Rendering Multi-Perspective Images with Trilinear Projection

Transcription

1 Rendering Multi-Perspective Images with Trilinear Projection Scott Vallance Paul Calder School of Informatics and Engineering Flinders University of South Australia, PO Box 2100, Adelaide, South Australia 5001, Abstract Non-linear projections of 3D graphical scenes can be used to compute reflections and refractions in curved surfaces, draw artistic images in the style of Escher or Picasso, and produce visualizations of complex data. Previously, most non-linear projections were rendered by ray tracing. This paper presents trilinear projection, a technique for rendering non-linear projections in a manner that achieves significant performance benefits by taking advantage of current rendering hardware and software. Trilinear projections are geometrically similar to Phong-shaded triangular patches, and like Phong patches they can be joined to represent more complicated shapes. The paper details how a single trilinear projection projects a scene point, how projections of scene triangles can be built up by considering the connectivity of projected points, and how multiple trilinear projections can be combined. Finally, it outlines a method for using trilinear projection to approximate reflections and refractions on curved surfaces. Keywords: Multi-Perspective Images, Non-Linear Projection, Reflections, Refractions, Rendering Algorithms. 1.1 Nonlinear Projections Nonlinear projections occur naturally as reflections and refractions on curved objects. The strange and distorted images seen in an amusement park mirror are a familiar example of the distortion nonlinear projections generate. These images have also been examined in art, photography and computer graphics where they have variously been named cubist images, multi-perspective images, multiple-centre-ofprojection images and multi-perspective panoramas. Traditional Chinese landscape paintings frequently contain different foci, or sub-images, which are seamlessly joined. For example, Figure 1 shows a scene in which the perspective shifts from left to right, following the path of the stream. German artist M. C. Escher frequently depicted views with multiple vanishing points, or perspectives. For example, Figure 2 has five different vanishing points: top left and right, centre, and bottom left and right. While the automatic generation of images like these from 3D geometry may not be practical, they illustrate the concept and the aesthetic potential. 1 Introduction Most computer graphics rendering of 3D scenes is linear. For example, a perspective projection simulates the physics of optics, mapping scene data back to a single point in space as if viewed through a camera lens. Nonlinear projections differ from linear projections in that straight lines in 3D may not be straight lines when projected. Such projections can be used to compute reflections and refractions in curved surfaces, draw artistic images in the style of Escher or Picasso, and produce visualistations of complex data. Previous techniques for nonlinear projection have used ray tracing or relied on distortion of the scene data. This paper presents a new technique, called trilinear projection, for rendering nonlinear projections in a manner analogous to perspective transformation matrix rendering. The technique is based on a trilinear interpolation similar to that used for Phongshaded trianglular patches. And like Phong patches, multiple trilinear projections can be joined together to represent complex projection surfaces while maintaining continuity across sub-projections. Copyright c 2006, Australian Computer Society, Inc. This paper appeared at Twenty-Ninth Australasian Computer Science Conference (ACSC2006), Hobart, Tasmania, Australia, January Conferences in Research and Practice in Information Technology, Vol. 48. Vladimir Estivill-Castro and Gill Dobbie, Ed. Reproduction for academic, not-for profit purposes permitted provided this text is included. Figure 1: Fishermans Evening Song by Xu Daoning, circa 11th Century Strip cameras are widely used in surveillance and mapping. These cameras have a continuous roll of film that slides past a slit as a picture is being taken. The camera may be moved whilst shooting, providing a change in point of view from one section of the film to another. For example, if used from a moving aeroplane a strip camera can capture a long section of curved earth as if it were flat. The technique has also been used for artistic purposes, such as the image in Figure 3 (Davidhazy 2001). 1.2 A Trilinear Projection Surface In geometric terms, a perspective projection can be defined by a set of rays that emanate from a single point in space, and an orthographic projection by a set of parallel rays that emanate from a plane. By extending this approach, a nonlinear projection can be defined as a projection created by a set of rays that emanate from an arbitrary surface, with the origin and direction of each ray a function of the surface. In computer graphics, complex curved surfaces are usually approximated as a mesh of triangles because triangle geometry is easy to compute and render. The

