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The concept of manifold hopping, a technique for navigating in manifold mosaics, which are 3d plenoptic functions constructed from rays taken along concentric circles. Manifold hopping allows users to observe an environment from the inside looking out, providing perceived continuous camera movement without rendering viewpoints at infinitesimal steps. The concepts of manifold mosaics, view interpolation, warping, and manifold hopping, using concentric mosaics as examples.

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Download Manifold Hopping: A Technique for Smooth Navigation in Manifold Mosaics and more Papers Computer Science in PDF only on Docsity! Rendering by Manifold Hopping H.-Y. Shum, L. Wang, J. Chai, X. Tong Microsoft Research, China Submitted to IJCV Special Issue on Video Computing Contact author: Harry Shum [email protected] Phone: 86-10-62617711x5780 Fax: 1-425-9367329 Abstract In this paper, we present a novel image-based rendering technique, which we call manifold hopping. Our technique provides users with perceptually continuous navigation by using only a small number of strategically sampled manifold mosaics or multiperspective panoramas. Manifold hopping has two modes of navigation: moving continuously along any manifold, and discretely be- tween manifolds. An important feature of manifold hopping is that significant data reduction can be achieved without sacrificing output visual fidelity, by carefully adjusting the hopping intervals. A novel view along the manifold is rendered by locally warping a single manifold mosaic using a constant depth assumption, without the need for accurate depth or feature correspondence. The rendering errors caused by manifold hopping can be analyzed in the signed Hough ray space. Ex- periments with real data demonstrate that we can navigate smoothly in a virtual environment with as little as 88k data compressed from 11 concentric mosaics. Keywords: Manifold mosaic, concentric mosaic, perceptually smooth navigation, warping, plenoptic functions, image-based rendering, video computing. 2 Overview of manifold hopping In this section, we introduce manifold mosaics, view interpolation using manifold mosaics, warping manifold mosaics, and manifold hopping. Throughout this section, concentric mosaics are used as examples of manifold mosaics to illustrate these concepts. 2.1 Manifold mosaics A multiperspective image is assembled from rays captured from multiple viewpoints (e.g., [25]). Multiperspective images have also been called MCOP images [18], multiperspective panoramas [24], pushbroom images [9], and manifold mosaics [17], among other names. In this paper, we de- fine a manifold mosaic as a multiperspective image where each pixel has a one-to-one mapping with a scene point1. Therefore, a conventional perspective image, or a single perspective panorama, can be regarded as a degenerate manifold mosaic in which all rays are captured at the same viewpoint. We adopt the term manifold mosaic from [17] because the viewpoints are generally taken along a continuous path or a manifold (surface or curve). For example, concentric mosaics are manifold mosaics constructed from rays taken along concentric circles [22]. Specifically, at each point on a circle, a slit image of single pixel width is taken. By assembling all slit images captured along a circle, a concentric mosaic is formed. Two kinds of concentric mosaics are shown in Figure 2 where rays are taken in the tangential direction (Figure 2(a)), and in the normal direction (Figure 2(b)), respectively. Concentric mosaics constitute a 3D plenoptic function because they are sampled natu- rally by three parameters: rotation angle, radius, and vertical elevation. Clearly there is a one-to-one mapping between pixels in a concentric mosaic and their corresponding scene points. Although many previous image-based rendering techniques (e.g., view interpolation, 3D warp- ing, etc.) are developed for perspective images, they can be applied to manifold mosaics as well. For example, 3D warping has been used to reproject a multiple-center-of-projection (MCOP) image in [18, 16] where each pixel of an MCOP image has an associated depth. Stereo reconstruction from multiperspective panoramas has also been studied [23]. It has been shown [22] that a novel view inside the capturing circle can be rendered from the concentric mosaics without any knowledge about the depth of the scene. From densely sampled con- centric mosaics, a novel view image can be rendered by linearly interpolating nearby rays from two 1By this definition, MCOP images are not manifold mosaics. 