effective pinhole, p=(0,c)

Transcription

1 Appeaed in the Poeedings of the 6th Intenational Confeene on Compute Vision, Bombay, Januay A Theoy of Catadiopti Image Fomation Simon Bake and Shee K. Naya Depatment of Compute Siene Columbia Univesity New Yok, NY 1007 Abstat Conventional video ameas have limited elds of view whih make them estitive fo etain appliations in omputational vision. A atadiopti senso uses a ombination of lenses and mios plaed in a aefully aanged onguation to aptue a muh wide eld of view. When designing a atadiopti senso, the shape of the mio(s) should ideally be seleted to ensue that the omplete atadiopti system has a single eetive viewpoint. In this pape, we deive the omplete lass of single-lens single-mio atadiopti sensos whih have a single viewpoint and an expession fo the spatial esolution of a atadiopti senso in tems of the esolution of the amea used to onstut it. We also inlude a peliminay analysis of the defous blu aused by the use of a uved mio. 1 Intodution Many appliations in omputational vision equie that a lage eld of view is imaged. Examples inlude suveillane, teleonfeening, and model aquisition fo vitual eality. Moeove, a numbe of othe appliations, suh as ego-motion estimation and taking, ould benet fom enhaned elds of view. Unfotunately, onventional imaging systems ae seveely limited in thei elds of view and so both eseahes and patitiones have had to esot to using eithe multiple o otating ameas in ode to image the entie sene. One eetive way to enhane the eld of view is to use mios in onjuntion with lenses (see, fo example, [Rees, 1970], [Chales et al., 1987], [Naya, 1988], [Yagi and Kawato, 1990], [Hong, 1991], [Goshtasby and Guve, 1993], [Yamazawa et al., 1993], [Bogne, 1995], [Nalwa, 1996], and [Naya, 1997]). We efe to the geneal appoah of using mios in ombination with onventional imaging systems as atadiopti 1 image fomation. As noted in [Rees, 1970], [Yamazawa et al., 1995], [Nalwa, 1996], and [Naya and Bake, 1997a], itis highly desiable that a atadiopti system (o, in fat, any imaging system) have a single viewpoint (ente of pojetion). The eason a single viewpoint is so desi- This wok was suppoted in pats by the DARPA/ONR MURI Gant N , an NSF National Young Investigato Awad, and a David and Luile Pakad Fellowship. 1 Dioptis is the siene of efating elements (lenses) wheeas atoptis is the optis of eeting sufaes (mios). The ombination of efating and eeting elements is theefoe efeed to as atadioptis [Heht and Zaja, 1974]. able is that it pemits the geneation of geometially oet pespetive images fom the image(s) aptued by the atadiopti ameas. These pespetive images an subsequently be poessed using the vast aay of tehniques developed in the eld of omputational vision whih assume pespetive pojetion. Moeove, if the image is to be pesented to a human, as in [Pei and Naya, 1997], it needs to be a pespetive image in ode to not appea distoted. In this pape, we begin in Setion by deiving the entie lass of atadiopti systems with a single eetive viewpoint and whih ae onstuted just using a single onventional lens and a single mio. As we will show, the -paamete family of mios whih an be used is exatly the lass of otated (swept) oni setions. Within this lass of solutions, seveal swept onis pove to be degeneate solutions and hene impatial, while othes lead to patial sensos. Duing ou analysis we will stop at many points to evaluate the meits of the solutions as well as the meits of losely elated atadiopti sensos poposed in the liteatue. An impotant popety of a senso that images a lage eld of view is its esolution. The esolution of a atadiopti senso is not, in geneal, the same as that of the sensos used to onstut it. In Setion 3 we study why this is the ase, and deive an expession fo the elationship between the esolution of a onventional imaging system and the esolution of a deived atadiopti senso. This expession should be aefully onsideed when onstuting a atadiopti imaging system in ode to ensue that the nal senso has suient esolution. Anothe optial popety whih is modied by the use of a atadiopti system is fousing. It is well known that a uved mio ineases image blu [Heht and Zaja, 1974], and so in Setion 4 we analyze this effet fo atadiopti sensos. Two fatos ombine to ause blu in atadiopti systems: (1) the nite size of the lens apetue, and () the uvatue of the mio. We st analyze how the inteation of these two fatos auses defous blu and then pesent some peliminay numeial esults fo one spei mio shape: the hypeboloid. The esults show that the foal setting of a atadiopti senso using a uved mio may be substantially dieent to that needed in a onventional senso. Moeove, ou analysis illustates a numeial method of omputing the elationship between the two 35

