinity. In these cases, potential minimizers are determined by numerically solving pertinent Euler–Lagrange equations. The actual mirror surface is then obtained by determining its profile curve, f r, from the relation
1.0
Fig. 2. Dilatation jμj for sphere and spindle with d 1∕3.
p −m − 1 2 1 m2 d − d ; yi y−m 1−m p m − 1 2 1 m2 d − d : yf ym m1 Our spindles will offer a field of view given by θd arcsinym − arcsiny−m: A straightforward computation confirms that θ0 d > 0, and so the overall field of view increases as d → 1. This means that, given a fixed amount of angle distortion, the spindle’s total field of view is improved as it becomes more spherical. The relationships between the dilatations for different surfaces in Eqs. (5)–(7) allow us to make analogous conclusions for the negatively curved surfaces. Similar analysis of the dilatations allow us to conclude the converse of Theorem 2.3, which we state as a corollary: Corollary 2.4. Of all (portions of) equi-areal mirrors that achieve a fixed field of view, θ0 , the sphere and pseudosphere minimize ‖μ‖∞ . 3. Distortion Functionals
Given that equi-areal mirrors destroy angles and that parabolic mirrors suffer highly nonuniform magnification, it is natural, then, to determine mirrors that do a fairer or more equitable job of managing both kinds of distortion. We call such a surface an equitable mirror. Our basic strategy for determining these mirrors involves a two-step process. First, we devise an appropriate functional, D:V → R, where V is a relevant space of functions. Second, we find and analyze the critical points of D. To interpret Dv as measuring combined angular and areal distortion, the functional must satisfy basic properties. The functional should vanish, Dv 0, if and only if v is associated to an ideal mirror, one with zero angle distortion and zero area distortion. In accordance with Gauss’ Theorem Egrigium, then, we first and foremost require that Dv > 0 for all v ∈ V whose mirrors offer a nontrivial field of view. Moreover, formulas for D should be well-motivated, either incorporating notions of angle
y
f 0 2 − 1 : f 0 2 1
(8)
When determining the best or least-distortive conic in Subsection 3.A, our functional is instead phrased in terms of f r and is minimized via other, more elementary techniques. In any event, to set up a tractable and realistic equitable mirror problem we first assume that the camera’s sensor lies along some finite interval. Without loss of generality we can take this interval to be −1; 1, and, as a result, the spaces V involve sufficiently smooth functions with common domain [0, 1]. A. Most Equitable Conic
Because they are inexpensive to manufacture and include both the paraboloid and the sphere, conic sections provide us with a natural family of surfaces to explore. In particular, it is reasonable to suspect that an ellipse or a hyperbola can serve as an equitable mirror. To rigorously assess this possibility, we first let V denote the space of functions that are continuous on [0, 1] and differentiable on (0, 1). Of particular interest are those functions f r ∈ V whose graphs correspond to portions of conic sections. We assume that our candidate functions satisfy f 0 0 0 and f 0 1 c ∈ 0; ∞ so that a full (180°) field of view is obtained if and only if c ∞. For this problem we can use the rather sensitive functional, D0 , given by D0 f K f − 1 ‖m0f ‖∞ ;
(9)
where K f is the quasi-conformal factor defined in Eq. (1). We only apply D0 to graphs of parabolas, hyperbolas, and ellipses, ignoring linear functions since they destroy areas and angles under orthographic projection. Straightforward calculations confirm that one can monotonically decrease the D0 distortion of a hyperbola (for any fixed c ∈ 0; ∞) by making it more parabolic. Similarly, the D0 distortion of an ellipse decreases as the ellipse becomes more circular. With respect to D0 -distortion, then, the most equitable conic can be found by comparing the D0 distortions of parabolas to the D0 distortions of circles. Such a comparison is depicted in Fig. 3 where it is shown that, for undesirably small fields of view, the parabola offers less D0 distortion than the circle. 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8475
r2 y4 4ry3 − r2 y00 − 2ry0 2y3 y2 − 2y 1 0: (13) Equation (13) is of fourth order, and so we impose initial conditions
Circle
0.25
y0 0 0;
y0 −1;
Parabola
y1 C and 0.25
0.48
Fig. 3. D0 distortion of spherical and parabolic mirrors.
p Indeed, when c > 2725 5 ≈ 0.48, a spherical mirror offers less distortion. For all practical purposes, then, a circle is the best conic. It is natural to regard D0 as a sort of “goldstandard” distortion-functional, one that remains finite if and only if mirrors are simultaneously quasi-conformal and quasi-equi-areal. However, minimizing D0 over a wider class of functions is a difficult problem. Moreover, requiring that our distortion be point-wise bounded is unnecessarily restrictive. As a result, we focus efforts on minimizing other, equally motivated but more easily managed functionals. B.
Manageable Distortion
The dilatation and magnification factor for a rotationally symmetric mirror can be expressed in terms of yr as follows: μ
ry0 − 1 y1 − y ; ry0 1 y1 − y
(10)
1 y0 r : mf 2 r
(11)
Expressions (10) and (11) are obtained by using formulas (2), (4), and (8). Note that a mirror will be conformal if the numerator for Eq. (10) is zero, and it will be equi-areal if and only if the derivative of Eq. (11) also vanishes. This motivates use of the following functional: Z D2 y
1 0
ry0 y2 − 12 ry00 − y0 2 dr;
The first two conditions ensure that yr and f r are smooth, while the values of C and D control the field of view and magnification factor at r 1, respectively. Although the second variation of D2 is difficult to work with, we can numerically solve Eq. (13) and, likewise, numerically verify that these solutions (locally) minimize D2 ; the Runga–Kutta method is used by our algorithms, and we carry this out for large values of C. For instance, Fig. 4 depicts the relationship between D2 y and the choice of the magnification parameter, D. In this plot, we have set y1 C 1 so that our mirrors each achieve a full field of view. The minimum occurs at approximately D ≈ 3.64 where D2 y ≈ 1.60. For comparison, we note that a sphere with height coordinate yS r and field of view restricted by yS 0 −1 and yS 1 C has D2 distortion D2 yS
1 C4 : 9
When C is near 1, then, the sphere’s distortion is near 16∕9 > D2 y. Hence, for large fields of view, this equitable mirror will feature less D2 distortion. Figure 5 shows the profile curve for this mirror, and Figs. 6 and 7 display plots of jμrj and jmf rj, respectively. Although this equitable mirror attains a smaller amount of distortion than the sphere, it nonetheless destroys angles (when r 0 and when r 1). Furthermore, the magnification factor changes at high rates; in fact, m0f becomes unbounded near r 0. Fortunately, these features occur relatively close to r 0 where the catadioptric sensor’s camera images itself and therefore has a blind spot. On average this blind spot occupies approximately 1% of the image
(12) 50
where y is a sufficiently differentiable function. One readily checks that the D2 distortions of a parabola and hyperbola become unbounded as their fields of view increase. The D2 distortion of an elliptical mirror is strictly greater than that of a spherical one, and, since these findings mirror those for the D0 functional, this suggests that our D2 functional is similarly informative. In order to minimize D2 , we solve the associated Euler–Lagrange equation, which is given by 8476
y0 1 D:
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
D2
40
30
20
10
D 2
4
6
8
Fig. 4. D2 y as a function of D with C 1.