2 (a) (b) Figure 4: A scene of columns (a) rendered with conventional perspective (b) rendered from a torus surface Figure 2: High and Low by M. C. Escher, an example of a nonlinear projection Figure 5: A nonlinear projection of an elephant Figure 3: A strip camera photograph of a man s head shape of a projection surface can thus be approximated by a suitable triangle mesh, and the directions of projection rays can be approximated by specifying the vertex normals of the triangles. The directions of rays internal to the triangles are defined implicitly by linear interpolation of the vertex rays. Using this approach, a complex non-linear projection can be computed by tiling smaller projections trilinear projections each defined by a triangle with specified vertex positions and normals. Because in general the vertex normals of a projection triangle do not converge to a single point, a trilinear projection will be nonlinear. And because adjacent triangles share vertex normals, the combined projection will be continuous across triangle boundaries. 1.3 Related Work Nonlinear projections can be rendered by ray tracing if a nonlinear projection can be expressed as a set of rays. For example, Löffelmann and Gröller (Löffelmann 1995) define an extended camera as a set of rays that start on a surface and point in the direction of the surface normal. The scene is then rendered with POVray [ovpl04], a widely available ray tracing implementation. Figure 4 shows a scene rendered with standard perpective projection and when rendered with a toroidal extended camera. Rademacher and Bishop (Rademacher & Bishop 1998) describe techniques for generating multiplecentre-of-projection (MCOP) images. The images are generated by moving a virtual camera through a scene and capturing a single line of pixels at regular intervals. The technique is effectively a virtual strip cam- era, and can generate images such as the one in Figure 5. Yu and McMillan (Yu & McMillan 2004) introduce general linear cameras (GLC) as a mathematical description for a class of nonlinear projections defined by three rays passing through two parallel planes. The authors define and name various special cases of these cameras, and implement them with ray tracing and a light field rendering system. The technique is similar to trilinear projection but more constrained in that the vectors must have equal magnitude in the direction normal to the view. In that sense, GLC projections are a subset of trilinear projections. Various authors have considered techniques for computing reflections from curved surfaces. For example, Glaeser (Glaeser 1999) presents equations for calculating the reflection of a space point on a sphere or cylinder of revolution, and environment mapping (originally described by Blinn and Newell (Blinn & Newell 1976)) approximates curved reflections by sampling the projection of a scene as if drawn from a point behind the reflection surface. Variations on environment mapping, such as extended environment maps (Cho 2000) and parameterised environment maps (Hakura, Snyder & Lengyel 2001), can produce more accurate images but at greater computational cost. Ofek and Rappoport (Ofek & Rappoport 1999) describe an intriguing approximation for rendering reflections on curved surface that involves distorting objects based on the reflective surface so that they may then be rendered using standard perspective projection. The technique requires an appropriate tessellation of both reflective surface and scene object to approximate the curvature of the reflected lines. The correspondence between a point in space and the reflective surface is approximated by computing an explosion map (the projection of the reflective surface to the surface of a surrounding sphere) and then computing where on the map a scene point falls. The performance of the technique is sufficient for real-time

