Designing Cat Imaging Systems - Swainathan_PAMI_03

8/10/2019 Designing Cat Imaging Systems - Swainathan_PAMI_03

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A Framework for Designing General Catadioptric

Imaging and Projection Systems

Rahul Swaminathan, Shree K. Nayar and Michael D. Grossberg

Department of Computer Science, Columbia University

New York, New York 10027

Email: {srahul, nayar, mdog}@cs.columbia.edu

This research was supported by the DARPA HID Program under Contract No. N00014-00-1-0916.

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Abstract

A key problem in designing catadioptric systems is finding the mirror shape that, together with

some knownprimary opticsprovides the required imaging geometry. Previous methods for mirror

design usually derived partial differential equations(PDEs) from the imaging geometry require-ments that constrained the mirror shape. Then, analytic or approximate solutions to these PDEs

for the mirror shape were sought. Therefore, previous mirror designs used case-specific methods

requiring considerable designer interaction and skill. In this paper we present we present a fully

automaticmethod to determine the mirror shape for a large class offlexiblecatadioptric imaging

systems. Flexibility is achieved by allowing the user to specify a map from pixels in the image to

points in the scene, called the image-to-scene map. Also theprimary opticsthat constitute the

catadioptric system can be general and are not restricted to perspective or orthographic projection

based lenses.

We model arbitrary mirror surfaces using tensor-product splinesand show that the parameters

of the spline can be efficiently computed by solving a set of linear equations. Although we focus

on the design of imaging systems, our framework is directly applicable to the design of projector

systems as well. Furthermore, the systems we can design, may not even have a single viewpoint.

In such cases a locus of viewpoints called a causticis formed. We also present a simple method to

compute these caustics using our framework. We demonstrate the effectiveness of our approach

by computing the mirror shapes for both catadioptric imaging and projector systems as well as

their caustics. These also include previously designed systems for comparison with ground truth.We believe that the method we propose is a powerful tool that is general and throws wide open

the space of imaging and projector systems that can be designed.

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Contents

1 Design of Catadioptric Imaging Systems 3

2 Image-to-scene Map using Catadioptrics 6

2.1 Modeling the Primary Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Modeling the Mirror Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 The Image-To-Scene Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Computing the Mirror Shape 9

3.1 Normals Known: A Linear Method . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 The General Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Caustics of Catadioptric Systems 12

5 Example Mirror Designs 13

5.1 Parabolic Mirror based Single Viewpoint System . . . . . . . . . . . . . . . . . . . 15

5.2 Elliptical Mirror based Single Viewpoint System, . . . . . . . . . . . . . . . . . . 16

5.3 Equi-Angular Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.4 Plane Rectifying Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.5 Cylindrical Panorama Imaging System . . . . . . . . . . . . . . . . . . . . . . . . 20

5.6 Conference Table Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.7 Skewed Plane Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Conclusions 24

A Weighting for Image/Scene Error Metric 25

A.1 Relating Weights to Mirror Normal . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A.2 Scene Error from Perturbation of Mirror Normal . . . . . . . . . . . . . . . . . . . 28

A.3 Mapping Errors to the Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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B Computing Caustics 29

C Practical Issues with Solving Large Linear Systems 30

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1 Design of Catadioptric Imaging Systems

Consider the light field associated with a scene. A perspective imaging system immersed in the

scene performs a very special type of sampling of the light field. In recent years, researchers in

the fields of computer vision and computer graphics have begun to realize that the light fieldcan be sampled in alternative ways to produce new forms of visual information. The time has

therefore arrived to broaden the notion of a camera. Such a notion should be more than just an

advanced implementation of the camera obscura. It should be viewed as a device that can sample

thelight-field[13]in a manner that is most suited to the application at hand.

consider for a moment the problem of acquiring panoramic images. This is usefull for many

applications including omnidirectional surveillance as well as for video-conferencing. Typical om-

nidirectional systems acquire images as shown in Fig. 1(a). However, this acquired image appears

distorted and necessitates the use of computational resources to create undistored cylindrical

panoramas. What if we could design a sensor that acquired such an image directly as shown

in Fig. 1(b), by optical warping the scene rays. Similarly, for surveillance applications, one can

develop a camera that makes the faces of people standing in a hallway or seated in an auditorium

have the same size in the image, irrespective of where the people are located in the scene (see

Fig. 1(c,d)). As our final example we consider the application of machine inspection of known ob-

jects such as golf balls. The area of the dimples in the image acquired with a perspective imaging

system, shown in Fig. 1(e), varies across the image due to foreshortening. We would ideally like to

image the golf-ball uniformly such that these foreshortening effects are minimized (see Fig. 1(f)).

Clearly, there are many applications that benefit from sampling the light-field in unconventional

ways. One might imagine that such sampling of the light field can be accomplished by designing

an appropriate imaging lens. In practice, however, this is not the case. Lenses with exotic shapes

generally produce strong undesirable optical aberrations. Furthermore, fabrication of such lenses

if very difficult. As a result, the class of light field samplings that can be done using just lenses

is very restricted. Mirrors however do not suffer from these shortcomings. For this reason, recent

years, there has been significant interest in catadioptric imaging systems which use a combination

of lenses and mirrors. The use of mirrors enables a designer to manipulate the sampling function

of the camera in a flexible manner without compromising image quality. Catadioptrics, therefore,

significantly broadens the class of cameras that can be realized in practice.

A variety of catadioptric imaging systems have been proposed over the last century. Much of

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TypicalView

(c)(e)

(d)(f)

DesiredView

Machine InspectionPanoramic Imaging

(a)

(b)

Surveillane

Figure 1: A comparison between the typical images acquired with conventional imaging systems and

desired flexible imaging systems. (a) An image acquired with a conventional para-catadioptric imaging

system[21]for omnidirectional imaging. Cylinfrical panoramic views can be computed, but necessitate

the use of computational resources. (b) We would rather directly acquire such an image using the optics

alone[15], to ensure image quality and freedom from computational devices. (c) A perspective view of

audience in an auditorium. Due to increasing depth from the imaging system, the peoples faces decrease

in size in the image. If the faces occupied the same number of pixels, as shown in (d), in the image,

applications such as face recognition could function just as well for people seated far from as well asclose to the sensor. (e) Another example of a perspective view of a golf ball. The image of the dimples

on the ball decrease in size due to the curvature induced foreshortening. In applications such as machine

inspection, we would like to view the ball surface at uniform resolution. Thus all the dimples should

occupy the same number of pixels in the image irrespective of their location, as in (f). We present a

novel framework to automatically design such flexible catadioptric imaging systems.