10
2.0
z
1.5
z
f r
1.0
0.5
r 0.2
0.4
0.6
0.8
1.0
Fig. 5. D2 -equitable mirror (13).
1.2 1.0 0.8 0.6 Circle 0.4 0.2
Equitable Mirror
r 0.2
0.4
0.6
0.8
angular distortion by examining how lines of latitude and longitude are imaged. Figures 9 and 10, for instance, show how parabolic and spherical mirrors perform in this room. As expected, the parabolic mirror images latitudinal lines as equally spaced, concentric circles. This corresponds to the fact that parabolic mirrors preserve angles. For comparison, Fig. 10 shows how a spherical mirror performs in this room: note that lines of latitude are imaged as unequally spaced, concentric circles. As shown in Fig. 11, the D2 -equitable mirror produces little angle distortion but offers a full field of view. The rectangular “checkerboard test room” is so named because, as seen in Fig. 12, it has walls that are checkered in different colors, and a sensor is located at its center. This also allows one to qualitatively determine how angles are distorted but also shows how lines are imaged by the catadioptric sensor. As a point of comparison, the performances of the parabolic and spherical mirrors are displayed in Figs. 13 and 14, respectively. As is seen in Fig. 15, the red-and-black colored floor of the checkerboard
1.0
Fig. 6. jμrj for the D2 -equitable mirror and sphere.
10
mf
8
6
4
Fig. 9. Parabolic mirror in the angular test room. 2
r 0
0.2
0.4
0.6
0.8
1.0
Fig. 7. jmf rj for the D2 -equitable mirror.
plane, and, as a result, our equitable mirror will, in actuality, feature very little angle distortion and mild area distortion. To better assess this equitable mirror’s strengths and weaknesses, we create three synthetic test rooms using POV-Ray, a ray tracing program. The first test room is spherical in shape and has a sensor located at its center, as shown in Fig. 8. This arrangement allows us to qualitatively assess a mirror’s
Fig. 8. Angular test room.
Fig. 10. Spherical mirror in the angular test room.
Fig. 11.
D2 -equitable mirror in the angular test room.
20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8477
test room is almost perfectly rescaled by the D2 equitable mirror. The third and final test room we employ has seven equal-sized spheres, each equidistant from the catadioptric sensor but of varying elevation, as seen in Fig. 16. This scene, as used in [3], allows the observer to determine how the sensor performs in terms of
preserving areas. With a true equi-areal sensor, then, the imaged spheres will be distorted in shape but will each contain the same amount of area. Performances of parabolic and spherical mirrors in this room are depicted in Figs. 17 and 18. Additionally, Fig. 19 shows that although there is variation in the areas of spheres imaged by the D2 equitable mirror, the change is mild. From these two qualitative analyses, we conclude that this D2 equitable mirror is superior to spherical and parabolic mirrors. Another functional that produces equitable mirrors is defined by Z DS;P y
Fig. 12. Checkerboard test room.
1 0
y − yS 2 y0 − y0P 2 dr;
(14)
where yS r and yP r denote the height coordinates for a sphere and parabola, respectively, with fields of view yS 1 yP 1 C. The Euler–Lagrange equation for DS;P y is y00 − y FC; r;
(15)
where Fig. 13. Parabolic mirror in the checkerboard test room.
Fig. 17. Parabolic mirror in the test room with equidistant spheres. Fig. 14. Spherical mirror in the checkerboard test room.
Fig. 15. D2 -equitable mirror in the checkerboard test room.
Fig. 16. Test room with equidistant spheres and a camera at the center. 8478
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
Fig. 18. Spherical mirror in the test room with equidistant spheres.
Fig. 19. D2 equitable mirror in the test room with equidistant spheres.
4C − 12 3C2 − 1r2 : FC; r 1 1 C −r2 1 r2 Cr2 − 13 We impose the boundary conditions y0 −1 and y1 C, and with extensive use of the exponential integral function, explicit solutions to this equation are available. Given their lengthy and cumbersome appearance, it is best to treat them as numerical solutions. The second variation for DS;P is easily computed and is always positive. This means that the critical points we find are local minima. Figure 20 depicts the profile curve for one such minimum. This second equitable mirror outperforms both the sphere and the paraboloid (in terms of DS;P distortion). In fact, the DS;P distortion of a sphere becomes unbounded as C → 1, while the distortion values for the paraboloid and the equitable mirror are shown in Fig. 21. Note that when C 1, Eq. (15) simplifies to
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
Fig. 22. jμrj for the DS;P -equitable mirror with C .95.
magnification factor behaves similarly to the magnification factor for the D2 -equitable mirror when C < 1, as shown in Fig. 23. The performance of the DS;P -equitable mirror in our three test rooms can be seen in Figs. 24–26. According to the angular and checkerboard test rooms, it is inferior to the D2 -equitable mirror, even
y00 − y 1 − 2r2 70
which can be solved explicitly:
mf
60
yr 3 2r2 C1 er C2 e−r :
50
Unfortunately, the DS;P functional is discontinuous at C 1. The total distortion of the solution above is approximately 4.12, which is significantly larger than that of a parabola. Moreover, as shown in Fig. 22, the complex dilatation remains bounded above by 1 (and below by −1), except when C 1 in which case μ1 1. The 3.0
40 30 20 10 0
r 0.2
0.4
0.6
0.8
1.0
Fig. 23. jmf rj for the DS;P -equitable mirror with C .95.
z
2.5
2.0
1.5
z
f r
1.0
0.5
r 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 20. Equitable mirror for DS;P with C .95.
