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Multiperspective Projection and Collineation

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Multiperspective Projection and Collineation Multiperspective Projection and Collineation Jingyi Yu Leonard McMillan Massachusetts Institute of Technology University of North Carolina Cambridge, MA 02139 Chapel Hill, NC 27599 [email protected] [email protected] Abstract We present theories of multiperspective projection and collineation. Given an arbitrary multiperspective imaging system that captures smoothly varying set of rays, we show how to map the rays onto a 2D ray manifold embedded in a (b) 4D linear vector space. The characteristics of this imaging system, such as its projection, collineation, and image dis- tortions can be analyzed by studying the 2-D tangent planes of this ray manifold. These tangent planes correspond to the recently proposed General Linear Camera (GLC) model. In this paper, we study the imaging process of the GLCs. (c) We show the GLC imaging process can be broken down into (a) two separate stages: the mapping of 3D geometry to rays Figure 1. (a) An image captured on a pear-shaped mirror. and the sampling of those rays over an image plane. We (b) Cross-slit type distortions. (c) Pushbroom type distor- derive a closed-form solution to projecting 3D points in a tions. scene to rays in a GLC. A GLC image is created by sam- pling these rays over an image plane. We develop a notion These mappings are unique down to a scale factor, and of GLC collineation analogous to pinhole cameras. GLC the same infrastructure can also be used to describe ortho- collineation describes the transformation between the im- graphic cameras. When the image plane of the pinhole cam- ages of a single GLC due to changes in sampling and image era changes, the corresponding images change according. plane selection. We show that general GLC collineations The transformation between images on two different planes can be characterized by a quartic (4th order) rational func- is often referred to as homography. In the pinhole case, it tion. GLC projection and collineation provides a basis is a projective transformation and, thus, lines are still pre- for developing new computer vision algorithms suitable for served as lines despite the change of the image plane. analyzing a wider range of imaging systems than current Recent developments have suggested alternative imag- methods, based on simple pinhole projection models, per- ing models such as pushbroom [10], cross-slit [16, 8], and mit. oblique [7] cameras. These cameras collect rays under dif- ferent geometric constraints. For instance, all rays pass through a particular line in a pushbroom camera. All rays of 1. Introduction a cross-slit cameras pass through two lines, and, in oblique cameras no two rays can intersect or be parallel. These The imaging process entails mapping 3D geometry onto camera models are often referred to as multiperspective a two dimensional manifold via some camera or imaging cameras. They provide alternate and potentially advanta- model. This projection depends on both the geometry of geous imaging systems for understanding the structure of the imaging system and the parametrization of the image observed scenes. The mapping from 3D points to pixels is plane. The most common imaging model is the pinhole no longer a projective transformation in these cameras, and camera, which collects rays passing through a point and is often difficult to calculate. For instance, the collineation organizes rays onto a image plane. This mapping can be is a 3x3x3 tensor in a cross-slit camera [16]. Interesting fully described using the classic 3 x 4 camera matrix [5]. multiperspective distortions are also observed on these cam- eras, as shown in Figure 1. When we model complex imaging systems such as cata- dioptric mirrors, it is difficult to compute the mapping from 3D points to 2D images because it requires an inverse map- ping from rays to points and a closed-form solution may not exist. The lack of a closed-form projection prohibits (a) (b) further analysis of the resulting images. In practice, how- ever, many catadioptric imaging systems do exhibit local distortions that are similar to pushbroom or cross-slit dis- tortions (Figure 1). In this paper, we show that these visual phenomenon are not coincidental. In fact, these imaging systems can be precisely modelled by a special class of lo- cal multiperspective camera models. Furthermore, we give a closed-form solution for finding the local multiperspective camera models and demonstrate that all possible images can (c) (d) be formed by these cameras. Given any imaging system, we first map the rays col- Figure 2. (a) and (c) are cross-slits images synthesized lected by the system to a two dimensional ray manifold em- from light fields. (b) and (d) are pushbroom images synthe- sized from light fields. The distortions of the curved isolines