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this work has focused on the design of telescopes [19] and microwave antennae [9]. In the last

decade, there has been considerable interest in developing catadioptric systems for computer

vision, computer graphics and photography. These systems are assumed to consist of some known

primary opticsand an unknown mirror. The key problem then is to determine the mirror shape

that implements the required image-to-scene map (see [23, 27, 28, 8, 3, 21, 10, 16, 6, 5, 12,

18]for examples). The complete class of single-mirror and two-mirror systems that have a single

viewpoint (center of projection) have been derived[1, 22]. In addition, several non-single viewpoint

systems[8, 3, 10, 16, 2, 11, 6, 20, 12, 18]. have been designed that perform specific image-to-scene

maps. In each of these cases, the approach has been to use the constraints imposed by the image-

to-scene map to derive partial differential equations(PDEs) that the mirror must satisfy. Then,

analytical solutions for the PDEs are sought. This approach is cumbersome and case specific,

making it hard to generalize.

It is therefore highly desirable to have a method for deriving mirror shapes for arbitrary image-to-

scene maps. Hicks was the first to pose this general problem[17]using geometric distributions.

He also presented a novel method to test if a mirror that implemented the desired map exists.

However, Hicks approach requires different analytical tools to be used and an appropriate numer-

ical method to be chosen for each case of mirror design. That is, although the formulation is a

general one, each mirror derivation requires the user to guide the process by imposing constraints

(generally PDEs) that are particular to the specific problem at hand.

The goal of this paper is to develop a completely automatic method to design catadioptric systems

for arbitrary image-to-scene maps with no human intervention or effort. There are cases where a

desired mapping simply cannot be accomplished by any mirror shape; that is, a solution does not

exist. In such cases, our method computes a mirror that is optimal in the sense that it minimizes

image distortions. Due to all these attributes, our method throws wide open the space of cameras

that can be implemented in practice.

At the core of our method is a linear and yet highly flexible and efficient representation of the

shape of the mirror. This representation is based on tensor product splines. It enables us to obtain

the optimal mirror shape for any given scene-image mapping using a iterative linear estimation

procedure. When the scene lies at infinity, a single iteration of the method produces the desired

mirror shape. As a result, the proposed method is very efficient. In addition, it can handle a

variety of models for the primary optics including orthographic, perspective, and the generalized

imaging model [14]. Furthermore, since our method is general, it the systems we design need

not always have a single viewpoint. In such cases a locus of viewpoints called a caustic [4,

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26]is formed. We also present a simple method to compute these caustics using our framework.

We demonstrate the power of the method by computing the mirror shapes for previously developed

catadioptric cameras as well as new ones. In the case of previously derived mirrors, we compare

our results with the known mirror shapes. In each case, we also compute the caustic of the imaging

system. In fact, we present the caustics of many previously designed non-single viewpoint cameras

whose caustics have not been known until this time.

While our focus in this paper is on designing cameras, it is worth noting that our method is

directly applicable to the design of novel projection systems as well. By simply defining the

mapping of image pixels to points on an arbitrary projection screen, we can compute the desired

mirror shape. In this case, the only difference is that we wish to minimize errors on the display

surface instead of in the image plane. This is shown to be an easy modification of the proposed

method.

We believe that the method we propose is a powerful tool that is general and throws wide open

the space of projector and imaging systems that can be designed.

2 Image-to-scene Map using Catadioptrics

The flexibility of our framework comes from the fact that the designer can define a map from pixels

in the image to points in the scene, which we call the image-to-scenemap. The catadioptric system

we design must therefore implement this image-to-scene map. As with previous approaches, we

assume the catadioptric system to consist of some known primary opticsand anunknown mirror.

Given the image-to-scene map and a model for the primary optics, we determine the mirror shape

that best implements the map. Throughout this paper we will address the problem as one of

designing catadioptric imaging systems. The same analysis applies directly to projector design as

well.

2.1 Modeling the Primary Optics

The catadioptric system is assumed to consist of some known primary opticsand a mirror. For

example, the para-catadioptric camera[21]uses a telecentric lens (orthographic projection) along

with a parabolic reflector. Our framework accommodates the use of primary optics possessing

any projection model including: (1) perspective, which has a fixed viewpoint Sl (Fig. 2(a)), (2)

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(a) Perspective (b) Orthographic (c) Generalized

V (u,v)l

V (u,v)l

V (u,v)l

S (u,v)l

S (u,v)l

S (u,v)l

Figure2: The different models of the primary optics for which our method can compute a mirror shape.

(a) A perspective imaging model found in most projectors and imaging systems. (b) Orthographic

projection, typically obtained with telecentric lenses. (c) The generalized imaging model (caustic model)

[14], wherein each pixel can have its own associated viewpoint and viewing direction. This last model is

most general and allows the most flexibility in designing catadioptric systems.

orthographic, which has a fixed viewing direction V l (Fig. 2(b)), and (3) the generalized imaging

model [14, 26], which can have a locus of viewpoints and viewing directions (Fig. 2(c)). In this

general case, pixel (u, v) in the image possesses a viewpoint Sl(u, v) and a viewing direction

Vl(u, v). Thus, the primary optics could be a simple perspective lens or a complex catadioptric

system itself.