2 settings. image of wold point The Fixed Viewpoint Constaint The xed viewpoint onstaint is a equiement that a atadiopti senso only measue the intensity of light passing though a single point in 3-D spae. In othe wods, the atadiopti senso an only sample the 5-D plenopti funtion [Adelson and Begen, 1991] at a single point. The xed 3-D point at whih a atadiopti senso samples the plenopti funtion will be efeed to as the eetive viewpoint. Suppose we use a single onventional amea as the only sensing element and a single mio as the only eeting sufae. If the amea is an ideal pespetive amea and we ignoe defous blu, it an be modeled by the point though whih the pespetive pojetion is pefomed; i.e. the eetive pinhole. Then, the xed viewpoint onstaint equies that eah ay of light passing though the eetive pinhole of the amea (whih was eeted by the mio) would have passed though the eetive viewpoint, if it had not been eeted by the mio..1 Deivation of the Constaint Equation Without loss of genealitywe an assume that the effetive viewpoint v of the atadiopti system lies at the oigin of a atesian oodinate system. Suppose that the eetive pinhole is loated at the point p. Then, again without loss of geneality, we an assume that the z-axis ^z lies in the dietion ~vp. Moeove, sine pespetive pojetion is otationally symmeti about any line though p, the mio an be assumed to be a sufae of evolution about the z-axis ^z. Theefoe, we wok in the -D atesian fame (v; ^; ^z) whee ^ is a unit veto othogonal to ^z, and ty to nd the - dimensional pole of the mio z() z(x; y) whee p x + y. Finally, if the distane fom v to p is denoted by the paamete, we have ^v (0; 0) and ^p (0;). See Figue 1 fo an illustation of the oodinate fame. We begin the tanslation of the xed viewpoint onstaint into symbols by denoting the angle between an inoming ay fom a wold point and the -axis by. Suppose that this ay intesets the mio at the point (z; ). Then, sine we assume that it also passes though the oigin v (0; 0) we have the elationship: tan z : (1) In Figue 1 we have dawn the as though it wee othogonal to the z-axis ^z indiating that the optial axis of the amea is (anti) paallel to ^z. In fat, the eetive viewpoint v and the axis of symmety of the mio pole z() need not neessaily lie on the optial axis. Sine pespetive pojetion is otationally symmeti with espet to any ay that passes though the pinhole p, the amea ould be otated about p so that the optial axis is not paallel to the z-axis, and moeove the an be otated independently so that it is no longe pependiula to ^z. effetive pinhole, p(0,) effetive viewpoint, v(0,0) z ˆ α nomal γ β γ+β θ wold point mio point, (,z) x + y Figue 1: The geomety used to deive the xed viewpoint onstaint equation. The viewpoint v (0; 0) is loated at the oigin of a -D oodinate fame (v; ^; ^z), and the pinhole of the amea p (0;) is loated at a distane fom v along the z-axis ^z. If a ay of light, whih was about to pass though v, is eeted at the mio point (;z), the angle between the ay of light and ^ is tan,1 z. If the ay is then eeted and passes though the pinhole p, the angle,1,z it makes with ^ is tan, and the angle it makes with ^z is 90,. Finally, if tan,,1, dz d is the angle between the nomal to the mio at (;z) and ^z, then by the fat that the angle of inidene equals the angle of eetion, we have the onstaint that : If we denote by the angle between the eeted ay and the (negative) -axis, we also have: tan, z () sine the eeted ay must pass though the pinhole p (0;). Next, if is the angle between the z-axis and the nomal to the mio at the point (;z), we have: dz, tan : (3) d Ou nal geometi elationship is due to the fat that we assume the mio to be speula. This means that the angle of inidene must equal the angle of eetion. So, if is the angle between the eeted ay and the z-axis, we have 90 o, and o. 36

