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Caustics of Catadioptric Cameras
 In Proc. International Conference on Computer Vision
, 2001
"... Conventional vision systems and algorithms assume the camera to have a single viewpoint. However, sensors need not always maintain a single viewpoint. For instance, an incorrectly aligned system could cause nonsingle viewpoints. Also, systems could be designed to specifically deviate from a single ..."
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Cited by 62 (10 self)
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Conventional vision systems and algorithms assume the camera to have a single viewpoint. However, sensors need not always maintain a single viewpoint. For instance, an incorrectly aligned system could cause nonsingle viewpoints. Also, systems could be designed to specifically deviate from a single viewpoint to tradeoff image characteristics such as resolution and field of view. In these cases, the locus of viewpoints forms what is called a caustic. In this paper, we present an indepth analysis of caustics of catadioptric cameras with conic reflectors. Properties of caustics with respect to field of view and resolution are presented. Finally, we present ways to calibrate conic catadioptric systems and estimate their caustics from known camera motion.
Flat refractive geometry
 In Proc. IEEE CVPR
, 2008
"... While the study of geometry has mainly concentrated on singleviewpoint (SVP) cameras, there is growing attention to more general nonSVP systems. Here we study an important class of systems that inherently have a nonSVP: a perspective camera imaging through an interface into a medium. Such systems ..."
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Cited by 29 (12 self)
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While the study of geometry has mainly concentrated on singleviewpoint (SVP) cameras, there is growing attention to more general nonSVP systems. Here we study an important class of systems that inherently have a nonSVP: a perspective camera imaging through an interface into a medium. Such systems are ubiquitous: they are common when looking into waterbased environments. The paper analyzes the common flatinterface class of systems. It characterizes the locus of the viewpoints (caustic) of this class, and proves that the SVP model is invalid in it. This may explain geometrical errors encountered in prior studies. Our physicsbased model is parameterized by the distance of the lens from the medium interface, beside the focal length. The physical parameters are calibrated by a simple approach that can be based on a singleframe. This directly determines the system geometry. The calibration is then used to compensate for modeled system distortion. Based on this model, geometrical measurements of objects are significantly more accurate, than if based on an SVP model. This is demonstrated in realworld experiments. 1.
Transparent and Specular Object Reconstruction
, 2010
"... This state of the art report covers reconstruction methods for transparent and specular objects or phenomena. While the 3D acquisition of opaque surfaces with lambertian reflectance is a wellstudied problem, transparent, refractive, specular and potentially dynamic scenes pose challenging problems ..."
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Cited by 20 (3 self)
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This state of the art report covers reconstruction methods for transparent and specular objects or phenomena. While the 3D acquisition of opaque surfaces with lambertian reflectance is a wellstudied problem, transparent, refractive, specular and potentially dynamic scenes pose challenging problems for acquisition systems. This report reviews and categorizes the literature in this field. Despite tremendous interest in object digitization, the acquisition of digital models of transparent or specular objects is far from being a solved problem. On the other hand, realworld data is in high demand for applications such as object modeling, preservation of historic artifacts and as input to datadriven modeling techniques. With this report we aim at providing a reference for and an introduction to the field of transparent and specular object reconstruction. We describe acquisition approaches for different classes of objects. Transparent objects/phenomena that do not change the straight ray geometry can be found foremost in natural phenomena. Refraction effects are usually small and can be considered negligible for these objects. Phenomena as diverse as fire, smoke, and interstellar nebulae can be modeled using a straight ray model of image formation. Refractive and specular surfaces on the other hand change the straight rays into usually piecewise linear ray paths, adding additional complexity to the reconstruction problem. Translucent objects exhibit significant subsurface scattering effects rendering traditional acquisition approaches unstable. Different classes of techniques have been developed to deal with these problems and good reconstruction results can be achieved with current stateoftheart techniques. However, the approaches are still specialized and targeted at very specific object classes. We classify the existing literature and hope to provide an entry point to this exiting field.
Analytical Forward Projection for Axial NonCentral Dioptric & Catadioptric Cameras
"... Abstract. Wepresentatechniqueformodelingnoncentralcatadioptric cameras consisting of a perspective camera and a rotationally symmetric conic reflector. While previous approaches use a central approximation and/or iterative methods for forward projection, we present an analytical solution. This allo ..."
