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The Radiance Equation

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Motivation Photo‑realistic image rendering is particularly difficult to compute because of the complexity of the physical nature of light. However, the radiosity global illumination methods approximates the physical nature of light and provides the necessary foundation for extremely high quality rendered photo‑realistic images. Radiosity has become established as the global illumination method for rendering the highest quality, view independent images for virtual environments and captures subtle lighting effects such as colour bleeding. The method is able to correctly compute shadows due to area light sources, producing accurate penumbra and umbra.

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Motovation Radiosity is a powerful tool for rendering photo‑realistic scenes. Once the radiosity of a scene has been calculated, a ‘virtual reality’ walkthrough of the scene is immediately available. However, this comes at a costly price as calculating the radiosity of a scene is anything but trivial.

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**Introduction Real-time walkthrough with global illumination**

Possible under limited conditions Radiosity (diffuse surfaces only) Real-time interaction Not possible except for special case local illumination Why is the problem so hard?

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Light Remember visible light is electromagnetic radiation with wavelengths approximately in the range from 400nm to 700nm 400nm 700nm

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**Light: Photons Light can be viewed as wave or particle phenomenon**

Particles are photons packets of energy which travel in a straight line in vaccuum with velocity c (300,000m.p.s.) The problem of how light interacts with surfaces in a volume of space is an example of a transport problem.

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Light: Radiant Power denotes the radiant energy or flux in a volume V. The flux is the rate of energy flowing through a surface per unit time (watts). The energy is proportional to the particle flow, since each photon carries energy. The flux may be thought of as the flow of photons per unit time.

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**Light: Flux Equilibrium**

Total flux in a volume in dynamic equilibrium Particles are flowing Distribution is constant Conservation of energy Total energy input into the volume = total energy that is output by or absorbed by matter within the volume.

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**Light: Equation (p,) denotes flux at pV, in direction **

It is possible to write down an integral equation for (p,) based on: Emission+Inscattering = Streaming+Outscattering + Absorption Complete knowledge of (p,) provides a complete solution to the graphics rendering problem. Rendering is about solving for (p,).

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**Simplifying Assumptions**

Wavelength independence No interaction between wavelengths (no fluorescence) Time invariance Solution remains valid over time unless scene changes (no phosphorescence) Light transports in a vacuum (non-participating medium) – ‘free space’ – interaction only occurs at the surfaces of objects

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

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

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**Radiance: Radiosity, Irradiance**

Radiosity - is the flux per unit area that radiates from a surface, denoted by B. d = B dA Irradiance is the flux per unit area that arrives at a surface, denoted by E. d = E dA

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**Radiosity and Irradiance**

L(p,) is radiance at p in direction E(p,) is irradiance at p in direction E(p,) = (d/dA) = L(p,) cos d

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**Recall Reflectance BRDF Relates Bi-directional Reflectance**

Distribution Function Relates Reflected radiance to incoming irradiance i r Incident ray Reflected ray Illumination hemisphere f(p, i , r )

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**Recall Reflectance: BRDF**

Reflected Radiance = BRDFIrradiance Formally: L(p, r ) = f(p, i , r ) E(p, i ) = f(p, i , r ) L(p, i ) cosi di In practice BRDF’s hard to specify Rely on ideal types Perfectly diffuse reflection Perfectly specular reflection Glossy reflection BRDFs taken as additive mixture of these

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The Radiance Equation Radiance L(p, ) at a point p in direction is the sum of Emitted radiance Le(p, ) Total reflected radiance Radiance = Emitted Radiance + Total Reflected Radiance

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**The Radiance Equation: Reflection**

Total reflected radiance in direction : f(p, i , ) L(p, i ) cosi di Radiance Equation: L(p, ) = Le(p, ) + f(p, i , ) L(p, i ) cosi di (Integration over the illumination hemisphere)

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**The Radiance Equation L(p, )**

p is considered to be on a surface, but can be anywhere, since radiance is constant along a ray, trace back until surface is reached at p’, then L(p, i ) = L(p’, i ) L(p, ) depends on all L(p*, i) which in turn are recursively defined. p* i L(p, ) p The radiance equation models global illumination.

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**Traditional Solutions to the Radiance Equation**

The radiance equation embodies totality of all 2D projections (view).

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**Irradiance Power per unit area incident on a surface. E = d /dA**

Unit: Watt / m2 arriving dA

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**Radiant Exitance Power per unit area leaving surface**

Also known as radiosity B = d /dA Same units as irradiance just direction changes. leaving dA

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**Basic Definitions Radiosity: (B) Energy per unit area per unit time.**

Emission: (E) Energy per unit area per unit time that the surface emits itself (e. g., light source). Reflectivity: (r) The fraction of light which is reflected from a surface. (0 <= r <=1) Form- Factor: (F) The fraction of the light leaving one surface which arrives to another. (0<=F<=1)

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**The Basic Radiosity Equation**

We will compute the light emitted from a single differential surface area dAi. It consists of: 1. Light emitted by dAi. 2. Light reflected by dAi. depends on light emitted by other dAj, fraction of it reaches dAi. The fraction depends on the geometric relationship between dAi and dAj: the formfactor .