3 rendering of moderate scenes, making it suited for visualisation tasks. 1.4 Organisation of the Paper The remainder of this paper is organised as follows: Section 2 presents the algorithms and techniques for computing the trilinear projection of a single scene point. Section 3 describes how the projection for a scene triangle can be computed from the projections of its vertices. Since the projection is non-linear, and since each point can have up to 3 images, the projection for a single scene triangle can have up to 9 vertices and may consist of several disconnected shapes. Section 4 shows how multiple trilinear projections can be combined to represent projection from a complex surface. Section 5 outlines how trilinear projection can be applied to the task of computing reflections and refractions from curved surfaces. Finally, Section 6 briefly examines the performance characteristics of trilinear projection. 2 Projecting a Point For purposes of this paper we define projecting a point as determining which location(s) on the projecting surface define a ray that intersects that point. A ray can be defined parametrically by a point, p and a normal n: r (t) := p + tn (1) If p and n are defined using barycentric coordinates u and v, vertex positions p 1..3 and n 1..3, then a ray on the surface of the trilinear projection becomes: r (u, v, t) := p (u, v) + tn (u, v) = (1 u v) p 1 + up 2 + vp 3 + t ((1 u v) n 1 + un 2 + vn 3 ) (2) So for a scene point p s : p s = r (u, v, t) = (1 u v) p 1 + up 2 + vp 3 + t ((1 u v) n 1 + un 2 + vn 3 ) (3) Solving for u, v and t in 3D means a system of three equations and three unknowns. The t and u, v terms are multiplied together making it a non-linear system of equations and an analytical solution is not readily apparent. 2.1 Treating the Trilinear Projection as a Parametric Triangle To analytically solve this problem, instead of representing the surface as a set of rays, we represent it as a parametric triangle. Each vertex has a point and a normal associated with it. Treating these as rays we can extend along them according to the parameter t giving three new points, which form the vertices of a triangle as shown in Figure 6. The three vertices of the parametric triangle are defined by the equations: r 1 := p 1 + tn 1 r 2 := p 2 + tn 2 r 3 := p 3 + tn 3 (4) A barycentrically defined point on the parametric triangle is: p (u, v, t) := (1 u v) r 1 + ur 2 + vr 3 = (1 u v) (p 1 + tn 1 ) + u (p 2 + tn 2 ) + v (p 3 + tn 3 ) (5) Figure 6: A parametric triangle shown at different values of t The task of projecting a scene point now becomes that of finding a point p (u, v, t) that coincides with the scene point. The u, v and t satisfying this constraint are the same as those satisfying p s = r (u, v, t) because the two equations are simply isomorphs. The advantage in representing the triangle as a parametric triangle is that the parameter t can be determined independently of u and v. First, for the parametric vertices defined in Equation 4, find the values of t such that the vertices and the scene point p s are coplanar, then solve for u and v in the plane of the parametric triangle. 2.2 Determining the Coplanarity of the Parametric Triangle and the Scene Point Any four points can be considered a tetrahedron; four coplanar points form a tetrahedron whose volume is 0. For a tetrahedron defined by four points A, B, C and D the volume of the tetrahedron is: volume := 1 AB (AC AD) (6) 4 This equation can also be expressed as the magnitude of the determinant of the matrix containing the three vectors. [ ] volume := 1 AB 4 det AC (7) AD Composing the three vectors from the parametric vertices defined in Equation 4 with the scene point p s and substituting into Equation 7 gives the volume of the tetrahedron defined by those four points. When these points are coplanar this volume is 0. Removing the unnecessary constant and magnitude gives: p 1 + tn 1 p s p 2 + tn 2 p s p 3 + tn 3 p s = 0 (8) This can be expanded for three dimensions, giving a cubic polynomial in terms of t. Either one or three real solutions for t exist, and each value of t defines a potentially different projection of the scene point. 2.3 Precalculating Partial Coefficient Values Computing the coefficients of Equation 8 involves substantial calculation not directly dependent on the scene point. These calculations depend only upon the values of the parametric triangle itself and therefore can be reused across scene points. This is useful because in a normal scene situation there are many