3 (a) (b) Lateral hopping (c) (d) Looming hopping Figure 1: Manifold hopping using concentric mosaics: a plan view. Manifold hopping has two modes of navigation: (a)(c) move continuously along any manifold, and (b)(d) discretely across manifolds. The arrows in (b)(d) indicate that the user can only hop to the viewpoints on the cir- cle, but not stop anywhere in between. Two classes of manifold hopping are shown here: lateral hopping whose discrete mode of navigation is perpendicular to the viewing direction, and looming hopping whose discrete mode of navigation is along the viewing direction. Lateral hopping uses tan- gent concentric mosaics (Figure 2(a)), while looming hopping employs normal concentric mosaics (Figure 2(b)). P' P (a) (b) Figure 2: Capturing two kinds of concentric mosaics: a plan view. A concentric mosaic is as- sembled by unit width slit images (a) tangent to the circle; and (b) normal to the circle. We call (a) tangent concentric mosaics and (b) normal concentric mosaics. Tangent concentric mosaics are called concentric mosaics in [22]. 4 neighboring concentric mosaics. In addition, a constant depth is assumed to find the best “nearby” rays for optimal rendering quality [4]. Figure 3(a) illustrates a rendering ray that is interpolated by two rays captured in nearby concentric mosaics. Despite the inevitable vertical distortion, concentric mosaics are very useful for wandering around (on a plane) in a virtual environment. In particular, concentric mosaics are easy to capture by using an off-centered video camera rotating along a circle. 2.2 Warping manifold mosaics View interpolation can create high quality rendering results when the sampling rate is higher than Nyquist frequency for plenoptic function reconstruction [4]. However, if the sampling interval be- tween successive camera locations is too large, view interpolation will cause aliasing artifacts, cre- ating double images in the rendered image. Such artifacts can be reduced by the use of geometric information (e.g., [8, 4]), or by pre-filtering the light fields [11, 4] (thus reducing output resolution). A different approach is to locally warp manifold mosaics, which is similar to 3D warping of a perspective image. An example of locally warping concentric mosaics using an assumed constant depth is illustrated in Figure 3(b). Any rendering ray that is not directly available from a concentric mosaic (i.e., not tangent to a concentric circle) can be retrieved by first projecting it to the constant depth surface, and then re-projecting it to the concentric mosaic. Therefore, a novel view image can be warped using the local rays captured on a single concentric mosaic, rather than interpolated by collecting rays from two or more concentric mosaics. For humans to perceive a picture correctly, it is essential that the image of an object should not contain any structural features that are not present in the object itself [26]. Double images, which are common artifacts from view interpolation with poor correspondence, unfortunately result in mistak- enly perceived structural features in the observed objects, e.g., more noticeable edges. On the other hand, locally warping a multiperspective image preserves structural features. An example of locally warping a concentric mosaic is shown in Figure 4, with images of different FOV’s. The projection error in the rendered image caused by warping the concentric mosaic with (incorrect) constant depth assumption increases as the field of view becomes larger. Note the distortion toward the right edge in Figure 4(b). The geometric distortions introduced by local warping methods because of imprecise geometric information are, however, tolerated by human visual perception when the field of view (FOV) of the rendering image is small (e.g., Figure 4(a)). 5 3 Analysis of lateral hopping using the signed Hough ray space The Hough transform is known to be a good representation for lines. However, it is not suitable for representing rays that are directional. The conventional Hough space can be augmented to a signed Hough ray space [3], or an oriented line representation [11], by using the following right-hand rule: a ray that is directed in a counter-clockwise fashion about the coordinate center is labeled positive, otherwise is labeled negative. A “positive” ray is represented by (r; ), whereas its “negative” coun- terpart is (r; ) where r is always a positive number. Figure 5 shows four different rays in a 2D space and their corresponding points in the signed Hough space. Figure 6 shows three typical viewing setups and their representations in the signed Hough space. For example, a panoramic image (i.e., rays collected at a fixed viewpoint in Cartesian space) is represented as a sampled sinusoidal curve in the parameter space, located at (r0; 0) as shown in Figure 6(a). A concentric mosaic shown in Figure 6(b) is mapped to a horizontal line, whereas par- allel projection rays (Figure 6(c)) are mapped to a vertical line in the signed Hough space. Thus, captured perspective images can be easily transformed into samples in the parameter space. Render- ing a novel view in the scene is equivalent to extracting a partial or complete sinusoidal curve from the signed Hough space. When the hopping direction is perpendicular to the viewing direction, as shown in Figure 1, we call it lateral hopping. In the signed Hough space, such a hopping is illustrated in Figure 7 where a segment of a sinusoidal curve is approximated by a line segment. Equivalently, at each rendering viewpoint, a perspective image is approximated by part of a concentric mosaic. Obviously, the smaller the hopping interval, the smaller the rendering error. On the other hand, the larger the hopping interval, the less data needed for wandering around an environment. We argue that a fairly large hopping interval for manifold hopping can be perceptually acceptable. 