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  Squaring the Circle in Panoramas Lihi Zelnik-Manor 1 Gabriele Peters 2 Pietro Perona 1 1. Department of Electrical Engineering 2. Informatik VII (Graphische Systeme)California Institute of Technology Universitat DortmundPasadena, CA 91125, USA Dortmund, Germany Abstract Pictures taken by a rotating camera cover the viewingsphere surrounding the center of rotation. Having a set of images registered and blended on the sphere what is left tobe done, in order to obtain a flat panorama, is projectingthe spherical image onto a picture plane. This step is unfor-tunately not obvious – the surface of the sphere may not be flattened onto a page without some form of distortion. Theobjective of this paper is discussing the difficulties and op- portunities that are connected to the projection from view-ing sphere to image plane. We first explore a number of al-ternatives to the commonly used linear perspective projec-tion. These are ‘global’ projections and do not depend onimage content. We then show that multiple projections maycoexist successfully in the same mosaic: these projectionsare chosen locally and depend on what is present in the pic-tures. We show that such multi-view projections can pro-duce more compelling results than the global projections. 1. Introduction As we explore a scene we turn our eyes and head andcapture images in a wide field of view. Recent advances instorage, computation and display technology have made itpossible to develop ‘virtual reality’ environments where theuserfeels‘immersed’inavirtualsceneandcanexploreitbymoving within it. However, the humble still picture, paintedor printed on a flat surface, is still a popular medium: itis inexpensive to reproduce, easy and convenient to carry,store and display. Even more importantly, it has unrivaledsize, resolution and contrast. Furthermore, the advent of in-expensive digital cameras, their seamless integration withcomputers, and recent progress in detecting and matchinginformative image features [4] together with the develop-ment of good blending techniques [7, 5] have made it possi-ble for any amateur photographer to produce automaticallymosaics of photographs covering very wide fields of viewand conveying the vivid visual impression of large panora-mas. Such mosaics are superior to panoramic pictures takenwith conventional fish-eye lenses in many respects: theymay span wider fields of view, they have unlimited reso-lution, they make use of cheaper optics and they are notrestricted to the projection geometry imposed by the lens.The geometry of single view point panoramas has longbeen well understood [12, 21]. This has been used for mo-saicing of video sequences (e.g., [13, 20]) as well as for ob-taining super-resolution images (e.g., [6, 23]). By contrastwhen the point of view changes the mosaic is ‘impossible’unless the structure of the scene is very special. When allpictures share the same center of projection C  , we can con-sider the viewing sphere, i.e., the unit sphere centered in C  , and identify each pixel in each picture with the ray con-necting  C   with that pixel and passing through the surfaceof the viewing sphere, as well as through the physical pointin the scene that is imaged by that pixel. By detecting andmatching visual features in different images we may regis-ter automatically the images with respect to each other. Wemay then map every pixel of every image we collected tothe corresponding point of the viewing sphere and obtaina spherical image that summarizes all our information onthe scene. This spherical image is the most natural repre-sentation: we may represent this way a scene of arbitraryangular width and if we place our head in  C  , the center of the sphere, we may rotate it around and capture the sameimages as if we were in the scene.What is left to be done, in order to obtain our panorama-on-a-page, is projecting the spherical image onto a pictureplane. This step is unfortunately not obvious – the surfaceof the sphere may not be flattened onto a page without someform of distortion. The choice of projection from the sphereto the plane has been dealt with extensively by painters andcartographers. An excellent review is provided in [9].The best known projection is  linear perspective  (alsocalled ‘gnomonic’ and ‘rectilinear’). It may be obtained byprojecting the relevant points of the viewing sphere onto atangent plane, by means of rays emanating from the cen-1  ter of the sphere  C  . Linear perspective became popularamongst painters during the Renaissance. Brunelleschi iscredited with being the first to use correct linear perspec-tive. Alberti wrote the first textbook on linear perspectivedescribing the main construction methods [1]. It is believedby many to be the only ‘correct’ projection because it mapslines in 3D space to lines on the 2D image plane and be-cause when the picture is viewed from one special point, the‘center of projection’ of the picture, the retinal image that isobtained is the same as when observing the srcinal scene.A further, somewhat unexpected, virtue is that perspectivepictures look ‘correct’ even if the viewer moves away fromthe center of projection, a very useful phenomenon called‘robustness of perspective’ [18, 22].Unfortunately, linear perspective has a number of draw-backs. First of all: it may only represent scenes that areat most 180 ◦ wide: as the field of view becomes wider,the area of the tangent plane dedicated to representing onedegree of visual angle in the peripheral portion of the pic-ture becomes very large compared to the center, and even-tually becomes