4 scene points that need to be projected by each parametric triangle. The most common drawing primitive, the triangle, is comprised of three such scene points. The minor additional memory requirements of storing these values is small in comparison to the reduction in computation. In Equation 7 the volume of a tetrahedron is defined as the determinant of three vectors formed from the four points. This can equally be expressed as the four by four determinant shown in Equation 9. p sx p sy p sz 1 p 1x + tn 1x p 1y + tn 1y p 1z + tn 1z 1 p 2x + tn 2x p 2y + tn 2y p 2z + tn 2z 1 = 0 (9) p 3x + tn 3x p 3y + tn 3y p 3z + tn 1z 1 The determinant in Equation 9 can be expanded to Equation 10 where E 1..4 are each three by three determinants. p sx E 1 p sy E 2 + p sz E 3 E 4 = 0 (10) Determinants E 1..3 are quadratics in t and E 4 is a cubic. This means the coefficient of t 3 depends only upon properties of the parametric triangle and not the scene point. Further, the other coefficients of the cubic defined by the full expansion of Equation 9 depend on the scene point in a useful way. Equations E 1..4 are defined by the general cubic equation, Equation 11, where A 1..3 = 0 E i = A i t 3 + B i t 2 + C i t + D i, i = 1, 2, 3, 4 (11) If the coefficients for the cubic equation defined by Equation 9 are represented by a, b, c and d, where at 3 + bt 2 + ct + d = 0, then the relationship between the coefficients and the properties (A, B, C, D) 1..4 can be described as: a b c d = A4 B1 B2 B3 B4 C1 C2 C3 C4 D1 D2 D3 D4 p sx p sy p sz 1 (12) All properties (A, B, C, D) 1..4 can be calculated independently of scene points. The full derivations of (A, B, C, D) 1..4 are provided elsewhere (Vallance 2005). Using these pre-calculated partial values allows for faster calculation across multiple scene points. 3 Drawing a Scene Triangle Each of the vertices of a scene triangle can be separately projected with the trilinear projection generating either one or three solutions for each vertex. The manner in which the projected vertices are connected can be understood by considering the sweep of the parametric triangle intersected with the scene triangle. 3.1 Computing the Projected Shapes At every value of t the parametric triangle is a standard triangle in scene space. The intersection of this triangle s plane with the scene triangle s plane forms a line. As the value of t changes the intersection line traverses the plane of the scene triangle. The motion of this intersection line is continuous because t is continuous, except where the intersection line is undefined because the scene triangle and parametric triangle are parallel or coplanar. A straight line intersects at most two sides of a triangle. Furthermore, the locus of a continuously defined line must intersect a vertex on the end of an edge before intersecting the edge itself. The order, from smallest to largest t value, in which projected scene triangle vertices are intersected determines the connectivity of those vertices. Initially the line of intersection crosses the two edges connected with the vertex projected by the smallest t value. Each vertex thereafter toggles which edges are being intersected by the parametric triangle. When all edges are toggled off the line of intersection is no longer traversing the scene triangle and the projected shape is complete. For strings of vertices, in order of t value, toggling the edge states reveals which vertices group together to form a shape. Even though the scene triangle has three vertices, it does not necessarily project shapes which have three vertices. Moreover, a single scene triangle can produce up to four shapes when projected with a parametric triangle. This particular case happens when the three scene vertices produce three t solutions each, and the nine projected vertices form three two-vertex shapes and one three-vertex shape. For example, for a list of vertices V =,,, v 1, which form a five-vertex shape the following diagrams show how these vertices are formed into a polygon. Each circle represents a processed vertex. An arrow between two circles represents the edge upon which those two vertics are connected. An unterminated arrow represents an open edge at that stage of the traversal. Only two arrows can be unterminated at a given stage in the traversal and the current unterminated arrows are named left and right. v 1 Step 1: Since the first vertex is the two edges v 1 and become active. If the next vertex was a v 1 then it would be placed on left because it has to be connected to the edge v 1. Similarly if the next vertex was a it would be placed on right. Finally, as a would be connected to both edges, it would finish the shape, and could be placed on either left or right. v 1 v 2 v 1 Step 2: The next vertex is in fact so it is placed on right. This toggles the active edges such that becomes inactive and v 1 active.