3.1 When is local warping good enough? When moving on a concentric mosaic, the horizontal field of view should be constrained within a certain range so that the distortion error introduced in local warping from a multiperspective image to a perspective image will not cause much visual discomfort to the user. The distortion threshold d is defined as the maximum allowable distance between point A and point B in Figure 9. These two points are projections of the rightmost pixel that are locally warped 8 y x l1 r l3 θ l4 l2 r θ0 2π l1 l3 l4 l2 0 Figure 5: Definition of the signed Hough ray space: each oriented ray in Cartesian space at the left is represented by a sampled point in the signed Hough space on the right. y x r0 r0 0θ r0 yy x x 0θ 0 0 0 (a) (b) (c) r r0 -r0 0 π2 θ Concentric mosaic Panorama Parallel mosaic (d) Figure 6: Three typical viewing setups and their respective sampled curves in the signed Hough space: (a) a panorama at a fixed point; (b) a concentric mosaic; (c) a parallel projection mosaic; and (d) their respective sampled curves in the signed Hough space. Two concentric mosaics (straight lines at r0 and r0) are shown in (d) to represent rays captured at opposite directions along the circle. Note that a perspective image is only part of a panorama, thus represented by a segment of a sinusoidal curve in the signed Hough space. 9 rn+1 rn 0 -rn -rn+1 Radial hopping Continuous angular rotation Figure 7: Hopping between concentric mosaics along a radial direction in the signed Hough space. Continuous rotation is achieved along any of the concentric circles, but hopping is necessary across any radial direction. r∆ θ∆ 0R rn+1 rn A B 0 π2 Figure 8: Analysis of hopping size: horizontal parallax change due to viewpoint change. r∆ θ∆ 2R rn A 0 π2 B FOVθ 1R Figure 9: Analysis of maximum FOV: warping error due to the incorrect depth value. 10 the average speed of rotation for a person to observe an environment is below 48Æ/second, then D0 should be 2Æ. In other words, a person can tolerate 2Æ of average pixel flow for two neighboring frames and still observe smooth and continuous motion. Consider a particular scene in which the radius of the outermost concentric mosaic is 1 unit and the objects are located at a distance of 4 units. If D0 is 1:5Æ, we have r = 0:1. Therefore, we need only 21 concentric mosaics (two for each concentric circle and one for the center). This is a significant reduction from 320 rebinned concentric mosaics needed in rendering with concentric mosaics [22]. 4 Analysis of looming hopping using the extended signed Hough space In the previous section, we have analyzed manifold hopping where the hopping direction is perpen- dicular to the viewing direction. If the hopping direction is along the viewing direction, i.e., if the user moves forward and backward, we cannot use the conventional concentric mosaics assembled by rays along the tangent lines of the concentric circle. Instead, hopping with a looming motion can be achieved if we construct normal concentric mosaics that are formed by slit images with unit pixel width along the normal directions of the concentric circle, as shown in Figure 2(b). A novel view at any point on a circle can be rendered by locally warping rays from the normal concentric mosaic near the viewpoint, as shown in Figure 1(c). The signed ray space is no longer adequate for analyzing looming hopping. For a looming motion, we need to represent points along the same ray differently. Therefore, we introduce the extended signed Hough space, defined by a 3-tuple (r; ; d) where d is the distance from the origin to the location where the ray is captured. Two points (P and P 0) along the same ray have identical (r; ) but different values of d, as shown in Figure 10. And d will take the same sign as r to differentiate a “positive” ray from a “negative” one, similar to the signed Hough space. Although rays captured at P and P 0 are the same in the plan view of Figure 2(b), slit images captured at these two points are different. Figure 10 also shows three different mosaics represented in the extended Hough space. A panorama: r = d sin( ); 13 )sin( ϕθ −= dr d r θ y x0 y x 0ϕ ϕ θ y 0 x y 0 x nrdr == ,0nrrd == d' PP' (a) (b) (c) (d) Figure 10: (a) The extended signed Hough ray space is defined by three parameters (r; ; d). Dif- ferent points on the same ray have different d values. (b)(c)(d) A panorama, a tangent concentric