unbounded. Second, there is an even morestringent limit to the size of the visual field that may berepresented successfully using linear perspective: beyondwidths of 30 ◦ -40 ◦ architectural structures (parallelepipedsand cylinders) appear to be distorted, despite the fact thattheir edges are straight [18, 14]. Furthermore, spheres thatare not in the center of the viewing field project to el-lipses onto the image plane and appear unnatural and dis-torted [18] (see Fig 1). Renaissance painters knew of theseshortcomings and adopted a number of corrective mea-sures [14], some of which we will discuss later.The objective of this paper is discussing the difficultiesand opportunities that are connected to the projection fromviewing sphere to image plane, in the context of digital im-age mosaics. We first argue that linear perspective is not theonly valid projections, and explore a number of alternativeswhich were developed by artists and cartographers. Theseare ‘global’ projections and do not depend on image con-tent. We explore experimentally the tradeoffs of these pro- jections: how they distort architecture and people and howwell do they tolerate wide fields of view. We then show thatmultiple projections may coexist successfully in the samemosaic: these projections are chosen locally and depend onwhat is seen in the pictures that form the mosaic. We showexperimentally that the such a content based approach oftenproduces more compelling results than the traditional ap-proach which utilizes geometry only. We conclude with adiscussion of the work that lies ahead.In this paper we do not address issues of image reg-istration and image blending. In all our experiments thecomputation of the transformations of the input images tothe sphere and the blending of the images was done usingMatthew Brown’s Autostitch software [4, 2]. Figure 1.  Perspective distortions.  Left: Five pho-tographs of the same person taken by a rotating cam-era, after rectification (removing spherical lens dis-tortion). Right: An overlay of the five photographs af-ter blackening everything but the person’s face showsthat spherical objects look distorted under perspec-tive projection even at mild viewing angles. For ex-ample, here, the centers of the faces in the cornersare at  ∼ 20 ◦ horizontal eccentricity. 2 Global Projections What are the alternatives to linear perspective?An important drawback of linear perspective is the ex-cessive scaling of sizes at high eccentricities. Considera painter taking measurements in the scene by using herthumb and using these measurements to scale objects onthe canvas. She takes angular measurements in the sceneand translates them into linear measurements onto the can-vas. This construction is called  Postel  projection [9]. Itavoids the ‘explosion’ of sizes in the periphery of the pic-ture. Along lines radiating from the point where the pictureplane touches the viewing sphere, it actually maps lengthson the sphere to equal lengths in the image. Lines that runorthogonal to those (i.e., concentric circles around the tan-gent point) will be magnified at higher eccentricities, butmuch less than by linear perspective. The Postel projectionis close to the cartographic  stereographic  projection. Thestereographic projection is obtained by using the pole oppo-site to the point of tangency as the center of projection.Consider now the situation in which we wish to repre-sent a very wide field of view. A viewer contemplating awide panorama will rotate his head around a vertical axis inorder to take in the full view. Suppose now that the viewhas been transformed into a flat picture hanging on a walland consider a viewer exploring that picture: the viewer willwalk in front of the picture with a translatory motion that isparallel to the wall. If we replace rotation around a verticalaxis with sideways translation in front of the picture we ob-tain a family of projections which are popular with cartog-raphers. Wrap a sheet of paper around the viewing sphereforming a cylinder that touches the sphere at the equator.One may project the meridians onto the cylinder by main-taining lengths along vertical lines, thus obtaining the  ge-  Perspective Geographic Mercator Transverse Mercator Stereographic Figure 2.  Spherical projections.  Figures taken out of Matlab’s help pages visualizing the distortions of various projections. Grid lines correspond to longitude and latitude lines. Small circles are placed at regular intervals acrossthe globe. After projection, the small circles appear as ellipses (called Tissot indicatrices) of various sizes, elongations,and orientations. The sizes and shapes of the ellipses reflect the projection distortions. ographic  projection. Alternatively, one may want to varylocally the scale of the meridians so that they keep in pro-portion with the parallels. This is the  Mercator  projection(for mathematical definitions of these projections see [16]).Figure 2 visualizes the properties of these projections.Grid lines correspond to longitude and latitude lines. Whenprojecting images onto the sphere, vertical lines are pro- jected onto longitude lines. Horizontal lines are not pro- jected onto latitude lines but rather onto tilted great circles,thus the visualization of the latitude lines does not conveywhat happens to horizontal image lines. All of these projec-tions are global and are independent of the image content.Figure 3 illustrates the above projections on a panoramaconstructed of images taken at an indoor scene. This is atypical example of panoramas of man-made environmentswhich usually contain many straight lines. Selecting fromthe above projections