RADIOSITY Submitted by CASULA BABUPRIYANK N Computer Graphics

RADIOSITY Submitted by CASULA, BABUPRIYANK. N

Computer Graphics Hardware & Architecture Application Computer Graphics Image Synthesis Animation

Image Synthesis Modeling • 2 d/3 d Rendering • Radiosity • Illumination models • Visibility • Ray Tracing • Texture Mapping Viewing 2 d/3 d

Radiosity l Surfaces in a scene reflect & emit light. – Some of this light reaches the viewer; this makes the surface visible. – But much of this reflected/emitted light will illuminate other surfaces. – This light will then reflect of these other surfaces; in fact, every surface in a scene will illuminate other surfaces in the scene.

Samples

Background needed… l Light Transport l Radiometry l Reflection Functions

Light l The visible light can be polarized l Optics is the area that studies about these radiations

Optics Geometric Shadows Physical Quantum Optical Interference Photons laws To study radiosity Geometric Optics is needed

Light Transportation l Light travels in the form of particles(photons) l Total number of particles in a small differential volume d. V is P(x) = p(x) d. V particle density P(x) = p(x) (v dt cos( )) d. A

Light Transportation contd. . Not all particles flow with the same speed and same direction. The particle density is now a function of two independent variables x, . Then we have P(x, ) = p(x, ) cos d d. A Here d is called the differential solid angle.

Angles 2 D angle 3 D/Solid Angle

Solid Angle Definition : The SA subtended by an object from a point P is the area of projection of the object onto the unit sphere centered at P. Area (d. A) = (r d ) (r sin d )= r 2 sin d d The differential solid angle : d = d. A cos / r 2 = cos sin d d

Radiant Energy Q Rendering systems consider the stuff that flows as radiant energy or radiant power( ) The radiant energy per unit volume is the photon volume density times the energy of a single photon(hc/ ). L(x, ) = p(x, , ) (hc/ ) d L is called radiance

Radiometry Science of Measuring light Analogous science called Photometry is based on human perception

Radiometry contd. . The radiometric quantities that characterize the distribution of light in the environment are: l Radiant Energy l Radiance l Radiant Power l Irradiance l Radiosity l Radiant Intensity

Radiance l Radiance (L) is the flux that leaves a surface, per unit projected area of the surface, per unit solid angle of direction. n d. A L

Radiance l For computer graphics the basic particle is not the photon and the energy it carries but the ray and its associated radiance. n L d. A Radiance is constant along a ray. d

Properties of Radiance 1)Fundamental quantity -all other quantities derived from it 2) Invariant along a ray - quantity used by ray tracers 3) Sensor response is proportional to radiance -eye/camera response depends on radiance

Radiant Power( ) Flow of energy. Power is the energy per unit time. Also called as radiant flux. = d. Q/dt. The differential flux is the radiance in small beam with cross sectional area d. A and solid angle d d = L(x, ) cos d d. A

Invariance of Radiance d = L 1 d. A 1 = L 2 d. A 2 d 1 =d. A 2 /r 2 and d 2 =d. A 1 /r 2 Throughput T = d d. A 1 1 2 2 2 = d. A / r 1 2

Irradiance: Radiant power per unit area incident on a surface E = Li(x, ) cos dw

Radiosity Official term : Radiant Exitance Radiosity: Radiant power per unit area exiting a surface B = Lo(x, ) cos dw

Radiant Intensity: Radiant power per solid angle of a point source I( ) = d( )/d( ) = I( ) d( ) For an isotropic point source: I( ) = /4 p

Irradiance due to a Point Light Irradiance on a differential surface due to an isotropic point light source is E = d / d. A = I( ) d( ) d. A cos( ) = 4 p |x – xs|2

Reflection Functions Reflection is defined as the process by which the light incident on a surface leaves the surface from the same side. The nomenclature and the general properties of reflection functions are discussed.

BRDF Bidirectional Reflection Distribution Function r f(x, i , r) =Lr(x, r)/ d. Ei(x, r) f(p, i , r ) i Reflected ray In short this is the ratio of radiance in a reflected direction to the differential irradiance that created Incident ray Illumination hemisphere

Properties of the BRDF l 1)Reciprocity f(x, i , r) = f(x, r , i) l 2)Anisotropy If the incident and the reflected light are fixed and the underlying surface is rotated about the surface normal, the percentage of light reflected may change.