2.2 Modeling the Mirror Shape

We now describe our model for the mirror shape. For convenience, we express the mirror shape

Sr in terms of the model for the primary optics as:

Sr(u, v) = Sl(u, v) D(u, v)Vl(u, v), (1)

where D(u, v) is the distance of the mirror from the viewpoint surface. We model D using tensor

product splinesin order to facilitate simple and efficient estimation of the mirror shape. We define

D(u, v) to be:

D(u, v) =

Kf

i=1

Kg

j=1

ci,jfi(u)gj(v), (2)

where fi(u) and gj(v) are 1-D spline basis functions, ci,j are the coefficients of the spline model,

andKf Kg are the number of spline coefficients.We now have a simple linear model for the mirror surface, which can be locally smooth and yet

flexible enough to model arbitrary mirror shapes.

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Scene

Mirror

M(u,v)

V (u,v)l

V(u,v)r

S (u,v)r

N(u,v)r

Primary Optics

(u,v)I

Figure3: A catadioptric imaging system consisting of known primary optics and a mirror. In general, a

pixel (u, v) in the image maps to the scene point M(u, v) after reflecting at Sr(u, v) on the mirror. This

forces constraints on the surface normals Nr(u, v) of the mirror.

2.3 The Image-To-Scene Map

Fig. 3 shows a catadioptric system used to image some known scene. The primary optics can

be perspective, orthographic or the generalized projection model. The user provides a mapMfrom points (u, v) in the image I to pointsM(u, v) in the scene. The mirror surface Sr(u, v)implements the mappingM by reflecting each scene pointM(u, v) along the scene rayVr(u, v)into the primary optics, where:

Vr(u, v) = Sr(u, v) M(u, v)|Sr(u, v) M(u, v)| . (3)

This constrains the surface normal of the mirror Nr as:

Nr(u, v) =

Vl(u, v)

Vr(u, v)

|Vl(u, v) Vr(u, v)| . (4)

We now have all the components needed by our framework to design any catadioptric system.

These include the general model for the primary optics, the spline based linear model for the

mirror surface and the image-to-scene map needed to describe the desired imaging system.

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3 Computing the Mirror Shape

We now present our method to compute the mirror shape for a general catadioptric imaging or

projection system. We begin by assuming that the surface normals of the mirror are known.

Later, we relax this constraint and present an iterative linear solution for the mirror shape forthe general case. In this case the normals are unknown and depend on the relative location of the

mirror and the scene.

3.1 Normals Known: A Linear Method

Many catadioptric systems are designed assuming the scene to be very distant (theoretically

at infinity) from the mirror. This is often used to design imaging systems (see [8, 16, 24], for

examples). In this scenario, the image-to-scene map essentially maps points in the image (u, v)to the reflected ray direction Vr(u, v). Since the primary optics are known, we can derive the

required mirror surface normals using Eq.(4) (see Fig.3).

From Eq.(1), the tangent vectors to the mirror surface are given by:

Tu=Sr

u , Tv =

Srv

. (5)

These tangents must be orthogonal to the normal in Eq.(4), providing two constraints on the

mirror shape:Sr(u,v)

u Nr(u, v) = 0,

Sr(u,v)v

Nr(u, v) = 0.(6)

Rearranging the terms and substituting Eq.(1) into Eq.(6), we get:

(Du

Vl+DVlu

) Nr = Slu Nr ,

(Dv

Vl+DVlv

) Nr = Slv Nr.(7)

Now, substituting Eq.(2) into Eq.(7), we get two new constraints:

Vl

i,j

ci,jf

i(u)gj(v) +Vl

u

i,j

ci,jfi(u)gj(v)

Nr =

Slu

Nr

Vl

i,j

ci,jfi(u)g

j(v) +Vl

v

i,j

ci,jfi(u)gj(v)

Nr =

Slv

Nr.

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where, fi(u) and gj(v) denote the partial derivatives offi(u) and gj(v), respectively. The above

constraints are linear in the spline coefficients ci,j and therefore can be re-written in the form:

A c= b, (8)

where,

A =

Vl Nrf0(u)g0(v) ... Vl Nrfi(u)gj(v) ... Vl NrfKf(u)gKj(v)+ + +

Vlu

Nrf0(u)g0(v) ... Vlu Nrfi(u)gj(v) ... Vlu NrfKf(u)gKj(v)

Vl Nrf0(u)g0(v) ... Vl Nrfi(u)gj(v) ... Vl NrfKf(u)gKj(v)

+ + +Vlv

Nrf0(u)g0(v) ... Vlv Nrfi(u)gj(v) ... Vlv NrfKf(u)gKj(v)

,

c =

c0,0 ... ci,j ... cKf,Kj

T,

b = Slu

Nr Slv NrT

.

Here, c represents the set of unknown coefficients c i,jof the spline, that models the mirror shape.

Every point (u, v) in the image provides two constraints. To solve forc, we need at least KfKg

2

image points at which the normals are known. We solve for the spline coefficients c, at the

resolution of the image which is an over-determined system of equations. This linear system is

formed by stacking the multiple constraints A, b, together to form A, b respectively, giving:

A c= b. (9)

The least-squares solution to Eq.(9) forc minimizes the algebraic errorin orthogonality between

the surface tangents and the desired normals. It however does not explicitly minimize the image

projection error

1

. Mirror computed using the above constraint minimize image errors only whenan exact solution to the mirror shape exists that implements the prescribed map. However, if no

exact solution for the mirror exists, then the computed mirror shape is not guaranteed to minimize

the image projection error. That is, minimizing the above algebraic error is not equivalent to

minimizing image projection errors.

1Note that for projector systems we must measure the scene projection error instead.

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Ideally, we should compute the mirror shape that minimizes error in the image. Generally speak-

ing, minimizing this metric makes the problem non-linear and unwieldy. For instance, a non-linear

search would require the estimation of hundreds of spline coefficients c. This is intractable and

not guaranteed to be free of local minima. We circumvent this problem by transforming the linear

system in Eqs.(8,9), into a new weighted system. We compute weights W for every equation at

every image point (see details of this computation in Appendix A). We then compute the mirror

shape by solving this set of linear weighted constraints for c as:

(WA)T(WA) c= (WA)TWb, (10)

So far we have derived a simple linear constraint to determine the mirror shape given the required

surface normals. The surface normals can be precisely known in cases when the scene is assumed

to be infinitely far away from the imaging system. This is a common case in many imaging system

designs, wherein we simply have a map from pixels in the image to viewing directions in the scene.

