Manufacturing and measurement of freeform optics

This survey is mainly motivated by the increased availability and use of panoramic image acquisition devices, in computer vision and various of its applications. Different technologies and different computational models thereof exist and algorithms and theoretical studies for geometric computer vision ("structure-from-motion") are often re-developed without highlighting common underlying principles. One of the goals of this survey is to give an overview of image acquisition methods used in computer vision and especially, of the vast number of camera models that have been proposed and investigated over the years, where we try to point out similarities between different models. Results on epipolar and multi-view geometry for different camera models are reviewed as well as various calibration and self-calibration approaches, with an emphasis on non-perspective cameras. We finally describe what we consider are fundamental building blocks for geometric computer vision or structure-from-motion: epipolar geometry, pose and motion estimation, 3D scene modeling, and bundle adjustment. The main goal here is to highlight the main principles of these, which are independent of specific camera models.

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Camera Models And Fundamental Concepts Used In Geometric Computer Vision

We designed a new omnidirectional image sensor COPIS (Conic Projection Image Sensor) to guide the navigation of a mobile robot. The feature of COPIS is passive sensing of the omnidirectional image of the environment, in real-time (at the frame rate of a TV camera), using a conic mirror. COPIS is a suitable sensor for visual navigation in a real world environment. We report here a method for navigating a robot by detecting the azimuth of each object in the omnidirectional image. The azimuth is matched with the given environmental map. The robot can precisely estimate its own location and motion (the velocity of the robot) because COPIS observes a 360/spl deg/ view around the robot, even when all edges are not extracted correctly from the omnidirectional image. The robot can avoid colliding against unknown obstacles and estimate locations by detecting azimuth changes, while moving about in the environment. Under the assumption of the known motion of the robot, an environmental map of an indoor scene is generated by monitoring azimuth change in the image. >

MapBased Navigation for a Mobile Robot with Ominidirectional Image Sensor COPIS

Conventional vision systems and algorithms assume the imaging system to have a single viewpoint. However, these imaging systems need not always maintain a single viewpoint. For instance, an incorrectly aligned catadioptric system could cause non-single viewpoints. Moreover, a lot of flexibility in imaging system design can be achieved by relaxing the need for imaging systems to have a single viewpoint. Thus, imaging systems with non-single viewpoints can be designed for specific imaging tasks, or image characteristics such as field of view and resolution. The viewpoint locus of such imaging systems is called a caustic.

In this paper, we present an in-depth analysis of caustics of catadioptric cameras with conic reflectors. We use a simple parametric model for both, the reflector and the imaging system, to derive an analytic solution for the caustic surface. This model completely describes the imaging system and provides a map from pixels in the image to their corresponding viewpoints and viewing direction. We use the model to analyze the imaging system's properties such as field of view, resolution and other geometric properties of the caustic itself. In addition, we present a simple technique to calibrate the class of conic catadioptric cameras and estimate their caustics from known camera motion. The analysis and results we present in this paper are general and can be applied to any catadioptric imaging system whose reflector has a parametric form.

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Non-Single Viewpoint Catadioptric Cameras: Geometry and Analysis

A computational camera uses a combination of optics and processing to produce images that cannot be captured with traditional cameras. In the last decade, computational imaging has emerged as a vibrant field of research. A wide variety of computational cameras has been demonstrated to encode more useful visual information in the captured images, as compared with conventional cameras. In this paper, we survey computational cameras from two perspectives. First, we present a taxonomy of computational camera designs according to the coding approaches, including object side coding, pupil plane coding, sensor side coding, illumination coding, camera arrays and clusters, and unconventional imaging systems. Second, we use the abstract notion of light field representation as a general tool to describe computational camera designs, where each camera can be formulated as a projection of a high-dimensional light field to a 2-D image sensor. We show how individual optical devices transform light fields and use these transforms to illustrate how different computational camera designs (collections of optical devices) capture and encode useful visual information.

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Computational Cameras: Convergence of Optics and Processing

The problem of controlling a single ray bundle with a single reflector is not generally solvable, but approximate solutions may often be found that are acceptable for applications. We introduce a new technique for finding such approximations and apply it to the design of a driver-side mirror for an automobile that has no blind spot and minimal distortion.

Controlling a ray bundle with a free-form reflector

I present a design technique for realizing given projections as catadioptric sensors. In general, these problems do not have solutions, but approximate solutions may often be found that are visually acceptable. The method described reduces the problem to solving a linear system. A given transformation from the image plane to an object surface is shown to determine a vector field that is normal to the surface in the case where the vector field is a gradient. For the case when the vector field is not a gradient, several functionals are presented that may be minimized to give approximate solutions. As an application several new designs are described, including a mirror that directly gives a full 360-deg cylindrical projection without the need for any digital processing.

Designing a mirror to realize a given projection.

We consider the problem of designing a passenger side mirror for an automobile that does not have a blind spot and that does not distort the image. Our model consists of a coupled pair of partial differential equations that do not have a common solution. Using a best mean-square-error functional, we find approximate solutions using nonlinear optimization. In one case a local minimum provides a mirror that solves the problem, but it does not reverse the image.

Blind-spot problem for motor vehicles

The problem of designing optical systems that contain free-form surfaces is a challenging one, even in the case of designing a single surface. Here we present a method for the coupled design of two free-form reflective surfaces that will have a prescribed distortion. On one hand, the method can be described using traditional vectors and matrices, which we do, but it is motivated by viewing the problem in the language of distributions from differential geometry and makes use of the exterior differential systems, which we relegate to an appendix. Example applications are given to the design of a mirror pair that increases the field of view of an observer, a similar mirror pair that also rotates the observer's view, and a pair of mirrors that give the observer a traditional panoramic strip view of the scene.

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Designing coupled free-form surfaces.

Optical design in the 19th century was largely empirical, and today design in the geometric realm is often
performed by optimizing a cost function which is defined via ray tracing. A natural question to ask then, is how
to perform optical design using a more direct method, such as solving partial differential equations or variational
problems. We consider the problem of writing down equations to model a single surface (mirror or lens) to
completely control a single bundle of rays. When this is done, with high probability the solution surface will not
be rotationally symmetric, but freeform. Although a bundle may not be completely controllable with a single
surface, approximate solution can be sometimes have applications. In particular, we will show how to compute
the shape of a driver-side mirror that has no blind-spot or distortion.

R. Andrew Hicks

Papers

Controlling a ray bundle with a free-form reflector

Designing a mirror to realize a given projection.

Blind-spot problem for motor vehicles

Designing coupled free-form surfaces.

Direct methods for freeform surface design