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Cited by 18 (8 self)
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Abstract. Wepresentatechniqueformodelingnoncentralcatadioptric cameras consisting of a perspective camera and a rotationally symmetric conic reflector. While previous approaches use a central approximation and/or iterative methods for forward projection, we present an analytical solution. This allows computation of the optical path from a given 3D point to the given viewpoint by solving a 6 th degree forward projection equation for general conic mirrors. For a spherical mirror, the forward projection reduces to a 4 th degree equation, resulting in a closed form solution. We also derive the forward projection equation for imaging through a refractive sphere (noncentral dioptric camera) and show that it is a 10 th degree equation. While central catadioptric cameras lead to conic epipolar curves, we show the existence of a quartic epipolar curve for catadioptric systems using a spherical mirror. The analytical forward projection leads to accurate and fast 3D reconstruction via bundle adjustment. Simulations and real results on single image sparse 3D reconstruction are presented. We demonstrate ∼ 100 times speed up using the analytical solution over iterative forward projection for 3D reconstruction using spherical mirrors. 1
Axial Light Field for Curved Mirrors: Reflect Your Perspective, Widen Your View
"... Mirrors have been used to enable wide fieldofview (FOV) catadioptric imaging. The mapping between the incoming and reflected light rays depends nonlinearly on the mirror shape and has been wellstudied using caustics. We analyze this mapping using twoplane light field parameterization, which pro ..."
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Cited by 9 (4 self)
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Mirrors have been used to enable wide fieldofview (FOV) catadioptric imaging. The mapping between the incoming and reflected light rays depends nonlinearly on the mirror shape and has been wellstudied using caustics. We analyze this mapping using twoplane light field parameterization, which provides valuable insight into the geometric structure of reflected rays. Using this analysis, we study the problem of generating a singleviewpoint virtual perspective image for catadioptric systems, which is unachievable for several common configurations. Instead of minimizing distortions appearing in a single image, we propose to capture all the rays required to generate a virtual perspective by capturing a light field. We consider rotationally symmetric mirrors and show that a traditional planar light field results in significant aliasing artifacts. We propose axial light field, captured by moving the camera along the mirror rotation axis, for efficient sampling and to remove aliasing artifacts. This allows us to computationally generate wide FOV virtual perspectives using a wider class of mirrors than before, without using scene priors or depth estimation. We analyze the relationship between the axial light field parameters and the FOV/resolution of the resulting virtual perspective. Real results using a spherical mirror demonstrate generating 140 ◦ FOV virtual perspective using multiple 30 ◦ FOV images. 1.
ImageSpace Caustics and Curvatures
"... Caustics are important visual phenomena, as well as challenging global illumination effects in computer graphics. Physically caustics can be interpreted from one of two perspectives: in terms of photons gathered on scene geometry, or in terms of a pair of caustic surfaces. These caustic surfaces are ..."
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Cited by 7 (2 self)
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Caustics are important visual phenomena, as well as challenging global illumination effects in computer graphics. Physically caustics can be interpreted from one of two perspectives: in terms of photons gathered on scene geometry, or in terms of a pair of caustic surfaces. These caustic surfaces are swept by the foci of light rays. In this paper, we develop a novel algorithm to approximate caustic surfaces of sampled rays. Our approach locally parameterizes rays by their intersections with a pair of parallel planes. We show neighboring ray triplets are constrained to pass simultaneously through two slits, which rule the caustic surfaces. We derive a ray characteristic equation to compute the two slits, and hence, the caustic surfaces. Using the characteristic equation, we develop a GPUbased algorithm to render the caustics. Our approach produces sharp and clear caustics using much fewer ray samples than the photon mapping method and it also maintains high spatial and temporal coherency. Finally, we present a normalray surface representation that locally parameterizes the normals about a surface point as rays. Computing the normal ray caustic surfaces leads to a novel realtime discrete shape operator.
Reflections of reality in Jan van Eyck and Robert Campin
 In Measuring Art: A Scientific Revolution in Art History
, 2003
"... There has been considerable debate about the perspectival / optical bases of the naturalism pioneered by Robert Campin and Jan van Eyck. Their paintings feature brilliantly rendered convex mirrors, which have been the subject of much comment, especially iconographical. David Hockney has recently arg ..."