3 (See Figue 1 fo an illustation of this onstaint.) Eliminating fom these two expessions and eaanging gives, : Then, taking the tangent ofboth sides yields: tan 1, tan tan, tan 1 + tan tan : (4) Substituting fom Equations (1), (), and (3) and eaanging yields the xed viewpoint onstaint equation: (,z) dz d,( +z,z ) dz +(z,) 0: (5). Geneal Solution of the Constaint The st step in the solution of the xed viewpoint onstaint equation is to solve it as a quadati to yield an expession fo the sufae slope: dz (z,, z) p +(z +, z) : (6) d (z, ) The next step is to substitute y z, and set b whih yields: dy (y,, b ) p 4 b +(y +, b ) : d y (7) Then, afte substituting x y +, b and eaanging we have: d 1 dx p b + 1 x d : (8) Integating both sides with espet to esults in: ln x + p b + x ln + C (9) whee C is the onstant of integation. Hene, x + p b + x k 1 (10) whee k e C > 0 is a onstant. By bak substituting, eaanging, and simplifying we aive at the two equations whih ompise the geneal solution of the xed viewpoint onstaint equation: z,, k, 1 k, (k ) (11) 4 k z, + 1+ k + (k >0): (1) k 4 In the st of these two equations the onstant paamete k is onstained by k (athe than k>0) sine 0 <k< leads to omplex solutions..3 Spei Solutions of the Constaint Togethe, the two elations in Equations (11) and (1) epesent the entie lass of mios that satisfy the xed viewpoint onstaint. A quik glane at the fom of these equations eveals that the mio poles ae all oni setions. Hene, the 3-D mios themselves ae swept oni setions. Howeve, as we shall see, although evey 3 oni setion is theoetially a solution of one of these two equations, a numbe of them pove tobe impatial and only some lead to ealizable sensos. We will now desibe eah of the solutions in detail..3.1 Plana Mios In Solution (11), if we set k and >0, we get the oss-setion of a plana mio: z : (13) This plane is the one whih bisets the line segment ~vp joining the viewpoint and the pinhole. Moeove, by examining the othe solutions it follows that the pependiula biseto of ~vp is the only plana solution. An immediate oollay of this esult is that, fo a single xed pinhole, no two dieent plana mios an shae the same viewpoint. Unfotunately, a single plana mio does not enhane the eld of view, sine, disounting olusions, the same amea moved fom p to v and eeted in the mio would have exatly the same eld of view. It follows that it is impossible to inease the eld of view by paking an abitay numbe of plana mios (pointing in dieent dietions) in font of a single onventional imaging system, while still obeying the xed viewpoint onstaint. On the othe hand, in appliations suh as steeo whee multiple viewpoints ae a neessay equiement, the multiple views of a sene an be aptued by a single amea using multiple plana mios. See, fo example, [Goshtasby and Guve, 1993]. This bings us to the panoami amea poposed by Nalwa [1996]. To ensue a single viewpoint while using multiple plana mios, Nalwa [1996] has aived at a design that uses fou sepaate imaging systems. Fou plana mios ae aanged in a squae-based pyamid and eah of the fou ameas is plaed above one of the faes of the pyamid. The eetive pinholes of the ameas ae moved until the fou eetive viewpoints (i.e. the eetions of the pinholes in the mios) oinide. The esult is a senso that has a single eetive viewpoint and a panoami eld of view of appoximately The panoami image is of elatively high esolution sine it is geneated fom the fou images aptued by the fou ameas. Nalwa's senso is 3 Thee is one oni setion whih is an exeption: the paabola. Although the paabola is not a solution of eithe Equation (11) o Equation (1) fo nite values of and k, it is a solution of Equation (11) in the limit that!1, k!1, and k h, a onstant. As shown in [Naya and Bake, 1997a], this limiting ase oesponds to othogaphi pojetion. Moeove, in that setting the paabola does yield a patial omnidietional senso with a numbe of advantageous popeties [Naya, 1997]. 37