1.8
Fig. 24. DS;P -equitable mirror in angular test room (with C .95).
DS,P
1.6 1.4 1.2 1.0
Parabola
Circle
0.8
C 0.92
0.94
0.96
0.98
1.00
Fig. 21. DS;P distortion as a function of C.
Fig. 25. DS;P -equitable mirror in checkerboard test room (with C .95). 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8479
where p ≥ 1. The ensuing Euler–Lagrange equations are significantly difficult, but for the sphere one readily computes Dp yS Fig. 26. DS;P -equitable mirror in the test room with equidistant spheres (with C .95).
though the angles are reasonably preserved. In particular, areas are severely distorted by this mirror as shown in Fig. 26, where the image of the center sphere is effectively reduced to a point. It should be noted that the analogously defined functional Z DP;S y
1 0
can be (locally) minimized via identical methods. However, the minima fail to feature less distortion than a spherical mirror. 4. Summary of Results and Future Work
We summarize our main results as follows. 1. Spherical and pseudospherical mirrors offer the widest fields of view among all equi-areal mirrors with restricted angle distortion (as established in Theorem 2.2). 2. Spherical and pseudospherical mirrors offer the least amount of angle distortion among all equi-areal mirrors with a specified field of view (as established in Corollary 2.4). 3. The unexplored problem of simultaneously minimizing multiple kinds of distortion can be elegantly posed and solved via distortion functionals and standard calculus-of-variations techniques. 4. The equitable mirror obtained by (numerically) minimizing the D2 functional is one solution to the problem of simultaneously managing angle and area distortion. This new mirror performs quite well in various test rooms. There are many directions in which to continue this research. Perhaps the most obvious concerns the development of other distortion functionals; different applications and problems will motivate different distortion and cost functionals. For instance, one can easily generalize the definition of D2 to the functional Dp defined by Z Dp y≔
8480
1 0
jry0 y2 − 1jp jry00 − y0 jp dr;
Since the Dp functionals become more sensitive as p → ∞, solutions to the associated Euler–Lagrange equations may produce equitable mirrors that significantly outperform current ones. It is possible that in order to solve certain problems both angle and area distortion need to be simultaneously minimized, but one kind of distortion is more important or highly prioritized than the other. In such a scenario, the functional Z
1 0
yp − y2 y0S − y0 2 dr
(16)
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
1 C2p : 1 4p
ry0 y2 − 12α ry00 − y0 2 dr
might be of service since it is more sensitive to angle distortion (for α > 1) than it is to area distortion. Alternatively, one may instead want to minimize different kinds of distortion altogether, including (but not limited to) distortions in distances and resolution. Additionally, the actual cost of producing a mirror can be factored in to functionals. For example, if we assume that smaller mirrors are less expensive to produce, then an expression like Z
1 0
s! 2 dr ry y − 1 ry − y 1−y 0
2
2
00
0 2
would be of use since it also takes into account the arc length of the mirror’s profile curve. This work was partially funded by the Center for Undergraduate Research (CURM) and NSF grant no. DMS-1148695. References 1. J. Chahl and M. Srinivasan, “Reflective surfaces for panoramic imaging,” Appl. Opt. 36, 8275–8285 (1997). 2. T. Conroy and J. Moore, “Resolution invariant surfaces for panoramic vision systems,” in Proceedings of the Seventh IEEE International Conference in Computer Vision (IEEE, 1999), pp. 392–397. 3. R. A. Hicks and R. K. Perline, “Equiresolution catadioptric sensors,” Appl. Opt. 44, 6108–6114 (2005). 4. G. I. Kweon, K. T. Kim, G. H. Kim, and H. S. Kim, “Folded catadioptric panoramic lens with an equidistance projection scheme,” Appl. Opt. 44, 2759–2767 (2005). 5. L. V. Ahlfors, Lectures on Quasiconformal Mappings (with additional chapters by C. J. Earle, I. Kra, M. Shishikura, and J. H. Hubbard, eds.), University Lecture Series (American Mathematical Society, 2006). 6. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Studies in Advanced Mathematics Series (Chapman & Hall/CRC, 2006).
St. Mary’s College of Maryland, 18952 E. Fisher Rd, St. Mary’s City, Maryland 20686, USA 2 3 4
e-mail: [email protected]
e-mail: [email protected]
e-mail: [email protected]
*Corresponding author: [email protected] Received 12 June 2014; revised 3 November 2014; accepted 10 November 2014; posted 14 November 2014 (Doc. ID 214027); published 17 December 2014
Mirror surfaces used in catadioptric sensors are often designed so as to minimize one particular kind of image distortion. In this article we explore some finer properties of equi-areal mirrors, those that feature no area distortion, and we propose novel ways to measure compound forms of distortion. Specifically, we develop new mirror surfaces with large fields of view that simultaneously minimize angular and areal distortion with respect to different cost functions. © 2014 Optical Society of America OCIS codes: (110.0110) Imaging systems; (230.4040) Mirrors; (060.3735) Fiber Bragg gratings. http://dx.doi.org/10.1364/AO.53.008471
1. Introduction
The main problem in catadioptric sensor design is to realize a given projection with a camera-mirror pair; that is, one seeks to determine one or multiple mirror surfaces that, when coupled with a camera, reflect the world in a desired way. A camera is a device that maps R3 to R2 via a specified projection. Our designs use simplistic but common models for cameras, namely perspective and orthographic models. A pinhole camera realizes perspective projection since all light rays pass through a common point to form an image. The distance between this pinhole and the image plane is the focal distance. Orthographic projection can then be viewed as a limiting case of perspective projection wherein the focal distance becomes infinite and light rays travel parallel to the optical axis. For the purposes of this paper, cameras are modeled via orthographic projection. Mirror surfaces can be designed so that the catadioptric sensor preserves certain geometrical 1559-128X/14/368471-10$15.00/0 © 2014 Optical Society of America
properties. For example, image-to-world mappings can be conformal, equi-areal, equidistant, equiresolution, etc., as in [1–4]. In efforts to achieve uniform resolution, for instance, Conroy and Moore designed catadioptric sensors that have solid angle-pixel density invariance [2], while Chahl and Srinivasan studied mirror surfaces that preserve a linear relationship between the angle of incidence of light onto the surface and the angle of reflection [1]. However, neither of these sensors feature a uniformly distributed correspondence between points in the image plane and points in the object sphere. Hicks and Perline address this in their work on the family of equi-areal mirrors, so named because they induce projections that uniformly rescale area [3]. In summary, then, catadioptric sensors are designed based on the needs of their users, and, as indicated by the preceding examples, when these needs involve minimizing a single kind of distortion much can be said. The equi-areal mirrors studied in [3] minimize areal distortion, for instance, while parabolic mirrors have long been known to minimize angle distortion—indeed, this result is re-derived and generalized in Theorem 2.1. Minimizing multiple types of distortion, however, has not been as popular. 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8471
In this paper we propose and analyze precise ways in which to measure and simultaneously minimize multiple kinds of distortion. The so-called equitable mirrors that we seek allow for both area and angle distortion but attempt to minimize their combined, overall impact. This problem overlaps with the “map maker’s problem” from cartography since a flat map of the earth cannot simultaneously preserve angles and rescale area, a fact that also follows from Gauss’ Theorem Egregium; indeed, a catadioptric generalization of this result appears in Theorem 2.2. As such, we first review useful and common notions of angle and area distortion. A.