5 v 1 vertices which is unlikely to be accurate. Figure 7 shows one scene triangle projected by a single trilinear projection resulting in two shapes, one with four vertices and the other with five. The left hand figure is the correct projection as computed by a ray tracing algorithm. The right hand figure shows the approximation introduced by trilinear projection. v 1 Step 3: The next vertex is also, indicating a transition to the edge. Accordingly the vertex is placed on the right and the right s edge is changed. v 1 v 1 v 1 v 1 Step 4: In a similar way, vertex v 1 is placed on the left and the left s edge is changed to v 1. (a) (b) Figure 7: A projected triangle resulting in two shapes: (a) ray trace (b) trilinear projection To more accurately represent the curves that connect projected vertices the solution can be tesselated. This can be done by sampling the scene triangle at intermediate t values. In the non-tesselated case the triangle is sampled at a value of t for each projected vertex. Extra values of t between the start and end of a shape can be inserted to increase the projected polygons accuracy. For each t value the parametric triangle is a triangle in scene space. The intersection of this triangle and the scene triangle results in a line. For every value of t from the smallest for a particular shape to the largest the intersection line crosses the scene triangle. This line, clipped to the overlapping sections of the scene and parametric triangle (for this particular t value), corresponds directly to a line on the final image. Figure 8 shows a scene triangle rendered with 5 extra t samples per shape. v 1 v 1 v 1 v 1 Step 5: The final vertex is which could be placed in either list; it is arbitrarily placed on left for convenience. Previously active edges v 1 and are toggled off and the shape is finished. 3.2 Drawing the Projected Polygons The result of projecting each vertex in a triangle and then assembling the vertices together is a set of polygons. One scene triangles may result in nine projected vertices, which can be arranged in up to four different polygons from two to nine vertices (though not all combinations are possible). Each projected vertex has a u, v and t value associated with it and these values correspond to post projection x, y and z coordinates. Using these coordinates a polygon can be drawn using standard rasterising techniques with visibility resolved by the depth coordinate. Rasterising assumes that the straight lines connect the projected Figure 8: Scene triangle sampled at 5 extra t levels per shape approximating a (4,5) shape configuration 4 Combining Multiple Projections A projection comprising a single trilinear projection has limited curvature. More complicated projections can be built with a mesh of trilinear triangles. In the same way that a triangular scene mesh can be approximated as smoothly curving by sharing surface normals, so can a projection mesh. When projecting with multiple trilinear triangles, issues of continuity arise that can be solved by clipping in scene space. The simplest approach to the problem of clipping is to clip each projection in view space. This can

6 result in discontinuity because the lines between projected vertices are drawn as straight when they are really curved. Discontinuities arising from this simplistic approach can be addressed by clipping scene triangles to the volume swept out by projecting the parametric triangle into scene space. The result is equivalent to tessellating the projected scene triangle at the boundary of the next trilinear projection. Each edge of a surface triangle forms a parametric line segment that traces out a surface through space which may intersect with edges in the scene triangles. To correctly clip to a parametric surface triangle region, the three parametric line segments defining the triangle edges must be traced through all the scene triangles, and the triangles clipped according to the intersections. A line segment of the parametric triangle at t is defined parametrically by: e (s, t) := p i + tn i + s (p j + tn j ) i, j = 1, 2, 3 i j (13) where s varies from 0 to 1. Consider the intersection between a parametric edge, defined by the points p i + tn i and p j + tn j, and a scene triangle edge, defined by the points p s1 and p s2. The value of t at the intersection as projected out of the trilinear projection, can be determined independently of s because for the two line segments to intersect they must lie on the same plane. According to Equation 7 the four points are coplanar when: p s1 p s2 p i + t.n i p s2 p j + t.n j p s2 = 0 (14) This can expanded giving a quadratic in terms of t. This quadratic may have two, or no real solutions. Each value of t defines a triangle and the scene line can be intersected with that triangle yielding a clipping point. 