4 staightfowad to implement, but equies fou of eah omponent: i.e. fou ameas and fou lenses..3. Conial Mios In Solution (11), if we set 0 and k, we get a onial mio with iula oss setion: k, z : (14) See Figue fo an illustation of this solution. The angle at the apex of the one is whee tan p (k, ): This might seem like a easonable solution, but sine 0 the pinhole of the amea must be at the apex of the one. This implies that the only ays of light enteing the pinhole fom the mio ae the ones whih gaze the one and so do not oiginate fom (nite extent) objets in the wold (see Figue.) Hene, the one (with the pinhole at the vetex) is a degeneate solution of no patial value. image of wold point effetive pinhole, p(0,0) τ τ ˆ mio effetive viewpoint, v(0,0) wold point Figue : The onial mio is a solution of the xed viewpoint onstaint equation. Sine the pinhole is loated at the apex of the one, this is a degeneate solution of little patial value. If the pinhole is moved away fom the apex of the one (along the axis of the one), the viewpoint isno longe a single point but athe lies on a iula lous. If is the angle at the apex of the one, the adius of the iula lous of the viewpoint ise os, whee e is the distane of the pinhole fom the apex along the axis of the one. Futhe, if > 60, the iula lous lies inside (below) the one, if < 60 the iula lous lies outside (above) the one, and if 60 the iula lous lies on the one. The one has been used fo wide-angle imaging a numbe of times [Yagi and Kawato, 1990] [Yagi and Yahida, 1991] [Bogne, 1995]. In these implementations the pinhole is plaed quite some distane fom the apex of the one. It is easy to show that in suh ases the viewpoint is no longe a single point [Nalwa, 1996]. If the pinhole lies on the axis of the one at a distane e fom the apex of the one, the lous of the eetive viewpoint is a ile. The adius of the ile is easily seen to be eos : If > 60, the iula lous lies inside (below) the one, if < 60 the iula lous lies outside (above) the one, and if 60 the iula lous lies on the one..3.3 Spheial Mios In Solution (1), if we set 0 and k > 0, we get the spheial mio: z + k : (15) Like the one, this is a solution with little patial value. Sine the viewpoint and pinhole oinide at the ente of the sphee, the obseve only sees itself. The sphee has also been used to enhane the eld of view seveal times [Hong, 1991] [Bogne, 1995] [Muphy, 1995]. In these implementations, the pinhole is plaed outside the sphee and so thee is no single effetive viewpoint. The lous of the eetive viewpoint an be omputed in a staightfowad manne using a symboli mathematis pakage, but it is a quite omplex funtion of the distane between the ente of the sphee and the pinhole. Sphees have also been used in steeo appliations [Naya, 1988], but as desibed befoe multiple viewpoints ae a equiement fo steeo..3.4 Ellipsoidal Mios In Solution (1), when k>0 and >0; we get the ellipsoidal mio: 1 z, (16) whee: a e a e k + 4 b e and b e k : (17) The ellipsoid is the st solution that an atually be used to enhane the eld of view. As shown in Figue 3, if the viewpoint and pinhole ae at the foi of the ellipsoid, and the mio is taken to be the setion of the ellipsoid that lies below the viewpoint (i.e. z<0), the eetive eld of view is the entie uppe hemisphee z 0. It is also possible to ut the ellipsoid with othe planes passing though v, but it appeas thee is little to be gained by doing so..3.5 Hypeboloidal Mios In Solution (11), when k> and >0, we get the hypeboloidal mio: 1 z, 1, 1 (18) whee: a h a h k, k b h and b h k : (19) 38

5 image of wold point image of wold point effetive pinhole, p(0,) p(0,) wold point wold point ˆ v(0,0) Figue 3: The ellipsoidal mio satises the xed viewpoint onstaint when the pinhole and viewpoint ae loated at the two foi of the ellipsoid. If the ellipsoid is teminated by the hoizontal plane passing though the viewpoint z 0, the eld of view is the entie uppe hemisphee z>0. As seen in Figue 4, the hypeboloid also yields a ealizable solution. The uvatue of the mio and the eld of view both inease with k. In the othe dietion, in the limit k!, the hypeboloid attens out to the plana mio of Setion.3.1. Rees [1970] appeas to have been st to use a hypeboloidal mio with a pespetive lens to ahieve a lage eld of view amea system with a single viewpoint. Late, Yamazawa et al. [1993] [1995] also eognized that the hypeboloid is indeed a patial solution and implemented a senso designed fo autonomous navigation. 