mosaic, and a normal concentric mosaic are represented in the extended signed Hough space. A tangent concentric mosaic: r = d = rn; A normal concentric mosaic: r = 0 and d = rn. Note that is the constant angle for the viewpoint, and rn is the diameter of one of the concentric circles. It becomes evident now why the location of the ray, which was ignored in lateral hopping (in the signed Hough space), should be considered in looming hopping because r is always zero under looming. Therefore, the (r; ; d) representation is necessary and sufficient to index rays in 2D (plan view in Figure 2(b)) to capture the looming effect as the user moves forward and backward. Figure 11 illustrates looming hopping in the extended signed Hough space. Similar to lateral hopping in the signed Hough space (Figure 7), rendering a novel view in looming hopping is also equivalent to approximating a partial sinusoidal curve by a line segment of a normal concentric mosaic. Unlike lateral hopping, however, each sinusoidal curve is constructed at a different d. For clarity, we skip the looming hopping interval analysis in the extended signed Hough space, which is similar to the analysis in the signed Hough space in the previous section. Lateral hopping is also illustrated in Figure 11. In the (r; ; d) space, the plane for lateral hopping is r = d, but r = 0 for looming hopping. The sinusoidal curve segment is approximated around the maximum r in lateral hopping, and around r = 0 for looming hopping. If we project the lateral hopping plane in (r; ; d) space onto the d = 0 plane, we obtain the (r; ) counterpart for lateral hopping. There is therefore a duality between lateral hopping (r; ) and looming hopping (d; ). 14 Looming hopping plane Lateral hopping plane Tangent concentric mosaic Normal concentric mosaic Panorama r d θ Figure 11: Looming hopping with normal concentric mosaics, and lateral hopping with tangent concentric mosaics in the extended signed Hough space. Rendering a novel perspective view is equivalent to approximating a sinusoidal curve segment by a straight line segment representing part of a concentric mosaic. In looming hopping, green segments are used to approximate the sine curve at different d values on the brown r = 0 plane. In lateral hopping, black segments are used to approximate the sine curve at different r (and d) values on the blue r = d plane. 15 = ( 2 1)D0: (10) For example, if D0 is 1Æ, and R = 3r, then is computed as 4Æ. In other words, we need to capture 90 images along the circle. 5.2 Hopping between parallel projection mosaics Another way to achieve continuous radial motion is to use parallel projection mosaics. Parallel mosaics are formed by collecting all parallel rays in the same direction. We call this angular hopping with parallel mosaics. Because parallel projection cameras are not commonly available, we rebin parallel mosaics by taking the parallel rays from a dense sequence of perspective images taken along a circle outside the object. Figure 15 shows a projective image from the original sequence and the rebinned parallel mosaic. Note that the rebinned mosaic is called 1D parallel mosaic because the vertical direction is still perspective, only the horizontal direction is under parallel projection. Assuming a constant depth for the object, we can reproject parallel mosaics to any novel view along the radial direction, as shown in Figure 14. Warping 1D parallel mosaic in Figure 15(b) using constant depth is shown in Figure 15(c). Even though warping errors are created, such as those around the boundary of the object, they are small enough to cause little visual distortion. 5.2.1 Close-up views Rendering a novel view with angular hopping using parallel mosaics can again be explained in the signed Hough space. Continuous motion along the angular direction is obtained by approximat- ing a cosine segment using a line segment. When the viewpoint is far away, the parallel mosaic approximates the perspective view very well. The reprojection or warping error increases as the viewpoint approaches the object. In addition, the image size of the parallel mosaic determines how closely the rendering camera can get to the object. Hopping using parallel mosaics and hopping using perspective images have similar warping errors, especially if constant depth is assumed. However, rebinned parallel mosaics can have a much higher resolution than the original image if a very dense sequence is captured. For example, we can obtain a 1D parallel mosaic of 640 240 from 640 original images with size 320 240. Close-up views rendered from rebinned parallel mosaics have better quality than simply zooming-in the original images. 