Introduction to Computer Vision CS / ECE 181B Tues, May 18, 2004 Ack: Matthew Turk (slides) Outline • Radiance and Irradiance • Lambertian and Specular surfaces • Bidirectional reflectance distribution function (BRDF) • Fundamental Radiometric Relation • Gradient Space • Reflectance Map • Photometric Stereo Geometry and Radiometry • In creating and interpreting images, we need to understand two things: – Geometry – Where scene points appear in the image (image locations) – Radiometry – How “bright” they are (image values) • Geometric enables us to know something about the scene location of a point imaged at pixel (u, v) • Radiometric enables us to know what a pixel value implies about surface lightness and illumination • This is relevant to both computer vision and computer graphics – Ray tracing – Illumination and shading models Radiometry • Radiometry is the measurement of light – Actually, electromagnetic energy • Imaging starts with light sources – Emitting photons – quanta of light energy – The sun, artificial lighting, candles, fire, blackbody radiators … • Light energy interact with surfaces – Reflection, refraction, absorption, fluorescence… – Also atmospheric effects (not just solid surfaces) • Light energy from sources and surfaces gets imaged by a camera – Through a lens, onto a sensor array, finally to pixel values – an image! Ray tracing Reversible: - From camera to light sources - From light sources to camera light source pixels visibility? reflection surface camera center What’s the intensity value (and color) at this pixel? Computer graphics examples CG example: Pixar Geri’s Game 1997 Oscar Award Best Animated Short Film Illumination and shading in CG • The illumination model describes how light interacts with the surface – A set of calculations for determining color and intensity at a given surface point, given a lighting model (model of light sources) – E.g., what color is that point on that surface? – Local vs. global Whether or not computation takes into account other elements in the scene (local model doesn’t) – May be empirical or experimental – Should model the effects of ambient light, incident light, surface reflection (diffuse and specular), etc. Illumination models • An illumination model is a function of – Time (exposure time) – Position and orientation of Object surface Lighting sources Camera – Wavelength of Lighting sources Surface reflectance function – Surface albedo () The fraction of incident light that is reflected by the surface, independent of wavelength Corresponds to intuitive notion of surface lightness Three surface reflectance functions/models Ideal diffuse (Lambertian) Ideal specular Directional diffuse Cook-Torrance Model Different parameters give different surface appearances Radiometry • Radiometry – the science of measurement of electromagnetic radiation – The complete EM spectrum (not just visible light) • To understand radiometry, we have to understand the notions of foreshortening and solid angle Foreshortening • Principal of foreshortening – When a patch of area A is tilted at an angle with respect to a distant view point p, it appears to have an area A cos • This is true for viewing a light source from the point of view of a surface patch; and also for viewing a surface patch from the point of view of a light source Which plate will heat faster? Irradiance cos Solid angle • Around any point on a surface is a hemisphere of directions – Parameterized by two angles, and • We’ll be considering light entering and exiting such hemispheres Solid angle • Solid angle of a cone of directions – the area cut out by the cone on the unit sphere – Solid angle () = surface area at r=1 – = A/r2 • Solid angle of a complete sphere = 4π steradians (sr) What’s the area of a circle? What’s the surface area of a sphere? Solid angle • The solid angle subtended by a patch of area dA is given by dA cos d r2 • The solid angle subtended by a patch of angular size (d, d) is d sin d d Radiance and irradiance • Radiance (L) – energy exiting a source or surface • Irradiance (E)– incoming energy E L E E L L E L Which (E or L) does a camera sensor array directly measure? Example • Luminance of common sources: – – – – – – surface of the sun: 2,000,000,000 cd/m2 sunlit clouds: 30,000 cd/m2 clear day: 3000 cd/m2 overcast day: 300 cd/m2 moonlight: 0.03 cd/m2 moonless sky: 0.00003 cd/m2 Photometric term • Luminance = commonly called “brightness” – Density of radiated power • Radiance = “scene brightness” • Irradiance = “image brightness” Radiometric terms Irradiance • A surface experiencing radiance L(x,,) coming in from solid angle d experiences irradiance E Lx, , cos d radiance solid angle foreshortening factor Calculating Irradiance • Integrate weighted incident radiance over hemisphere solid angle Since d sin d d Light and surfaces • When light energy hits a surface, several things can happen, depending on the surface properties (texture, color, opacity, material, …) – – – – – – Reflection Absorption Refraction Transmission Scattering Various combinations • The result is called reflectance, and it is a function of the incoming (incident) and outgoing angles of the light Reflectance example Irradiance (E) Radiance (L) ? Surface Reflectance K Bidirectional Reflectance Distribution Function • The BRDF tells us how bright a surface appears when viewed from one direction while light falls on it from another one – General model of local reflection • More precisely, it is the ratio of reflected radiance dLr in the direction toward the viewer to the irradiance dEi in the direction toward the light source Reflected energy N Incident energy BRDF : f ( i , i , o , o ) 0 i o BRDF models For many surfaces, a simple BRDF suffices • Specular surface (e.g., a mirror) 1 if i o and i o f ( i , i , o , o ) cos sin 0 otherwise • Lambertian (diffuse, matte) surface (e.g., white powder) – Independent of exit angle f (i , i ,o , o ) L • Combinations (Phong, Lambertian+Specular, …) Surface Brightness How bright will a Lambertian surface be when it is illuminated by a point source of radiance E? Cosine or Lambert’s law of reflection A point source located in direction The term has radiance E: ensures that the integral of this expression is just E, that is: Examples (again) Ideal diffuse (Lambertian) Ideal specular Directional diffuse Image formation by thin lens Image irradiance on the image plane • The image irradiance (E) is proportional to the object radiance (L) Lens diameter Angle off optical axis d 2 E cos 4 L 4 f Focus distance What the image reports to us via pixel values What we really want to know Power captured by the lens • Power captured by the lens from the surface patch is the product of the following three terms – The radiance L of the surface patch in the direction of the lens – The foreshortened area of the patch in the direction of the lens – The solid angle subtended by the aperture of the lens at the surface patch Foreshortened area of the aperture in the direction of the patch Square of the distance of the aperture from the patch Fundamental Radiometric Relation Irradiance Radiometry summary • Radiometry tells us how light energy from sources interacts with object surfaces and eventually gets collected (by photons releasing electrons) in the image sensor array • Light is additive and the process is linear – Double the power of the light source and the image value doubles (sort of…) • Modeling the true paths of all light sources (including surface interreflections, atmospheric effects, etc.) is impossible – our models are approximations • Can we determine the shape of an object by its image? By detecting shadows? • What about color?