3 Resolution of a Catadiopti Senso In this setion, we assume that the onventional amea used in the atadiopti senso has a fontal image plane loated at a distane u fom the pinhole, and that the optial axis of the amea is aligned with the axis of symmety of the mio. See Figue 5 fo an illustation of this senaio. Then, the denition of esolution whih we will use is the following. Conside an innitesimal aea on the. If this innitesimal pixel images an innitesimal solid angle d of the wold, the esolution of the senso (as a funtion of the point on the at the ente of the innitesimal aea v(0,0) Figue 4: The hypeboloidal mio satises the xed viewpoint onstaint when the pinhole and the viewpoint ae loated at the two foi of the hypeboloid. This solution does podue the desied inease in eld of view. ) is: ˆ d : (0) If is the angle made between the optial axis and the line joining the pinhole to the ente of the innitesimal aea (see Figue 5), the solid angle subtended by the innitesimal aea at the pinhole is: d! os u os os3 u : (1) Theefoe, the esolution of the onventional amea is: u d! os 3 : () Then, the aea of the mio imaged by the innitesimal aea is: ds d! (, z) os os (, z) os u os (3) whee is the angle between the nomal to the mio at (;z) and the line joining the pinhole to the mio point (;z). Sine eetion at the mio is speula, 39

6 pixel aea, u solid angle, dω pinhole, p(0,) solid angle, dω mio point, (,z) image of wold point ψ optial axis ˆ viewpoint, v(0,0) nomal φ φ foal plane mio aea, ds solid angle, dυ wold point Figue 5: The geomety used to deive the spatial esolution of a atadiopti senso. We assume that the onventional senso has a fontal whih is loated at a distane u fom the pinhole and that the optial axis is aligned with the z-axis ^z. the solid angle of the wold imaged by the atadiopti amea is: d ds os + z (, z) os u ( + z ) : (4) Hene, the esolution of the atadiopti amea is: u ( + z ) ( + z ) os d (, z) os (, z) d! : (5) But, sine: we have: os d (, z) (, z) + (6) + z (, z) + d! : (7) Hene, the esolution of the atadiopti amea is the esolution of the onventional amea used to onstut it multiplied by a fato of: + z (, z) + (8) whee (;z) is the point on the mio being imaged. The st thing to note fom Equation (7) is that fo the plana mio z, the esolution of the atadiopti senso is the same as that of the onventional senso used to onstut it. This is as expeted by symmety. Seondly, note that the fato in Equation (8) is the squae of the distane fom the point (;z) to the eetive viewpoint v divided by the squae of the distane to the pinhole p. Using simple popeties of ellipsoids and hypeboloids it follows that fo these two mio shapes, the fato in Equation (8) ineases with. Hene both hypeboloidal and ellipsoidal atadiopti sensos will have highest esolution aound the peiphey, a useful popety fo etain appliations suh as teleonfeening. 4 Defous Blu of a Catadiopti Senso Two fatos ombine to ause defous blu in atadiopti imaging systems (that is, in addition to the nomal auses in onventional diopti systems, suh as diation and lens abeations): (1) the nite size of the lens apetue, and () the uvatue of the mio. In this setion we investigate this eet fo the hypeboloid mio. (Genealization to the othe mios is staightfowad.) We poeed by onsideing a xed point in the wold and a xed point in the lens. We nd the point on the uved mio whih eets a ay of light fom the wold point though the lens point. Then, we ompute whee on the this mio point is imaged. By onsideing the lous of imaged mio points as the lens point vaies, we an ompute the aea of the onto whih a xed wold point is imaged. To pefom this analysis we need to wok in 3-D. We use the 3-D atesian fame (v; ^x; ^y; ^z) whee v is the loation of the eetive viewpoint, p is the loation of the eetive pinhole, ^z is a unit veto in the dietion ~vp, and the vetos ^x and ^y ae othogonal unit vetos in the plane z 0. As befoe, we assume that the eetive pinhole is loated at a distane fom the eetive viewpoint. Moeove, as in Setion 3, we assume that the onventional amea used in the atadiopti senso has a fontal loated at a distane u fom the pinhole and that the optial axis of the amea is aligned with the z-axis. Finally, we assume that the eetive pinhole of the lens is loated at the ente of the lens and that the lens has a iula apetue. See Figue 6 fo an illustation of this onguation. Conside a point m (x; y; z) on the mio and a point w l (x; y; z) in the wold, whee l > kmk. kmk Then, sine the hypeboloid mio satises the xed viewpoint onstaint, a ay of light fom w whih is eeted by the mio at m passes dietly though the ente of the lens (i.e. the pinhole.) This ay oflight is known as the pinipal ay [Heht and Zaja, 1974]. Next, suppose a ay of light fom the wold point w is eeted at the point m 1 (x 1 ;y 1 ;z 1 ) on the mio and then passes though the lens point l (dos ; d sin ; ). In geneal, this ay of light will not be imaged 40