Angle Distortion
As detailed in [5], there is a natural way in which to measure angle distortion for mappings between planar domains. A bijective mapping w:U⊆C → C from a subset of the complex plane into the complex plane is said to be K quasi-conformal if ∂w ∂w sup ∂z ∂¯z ≤ K: z∈U ∂w ∂w − ∂z ∂¯z
(1)
In Eq. (1) the quantity K is a fixed real number that satisfies K ≥ 1, and we have used the complex notation ∂w 1 ∂w ∂w −i ∂z 2 ∂x ∂y
and
∂w 1 ∂w ∂w i ; ∂¯z 2 ∂x ∂y
where i2 −1 and z¯ x − iy denotes the complex conjugate of z x iy. The theory of quasi-conformal mappings is typically developed for orientation-preserving functions but can be extended to orientation-reversing functions in a simple way: we say that an orientationreversing mapping, wz, is K quasi-conformal if ¯ its orientation-preserving conjugate, wz, is K quasi-conformal. The prototypical example of a K-quasi-conformal mapping is obtained by “shearing” the complex plane in a fixed direction. For instance, the function wz 2x iy stretches the complex plane by a factor of 2 along lines parallel to the x axis. In this example, the fraction appearing in Eq. (1) equals the constant 2 at every input z, making the mapping 2-quasiconformal, and intuition for this fact can be gained in several ways. First, one can note that at every point z the function wz stretches infinitesimal squares into infinitesimal rectangles with a lengthto-height ratio of 2. Alternatively, one can also note that at every point z the function wz stretches infinitesimal circles into infinitesimal ellipses whose major-axis is twice as long as its minor-axis. 8472
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
In general, angle-preserving or conformal mappings take infinitesimal circles to infinitesimal circles, but quasi-conformal mappings take infinitesimal circles to infinitesimal ellipses. In fact, the fraction appearing in definition (1) is directly related to the eccentricity of an infinitesimal ellipse centered at the point wz. If this quantity is one, then w preserves angles at z. If it is infinite, then w effectively destroys angles at z, and therefore fails to be quasi-conformal. Definition (1) can be rephrased using the complex dilatation of a mapping, μ, which is defined by
∂w ∂¯z μz : ∂w ∂z A mapping w is K quasi-conformal, then, if its complex dilatation satisfies ‖μ‖∞ k, and 1k ≤ K; 1−k where ‖μ‖∞ denotes the suprema of jμzj. Furthermore, angles are preserved when μ 0 and destroyed when jμj 1. The complex dilatation μ need only be a measurable, essentially bounded function, which allows the notion of quasi-conformality to apply to a much wider class of functions. For the purposes of this paper, though, we can restrict the definition to orientation-reversing and orientation-preserving diffeomorphisms—differentiable functions with differentiable inverses. One readily checks that the dilatation for an orientation-reversing map, w, is the ¯ reciprocal of the dilatation for w. A rotationally symmetric mirror surface with profile curve f r induces a planar mapping between the image plane and stereographically projected object spheres; this is accomplished by tracing reversed, orthographically projected light rays from the image plane onto an object sphere. As in [3], we use the sphere at infinity to model distant object spheres, referring to it as a “reflection sphere” since it keeps track of the directions of reflected light rays. Moreover, because our mirrors are rotationally symmetric, we can replace this sphere with the circle at infinity, parameterizing it as a unit circle. The dilatation associated to such a mapping under orthographic projection is then given by the formula μr
rf 00 r − f 0 r : rf 00 r f 0 r
(2)
More generally, if we produce a mirror surface by revolving the planar curve γt rt; zt about the vertical z axis, then the dilatation of its induced mapping is given by
μt
z00 r0 − z0 r00 r0 r − r0 3 z0 ; z00 r0 − z0 r00 r0 r r0 3 z0
(3)
where differentiation in Eq. (2) is taken with respect to r but in Eq. (3) it is taken with respect to t. For instance, surfaces of the form f r arp have constant μ p − 2∕p and therefore have K p − 1 (for p ≥ 2). These uniformly quasi-parabolic mirrors distort angles by a factor of p − 1 at every point. B.
