Camera Models and Optical Systems Used in Computer Graphics: Part I, Object-Based Techniques Brian A. Barsky1;2;3, Daniel R. Horn1, Stanley A. Klein2;3 Je®rey A. Pang1, and Meng Yu1 1 Computer Science Division 2 School of Optometry 3 Bioengineering Graduate Group University of California, Berkeley, California, 94720-1776, USA http://www.cs.berkeley.edu/optical Contact author: [email protected] Abstract. Images rendered with traditional computer graphics tech- niques, such as scanline rendering and ray tracing, appear focused at all depths. However, there are advantages to having blur, such as adding realism to a scene or drawing attention to a particular place in a scene. In this paper we describe the optics underlying camera models that have been used in computer graphics, and present object space techniques for rendering with those models. In our companion paper [3], we survey im- age space techniques to simulate these models. These techniques vary in both speed and accuracy. 1 Introduction The goal of increasing realism fuels progress in computer graphics. However, perfect realism is impossible to achieve simply due to the number of interactions that occur when sending light into a scene and collecting the results on a discrete plane. The human eye can discern many details; thus, humans can notice the surreal clarity of most computer-generated scenes. In particular, images often lack depictions of focus, blur, optical aberrations, and depth of ¯eld. This paper illustrates the concepts behind modeling computer graphics with a lens system. The rendering of a 3D scene requires transforming from object space to image space. In object space, the entire geometry of the scene, with all the materials and surfaces, is present. This transformation is parameterized by a camera angle and location. There are various algorithms that can be used to translate the 3D geometry into a ¯nal, 2D image. When the scene is transformed into a 2D image, it often retains a depth map which contains distance measurements to the camera at any given pixel; this may be used for additional post-processing algorithms. Two ubiquitous methods to translate a 3D model in object space into a 2D image are scanline rendering and ray tracing. However, images generated with these methods are rendered in focus at all distances, as if photographed through a pinhole camera with an in¯nitely tiny aperture. Thus, these standard rendering techniques cannot convey focus nor represent arbitrary lens con¯gurations. The rendering of images with focal blur is an interesting and important ¯eld of research in computer graphics. Several techniques have been proposed to model camera lens systems to varying degrees of accuracy. They range from object-based modi¯cations to ray tracing and scanline rendering to image-based techniques that convolve and otherwise distort an image after it has been ren- dered. We discuss object based techniques in this paper, and image based tech- niques in our companion paper [3], published later in these proceedings. Cur- rently, these techniques o®er varying realism and speed in rendering images with camera models. Before studying them, some background on optics will be reviewed. 2 Camera Models Several camera models have been employed in computer graphics to approximate physical optical systems. Although each has distinctive qualities and produces characteristically di®erent images, they all share some common traits. A camera model simulates the capture of light from a three-dimensional scene in object space onto a two-dimensional image, or image space. Most models contain or approximate a system of parallel lenses such as that of a camera or the eye. An axis that passes through the geometric center of the system of lenses is called the optical axis. In computer graphics, the center of a single lens system is sometimes called the center of projection (COP). Objects in the scene are projected through the lens system to form an image located on the opposite side of the system. Each lens has an aperture which de¯nes the area through which light is allowed to pass to the image. Although rendering models usually consider an image plane in front of the lens system when forming an image, camera models consider a ¯lm plane behind the lens system. Like physical optical systems, the image formed on the ¯lm plane is inverted. An appropriate image plane can be derived from the location of the ¯lm plane in some camera models. The ¯eld of view is parameterized by the size of the ¯lm plane. In optics, the standard unit of lens power is the diopter [1], a unit which is measured in inverse meters. Although the amount blur is not linear in distance, it is approximately linear in diopters. 2.1 Camera Obscura (Pinhole Camera) The standard rendering algorithms in computer graphics are equivalent to mod- eling a camera obscura or pinhole camera. In this model, all the light rays from the scene pass through a single point, called the center of projection, or through a lens with an in¯nitesimal aperture. Since only a single light ray cast from each point in the scene can pass through the COP regardless of its location in space, each point will be rendered exactly l l0 f f0 object diam 0 F a F optical axis principal plane image Fig. 1. Thin lens model. once on the ¯lm plane. This causes all objects in the scene to appear in sharp focus on the image produced. The pan of the camera can be parameterized by the location of the ¯lm plane along the axis. This model could not be realized physically since in reality, a point-sized aperture would produce an image on the ¯lm plane that would be too dim to observe. However, simply reversing this model yields a ray tracer, where a ray is projected from the COP through each pixel on the image plane and then the colors of the materials it intersects determine the color of the particular pixel. Analytic techniques have also been developed using this model to project objects to image space e±ciently [7]. 2.2 Thin Lens Approximation In reality, all lenses have a ¯nite aperture, and hence each point in the scene emits a cone of light which is visible to the lens. In geometric optics, when light rays encounter the boundary between two media, the angle through which light is refracted can be calculated using Snell's law. Since real lenses are composed of a refractive material that has a ¯nite and non-constant thickness, Snell's law can be used to project light rays through a lens. The thin lens approximation instead assumes that even though lenses have a ¯nite aperture, they have an in¯nitesimal thickness. In general, thin lenses are also spherical in form, either convex or concave, and ideal. A lens is ideal if it has the property that the change in slope for a ray passing through the lens is proportional to the distance from the center of the lens to the point at which the ray encounters it [1]. The plane that is normal to the optical axis at the lens is called the principal plane (Figure 1). Along the optical axis of the lens is the focal point, F . Formally, F is an axial point having the property that any ray emanating from it or proceeding toward it travels parallel to the axis after refraction by the lens [8]. A secondary focal point, F 0 exists on the opposite side of the lens. Analogously, this point is de¯ned as an axial point having the property that any incident ray traveling parallel to the axis will proceed toward F 0 after refraction (Figure 1). The distance between the center of the lens and a focal point is called the focal length. If the medium is same on both sides of the lens (hence the indices of refraction, n and n0, are equal) and if the lens is symmetric, then these distances, denoted by f and f 0 are the same. The law governing image formation through a thin lens in Gaussian form is: 1 1 1 + = (1) l l0 f where l is the distance from the object to the lens, l0 is the distance from the lens to the image, and f is the focal length (Figure 1). By solving for l0 in equation (1), we can derive that the projection of an object at a distance l in front of the lens will converge at a distance fl l0 = (2) l ¡ f behind the lens. The f-stop (or F number), which we denote as fstop, is de¯ned as the ratio of the focal length to the diameter of the aperture of the lens, denoted by adiam: f fstop = : (3) adiam When focusing at a particular distance dfocus in front of the principal plane, 0 there is a speci¯c location dfocus where the ¯lm plane must be located such that each point that is at distance dfocus forms its image on that plane (Figure 2). The plane perpendicular to the axis at dfocus is called the focal plane. To derive 0 the distance dfocus behind the lens to place the ¯lm plane, we substitute dfocus 0 0 and dfocus for l and l , respectively, in equation (2), yielding: 0 fdfocus dfocus = : (4) dfocus ¡ f l l0 optical axis θ c diam diam a object o image of o focal plane principal plane ¯lm plane 0 dfocus dfocus Fig. 2. The circle of confusion cdiam and blur angle θ in the thin lens model. A point that is closer to the lens than the focal plane projects behind the ¯lm plane, whereas the projection of a point farther from the lens than the focal plane converges in front of the ¯lm plane.