Also, we have presented a weighting scheme to bias the linear constraints in order to compute a

mirror shape that causes the least image distortion.

3.2 The General Iterative Method

In the previous section, we derived a simple linear method to compute the mirror shape given a

set of known surface normals. This is the case when the scene is very far from the sensor. We now

extend this method for the general case of arbitrary image-to-scene maps and scenes that can be

close to the sensor.

We first observe the mirror can lie anywhere within the field of view of the primary optics. Fig. 4

shows two mirrors Sr(1) and Sr

(2) at different locations, reflecting the scene pointM(u, v) intothe primary optics. As seen, the directionVr(u, v) along which the scene point M(u, v) is vieweddepends on the mirror location. This in turn influences the surface normals on the mirror and

hence its shape.

We resolve this cyclical dependency using an iterative method. We first approximate the mirrorby a set of facets (say, one for each pixel) on a plane, whose distance from the primary optics is

chosen by the designer. We then estimate the initial set of surface normals using Eqs.(3,4). These

normals are used to linearly solve for the mirror shape using Eq.(10). We now iterate as shown

in Fig. 6, by using the computed mirror shape in each iteration to obtain a better estimate of the

surface normals for the next iteration, until the mirror shape converges. Typically, convergence is

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MScene

Mirror 1

Mirror 2

Sr

Sr(2)

(1)

Vl

Vr(2)

(1)Vr(1)Nr

(2)Nr

Primary Optics

(u,v)I

Figure 4: The direction along which a prescribed scene point M(u, v) is viewed by the sensor depends

on the position of the point of reflection on the mirror. As shown here, changing the location from S(0)

to S(1) alters the reflection direction, thus changing the surface normal.

achieved within 10 iterations. An example of how the mirror shape evolves from the initial planar

guess to its final shape is shown in Fig. 5. In most cases, the mirror shape is already close to its

final shape after the first iteration.

4 Caustics of Catadioptric Systems

The catadioptric systems our framework can design are not restricted to single viewpoint systems

alone. Due to the generality of the framework, the designed system can actually possess a locus

of viewpoints described by their caustics [26]is formed. The caustic is essentially the envelope to

all scene rays imaged by our system.

Caustics are important as they completely describe the geometry of the catadioptric system. With

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Iteration: 5

X

48

12

5

10

10

20

Z

Initial guess (planar)

8

14

5

10

0

1

X

Z

Y

Iteration: 1

48

12

5

10

20

0

20

X

Z

Y

X

Iteration: 10 (Final iteration)

48

12

5

10

10

20

Z

YY

Figure 5: Convergence of the mirror shape designed for the skewed plane projection in Fig. ??(row 3).

(a) Initial planar guess for the mirror with facets having different surface normals. (b) After the first

iteration, the mirror shape is already close to the final solution. (c) After five iterations, the mirrors

changes a negligible amount showing fast convergence. (d) The final mirror shape obtained after 10

iterations.

respect to imaging systems they describe the effective viewpoint locus. For projector systems

caustics describe the effective projection model of the catadioptric projector (see Fig.2).

Our framework makes it very easy to compute the caustic of arbitrary imaging and projector

systems. In the Appendix B, we present details on computing caustics for general imaging and

projector systems using our framework.

5 Example Mirror Designs

We now use our method to compute the mirror shapes for different catadioptric imaging and

projector systems. These include previously designed systems as well as two novel designs. The

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START

Compute :

Estimate of mirror shape : S(2)

Input:Scene-Image map, Primary Optics Model,

Initial mirror depths (default planar)S(1)=

Compute :Change in mirror shape

S(1)- S

(2)

=

< Threshold

Yes

Output:

Mirror shape, spline parameters.Error in image pixels, Error in scene points.

STOP

Assign :

S(2)

S(1)

No

Compute :

Set of "desired" surface-normals : Nr

Figure 6: Flow-chart for the spline-based method to compute mirror shapes for general catadioptric

imaging systems. The user specifies a map between pixels in the image and corresponding scene points

as well as the geometry of the primary optics (shown in Figs. 2(a,b,c)). Using these as inputs, our method

computes the required mirror shape automatically.

previously designed systems we present include the parabolic and elliptic reflector based single

viewpoint systems [21, 1], the equi-angular system [8], the plane rectifying system [16] and the

cylindrical panorama imaging system[

15, 24]. The new systems we present include a novel imaging

system for tele-conferencing applications and a new catadioptric projector design.

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0 1

2

1

1

2

Ideal mirror.

Spline mirror.

(a) Comparison between Spline and Ideal Mirror (b) Computed Spline Mirror in 3D

(c) Image Projection Errors in Degress (view angle) (d) Caustic of Spline Mirror based Sensor

No Error

Telecentric

optics

40

4x 108 4

04

x 108

1

0 2

210-1-2

1

0

X

Y

Z

N/A

Figure7: Results of applying our general mirror design method for the para-catadioptric imaging system

[21]. (a) The mirror shape computed using our approach is identical to the ground truth solution

(parabolic mirror). (b) The 3D shape of the mirror shown for visualization. (c) The mirror design being

an accurate fit to ground truth, produced no image projection error. (d) The caustic, essentially a very

compact point cloud, conforms to the required single viewpoint.

5.1 Parabolic Mirror based Single Viewpoint System

We computed the mirror for a para-catadioptric [21]imaging system using a telecentric lens for

the primary optics (orthographic projection). The theoretical mirror shape for this system is

known to be parabolic. The mirror was designed by specifying an image-to-scene map such that

all the imaged rays pass through a virtual viewpoint located 1cmbelow the apex of the reflector.