Reflectance Equation The BRDF allows us to calculate outgoing light, given incoming light: Lr(x, r)= f(x, i , r) * d. Ei(x, r) = f(x, i , r) * Li(xi, ) cos dwi Integrating over the hemisphere gives the reflectance equation: Lr(x, r)= f(x, , ) * Li(x , ) cos i r i dwi

Reflectance l Reflectance: ratio of reflected flux to incident flux L ( ) cos r dw r = d r/ d o= r r r L ( ) cos i dw i i i r i l Reflectance is always between 0 and 1 but depends on incident radiance distribution

Lambertian Diffuse Reflection is equal in all directions f r , diffuse (x, i , r) is constant. l Lr(x, r)= f r , diffuse(x, , ) * Li(x , ) cos i r i = f r , diffuse(x, i , r) Li(xi, ) * cos dwi = f r , diffuse(x, i , r) E dwi

Lambertian Diffuse Reflection Reflected radiance is independent of direction Therefore the radiosity is simply: B = Lr, diffuse(x, ) cos dw =p Lr, diffuse = p f r , diffuse(x, i , r) E

Lambertian Diffuse Reflection L ( ) cos r dw r = d r/ d o= r r r L ( ) cos i dw i = i Lr, diffuse cos r dwr r = i p f r , diffuse E r i

Global Illumination Radiance is invariant along a ray Li(x`, i) = Lr(x, r) V(x, x`) is the visibility from point x to x’ Converting the directional integral into a surface integral dwi = cos o d. A |x-x`| 2 The Projected solid angle is cos i dwi = cos i cos o d. A |x-x`| 2

Global Illumination Geometry term: G(x, x 1) = cos i cos o |x-x`| 2 cos i dwi = G(x, x 1) d. A l Rewriting the reflectance equation: Lr(x`, `)= f(x, - , `)L(xi, ) G(x, x’)V(x, x`)d. A s

Global Illumination Reparameterizing gives: Lr(x`, `)= f(x, - , `)L(xi, ) G(x, x’)V(x, x`)d. A s Lr(x`, x``)= f(x x``)L(x x`) G(x, x’)V(x, x`)d. A Lr(x`, x``)=Lr(x`, x``)+ s r(x )/p * L (x x`) G(x, x’)V(x, x`)d. A s The radiance sent from x’ to x’’ is simply the amount of radiance sent from all other visible points x in the scene and then reflected to x’’

Rendering Equation l Adding in the radiance directly emitted from x’ to x’’ yields the rendering equation: Lr(x`, x``)=Lr(x`, x``)+ f(x x``)L(x x`) G(x, x’)V(x, x`)d. A s The radiance sent from x’ to x’’ is simply the amount of radiance directly emitted from x’ to x’’ plus the radiance sent from all other visible points x in the scene and then reflected to x’’

Radiosity Equation More importantly the outgoing radiance is same in all directions and in fact equals B/ p. B(x) = E(x) +r(x ) B(x`) G(x, x’)V(x, x`)d. A s p

Advantages 1)Highly realistic quality of the resulting images by calculating the diffuse interreflection of light energy in an environment. 2)Accurate simulation of energy transfer. 3)The viewpoint independence of the basic radiosity algorithm provides the opportunity for interactive "walkthroughs" of environments. 4)Soft shadows and diffuse interreflection.

Disadvantages 1)Large computational and storage costs form factors. 2)Must preprocess polygonal environments. 3)Non-diffuse components of light not represented. 4)Will be very expensive if object(s) is moving in the scene.

References Radiosity Papers are available here 1)http: //www. scs. leeds. ac. uk/cuddles/rover/radpap. htm 2)SIGGRAPH 1993 Education Slide Set, by Stephen Spencer http: //www. education. siggraph. org/materials/Hyper. Graph/radioity/o verview_1. htm Books: 1)Radiosity and Global Illumination Sillion and Puech ISBN 1 -55860 -277 -1 2)Radiosity and Realistic Image Synthesis. Cohen and Wallace. ISBN 0 -12 -178270 -0 Software: 1)www. acurender. com