01/21/05© 2005 University of Wisconsin Last Time Course introduction A simple physically-based rendering example

01/21/05© 2005 University of Wisconsin Today Raytracing –Chapter 1 of the PBR Radiometry – measuring light –Chapter 5 of PBR Chapter 2-4 of PBR are about geometric issues

01/21/05© 2005 University of Wisconsin Raytracing (PBR Sect. 1.2) Cast rays out from the eye, through each pixel, and determine what the rays hit –Builds the image pixel by pixel, one at a time Cast additional rays from the hit point to determine the pixel color Rays test visibility – what do I see from this point in this direction? –Ray casting is widely used in graphics to test visibility

01/21/05© 2005 University of Wisconsin Raytracing Shadow rays Reflection ray Transmitted ray

01/21/05© 2005 University of Wisconsin Recursive Ray Tracing When a reflected or refracted ray hits a surface, repeat the whole process from that point –Send out more shadow rays –Send out new reflected ray (if required) –Send out a new refracted ray (if required) –Generally, reduce the weight of each additional ray when computing the contributions to surface color –Stop when the contribution from a ray is too small to notice The result is a ray tree

01/21/05© 2005 University of Wisconsin Raytracing Implementation (PBR Sect. 1.3) Raytracing breaks down into several tasks: –Constructing the rays to cast –Intersecting rays with geometry –Determining the “color” of the thing you see –PBRT breaks these tasks into several steps, because it makes the system very general Intersection testing is “geometry” and we won’t talk about it –Chapters 3 and 4 of the book –You have to read it to understand how to use the provided code, but ignore how it’s implemented

01/21/05© 2005 University of Wisconsin PBRT main() function Loads the scene Initializes things Renders the scene Cleans up

01/21/05© 2005 University of Wisconsin PBRT Render() function Basic raytracer interpretation: –Sampler chooses which pixel –Camera constructs the ray through that pixel –Integrator determines the color coming back along the ray –Film records that pixel Loop, getting all the pixels from sampler Then tell Film to save file

01/21/05© 2005 University of Wisconsin PBRT Integrator() functions Integrators do most of the interesting work –Although there is plenty of interesting stuff for us in Sampler and Camera and Film Basic Raytracing Integrator: –Determine what is hit –Get the reflectance function there –Find the incoming light and push through the reflectance function –Use the reflectance function to get reflection/transmission directions –Recurse on those directions

01/21/05© 2005 University of Wisconsin Measuring Light (PBR Chap. 5) To go much further, we need to talk about measuring properties of light We will consider only the particle nature of light –No diffraction or interference effects, no polarization At any place, at any moment, you can measure the “flow” of light through that point in a given direction –The plenoptic function describes the light in a region:  (x, , ,t) –The plenoptic function over a region defines the light field in that region –We will return to this for image based rendering

01/21/05© 2005 University of Wisconsin Light and Color The wavelength,, of light determines its “color” –Frequency, , is related: Describe light by a spectrum –“Intensity” of light at each wavelength –A graph of “intensity” vs. wavelength We care about wavelengths in the visible spectrum: between the infra-red (700nm) and the ultra-violet (400nm)

01/21/05© 2005 University of Wisconsin White Note that color and intensity are technically two different things However, in common usage we use color to refer to both –White = grey = black in terms of color We will be precise in this part of the course, and only use the words in their physical sense # Photons Wavelength (nm) White Less Intense White (grey)

01/21/05© 2005 University of Wisconsin Helium Neon Laser Lasers emit light at a single wavelength, hence they appear colored in a very “pure” way # Photons Wavelength (nm)

01/21/05© 2005 University of Wisconsin Normal Daylight # Photons Wavelength (nm) Note the hump at short wavelengths - the sky is blue Other bumps came from solar emission spectra and atmospheric adsorption

01/21/05© 2005 University of Wisconsin Tungsten Lightbulb Most light sources are not anywhere near white It is a major research effort to develop light sources with particular properties # Photons Wavelength (nm)

01/21/05© 2005 University of Wisconsin PBRT and Spectra (PBR Sect. 5.1) Represent spectra using 3 piecewise constant basis function –RGB, basically –You won’t get good rainbows with the default You could change it –You would have to change some reflectance function implementations –You would have to recompile everything To standardize the writing of images, conversions to XYZ color space are required –If you don’t know what XYZ color space is, read the notes from CS559 last semester (lectures 2 and 3)

01/21/05© 2005 University of Wisconsin Radiometric Quantities (PBR Sect. 5.2) Quantities that measure the amount of light They differ mostly not in what they measure, but in what you measure it over –Flux: Total in some domain –Irradiance: Per unit area –Intensity: Per unit angle –Radiance: Per unit angle per unit projected area Basically, all the latter are differential quantities that have no meaning until you integrate them over some domain –Like you have to integrate speed to get distance traveled

01/21/05© 2005 University of Wisconsin Radiant Flux Total amount of energy passing through a surface or region of space per unit time –Typically denoted by  (Phi) –Also called power –Measured in watts (W) or joules/second (J/s) These are metric quantities (rather than eg. calories/second) Typically used for a light’s total output –You talk about a 60W light bulb –Only part of the story, also need distribution of power over space A camera integrates, over the time the shutter is open, the power arriving at the film

01/21/05© 2005 University of Wisconsin Irradiance Irradiance is the power arriving at a surface, per unit area on the surface –Denoted E in PBR, or sometimes Ir or sometimes I –Units: Wm -2 r E(p)=  /4  r 2  A E(p)=  cos  /A p p

01/21/05© 2005 University of Wisconsin Irradiance to Flux Integrate over area: Hence, we can also say (d  is flux density, varies over area):

01/21/05© 2005 University of Wisconsin Measuring Angle The solid angle subtended by an object from a point P is the area of the projection of the object onto the unit sphere centered at P –Measured in steradians, sr –Definition is analogous to projected angle in 2D –If I’m at P, and I look out, solid angle tells me how much of my view is filled with an object

01/21/05© 2005 University of Wisconsin Intensity Flux density per unit solid angle: –Used to describe the directional distribution of light Depends on the direction,  –Only meaningful for point light sources (otherwise we would need some concept of the area of the light)

01/21/05© 2005 University of Wisconsin Radiance Flux density per unit area perpendicular to the direction of travel, per unit solid angle: –Units Wm -2 sr -1, power per unit area per unit solid angle Radiance is the fundamental measurement –All others can be computed from it via integrals over area and/or directions Radiance is constant along lines –No r 2 falloff, the per unit solid angle takes care of it

01/21/05© 2005 University of Wisconsin Incident and Exitant Radiance We want to talk about radiance arriving at a point (incident radiance) and radiance leaving a point (exitant radiance) Denote with L i (p,  ) and L o (p,  ) –  always points away from the p In general, L i (p,  )  L o (p,  ) Also, because radiance is constant along a line, at a point in space (not on a surface) L o (p,  ) = L i (p,-  )

01/21/05© 2005 University of Wisconsin Irradiance from Radiance (PBR Sect. 5.3) Integrate radiance over directions in the upper hemisphere: –cos term deals with projected solid angle.  is angle between  and n (the normal) Next time, how to evaluate integrals like this, and reflectance functions