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As seen in Fig. 7(a), the profile of the computed mirror using our method matches precisely with

the analytic parabolic profile. We also show the three dimensional shape of the computed mirror

in Fig. 7(b). Due to the precision of the recovered mirror shape, errors in terms of the reflected

ray directions are practically zero. The small perturbations in error we observed (on order of

108) were due to numerical imprecision issues. Therefore, we do not present an actual error plot

in fig. 7(c). Finally, the caustic surface (viewpoint locus) for this system was computed and found

to be a very compact cluster of points (essentially a single viewpoint).

5.2 Elliptical Mirror based Single Viewpoint System,

The previous design used a telecentric lens for its primary optics. However, one can also use

perspective lens based primary optics to design single viewpoint systems (see [1] for details).

Depending on the location of the mirror apex with respect to the effective viewpoint, the mirroris either hyperbolic or elliptical. If the viewpoint is behind the mirror, we expect a hyperbolic

mirror. Conversely, if the viewpoint lies in front of the mirror, we expect an elliptical mirror. We

now present the latter design.

The perspective lens based primary optics were assumed to lie 10 inches away from the mirror

apex. The effective viewpoint was constrained to lie behind the entrance pupil of the perspective

primary optics. Thus, entrance pupil of the primary sensor lies between the mirror surface and

the effective viewpoint. The computed mirror surface using our general method closely matches

the ground truth (see Fig. 8(a)). The computed mirror surface is also shown in Fig. 8(b). As

shown in Fig. 8(c) the errors in the resulting viewing directions of this sensor are quite small. The

effective viewpoint locus (see Fig. 8(d)) although not a single point is still small in extent.

5.3 Equi-Angular Sensors

We now present our results on computing the mirror shape for the equi-angular imaging system

developed by Chahl and Srinivasan [8]. This mirror provides a linear map between scene-ray

and reflected ray entering into the primary optics by controlling a single parameter, the angular

magnification.

The authors derived a family of mirror surfaces that conform to the required constraint by solving

a set of PDEs as:

We tested our method on various values of = 3, 5, 7 off which we present results for = 5. The

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Ideal mirror.

Spline mirror.


(c) Image projection errors in degress (view angle) (d) Caustic of Spline Mirror based Sensor

Max Error = 0.0028deg. Min Error = 0.0 deg

4.4 4.8

0.4

0.2

0

0.2

0.4

0.40.20-0.2-0.4

4.98

4.9

00.4

.005

.005

.005

-.005

4.05

4.3

X

Y

Z

X

Y

Z

Figure 8: Result of applying our general mirror design method for the elliptical mirror based single

viewpoint catadioptric imaging system[1]. (a) The mirror shape computed using our approach is identical

to the ground truth solution (elliptic mirror). (b) The 3D shape of the mirror shown for visualization.

(c) There were very small almost negligible errors in the viewing direction at every pixel as shown. (d)

We computed the caustic and although not a single point, is quite small in extent.

analytic solution in this case becomes:

x

x2 3y2

= x30. (11)

As shown in Fig. 9(a), the plot of the mirror computed using our general technique closely matches

the ground truth mirror shape (see Eq.(11)). The computed mirror shape in three dimensions is

shown in Fig. 9(b). The errors in viewing direction (scene-ray) across the entire image plotted

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7 9 11 13 15

3

1

1

3

Ideal mirror.

Spline mirror.



0 2

202

11

10

0.5

0

0.5

10.5

00.5

1

11.8

11.6

11.4


Figure 9: Results of applying our general mirror design method for the equi-angular imaging system

[8]. (a) The mirror shape computed using our approach is identical to the ground truth solution (see

Eq.(11)). (b) The 3D shape of the mirror is also shown for visualization. (c) The mirror design being an

accurate fit to ground truth, produced no image errors. (d) The caustic was computed using our general

frameowrk. Note this viewpoint locus as not computed until now.

in degrees is also shown to be negligible (see Fig. 9(c)). The regular pattern in the error plot isconjectured to be due to the piece-wise spline modeling. This catadioptric system does not have

a single viewpoint but rather a locus of viewpoints described by its caustic. We computed the

caustic using our general framework as shown in Fig. 9(d).

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(a) Setup for panoramic imaging system (b) Computed Spline Mirror in 3D

(c) Image projection errors in pixels. (d) Caustic of Spline Mirror based Sensor

Max Error = 0.03 pixels RMS Error = 0.01 pixles

X

Y

Z

X

Y

Z

02

210-1-2

1

0

6

0

6

60

-6

4

1

10 40 70 100

10

40

70

100

Primary Optics

(u,v)I

Ground plane

M(u,v)

Sought

mirror

Figure10: Results of applying our mirror design method for the plane-rectifyingimaging system[16]. (a)

The setup showing the mirror reflecting the ground plane into the primary optics. The image-to-scene

map is defined in Eq.(12). (b) The 3D shape of the mirror is shown for visualization. (c) The computed

mirror is very accurate and produces almost no image error. (d) The caustic of this system computed

using our general framework was alao was not computed until now.

5.4 Plane Rectifying Imaging System

Consider the scenario of imaging the ground plane with some imaging system. Most imaging

systems including those designed above, image the ground plane in a non-linear fashion. Thus,

distances in the image are not linearly related to distances in the ground plane. This property

is however desirable in applications such as autonomous robot navigation. We now present a

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catadioptric imaging system called the plane rectifying system[16]that achieves this goal.

As shown in Fig. 10(a), the mirror faces down towards the ground plane. The primary optics are

located below the mirror facing upwards. We know the ground plane to be precisely 34 inches

below the mirror as in[16]. Assuming the mirror apex to lie at the origin, and a scale factor of

= 54, we define the image-to-scene map as:

X= 54u, Y = 54v , Z= 34 , (12)

where u, v represent the image coordinates and X, Y, Z the scene coordinates.

Using the iterative method described in Section ??, we computed the mirror shape as shown in

Fig. 10(b). The errors in image projections of scene points are negligible as shown in Fig. 10(c).

Furthermore, the caustic of this system, shown in Fig. 10(d), was computed using our framework.