Area Distortion
A catadioptric sensor is called equi-areal if its magnification factor, denoted by mf , is constant. This quantity compares the area of a region on the reflection sphere to the area of its projection on the image plane. Hicks and Perline compute the magnification factor of a sensor under orthographic projection by considering two concentric circles of radii r and r Δr around the optical axis [3]. The infinitesimal annulus with area πr Δr2 − r2 is then mapped to the topological annulus on the sphere, bounded by ϕ and ϕ Δϕ, whose area is π1 − cosϕ Δϕ− 1 − cosϕ. The limiting ratio of these two areas, taken as Δr → 0, is the magnification factor mf lim
Δr→0
πcosϕ − cosϕ Δϕ sinϕ dϕ : 2r dr πr Δr2 − r2
Using the fact that ϕ 2 arctanf 0 r, we obtain the following expression for the magnification factor: mf
2f 0 rf 00 r ; r1 f 0 r2 2
(4)
which, as noted in [3], is a multiple of the revolved surfaces’ Gaussian curvature. 2. Preserving Area or Angles
For completeness sake, we begin with a brief but thorough treatment of conformal mirrors and then discuss equi-areal mirrors. Although it is well known that the only rotationally symmetric, conformal mirror is the paraboloid, Theorem 2.1 classifies all conformal mirror surfaces. Theorem 2.1. The only angle-preserving mirrors (with respect to orthographic projection) are graphs of harmonic functions z f x; y and graphs of the quadric surfaces z ax − x0 2 y − y0 2 d; where a, d ∈ R and x0 ; y0 ∈ R2 are fixed. Proof. After parameterizing a portion of the mirror surface as a graph, z f x; y, one finds that the angle-preserving properties are determined by the mapping gx; y ∇f x; y. This follows since stereographic projection is a conformal mapping between the sphere at infinity and the plane. In particular, our mirror preserves angles if and only if
∂g 0 ∂¯z
∂g 0; ∂z
or
where the first equation implies g is orientationpreserving, and the second one implies it is orientation-reversing. For the orientation-preserving case, we find ∂g 0⇔f xx f yy ∂¯z
and
f xy −f yx :
From these equations it easily follows that f x; y ax − x0 2 ay − y0 2 d; where x0 ; y0 ; d ∈ R are fixed. Analogous computations confirm that when g is an orientation-reversing, angle-preserving map the function f x; y must satisfy f xx −f yy ; f xy f yx : In other words, f x; y is harmonic. □ Of course, large portions of harmonic graphs are not viable as mirror surfaces since they can cause self-reflections. In addition, the only rotationally symmetric harmonic functions are unbounded near zero as they are given by f r c ln r, where c ∈ 0; ∞ determines the mirror’s field of view at r 1. As a result, we will use paraboloids as the premiere example of angle-preserving mirrors, and after vertically and horizontally shifting a parabolic mirror, it can be assumed to have the more familiar form f r ar2 . For the remainder of the paper, then, we use r ∈ 0; ∞ as our independent variable and use y to denote the height coordinate of the reflected light ray on the circle at infinity. Before developing equitable mirrors that efficiently interpolate between parabolic mirrors and equi-areal mirrors, we first collect some facts about the “extreme cases.” That is, we analyze the area distortion caused by a large field-of-view paraboloid and then summarize the angle distortions caused by large field-of-view equi-areal mirrors. The magnification factor for a parabolic mirror, f r ar2 , is given by mf r
8a2 : 1 4a2 r2 2
If our parabolic mirror is to have a large field of view over a finite interval r ∈ 0; r1 , then it will suffer significant area distortion. This follows from the fact that mf 0 8a2 → ∞
as a → ∞:
In order for f r to achieve a full field of view at r r1 the parameter a necessarily tends to infinity. Moreover, as mf 0 becomes unbounded, mf r → 0 for all other values of r. In summary, then, conformal 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8473
mirrors with large fields of view have highly nonuniform magnification. At the other end of the spectrum lie the constantGaussian curvature surfaces of revolution, mirror surfaces that Hicks and Perline showed are equiareal (see [3]). As categorized in [6], these surfaces fall into one of six families: spheres, bulges, spindles, pseudospheres, hyperboloid types, and conic types. The first three have positive Gaussian curvature and are generated by the curve
plane or a singularity. For instance, μB and μS have unit length when y 1, while μSp has unit length when y 1 − 2d. By restricting the field of view to avoid those points, where Eqs. (5)–(7) have unit length, one can limit the amount of angle distortion caused by an equi-areal mirror. These considerations lead us to the following Theorem 2.3. Let m ∈ 0; 1 be fixed. Of all (portions of) equi-areal mirrors with
s Z t∕a t s 2 2 γt b cos ; ds ; a − b sin a a 0
‖μ‖∞ ≤ m;
where a; b ∈ 0; ∞ are fixed. The latter three are orientation reversing mirrors generated by hyperbolic versions of γt; these surfaces are similarly determined by two, positive parameters a and b. The size of d b2 ∕a2 determines a particularly oriented surface’s type, with d 1 corresponding to the sphere and pseudosphere, d > 1 corresponding to the bulge and hyperboloid type, and d ∈ 0; 1 corresponding to the spindle and conic type. Using Eq. (3) and the explicit profile curves, γt, the dilatation μ is easily expressed in terms of the parameter d and the height coordinate, y, on the circle at infinity. These expressions are related in elementary ways. For the sphere and the pseudosphere one has μS y
1y μPs −y; 3−y
(5)
with y ∈ −1; 1 for the sphere and y ∈ −1; 1 for the pseudosphere. The dilatation for the bulge and hyperboloid type are given by μB y; d
4d − 3 2y y2 μH −y; d 1 4d − 1 2y − y2
(6)
with d > 1 and y ∈ −1; 1. Lastly, the spindle and the conic type have μSp y; d
4d − 3 2y y2 −μC −y; 1 − d 4d − 1 2y − y2
(7)
with d ∈ 0; 1 and y ∈ 1 − 2d; 1 for the spindle and y ∈ −1; 1 − 2d for the conic type. Note that neither the spindle nor the conic type is capable of achieving a full field of view, i.e., image a full hemisphere. These expressions allow us to easily prove a surprising generalization of the map maker’s problem that we call Theorem 2.2. (The mirror maker’s problem). There is no quasi-conformal, equi-areal mirror surface (of revolution). Proof. In each of the expressions above one can determine values of y where jμj 1. These occur at the specified end points for y and correspond to points where our mirrors either have a vertical tangent 8474
the sphere and pseudosphere offer the largest field of view. Proof. An elementary but careful analysis of the dilatations in Eqs. (5)–(7) lies at the heart of our proof. As indicated in Fig. 1, one first observes that to achieve ‖μ‖∞ m, values of y near the end points should be discarded. The key observation is that as d → 1, the surfaces of constant positive (resp. negative) Gaussian curvature tend to a sphere (resp. pseudosphere), and their fields of view improve as they do so. This phenomenon is easy to establish when comparing spheres and bulges (resp. pseudospheres and hyperboloid types). No matter what value we set for ‖μ‖∞ m, the sphere will always offer a larger field of view than the bulge. This is depicted in Fig. 1 where we have set ‖μ‖∞ 0.9. As indicated in Fig. 2 though, comparing the sphere and the spindle (resp. the pseudosphere and the conic type) is more subtle. The spindle can offer a larger field of view when y is sufficiently close to yf where jμyf j ‖μ‖∞ m. Solving Eq. (7) for y yields
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
y yμSp
q μSp − 1 2 1 μSp 2 d − d μSp 1
;
and from this we can conclude that the largest total field of view for the spindle will occur when we set Μ
0.9
1.0
Μ
0.8 0.6
Bulge
0.4 Sphere 0.2
y 1.0
0.5
0.5
1.0
Fig. 1. Dilatation jμj for sphere and bulge with d 1.5.