5 Applications Non-linear projections have been explored in art and visualisation because of their ability to represent 3D objects in unusual ways. Though not as readily interpreted as perspective projections they may have the potential to illustrate complicated relations that are hidden in perspective projections. Finding nonlinear projections that genuinely improve the task of visualisation is an unsolved problem. This paper instead examines the use of trilinear projection as way of approximating reflections and refractions on curved surfaces. 5.1 Reflection The specular reflection of a ray is governed by the equation: θ r = θ i (15) where θ r is the angle of reflection and θ i is the angle of incidence. This leads to following equation: r = 2 (n i) n i (16) where r, n and i are unit vectors for the reflection, surface normal and incidence direction. 5.2 Refraction Refraction is the bending of light due to the difference in refractive index between two materials. Simple refraction can be described by Snell s law, which shows the relation between angles of incidence, refraction and the refractive index of the materials. Snell s law is described by the following relation: sin θ i sin θ t = η t η i (17) where θ i is the angle of incidence, θ t is the angle of transmission, η t is the refractive index of the material the ray is entering and η i is the refractive index of the material the ray is leaving. This leads to the following equation: q = ηi (cos θ t + η cos θ i )n (18) where q, i and n are the vectors of transmission, incidence and the surface normal, and η = ηt η i Approximating Reflections and Refractions For a given scene surface, represented by a polygon mesh with shared normal vectors (a Phong-shaded mesh), the reflected or refracted image in each segment can be approximated by a trilinear projection. The points p 1..3 and vectors r 1..3 in Figure 9 form a trilinear projection approximating the reflection seen from the eyepoint e in the triangle defined by the points p 1..3 and the normals n The projection is exactly correct at the vertices, but the reflection vectors across the surface will in general only approximate the correct solution. This is because instead of interpolating the surface normal and then calculating the reflection vector, the reflection vector is interpolated from the vertex reflection vectors. In Figure 9 the reflection at point p should be along the ray r but in the trilinear projection it is along ray r. Despite the inaccuracy the computed reflection has several desirable characteristics: it is nonlinear, which means that the reflection is curved, as is expected, and it is continuous across multiple reflective facets. Refractions, inter-reflections and inter-refractions can all be approximated by using the vector equations for reflection and refraction at the vertices of the face to generate a trilinear projection. r 1 e n 1 p 1 i 2 i 1 i 3 i Figure 9: An approximation of a reflection with a trilinear projection n p r r' p 2 n 2 p 3 r 2 n 3 r 3

7 5.3 Example Projections Figure 10 shows a scene with a reflective sphere and a cube that is being rendered from a view point close to the sphere s surface. The rays at the four corners of the view intersect the sphere and are reflected off at various angles. Figure 11 (a) shows a ray traced image similar to that which would be seen in the section of reflective sphere shown in Figure 10. Figures 11 (b) to (f) show a trilinear projection approximation of the image with increasing projection mesh resolution. In the 1x1 surface, Figure 11 (b), the trilinear projection is two coplanar triangles whose normals are the reflection vectors show in Figure 10. Figures 11 (c) to (f) are rendered from projection surfaces that are increasingly accurate approximations of the surface of the sphere and the reflection vectors off the sphere s surface. The increasing mesh resolution means more trilinear projections are used to approximate the reflection, resulting in more accurate rendering. Image Projections Ray(ms) Trilinear(ms) (a) (b) (c) (d) (e) (f) Table 1: Execution time for rendering examples in Figure 11 Image Projections Ray(ms) Trilinear(ms) (a) (b) (c) (d) (e) (f) Table 2: Execution time for rendering examples in Figure 12 the cubic equation, solving the cubic and rejoining the projected vertices. The performance of rendering with multiple trilinear projections scales in a linear fashion. 