18 Figure 14: Reprojecting a parallel mosaic to different perspective images along the radial direction using constant depth assumption. (a) (b) (c) Figure 15: Warping parallel projection images: (a) a perspective image; (b) a 1D parallel projection mosaic; (c) 1D mosaic of (b) warped with constant depth; 19 6 Experiments 6.1 Synthetic environments We represent a synthetic environment with 41 concentric mosaics (with size 2400 288) on 11 concentric circles. There are 21 tangent concentric mosaics, and 21 normal concentric mosaics. Note that the center mosaic degenerates to a single perspective panorama, as shown in Figure 17(a). At the outermost circle, the tangent concentric mosaic is shown in Figure 17(b), while the normal concentric mosaic is shown in Figure 17(c). By hopping between these mosaics, we render five images from the left, right, center, front and back viewpoints shown in Figure 17(d). Parallax effects (both lateral and looming) are clearly visible from the rendered images. And hopping between these mosaics provides a smooth navigation experience. However, one can only switch lateral motion and looming motion at the center. In conventional rendering with concentric mosaics, we would have used 720 such mosaics. Therefore, manifold hopping requires much less data for a similar viewing experience. A much larger environment can be constructed by combining more mosaics captured at different locations. By carefully adjusting constant depths used for different sets, we can hop smoothly from one circle to another, in addition to inside a circle. 6.2 Real environments We have used a Sony Mini DV digital video camera to capture concentric mosaics of a real environ- ment. The camera rotates along a circle. The video is digitized at the resolution of 720 576. A total of 5726 frames are captured for a full circle. The raw data for the video sequence amounts to a total of 7 Gigabytes. Instead of using 720 rebinned concentric mosaics of size 5726 576, we select only a small subset (typically 21) of resampled concentric mosaics. Three rebinned concentric mosaics are shown in Figure 18(a). Two high resolution images (with display size 500 400) rendered from 21 concentric mosaics are shown in Figures 18(b) and (c). Horizontal parallax around the tree and lighting change reflected from the window can be clearly observed. Constant depth correction is used in all our experiments. To reduce the amount of data used in manifold hopping, we can resize the original concentric mosaics. As shown in Figures 18(d) and (e), two images with low resolution 180 144 are rendered 20 There are two major differences between manifold hopping with concentric mosaics and hopping with panoramas. The first difference is in capturing. As shown in Figure 16, panoramas can capture similar rays to concentric mosaics as the number of panoramas increases. However, the same result will require capturing panoramas many times at different locations, as opposed to rotating the camera only once for capturing concentric mosaics. The second and perhaps more important difference is in sampling. Each manifold mosaic is mul- tiperspective, while each panorama has only a single center of projection. Since different viewpoints can be selected as the desired path for the user, a multiperspective panorama could be more repre- sentative of a large environment than a single perspective panorama. If the multiperspective image is formed by rays taken along the desired path of the user, the warping error from a multiperspective image is, on average, smaller than that from a perspective image (e.g., a panorama). Concentric mosaics are suitable for the inside looking out. To observe objects from the outside looking in, parallel mosaics can be used for manifold hopping. For concentric mosaics, the mani- fold is a cylindrical surface. For parallel mosaics, the manifold is a plane originating from the object center. In this paper, we have discussed manifold hopping in two dimensional space by constraining the rendering camera on a plane. The concept of manifold hopping can be generalized to higher di- mensions. The analysis in higher dimensions is very similar to the two-dimensional cases. However, it is difficult to capture such manifold mosaics in practice. 8 Conclusion and future work We have described a new image-based rendering technique which we call manifold hopping. In summary, our technique has the following properties: It does not require a large amount of image data, and yet the user can perceive continuous camera movement. It requires neither accurate depth nor correspondence, yet generates perceptually acceptable rendered images. Specifically, manifold hopping renders a novel view by locally warping a single manifold mo- saic, without the need for interpolating from several images. We have shown that warping a sin- gle multiperspective image to a perspective image with a regular field of view causes insignificant 23 Geometry Images Rendering Viewpoints Perceived Motion Light fields [11, 8, 22] no/approximate very large (100 10000+) continuous continuous 3D Warping [13, 14, 21, 5] accurate small (1 10+) continuous continuous View interpolation [6, 15, 20, 2] accurate small (2 10+) continuous continuous Hopping [12, 7] no moderate (10 100+) discrete discrete Manifold hopping no/approximate