0.9
1.0 0.8 0.6
Spindle
0.4 Sphere 0.2
y 1.0
0.5
0.0
0.5
distortion, μ, and area distortion, mf , or comparing mirrors to those with desirable properties, like spheres and paraboloids. In much of what follows, our functionals are defined in terms of yr ∈ −1; 1, the height coordinate on the circle at infinity. In these cases, potential minimizers are determined by numerically solving pertinent Euler–Lagrange equations. The actual mirror surface is then obtained by determining its profile curve, f r, from the relation
1.0
Fig. 2. Dilatation jμj for sphere and spindle with d 1∕3.
p −m − 1 2 1 m2 d − d ; yi y−m 1−m p m − 1 2 1 m2 d − d : yf ym m1 Our spindles will offer a field of view given by θd arcsinym − arcsiny−m: A straightforward computation confirms that θ0 d > 0, and so the overall field of view increases as d → 1. This means that, given a fixed amount of angle distortion, the spindle’s total field of view is improved as it becomes more spherical. The relationships between the dilatations for different surfaces in Eqs. (5)–(7) allow us to make analogous conclusions for the negatively curved surfaces. Similar analysis of the dilatations allow us to conclude the converse of Theorem 2.3, which we state as a corollary: Corollary 2.4. Of all (portions of) equi-areal mirrors that achieve a fixed field of view, θ0 , the sphere and pseudosphere minimize ‖μ‖∞ . 3. Distortion Functionals
Given that equi-areal mirrors destroy angles and that parabolic mirrors suffer highly nonuniform magnification, it is natural, then, to determine mirrors that do a fairer or more equitable job of managing both kinds of distortion. We call such a surface an equitable mirror. Our basic strategy for determining these mirrors involves a two-step process. First, we devise an appropriate functional, D:V → R, where V is a relevant space of functions. Second, we find and analyze the critical points of D. To interpret Dv as measuring combined angular and areal distortion, the functional must satisfy basic properties. The functional should vanish, Dv 0, if and only if v is associated to an ideal mirror, one with zero angle distortion and zero area distortion. In accordance with Gauss’ Theorem Egrigium, then, we first and foremost require that Dv > 0 for all v ∈ V whose mirrors offer a nontrivial field of view. Moreover, formulas for D should be well-motivated, either incorporating notions of angle
y
f 0 2 − 1 : f 0 2 1
(8)
When determining the best or least-distortive conic in Subsection 3.A, our functional is instead phrased in terms of f r and is minimized via other, more elementary techniques. In any event, to set up a tractable and realistic equitable mirror problem we first assume that the camera’s sensor lies along some finite interval. Without loss of generality we can take this interval to be −1; 1, and, as a result, the spaces V involve sufficiently smooth functions with common domain [0, 1]. A. Most Equitable Conic
Because they are inexpensive to manufacture and include both the paraboloid and the sphere, conic sections provide us with a natural family of surfaces to explore. In particular, it is reasonable to suspect that an ellipse or a hyperbola can serve as an equitable mirror. To rigorously assess this possibility, we first let V denote the space of functions that are continuous on [0, 1] and differentiable on (0, 1). Of particular interest are those functions f r ∈ V whose graphs correspond to portions of conic sections. We assume that our candidate functions satisfy f 0 0 0 and f 0 1 c ∈ 0; ∞ so that a full (180°) field of view is obtained if and only if c ∞. For this problem we can use the rather sensitive functional, D0 , given by D0 f K f − 1 ‖m0f ‖∞ ;
(9)
where K f is the quasi-conformal factor defined in Eq. (1). We only apply D0 to graphs of parabolas, hyperbolas, and ellipses, ignoring linear functions since they destroy areas and angles under orthographic projection. Straightforward calculations confirm that one can monotonically decrease the D0 distortion of a hyperbola (for any fixed c ∈ 0; ∞) by making it more parabolic. Similarly, the D0 distortion of an ellipse decreases as the ellipse becomes more circular. With respect to D0 -distortion, then, the most equitable conic can be found by comparing the D0 distortions of parabolas to the D0 distortions of circles. Such a comparison is depicted in Fig. 3 where it is shown that, for undesirably small fields of view, the parabola offers less D0 distortion than the circle. 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8475
r2 y4 4ry3 − r2 y00 − 2ry0 2y3 y2 − 2y 1 0: (13) Equation (13) is of fourth order, and so we impose initial conditions
Circle
0.25
y0 0 0;
y0 −1;
Parabola
y1 C and 0.25
0.48
Fig. 3. D0 distortion of spherical and parabolic mirrors.
p Indeed, when c > 2725 5 ≈ 0.48, a spherical mirror offers less distortion. For all practical purposes, then, a circle is the best conic. It is natural to regard D0 as a sort of “goldstandard” distortion-functional, one that remains finite if and only if mirrors are simultaneously quasi-conformal and quasi-equi-areal. However, minimizing D0 over a wider class of functions is a difficult problem. Moreover, requiring that our distortion be point-wise bounded is unnecessarily restrictive. As a result, we focus efforts on minimizing other, equally motivated but more easily managed functionals. B.