5.5 Cylindrical Panorama Imaging System

The omnidirectional catadioptric cameras designed above, acquire a panoramic image but need

computational resources to unwarp it into a panorama. Hicks and Srinivasan [15, 24] indepen-

dently presented a design of a catadioptric camera that acquires an optically unwarped panoramic

image.

In Fig. 11(a) we show a schematic to describe the desired imaging system. A telecentric lens is

used as the primary optics of the catadioptric system. The mirror then reflects the surroundingscene into the image directly to form a cylindrical panorama. Every image pixel (u, v) is mapped

directly to the required viewing direction Vr(u, v) of a panorama assuming the scene to be at

infinity. The horizontal field of view is a full 360 degrees, while the vertical field of view is 60

degrees.

Since the scene is assumed to lie at infinity, it is possible to compute the mirror shape using a

single iteration of our method. This shape is shown in Fig. 11(b). Only for certain aspect ratios

of the image does this image-to-scene posses an exact mirror. The aspect ratio we chose was close

enough to provide very small image distortions as shown in Fig. 11(c). Note we show the error in

the angle of the scene-ray being imaged. Also, we computed the caustic for this new and complex

catadioptric system using our general framework (see Fig. 11(d)).

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0

0.4

-0.4

0

-0.5

0.5 0.2

-0.2

0100

0 575

X

Y

0

0.2

-0.4

-0.8

0

0.8

-0.8

X

Y

Z

(a) Setup for panoramic imaging system (b) Computed Spline Mirror in 3D

(c) Image projection errors in degress (view angle) (d) Caustic of Spline Mirror based Sensor


X

Y

Z

0 2

0

/2

3/2

Primarysensor

Field of view(panorama)

Figure 11: Results of applying our general mirror design method for the cylindrical panoramic imaging

system[15, 24]. (a) The setup shows the mirror reflecting the entire panorama around the system into

the primary optics to directly form the cylindrical panorama in the image. (b) The 3D shape of the

mirror shown here closely resembles those obtained previously [15]. (c) The computed mirror exhibits

mimnimal viewing direction errors across the image. (d) The caustic was again coputed using our general

method. Again the caustic of this complex catadioptric system was not known until now.

5.6 Conference Table Rectification

We now present two new designs for catadioptric systems. Consider the scenario of imaging people

seated at an elliptic conference table. We would like to display the acquired image directly, such

that the curved edge of the table appears straight. Thus, all the people would appear as if seated

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0

0.4

-0.4

0

-0.5

0.5 0.2

-0.2

0100

0 575

X

Y

0

0.2

-0.4

-0.8

0

0.8

-0.8

X

Y

Z

(a) Setup for Panoramic Imaging System (b) Computed Spline Mirror in 3D



X

Y

Z

0 2

0

/2

3/2

Primarysensor

Field of view(panorama)

Figure 12: A novel imaging system designed using our mirror design method. (a) The mirror shape is

chosen so as to image people seated around a specified conference table so that thay appear as if seated

along a straight bench. In theory, a mirror hat implements this map need not exist. (b) We therefore

used the weighted scheme (see ?? for details) to compute the optimal mirror shape. (c) The computed

mirror shape does not exist and exhibits errors as shown. (d) Again we used our framework to also

compute the caustic surface.

along a desk. In general, we might wish to acquire images or video with some pre-determined

warp, so as to display them directly without the use of any computational resources.

We call the sensor used in such a conference table scenario, as the conference table rectifying

sensor. The setup of the mirror and table for the rectifying system are shown in Fig. 12(a).

The table consists of a semi-circle section of radius 30 with two extended straight sides, each 30

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long. The camera is assumed to lie roughly 30 behind this table facing away from the scene into

a mirror.

The computed mirror shape, is shown in Fig. 12(b). Note that no mirror shape exists that

provides the required image-to-scene map. The computed mirror shape approximates such a map

by minimizing image projection errors. It was therefore important to use the weighting scheme

discussed in Appendix (A) to design the mirror. The resulting image projection errors are shown

in Fig. 12(c). This imaging system also does not have a single effective viewpoint but rather a

locus of viewpoints shown in Fig. 12(d).

5.7 Skewed Plane Mapping

We now present a catadioptric projector design geared towards avoiding occlusions. Typical front

projection systems suffer from occlusions due to the user coming between the projecor and the

display surface. Techniques to reduce distortions [25] work to a limited extend and produce

artifacts at the shadow edges.

We propose to eliminate occlusions by positioning the projector very close to the display surface.

However, in such a configuration, the image needs to unwarped prior to projection. Digitally

warping the image leads to image quality degradation due to re-sampling. Furthermore, to project

onto large areas from such proximity is impossible using conventional projectors due to their

restricted field of view. We therefore have to optically warp the image using catadioptrics.

Referring to Fig. 13 (a), we note that the display surface lies 1 below and away from the projector

and spans a 10 10 square region. The projectors image plane was assumed to be parallel tothe XY-plane. The computed mirror shape using the weighted method, its associated scene

projection errors and caustic are shown in Fig. 13 (b,c,d). As expected from the setup, the mirror

is symmetric about a single plane. Note that, in spite of the projector being only a foot away

from a large screen, the errors in projection are negligible.

A point to note is that the same method was used to estimate all the mirror shapes. We could

compute single viewpoint as well as non-single viewpoint systems just as easily. Thus, makingour method truly flexible and applicable to a large space of mirror design problems.

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(a) Setup for Projector System. (b) Computed Spline Mirror in 3D

(c) Scene Projection Errors in Inches. (d) Caustic of Spline Mirror based Projector

Error : MAX = .02 (inches) RMS = .014 (inches)

1

1

Mirror-X

Z

Y

Desired Projector SystemImage

10

1010.5

0-0.5

1

0

0.4

0

0.50-0.5

4.5

3.5

2.5

4

3

2

Z

X

Y

Z

X

Y

10 40 70 100

10

40

70

100

Projection surface

Figure13: We used our mirror design method to also design a catadioptric projection system. Note the

same method can be directly used for projectors as well. Here we design a occlusion minimizing projector

by placing it very close to the wall. (a) We design a catadioptric projector system located 1 above and 1

away from a 10 10 screen. (b) The computed mirror shape is shown for visualization. (c) Projection

errors were measured in the scene (on the display) and are shown to be negligible. (d) The point of

projection is no longer a point byt rather a locus called a caustic. We computed this locus as well, using

pur general framework.