bedded in a 4D ray space. We benefit from the recently on the objects illustrate various multi-perspective effects of proposed General Linear Camera (GLC) model [17], which GLC cameras in (a) and (b). The pushbroom camera (b) describes all possible linear manifolds. We use the GLC and (d) exhibits apparent stretching/duplications of objects model to locally analyze the ray manifold using its tan- lying far away from the image plane. gent plane which can be completely characterized as one of the eight GLCs. We provide a closed-form solution to in which each pair of rays are oblique. The resulting im- Projection for all GLCs. Next, we show how to deter- ages captured by these cameras are easily interpretable, yet mine the Collineation using the GLC intrinsics. We show they exhibit interesting multiperspective distortions. For in- the Collineation between two image planes is, in general, stance, lines often project to curves as shown in Figure 2 and a quartic (4th order) rational transformation. Most imag- a single 3D point might project to multiple points, causing ing systems, such as the catadioptric mirrors, can be easily feature duplications shown in Figure 2(d). analyzed using our framework and distortions can be in- Multiperspective imaging techniques have also been ex- terpreted using the local GLC Projection and Collineation plored in the field of computer graphics. Example images model. include multiple-center-of-projection images [2], manifold mosaics [9], and multiperspective panoramas [15]. Multi- 2. Previous Work perspective distortions have also been studied on real cata- dioptric imaging systems [1, 13] and are often characterized Pinhole cameras collect rays passing through a single using caustics [13]. Zorin and Barr [19] studied the use of point. Because of its simplicity, the projection is the clas- multiperspective and other geometric distortions to improve sic 3 x 4 camera matrix [5], which combines six extrin- perceptual qualities of images. Swaminathan et al [14] pro- sic and five intrinsic camera parameters into a single op- posed a method to compute minimally distorted images us- erator that maps homogenous 3D points to a 2D image ing simple geometry priors on scene structure. plane. These mappings are unique down to a scale factor, Seitz [11] has analyzed the space of multiperspective and the same infrastructure can also be used to describe or- cameras to determine those with a consistent epipolar ge- thographic cameras. When only the image plane changes ometry. His work suggests that only a small class of while the pinhole remains constant, the transformation be- multiperspective images can be used to analyze three- tween the two images is a projective transformation, which dimensional structures, mainly because the epipolar con- is called a homography. straints cannot be established over the these images. Recently, several researchers have proposed alternative General projection and imaging models have also been multiperspective camera models, which capture rays origi- studied in terms of rays. Pajdla proposed the ray-closure nating from different points in space. These multiperspec- model to describe a camera and its projection. Gu et al [4] tive cameras include pushbroom cameras [10], which col- explicitly parameterized rays under a particular 4D map- lect rays along parallel planes from points swept along a ping known as a two-plane parametrization. Most recently, linear trajectory, two-slit cameras [8, 16], which collect all we characterize all linear manifolds in the 4D ray space de- rays passing through two lines, and oblique cameras [7], fined by a two-plane parametrization, which we call Gen- r3 r 2 r2 r3 t (s3, t3) r3 (s2, t2) r2 v r1 r Π1 1 Π C 2 Π3 Π4 (u2, v2) s r (b) (c) (d) 1 u (e) (u3, v2) r1 (s1, t1) (u1, v1) r2 r r2 3 r1 r3 α⋅ r1 +β⋅ r 2 + (1−α−β )⋅ r 3 Π 1 r1 r3 Π2 z Π 3 r r2 (a) (f) (g) (h) (i) Figure 3. General Linear Camera Models. (a) A GLC Model collects radiance along all possible affine combination of three rays. The rays are parameterized by their intersections with two parallel planes. There are precisely eight GLCs, shown in (b) to (i). (b) In a pinhole camera, all rays pass through a single point. (c) In an orthographic camera, all rays are parallel. (d) In a pushbroom, all rays lie on a set of parallel planes and pass through a line. (e) In a cross slit camera, all rays pass through two non-coplanar lines. (f) In a pencil camera, all coplanar rays originate from a point on a line and lie on a specific plane through the line. (g) In a twisted orthographic camera, all rays lie on parallel twisted planes and no rays intersect. (h) In an bilinear camera, no two rays are coplanar and no two rays intersect. (i) In an EPI camera, all rays lie on a 2D plane. eral Linear Cameras. Most of these cameras satisfy Seitz’s (lines) in 3D space that all rays will pass through: criterion [11].
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