moderate (10 100+) discrete continuous Table 1: A table of comparison for different IBR techniques: geometry requirements, number of images, rendering viewpoints and perceived camera movement. The citations are for reference only, not meant to be complete. distortion to human beings, even with warping errors resulting from incorrect depth information. Furthermore, local warping does not introduce structural errors such as double images which are perceptually disturbing. Most importantly, manifold hopping requires relatively little input data. Capturing manifold mosaics such as concentric mosaics is also easy. By sparsely sampling the concentric mosaics, we can reduce the amount of data from the original concentric mosaics by more than 10-fold. While manifold hopping provides only discrete camera motion in some directions, it provides reasonably smooth navigation by allowing the user to move in a circular region and to observe significant horizontal parallax (both lateral and looming) and lighting changes. The ease of capture and the very little data requirement make manifold hopping very attractive and useful for many virtual reality applications, in particular those on the Internet. Table 1 compares how manifold hopping differs from previous IBR systems, in terms of their geometric requirements, number of images, rendering viewpoints and perceived camera movement. Manifold hopping stands out in that it ensures a perceived continuous camera movement even though rendering viewpoints are discrete. It builds on the observation that a fairly large amount of view- point change is allowed, while maintaining perceptually continuous camera movement to humans. This observation of “just-enough hopping” for reducing image samples is, in spirit, similar to the “just-necessary effort” adopted by perceptually based techniques [19] on realistic image synthesis to reduce computational cost. While we have experimentally demonstrated the feasibility of our choices (e.g., 21 concentric mosaics used in most of our experiments), we plan to conduct a more comprehensive study on the psychophysics of visualization for our technique. 24 References [1] E. H. Adelson and J. Bergen. The plenoptic function and the elements of early vision. In Computational Models of Visual Processing, pages 3–20. MIT Press, Cambridge, MA, 1991. [2] S. Avidan and A. Shashua. Novel view synthesis in tensor space. In Proc. Computer Vision and Pattern Recognition, pages 1034–1040, 1997. [3] J.-X. Chai, S. B. Kang, and H.-Y. Shum. Rendering with non-uniform approximate concentric mosaics. In Proc. ECCV2000 Workshop SMILE2, Dublin, Ireland, 2000. [4] J.-X. Chai, X. Tong, S.-C. Chan, and H.-Y. Shum. Plenotpic sampling. In Proc. SIGGRAPH 2000, 2000. [5] C. Chang, G. Bishop, and A. Lastra. Ldi tree: A hierarchical representation for image-based rendering. SIGGRAPH’99, pages 291–298, August 1999. [6] S. Chen and L. Williams. View interpolation for image synthesis. Computer Graphics (SIGGRAPH’93), pages 279–288, August 1993. [7] S. E. Chen. QuickTime VR – an image-based approach to virtual environment navigation. Computer Graphics (SIGGRAPH’95), pages 29–38, August 1995. [8] S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen. The lumigraph. In Computer Graphics Proceedings, Annual Conference Series, pages 43–54, Proc. SIGGRAPH’96 (New Orleans), August 1996. ACM SIGGRAPH. [9] R. Gupta and R. Hartley. Linear pushbroom cameras. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(9):963–975, September 1997. [10] S. B. Kang. A survey of image-based rendering techniques. In VideoMetrics, SPIE Vol. 3641, pages 2–16, 1999. [11] M. Levoy and P. Hanrahan. Light field rendering. In Computer Graphics Proceedings, Annual Confer- ence Series, pages 31–42, Proc. SIGGRAPH’96 (New Orleans), August 1996. ACM SIGGRAPH. [12] A. Lippman. Movie maps: An application of the optical videodisc to computer graphics. Computer Graphics (SIGGRAPH’80), 14(3):32–43, July 1980. [13] W. Mark, L. McMillan, and G. Bishop. Post-rendering 3d warping. In Proc. Symposium on I3D Graph- ics, pages 7–16, 1997. [14] L. McMillan. An image-based approach to three-dimensional computer graphics. Technical report, Ph.D. Dissertation, UNC Computer Science TR97-013, 1999. [15] L. McMillan and G. Bishop. Plenoptic modeling: An image-based rendering system. Computer Graph- ics (SIGGRAPH’95), pages 39–46, August 1995. 25 (b) (c) (d) (e) (f) (g) (a) Figure 18: Hopping between concentric mosaics: (a) three concentric mosaics projected onto cylin- ders; (b)(c) two rendered images at a high resolution 500 400; (d)(e) rendered images with a low resolution 180 144; (f)(g) low resolution rendered images using 88k compressed data. 28 (a) (b) (c) (d) (e) (f) Figure 19: Hopping between parallel mosaics: (a) a perspective image from the original sequence; (b) a rebinned 1D parallel mosaic with higher resolution; (c)(d) two rendered images from different viewing directions; (e)(f) close-up views along the viewing direction of (c). 29