6 Conclusions

In this paper we presented a framework to design general catadioptric imaging and projector sys-

tems. Such systems are assumed to consist of some known primary optics and an unknown mirror.

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Using our framework, the designer has complete freedom in designing arbitrary catadioptric sys-

tems. One can define any desired imaging geometry, by specifying a map from pixels in the image

to points in the scene, which we called the image-to-scene map. Our method then computes the

optimal mirror shape that implements the desired image-to-scene map.

A major advantage of our method is that it is flexible enough to be used with all possible models

for the primary optics. These include the perspective, orthographic and the generalized imaging

model [14]. Furthermore, the same method can be used to design a very large class of mirror

designs including rotationally symmetric, asymmetric, single viewpoint or non-single viewpoint

systems.

We proposed the use of tensor-product splines to model mirrors. These are locally smooth as

well as flexible enough to model arbitrary mirror shapes. Using this framework, we presented a

simple linear solution to the mirror shape for a common class of image-to-scene maps, assuming

the scene to be at infinity. We also presnted a generalized iterative method for arbitrary image-

to-scene maps. Furthermore, linear constraints were weighted in order to determine the mirror

shape that minimized image or scene projection error. We also used our framework to compute

thecausticsof such general catadioptric systems. The caustics completelty describe the geometry

of the imaging or projection system. The caustic is needed for all vision tasks including stereo

and structure from motion.

We finally presented results by designing many catadioptric systems using our general method.

The systems included previously designed imaging systems for comparison with ground truth whenapplicable. Also, we presented a new imaging system with applications to video conferencing as

well as a new projector system for occlusion elimination. For each designed system we also

computed its caustic surface.

We believe that the method we proposed is a powerful tool that is general and throws wide open

the space of projector and imaging systems that can be designed.

A Weighting for Image/Scene Error Metric

The linear constraints derived in Eq.(9) measured an algebraic error between the surface tangents

(Tu, Tv) and the known surface normal Nr. However, in designing an imaging system we must

determine the mirror that minimizes image distortion rather than this algebraic error in sur-

face normals. Similarly, for catadioptric projector systems, the computed mirror must minimize

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projection errors on the display surface.

In general, the computed mirror shape Sr(u, v) approximates the desired image-to-scene map

M(u, v). We denote the map achieved using the mirror by(u, v). Then, the distortion (projectionerror) in the scene (u, v) for projector systems is:

(u, v) = |(u, v) M(u, v)|. (13)

Similarly, for imaging systems the image projection error is given by:

I(u, v) = |M1((u, v)) (u, v)|. (14)

Ideally our algorithm should be minimize these errors.

The above metric is extremely non-linear and intractable. Recall that we potentialy have to

estimate hunbdreds of spline coefficients. We therefore linearize the metric and solve for the

optimal2 mirror shape. The linearization is achieved by weighting the constraints in Eq.(9) as

used in Eq.(10).

Every image point provides two constraints on the mirror shape (see Eq.(7)). Our goal is to

compute two weights wu and wv to bias each of the constraint equations. We denote the weights

for projector systems by w,u, w,v, and for imaging systems by wI,u, wI,v respectively. These

weights scale the errors in Eq.(9) so that they approximate the scene error and the image error

Iat a point.

A.1 Relating Weights to Mirror Normal

Recall that Eq.(7) expresses the error in the direction of the mirror surface normals. To relate

this error to the image or scene error we consider small perturbations of the normal vector at a

point on the mirror. A change in the mirror normalNrvector results in a change in the scene-ray

Vr direction. In turn this causes the scene-ray to intersect the scene at a point other than the

desired point. If we consider a perturbation of the normal vector from the desired mirror normal

vectorNr(u, v) then the displacement in the scene is the scene error . For imaging systems, we

map this error onto the image to obtain the image error .

Revisiting Eq.(7), we observe that for an orthographic primary sensor, both the termsVl(u, v)/u

andVl(u, v)/v vanish. This implies that independent of the mirror shape specified by D(u, v),

2By optimal we mean the mirror minimizes the RMS image distortion or scene projection error.

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the tangent vectors Tu andTv lie in planes normal tou andv respectively, where:

u= Vl Slu

, v=Vl Slv

. (15)

Similarly, in the case of a perspective primary sensor, the partial derivatives ofSl(u, v) vanish. In

this case, the tangent vectors Tu and Tv are normal to:

u=Vl Vlu

, v=Vl Vlv

. (16)

Thus, the tangent vectors lie in two unique planes independant of the mirror shape. Furthermore,

the scene or image error due to purturbations of the normal vector can also be decoupled into two

independent components. In particular, the error given by the dot product ofNrwithTuand Tv

depend only on the projections ofNr onto the planes orthogonal to the vectors u and v given

byu andv, respectively. We therefore have:

Tu Nr = |Tu||u(Nr)| cos u , (17)Tv Nr = |Tv||v(Nr)| cos v . (18)

where, u and v are the angles between the tangent vectors and the normal vector, and u and

v are the projections onto u and

v respectively . We only present analysis for errors related

toTu. as the analysis of errors related to Tu is exactly the same.

For small perturbations in u, we have u= 2 + , where 0. Thus, Eq.(17) reduces to:

Tu Nr |Tu||u(Nr)|u ,

The weightwu,that relates the error in Eq.(19) to the scene or image error must satisfywu|u(Nr)|u, where the error , is either the scene errror or image error I. Therefore we define the

weights as:

wu = 1

|Tu|1

|u(Nr)|

u,

wv =

1

|Tv|1

|v (Nr)|

v . (19)

Notice that the above weight depends on three components: (1) the length of the tangent vector

|Tu|, (2) the projection ofNr onto u , and (3) dependence of error /u.The magnitude of|Tu| depends on the spline parameters that are yet to be determined. However,beyond the first iteration, the changes in the computed mirror shapes are negligible (see Section 3).