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Cited by 5 (1 self)
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There has been considerable debate about the perspectival / optical bases of the naturalism pioneered by Robert Campin and Jan van Eyck. Their paintings feature brilliantly rendered convex mirrors, which have been the subject of much comment, especially iconographical. David Hockney has recently argued that the Netherlandish painters exploited the imageforming capacities of concave mirrors. However, the secrets of the images within the painted mirrors have yet to be revealed. Using novel, rigorous techniques to analyse the geometric accuracy of the mirrors, unexpected findings emerge, which radically affect how we see the paintings as being generated. We focus on Jan van Eyck’s Arnolfini Portrait, and the Heinrich von Werl Triptych, here reattributed to Robert Campin. The accuracy of the convex mirrors depicted in these paintings is assessed by applying mathematical techniques drawn from computer vision. The proposed algorithms allow us also to “rectify ” the image in the mirror so that it becomes a normalised projection, thus providing us with a second view from the back of the painted room. The plausibility of the painters ’ renderings of space in the convex mirrors can be assessed. The rectified images can be used for purposes of threedimensional reconstruction as well as measuring accurate dimensions of objects and people. The surprising results presented in this paper cast a new light on the understanding of the artists ’ techniques and their optical imitation of seen things, and potentially require a rethinking of the foundations of Netherlandish naturalism. They also suggest that the von Werl panels should be reinstated as autograph works by R. Campin. Additionally, this research represents a further attempt to build a constructive dialogue between two very different disciplines: computer science and history of art. Despite their fundamental differences, the procedures followed by science and art history can learn and be enriched by each other.
Planebased calibration of central catadioptric cameras
 in Proc. IEEE 12th Int. Conf. Comput. Vis., 2009
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Beyond Alhazen’s Problem: Analytical Projection Model for NonCentral Catadioptric Cameras with Quadric Mirrors
"... Catadioptric cameras are widely used to increase the field of view using mirrors. Central catadioptric systems having an effective single viewpoint are easy to model and use, but severely constraint the camera positioning with respect to the mirror. On the other hand, noncentral catadioptric system ..."
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Cited by 4 (0 self)
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Catadioptric cameras are widely used to increase the field of view using mirrors. Central catadioptric systems having an effective single viewpoint are easy to model and use, but severely constraint the camera positioning with respect to the mirror. On the other hand, noncentral catadioptric systems allow greater flexibility in camera placement, but are often approximated using central or linear models due to the lack of an exact model. We bridge this gap and describe an exact projection model for noncentral catadioptric systems. We derive an analytical ‘forward projection’ equation for the projection of a 3D point reflected by a quadric mirror on the imaging plane of a perspective camera, with no restrictions on the camera placement, and show that it is an 8 th degree equation in a single unknown. While previous noncentral catadioptric cameras primarily use an axial configuration where the camera is placed on the axis of a rotationally symmetric mirror, we allow offaxis (any) camera placement. Using this analytical model, a noncentral catadioptric camera can be used for sparse as well as dense 3D reconstruction similar to perspective cameras, using wellknown algorithms such as bundle adjustment and plane sweeping. Our paper is the first to show such results for offaxis placement of camera with multiple quadric mirrors. Simulation and real results using parabolic mirrors and an offaxis perspective camera are demonstrated. 1.
AutoCalibration and 3D Reconstruction with NonCentral Catadioptric Sensors using Polarization Imaging
"... Abstract — Noncentral catadioptric sensors became a mandatory effective tool in a broad range of applications and hence robust modeling and calibration for such sensors are required. One simple and efficient method is to model the catadioptric sensor by its catacaustics. Calibrating the sensor is e ..."
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Cited by 3 (1 self)
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Abstract — Noncentral catadioptric sensors became a mandatory effective tool in a broad range of applications and hence robust modeling and calibration for such sensors are required. One simple and efficient method is to model the catadioptric sensor by its catacaustics. Calibrating the sensor is equivalent to computing its catacaustics which fully geometrically describe the sensor. In this work, we calibrate noncentral catadioptric sensors by computing their catacaustics using polarization imaging. We provide stable and accurate results by applying our novel practical method to extract required parameters. We have applied our method on a parabolic, a hyperbolic, and two spherical reflectors. In order to evaluate the calibration quality, 3D reconstructions for multiple real world points were simulated assuming to have configurations with a perspective lens or a telecentric lens. 3D reconstruction simulation from caustics using polarization for various configurations of catadioptric sensors is firstly presented in this work. I.