7 u l(d osλ,d sinλ,) v blu egion mio m (x,y,z ) viewpoint, v(0,0,0) pinhole, p(0,0,) lens wold point, w pinipal ay nomal, n, z+u foal plane, z l m m(x,y,z) (x,y,z) foused plane, z-v plane, z0 Figue 6: The geomety used to analyze the defous blu. We wok in the 3-D atesian fame (v; ^x; ^y; ^z) whee ^x and ^y ae othogonal unit vetos in the plane z 0. In addition to the assumptions of Setion 3, we also assume that the eetive pinhole is loated at the ente of the lens and that the lens has a iula apetue. at the same point on the as the pinipal ay. When this happens thee is defous blu. The lous of the intesetion of the inoming ays though l and the as l vaies ove the lens is known as the blu egion o egion of onfusion [Heht and Zaja, 1974]. Fo an ideal thin lens, the blu egion is iula and so is often efeed to as the blu ile [Heht and Zaja, 1974]. As is shown in [Naya and Bake, 1997b], fo a atadiopti senso the shape of the blu egion is not, in geneal, iula. If we know the points m 1 and l, we an nd the point on the whee the ay of light though these points is imaged. Fist, the line though m 1 in the dietion lm ~ 1 is extended to inteset the foused plane. By the thin lens law [Heht and Zaja, 1974] the foused plane is: z, v, f u u, f (9) whee f is the foal length of the lens and u is the distane fom the foal plane to the. Sine all points on the foused plane ae pefetly foused, the point of intesetion on the foused plane an be mapped onto the using pespetive pojetion. Hene, the x and y oodinates of the intesetion of the ay though l and the ae the x and y oodinates of:, u v l + v (m 1, l), z 1 (30) and the z oodinate is the z oodinate of the image plane + u. Given the lens point l (d os ; d sin ; ) and the wold point w l (x; y; z), thee ae thee onstaints on the point m 1 (x 1 ;y 1 ;z 1 ). Fist, m 1 must kmk lie on the mio and so we have: z 1,,, x 1 + y1 k, 1 k, : (31) 4 k Seondly, the inident ay(w, m 1 ), the eeted ay (m 1, l), and the nomal to the mio at m 1 must lie in the same plane. The nomal to the mio at m 1 lies in the dietion: n ([k, ]x 1 ; [k, ]y 1 ;, z 1 ) : (3) Hene, the seond onstaint is: n (w, m 1 ) ^ (l, m 1 ) 0: (33) Finally, the angle of inidene must equal the angle of eetion and so the thid onstaint on the point m 1 is: n (w, m 1 ) kw, m 1 k n (l, m 1) kl, m 1 k : (34) These thee onstaints on m 1 ae all multivaiate polynomials in x 1, y 1, and z 1 : Equation (31) and Equation (33) ae both of ode, and Equation (34) is of ode 5. We wee unable to nd a losed fom solution to these thee equations (Equation (34) has 5 tems in geneal and so it is pobable that none exists) but we did investigate numeial solutions. Some of the esults ae pesented in Figue 7. (The inteested eade is efeed to [Naya and Bake, 1997b] fo a moe omplete pesentation of the esults, inluding an explanation of the loal minima in Figue 7.) Fo the numeial solutions we set 1 mete, used the hypeboloid mio with k 4, and assumed the adius of the lens to be 5 entimetes. We onsideed the point m (0:15; 0:0; 0:15) on the mio and set the distane fom the viewpoint tothe wold point w to be l 5 metes. In Figue 7 we plot the aea of the blu egion (on the odinate) against the distane to the foused plane v (on the absissa). The smalle the aea of the blu egion, the bette foused the image will be. We see fom Figue 7 that the aea neve eahes exatly 0, and so an image fomed using this atadiopti senso an neve be pefetly foused. Howeve, it should be emphasized that the minimum aea is vey small, and in patie thee is no poblem fousing the image fo a single wold point. Moeove, is is possible to use additional oetive lenses to ompensate fo most of this eet [Heht and Zaja, 1974]. 41