7 Conclusion Figure 10: A cube reflected in a sphere Figures 12 (a) to (f) show a refraction of a cube scene through a plane whose normals are coincident. This simulates the effect of viewing an object through a convex lens. Figure 12 (a) is the correct ray traced solution and figures 12 (b) through (f) show trilinear projections to approximate this using 1x1, 2x2, 3x3, 4x4 and 5x5 resolution meshes respectively. 6 Performance Characteristics An implementation of the trilinear projection algorithms and a ray tracer was developed using OpenGL (Segal & Akeley 1998). The prototype can draw single and multiple trilinear projections, with or without clipping and tessellation. Reflective or refractive projection surfaces can be generated automatically, and the ray tracing implementation can render correct reflection and refraction solutions for comparison. The ray tracing code is naive, and performs no scene organisation optimisation to decrease the rendering time. The results here were obtained on a 800 MHz Athlon with a GeForce 2mx graphics card. Execution speeds were averaged across multiple runs. Table 1 shows the speed in milliseconds to render a single frame of the scene presented in Figure 11 at resolutions of pixels. The scene consists of 686 untextured triangles. Table 2 shows the time taken to render the images in Figure 12, also at pixels with same scene rendered through a refractive sphere. As noted before ray tracing can be speeded up by organising the scene so that for any particular ray only a subset of triangles need be intersected. Trilinear projection can also use scene organisation to improve speed because any particular trilinear projection may only image a subset of the scene. For a single trilinear projection there is only a linear performance cost over scanline rendering. This performance cost relates to determining the coefficients of The problem of rendering nonlinear projections with scanline-style algorithms has not previously been thoroughly examined. This paper presents an overview of a trilinear projection algorithm that projects scene points and triangles in a manner compatible with scanline rendering algorithms. The algorithms have been implemented and used to demonstrate a range of applications. More details of the algorithms and performance benchmarks appear elsewhere (Vallance 2005). The performance characteristics of scanline rendering have made it very useful in interactive and animated computer graphics, and algorithms compatible with scanline rendering have a substantial base of hardware and software to draw upon. A limitation of scanline rendering has been its difficulty in modeling certain complicated optical interactions. The trilinear projection algorithm provides a new technique for rendering optical interactions that presents developers with more tools to render unusual visualisations and optical phenomena with acceptable performance. References Blinn, J. F. & Newell, M. E. (1976), Texture and reflection in computer generated images, Communications of the ACM 19(10), Cho, F. (2000), Towards Interative Ray Tracing in Two- and Three-Dimensions, PhD thesis, University of California at Berkeley. Davidhazy, A. (2001), Peripheral portraits and other strip camera photographs, Retrieved October, 2001 from the World Wide Web rit.edu/~andpph/exhibit-6.html. Glaeser, G. (1999), Reflections on spheres and cylinders of revolution, Journal for Geometry and Graphics 3(2), Hakura, Z., Snyder, J. & Lengyel, J. (2001), Parameterized environment maps, in Proc. of the 2001 Symposium of Interactive 3D Graphics.

8 Löffelmann, H. (1995), Extended cameras for ray tracing, Master s thesis, Vienna Technical Institute. Ofek, E. & Rappoport, A. (1999), Interactive reflections on curved objects, in Proc of SIGGRAPH 99. Rademacher, P. & Bishop, G. (1998), Multiple-centerof-projection images, in Proc. of SIGGRAPH 98. Segal, M. & Akeley, K. (1998), The OpenGL Graphics System: a Specification (Version 1.2). Vallance, S. (2005), Trilinear Projection, PhD thesis, Flinders University. Yu, J. & McMillan, L. (2004), General linear cameras, in T. Pajdla & J. Matas, eds, Computer Vision - ECCV 2004, 8th European Conference on Computer Vision, Prague, Czech Republic, May 11-14, Proceedings, Part II, Vol of Lecture Notes in Computer Science, Springer.

9 (a) (b) (a) (b) (c) (d) (c) (d) (e) Figure 11: A cube reflected on a sphere: (a) ray traced, (b) 1x1 surface, (c) 2x2 surface, (d) 3x3 surface, (e) 4x4 surface, (f) 5x5 surface (f) (e) Figure 12: A cube refracted through a plane with spherical normals: (a) ray traced, (b) 1x1 surface, (c) 2x2 surface, (d) 3x3 surface, (e) 4x4 surface, (f) 5x5 surface (f)