implies bending either the horizon-tal lines, the vertical lines, or both. In most cases a bet-ter choice is to keep vertical lines straight as this results ina panorama where narrow vertical slits look correct. Thismatches the observations in [22], which shows that our per-ception of a picture is affected by the fact that normally peo-ple shift their gaze horizontally and rarely shift it vertically.Shifting one’s gaze horizontally across a panorama looksbest when vertical lines are not bent. This motivates theuse of either the Geographic or the Mercator projections. Inboth these projections the rotation of the camera is trans-formed into sideways motion of the observer.When the camera performs mostly pan motion, i.e.,when the vertical angle is small, both projections producepractically the same result. However, for larger tilt an-gles the Geographic projection distorts circles, i.e., it doesnot maintain correct proportions, while the Mercator doesmaintainconformality, thustheMercatorprojectionisabet-ter option (see Figure 4). Note, that the conformality im-plies that in the Mercator projection spherical and cylindri-cal objects, such as people, are not distorted but the back-ground is, see for example Figure 7.An important issue in all cylindrical projections is thechoice of equator. Once the images are on the sphere onecan rotate the sphere in any desired way before projecting tothe plane. In other words, the cylinder wrapping the spherecan touch the sphere along an equator of choice. When awrong equator is selected, vertical lines in 3D space willnot be projected onto vertical lines in the panorama (see leftpanel of Figure 5). Finding the correct equator is easy. Theuser is requested to mark a single vertical line and a horizonpoint in one (or two) of the input images. The sphere isthen rotated so that projection of the marked vertical linealigns with a longitude line and the equator goes throughthe selected horizon point. This results in a straightenedpanorama, see for example, right panel of Figure 5.Should other projections be considered? Yes, we think so. The  Transverse Mercator  projection is known inthe mapping world as an excellent choice for mapping ar-eas that are elongated north-to-south. This corresponds topanoramas with little pan motion and large tilt motion. Thebending of vertical lines is small near the meridian, thus,when the pan angle is small we are better off using theTransverse Mercator projection which keeps the horizontallines straight. This is illustrated in Figures 4, 6.Forfarawayoutdoorsscenesalmostanyprojectionlooksgood as the scenes rarely contain any straight lines. Never-theless, too much bending might disturb the eye even onfree form objects like clouds. This implies the usage of thestereographicprojection, whichbendsbothverticalandhor-izontal lines but less than the cylindrical projections.So, which is the best option? Unfortunately, there is nosingle answer. As discussed in above, each photographingsetup will have its own best solution. Instead, in the nextsection we suggest a multi-view approach and show exper-imentally that it produces better looking mosaics. 3 Multi View Projection The projections explored in Section 2 are ‘global’, in thatonce a tangent point or a tangent line is chosen, the projec-tion is completely determined by this parameter. As can be  Perspective Transverse MercatorMercator StereographicGeographic Multi-Plane Figure 3.  Spherical projections.  There are many spherical projections. Each has its pros and cons. Mercator With Wrong Equator Mercator With Correct Equator Figure5. Choiceofequator (ThePantheon). Awrongchoiceoftheequatorresultsintiltedverticallines. Thecolumnson the right and left appear converging. Correcting the equator selection results in columns standing up-right. seen in the example figures, they all introduce some dis-tortions. Zorin and Barr [24] suggested an approach to re-move distortions from perspective images by compromis-ing between bending lines and distorting spheres. Theypresent impressive results, however, their solution is lim-ited to fields of view which can be covered by the linearperspective projection. Using a single global projection isby no means a necessary property for a good projection. Wemay instead tailor the projection locally to the content of theimages in order to improve the final effect. We next explorea few options for such multi-view projections. 3.1 Multi-Plane Perspective Projection As was shown in Section 2, a global projection of widepanoramas bends lines, which is unpleasant to the eye. Toobtain both a rectilinear appearance and a large field of viewwe suggest using a multi-plane perspective projection. Suchmulti-plane projections were suggested by Greene [11] forrendering textured surfaces. Rather than projecting thesphere onto a single plane, multiple tangent planes to thesphere are used. Each projection is linear perspective. Thetangent planes have to be arranged so that they may be un-folded into a flat surface without distortion, e.g., the pointsof tangency belong to a maximal circle. One may think of the intersections of the tangent planes being fitted withhinges that allow flattening. The projection onto each planeis perspective and covers only a limited field of view, thus itis pleasant to the eye.This process introduces large orientation discontinuitiesat the intersection between the projection planes, however,in many man-made environment these discontinuities willnot be noticed if they occur along natural discontinuities.The tangent planes must therefore be chosen in a way that
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