8 Aea of Blu Region at Unit Distane (metes squaed) e-05 6e-05 4e-05 e-05 Hypeboloid Mio f u Distane to the Foused Plane v (in metes) u - f Figue 7: The aea of the blu egion plotted against the distane to the foused plane v fu fo a point m u,f (0:15; 0:0; 0:15) on the hypeboloid mio with k 4. In this example, we have 1 mete, the adius of the lens 5 entimetes, and the distane fom the viewpoint tothe wold point l 5 metes. The aea neve beomes exatly 0 and so the image an neve be pefetly foused. Howeve, the aea does beome vey small and so fousing on a single wold point is not a poblem in patie. It is inteesting to note that the distane at whih the image of the wold point will be best foused (i.e. somewhee in the ange 1.05{1. metes) is muh less than the distane fom the pinhole to the wold point (appoximately 1 mete fom the pinhole to the mio and then 5 metes fom the mio to the wold point). The eason fo this phenomenon is that the mio is onvex and so tends to inease the divegene of ays emanating fom the wold point. 5 Summay In this pape we have studied thee design iteia of atadiopti sensos: (1) the shape of the mios, () the esolution of the ameas, and (3) the fous settings of the ameas. In patiula, we have deived the omplete lass of mios that an be used with a single amea, found an expession fo the esolution of a atadiopti senso in tems of the esolution of the onventional amea used to onstut it, and pesented a peliminay analysis of defous blu. Refeenes [Adelson and Begen, 1991] E.H. Adelson and J.R. Begen. The plenopti funtion and elements of ealy vision. In Landy and Movshon, editos, Computational Models of Visual Poessing, hapte 1. MIT Pess, [Bogne, 1995] S. Bogne. Intodution to panoami imaging. In Poeedings of the IEEE SMC Confeene, pages 3100{3106, Otobe [Chales et al., 1987] J.R. Chales, R. Reeves, and C. Shu. How to build and use an all-sky amea. Astonomy Magazine, Apil [Goshtasby and Guve, 1993] A. Goshtasby and W.A. Guve. Design of a single-lens steeo amea system. Patten Reognition, 6(6):93{937, [Heht and Zaja, 1974] E. Heht and A. Zaja. Optis. Addison-Wesley, [Hong, 1991] J. Hong. Image based homing. In Poeedings of the IEEE Intenational Confeene on Robotis and Automation, May [Muphy, 1995] J.R. Muphy. Appliation of panoami imaging to a teleopeated luna ove. In Poeedings of the IEEE SMC Confeene, Otobe [Nalwa, 1996] V.S. Nalwa. A tue omnidietional viewe. Tehnial epot, Bell Laboatoies, Holmdel, NJ 07733, USA, Febuay [Naya and Bake, 1997a] S.K. Naya and S. Bake. Catadiopti image fomation. In Poeedings of the 1997 DARPA Image Undestanding Wokshop, pages 1431{1437, New Oleans, Louisiana, May [Naya and Bake, 1997b] S.K. Naya and S. Bake. A theoy of atadiopti image fomation. Tehnial Repot CUCS , Depatment of Compute Siene, Columbia Univesity, USA, Apil [Naya, 1988] S.K. Naya. Spheeo: Reoveing depth using a single amea and two speula sphees. In Poeedings of SPIE: Optis, Illumination, and Image Sensing fo Mahine Vision II, Novembe [Naya, 1997] S.K. Naya. Catadiopti omnidietional amea. In Poeedings of the 1997 Confeene on Compute Vision and Patten Reognition, pages 48{488, June [Pei and Naya, 1997] V. Pei and S.K. Naya. Geneation of pespetive and panoami video fom omnidietional video. In Poeedings of the 1997 DARPA Image Undestanding Wokshop, New Oleans, May [Rees, 1970] D.W. Rees. Panoami television viewing system. United States Patent No. 3,505,465, Apil [Yagi and Kawato, 1990] Y. Yagi and S. Kawato. Panoami sene analysis with oni pojetion. In Poeedings of the Intenational Confeene on Robots and Systems, [Yagi and Yahida, 1991] Y. Yagi and M. Yahida. Real-time geneation of envionmental map and obstale avoidane using omnidietional image senso with oni mio. In Poeedings of the 1991 Confeene on Compute Vision and Patten Reognition, pages 160{165, June [Yamazawa et al., 1993] K. Yamazawa, Y. Yagi, and M. Yahida. Omnidietional imaging with hypeboloidal pojetion. In Poeedings of the Intenational Confeene onrobots and Systems, [Yamazawa et al., 1995] K. Yamazawa, Y. Yagi, and M. Yahida. Obstale avoidane with omnidietional image senso HypeOmni Vision. In Poeedings of the IEEE Intenational Confeene on Robotis and Automation, pages 106{1067, May