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We therefore remove dependence on |Tu| iteratively, by using the value of |Tu|, after one iterationin the next iteration. Also, we compute|u(Nr)| using Eqs.(15,16) as:

|u(Nr)| = Nr u Nr

|u

|2

u .

We must now determine the dependence of error /u on the angle u.

A.2 Scene Error from Perturbation of Mirror Normal

We first relate the angle u with the displacement in the scene using /u d/du. Wenote that u represents a rotation around the axis v. Let G(u) be the one parameter family

of rotations. The infinitesimal rotation on the normal vector is:

d

duG(u)Nr= u Nr. (20)

The change in the reflected ray due to rotation around the axis v is:

u = d

du(Vl 2(Vl G(u)Nr)G(u)Nr),

= 2(Vl (u Nr))Nr 2(Vl Nr)(u Nr) .

The change in the reflected ray produces a change in the projected scene point. This change is

computed by projecting u onto the tangent plane of a scene point and scaling by the distance

DM of the mirror to the scene. If the unit normal to the surface of the scene is NM then theinfinitesimal vector displacement in the scene is:

u= (u (NM u)NM)DM (21)

Simlarly, the error corresponding to Tv is:

v= (v (NM v)NM)DM (22)

For a projection system the change in errors d/du andd/dv are simply the lengths|u| and|v|, respectively.

A.3 Mapping Errors to the Image

For an imaging system we must compute weights that represent image error rather than scene

error. We must therefore inverse map the scene error component vectors u and v, onto the

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image plane. The image error vectors [a, b], [c, d] are related to the scene error as:

u= aMu+bMu , v=cMu+dMv ,

where,

Mis the image-to-scene map and

Mu=

Mu

and

Mv =

Mv

.

Assuming the mappingM is non-degenerate these equations can be solved since the scene dis-placement lies in the tangent plane to the scene, spanned by Muand Mv. To solve these equationwe take the dot products withMu andMv to get four scaler equations in four unknowns:

u Mu = a|Mu|2 +bMv Mu ,v Mu = c|Mu|2 +dMv Mu ,u Mv = aMu Mv+b|Mv|2 ,v

Mv = c

Mu

Mv+d

|Mv

|2 .

Re-writing these cosnstraints as a matrix, we solve for a, b, c, d as:

a c

b d

=

|Mu|

2 Mv MuMu Mv |Mv|2

1 u Mu v Mu

u Mv v Mv

. (23)

The change in the error dI/du is the length of the displacement associated to the change in

angle. Using Eq.(25) we see that:

wI,u

= (|

u

(Nr)|)1(

a2 +b2) (24)

wI,v = (|v(Nr)|)1(

c2 +d2) (25)

where, is a normalization factor,given by the sum of all the weights. Referring to Eq.(10),

W = [wI,u wI,v ]T.

B Computing Caustics

We now present a simple method to compute the caustics of catadioptric systems designed usingthe proposed spline based method. The use of splines to model the mirror shape also simplifies

the estimation of its caustic surface. We begin by deriving the caustic surface for the general

imaging system, and then describe ways to compute it numerically.

The caustic can be defined in terms of the reflector surface Sr(u, v) and the set of incoming light-

rays Vr(u, v) as it lies along Vr(u, v) at some distance rc from the point of reflection Sr(u, v)

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given by:

L(u,v,rc) =Sr(u, v) +rc Vr(u, v). (26)

In order to determine rc we employ the Jacobian method [7] by constraining the determinant of

the Jacobian of Eq.(26) to vanish:

det

Sr(x)(u,v)u

+rcVr

(x)(u,v)u

Sr(y)(u,v)u

+rcVr

(y)(u,v)u

Sr(z)(u,v)u

+rcVr

(z)(u,v)u

Sr(x)(u,v)v

+rcVr

(x)(u,v)v

Sr(y)(u,v)v

+rcVr

(y)(u,v)v

Sr(z)(u,v)v

+rcVr

(z)(u,v)v

Vr(x)(u, v) Vr

(y)(u, v) Vr(z)(u, v)

= 0 , (27)

where the superscripts of (x), (y) and (z) denotes the X-axis, Y-axis and Z-axis components,

respectively.

Note that, we do not know the analytic forms for Sr(u, v) andVr(u, v). To compute their partial

derivatives, we first fit splines to Sr(u, v) and Vr(u, v). Then, we compute the required partial

derivatives numerically. Thus, using the framwork of spline, we can easily determine the caustic

of any general projector or imaging system.

C Practical Issues with Solving Large Linear Systems

The size of the matrix formed by ATAin Eq.(10) isKf Kg, whereKfandKg are the number ofspline coefficients. Typically the system of equations can be very large, thereby inhibiting memory

efficient and fast computations. For instance, using a spline with 30 coefficients, produces a matrix

of size 900 900. However, as seen from Fig. 14, the matrix ATA is sparse. We therefore use asparse representation for the matrices to efficiently compute the spline parameters. Also, to solve

for the spline coefficients we cannot use typical techniques such assingular value decompositiondue

to the large non-sparse matrices formed. Hence we use iterative methods such as theBiConjugate

Gradients methodfound in Matlab.

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3542, 1998.

[2] R. Benosman, E. Deforas, and J. Devars. A New Catadioptric Sensor for the Panoramic

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0 50 100 150 200

0

50

100

150

200

Coefficients

Coefficients

Number of Zeros = 82%

Figure14: A binary image view of the matrix A Ashowing the non-zero elements as black points. Most

of this image is white indicating a majority of zero valued elements. We therefore use the sparse

representation to do our computations. This not only optimizes the use of memory, but facilitates faster

computations. This memory optimization is critical for large scale problems, where we need to compute

the mirror shapes at a high resolution.

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Designing Cat Imaging Systems - Swainathan_PAMI_03