Manageable Distortion
The dilatation and magnification factor for a rotationally symmetric mirror can be expressed in terms of yr as follows: μ
ry0 − 1 y1 − y ; ry0 1 y1 − y
(10)
1 y0 r : mf 2 r
(11)
Expressions (10) and (11) are obtained by using formulas (2), (4), and (8). Note that a mirror will be conformal if the numerator for Eq. (10) is zero, and it will be equi-areal if and only if the derivative of Eq. (11) also vanishes. This motivates use of the following functional: Z D2 y
1 0
ry0 y2 − 12 ry00 − y0 2 dr;
The first two conditions ensure that yr and f r are smooth, while the values of C and D control the field of view and magnification factor at r 1, respectively. Although the second variation of D2 is difficult to work with, we can numerically solve Eq. (13) and, likewise, numerically verify that these solutions (locally) minimize D2 ; the Runga–Kutta method is used by our algorithms, and we carry this out for large values of C. For instance, Fig. 4 depicts the relationship between D2 y and the choice of the magnification parameter, D. In this plot, we have set y1 C 1 so that our mirrors each achieve a full field of view. The minimum occurs at approximately D ≈ 3.64 where D2 y ≈ 1.60. For comparison, we note that a sphere with height coordinate yS r and field of view restricted by yS 0 −1 and yS 1 C has D2 distortion D2 yS
1 C4 : 9
When C is near 1, then, the sphere’s distortion is near 16∕9 > D2 y. Hence, for large fields of view, this equitable mirror will feature less D2 distortion. Figure 5 shows the profile curve for this mirror, and Figs. 6 and 7 display plots of jμrj and jmf rj, respectively. Although this equitable mirror attains a smaller amount of distortion than the sphere, it nonetheless destroys angles (when r 0 and when r 1). Furthermore, the magnification factor changes at high rates; in fact, m0f becomes unbounded near r 0. Fortunately, these features occur relatively close to r 0 where the catadioptric sensor’s camera images itself and therefore has a blind spot. On average this blind spot occupies approximately 1% of the image
(12) 50
where y is a sufficiently differentiable function. One readily checks that the D2 distortions of a parabola and hyperbola become unbounded as their fields of view increase. The D2 distortion of an elliptical mirror is strictly greater than that of a spherical one, and, since these findings mirror those for the D0 functional, this suggests that our D2 functional is similarly informative. In order to minimize D2 , we solve the associated Euler–Lagrange equation, which is given by 8476
y0 1 D:
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
D2
40
30
20
10
D 2
4
6
8
Fig. 4. D2 y as a function of D with C 1.
10
2.0
z
1.5
z
f r
1.0
0.5
r 0.2
0.4
0.6
0.8
1.0
Fig. 5. D2 -equitable mirror (13).
1.2 1.0 0.8 0.6 Circle 0.4 0.2
Equitable Mirror
r 0.2
0.4
0.6
0.8
angular distortion by examining how lines of latitude and longitude are imaged. Figures 9 and 10, for instance, show how parabolic and spherical mirrors perform in this room. As expected, the parabolic mirror images latitudinal lines as equally spaced, concentric circles. This corresponds to the fact that parabolic mirrors preserve angles. For comparison, Fig. 10 shows how a spherical mirror performs in this room: note that lines of latitude are imaged as unequally spaced, concentric circles. As shown in Fig. 11, the D2 -equitable mirror produces little angle distortion but offers a full field of view. The rectangular “checkerboard test room” is so named because, as seen in Fig. 12, it has walls that are checkered in different colors, and a sensor is located at its center. This also allows one to qualitatively determine how angles are distorted but also shows how lines are imaged by the catadioptric sensor. As a point of comparison, the performances of the parabolic and spherical mirrors are displayed in Figs. 13 and 14, respectively. As is seen in Fig. 15, the red-and-black colored floor of the checkerboard
1.0
Fig. 6. jμrj for the D2 -equitable mirror and sphere.
10
mf
8
6
4
Fig. 9. Parabolic mirror in the angular test room. 2
r 0
0.2
0.4
0.6
0.8
1.0
Fig. 7. jmf rj for the D2 -equitable mirror.
plane, and, as a result, our equitable mirror will, in actuality, feature very little angle distortion and mild area distortion. To better assess this equitable mirror’s strengths and weaknesses, we create three synthetic test rooms using POV-Ray, a ray tracing program. The first test room is spherical in shape and has a sensor located at its center, as shown in Fig. 8. This arrangement allows us to qualitatively assess a mirror’s
Fig. 8. Angular test room.
Fig. 10. Spherical mirror in the angular test room.
Fig. 11.
D2 -equitable mirror in the angular test room.
20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8477
test room is almost perfectly rescaled by the D2 equitable mirror. The third and final test room we employ has seven equal-sized spheres, each equidistant from the catadioptric sensor but of varying elevation, as seen in Fig. 16. This scene, as used in [3], allows the observer to determine how the sensor performs in terms of
preserving areas. With a true equi-areal sensor, then, the imaged spheres will be distorted in shape but will each contain the same amount of area. Performances of parabolic and spherical mirrors in this room are depicted in Figs. 17 and 18. Additionally, Fig. 19 shows that although there is variation in the areas of spheres imaged by the D2 equitable mirror, the change is mild. From these two qualitative analyses, we conclude that this D2 equitable mirror is superior to spherical and parabolic mirrors. Another functional that produces equitable mirrors is defined by Z DS;P y
Fig. 12. Checkerboard test room.
1 0
y − yS 2 y0 − y0P 2 dr;
(14)
where yS r and yP r denote the height coordinates for a sphere and parabola, respectively, with fields of view yS 1 yP 1 C. The Euler–Lagrange equation for DS;P y is y00 − y FC; r;
(15)
where Fig. 13. Parabolic mirror in the checkerboard test room.
Fig. 17. Parabolic mirror in the test room with equidistant spheres. Fig. 14. Spherical mirror in the checkerboard test room.
Fig. 15. D2 -equitable mirror in the checkerboard test room.
Fig. 16. Test room with equidistant spheres and a camera at the center. 8478
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
Fig. 18. Spherical mirror in the test room with equidistant spheres.
Fig. 19. D2 equitable mirror in the test room with equidistant spheres.
4C − 12 3C2 − 1r2 : FC; r 1 1 C −r2 1 r2 Cr2 − 13 We impose the boundary conditions y0 −1 and y1 C, and with extensive use of the exponential integral function, explicit solutions to this equation are available. Given their lengthy and cumbersome appearance, it is best to treat them as numerical solutions. The second variation for DS;P is easily computed and is always positive. This means that the critical points we find are local minima. Figure 20 depicts the profile curve for one such minimum. This second equitable mirror outperforms both the sphere and the paraboloid (in terms of DS;P distortion). In fact, the DS;P distortion of a sphere becomes unbounded as C → 1, while the distortion values for the paraboloid and the equitable mirror are shown in Fig. 21. Note that when C 1, Eq. (15) simplifies to
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
Fig. 22. jμrj for the DS;P -equitable mirror with C .95.
magnification factor behaves similarly to the magnification factor for the D2 -equitable mirror when C < 1, as shown in Fig. 23. The performance of the DS;P -equitable mirror in our three test rooms can be seen in Figs. 24–26. According to the angular and checkerboard test rooms, it is inferior to the D2 -equitable mirror, even
y00 − y 1 − 2r2 70
which can be solved explicitly:
mf
60
yr 3 2r2 C1 er C2 e−r :
50
Unfortunately, the DS;P functional is discontinuous at C 1. The total distortion of the solution above is approximately 4.12, which is significantly larger than that of a parabola. Moreover, as shown in Fig. 22, the complex dilatation remains bounded above by 1 (and below by −1), except when C 1 in which case μ1 1. The 3.0
40 30 20 10 0
r 0.2
0.4
0.6
0.8
1.0
Fig. 23. jmf rj for the DS;P -equitable mirror with C .95.
z
2.5
2.0
1.5
z
f r
1.0
0.5
r 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 20. Equitable mirror for DS;P with C .95.
1.8
Fig. 24. DS;P -equitable mirror in angular test room (with C .95).
DS,P
1.6 1.4 1.2 1.0
Parabola
Circle
0.8
C 0.92
0.94
0.96
0.98
1.00
Fig. 21. DS;P distortion as a function of C.
Fig. 25. DS;P -equitable mirror in checkerboard test room (with C .95). 20 December 2014 / Vol. 53, No. 36 / APPLIED OPTICS
8479
where p ≥ 1. The ensuing Euler–Lagrange equations are significantly difficult, but for the sphere one readily computes Dp yS Fig. 26. DS;P -equitable mirror in the test room with equidistant spheres (with C .95).
though the angles are reasonably preserved. In particular, areas are severely distorted by this mirror as shown in Fig. 26, where the image of the center sphere is effectively reduced to a point. It should be noted that the analogously defined functional Z DP;S y
1 0
can be (locally) minimized via identical methods. However, the minima fail to feature less distortion than a spherical mirror. 4. Summary of Results and Future Work
We summarize our main results as follows. 1. Spherical and pseudospherical mirrors offer the widest fields of view among all equi-areal mirrors with restricted angle distortion (as established in Theorem 2.2). 2. Spherical and pseudospherical mirrors offer the least amount of angle distortion among all equi-areal mirrors with a specified field of view (as established in Corollary 2.4). 3. The unexplored problem of simultaneously minimizing multiple kinds of distortion can be elegantly posed and solved via distortion functionals and standard calculus-of-variations techniques. 4. The equitable mirror obtained by (numerically) minimizing the D2 functional is one solution to the problem of simultaneously managing angle and area distortion. This new mirror performs quite well in various test rooms. There are many directions in which to continue this research. Perhaps the most obvious concerns the development of other distortion functionals; different applications and problems will motivate different distortion and cost functionals. For instance, one can easily generalize the definition of D2 to the functional Dp defined by Z Dp y≔
8480
1 0
jry0 y2 − 1jp jry00 − y0 jp dr;
Since the Dp functionals become more sensitive as p → ∞, solutions to the associated Euler–Lagrange equations may produce equitable mirrors that significantly outperform current ones. It is possible that in order to solve certain problems both angle and area distortion need to be simultaneously minimized, but one kind of distortion is more important or highly prioritized than the other. In such a scenario, the functional Z
1 0
yp − y2 y0S − y0 2 dr
(16)
APPLIED OPTICS / Vol. 53, No. 36 / 20 December 2014
1 C2p : 1 4p
ry0 y2 − 12α ry00 − y0 2 dr
might be of service since it is more sensitive to angle distortion (for α > 1) than it is to area distortion. Alternatively, one may instead want to minimize different kinds of distortion altogether, including (but not limited to) distortions in distances and resolution. Additionally, the actual cost of producing a mirror can be factored in to functionals. For example, if we assume that smaller mirrors are less expensive to produce, then an expression like Z
1 0
s! 2 dr ry y − 1 ry − y 1−y 0
2
2
00
0 2
would be of use since it also takes into account the arc length of the mirror’s profile curve. This work was partially funded by the Center for Undergraduate Research (CURM) and NSF grant no. DMS-1148695. References 1. J. Chahl and M. Srinivasan, “Reflective surfaces for panoramic imaging,” Appl. Opt. 36, 8275–8285 (1997). 2. T. Conroy and J. Moore, “Resolution invariant surfaces for panoramic vision systems,” in Proceedings of the Seventh IEEE International Conference in Computer Vision (IEEE, 1999), pp. 392–397. 3. R. A. Hicks and R. K. Perline, “Equiresolution catadioptric sensors,” Appl. Opt. 44, 6108–6114 (2005). 4. G. I. Kweon, K. T. Kim, G. H. Kim, and H. S. Kim, “Folded catadioptric panoramic lens with an equidistance projection scheme,” Appl. Opt. 44, 2759–2767 (2005). 5. L. V. Ahlfors, Lectures on Quasiconformal Mappings (with additional chapters by C. J. Earle, I. Kra, M. Shishikura, and J. H. Hubbard, eds.), University Lecture Series (American Mathematical Society, 2006). 6. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Studies in Advanced Mathematics Series (Chapman & Hall/CRC, 2006).