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**Mesh Surfaces into Elements Reconstruct and Display Solution**

Classic Radiosity Algorithm Mesh Surfaces into Elements Compute Form Factors Between Elements Solve Linear System for Radiosities Reconstruct and Display Solution

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**The Descrete Radiosity Equation**

Total power leaving an element i and reflected light. weighted by geometric coupling j->i and reflectivity is sum of emitted light by element i Reflected light depends on contribution from every other element j

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The Form Factor: the fraction of energy leaving one surface that reaches another surface It is a purely geometric relationship, independent of viewpoint or surface attributes Surface j Surface i

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**The Reciprocity Relationship**

If we had equal sized emitters and receivers, the fraction of energy emitted by one and received by the other would be identical to the fraction of energy going the other way. Thus, the formfactors from Ai to Aj and from Aj to Ai are related by the ratios of their areas: Thus: The radiosity equation is now:

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Patches and Elements Patches are used for emitting light. Some patches are divided into elements, which are used to more accurately compute the received light after the patch solution have been computed.

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**Next Step: Learn ways of computing form factors**

Needed to solve the Descrete Radiosity Equation: Form factors Fij are independent of radiosities (depend only on scene geometry)

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**The overall form factor between i and j is found by integrating**

Between differential areas, the form factor equals: The overall form factor between i and j is found by integrating Surface j Surface i

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**Form Factors in (More) Detail**

where Vij is the visibility (0 or 1)

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**We have two integrals to compute:**

Surface j Area integral over surface i Area integral over surface j Surface i

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The Nusselt Analog Integration of the basic form factor equation is difficult even for simple surfaces! Nusselt developed a geometric analog which allows the simple and accurate calculation of the form factor between a surface and a point on a second surface.

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The Nusselt Analog The "Nusselt analog" involves placing a hemispherical projection body, with unit radius, at a point on a surface. The second surface is spherically projected onto the projection body, then cylindrically projected onto the base of the hemisphere. The form factor is, then, the area projected on the base of the hemisphere divided by the area of the base of the hemisphere.

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**Numerical Integration: The Nusselt Analog**

This gives the form factor FdAiAj Aj dAi

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Method 1: Hemicube Approximation of Nusselt’s analog between a point dAi and a polygon Aj Polygonal Area (Aj) Infinitesimal Area (dAi)

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The Hemi-cube We compute the delta formfactor of each grid cells DF and store in a table. Project all patches onto the ‘ hemi- cube ’ screen, drawing a patch- id instead of color. Sum the delta form factors of all grid cells covered by the patch’s id. Delta form factor

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The Hemicube In Action

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The Hemicube In Action This illustration demonstrates the calculation of form factors between a particular surface on the wall of a room and several surfaces of objects in the room.

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**Projecting all other surfaces onto the hemicube**

Compute the form factors from a point on a surface to all other surfaces by: Projecting all other surfaces onto the hemicube Storing, at each discrete area, the identifying index of the surface that is closest to the point.

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**Discrete areas with the indices of the surfaces which are ultimately visible to the point.**

From there the form factors between the point and the surfaces are calculated. For greater accuracy, a large surface would typically be broken into a set of small surfaces before any form factor calculation is performed.

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**Hemicube Method Scan convert all scene objects onto hemicube’s 5 faces**

Use Z buffer to determine visibility term Sum up the delta form factors of the hemicube cells covered by scanned objects Gives form factors from hemicube’s base to all elements, i.e. FdAiAj for given i and all j

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**Hemicube Algorithms Advantages + First practical method**

+ Use existing rendering systems; Hardware + Computes row of form factors in O(n) Disadvantages - Computes differential-finite form factor Aliasing errors due to sampling - Proximity errors - Visibility errors - Expensive to compute a single form factor

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**We have found the Radiosity Matrix Elements**

Ei Bi

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Radiosity Matrix The "full matrix" radiosity solution calculates the form factors between each pair of surfaces in the environment, as a set of simultaneous linear equations. This matrix equation is solved for the "B" values, which can be used as the final intensity (or color) value of each surface.

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Radiosity Matrix This method produces a complete solution, at the substantial cost of first calculating form factors between each pair of surfaces and then the solution of the matrix equation. Each of these steps can be quite expensive if the number of surfaces is large: complex environments typically have above ten thousand surfaces, and environments with one million surfaces are not uncommon. This leads to substantial costs not only in computation time but in storage.

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**Solve [F][B] = [E] Direct methods: O(n3) Iterative methods: O(n2)**

Gaussian elimination Goral, Torrance, Greenberg, Battaile, 1984 Iterative methods: O(n2) Energy conservation ¨diagonally dominant ¨ iteration converges Gauss-Seidel, Jacobi: Gathering Nishita, Nakamae, 1985 Cohen, Greenberg, 1985 Southwell: Shooting Cohen, Chen, Wallace, Greenberg, 1988

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Gathering In a sense, the light leaving patch i is determined by gathering in the light from the rest of the environment

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Gathering Gathering light through a hemi-cube allows one patch radiosity to be updated.

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Gathering

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Shooting Radiosity Shoot the radiosity of patch i and update the radiosity of all other patches.

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Shooting Shooting light through a single hemi-cube allows the whole environment's radiosity values to be updated simultaneously. For all j where

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Shooting

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**Progressive Radiosity**

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**Next Accuracy from meshing**

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Artifacts

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**Increasing Resolution**

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Adaptive Meshing

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**Some Radiosity Results**

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**Discontinuity Meshing**

Dani Lischinski, Filippo Tampieri and Donald P. Greenberg created this image for the 1992 paper Discontinuity Meshing for Accurate Radiosity. It depicts a scene that represents a pathological case for traditional radiosity images, many small shadow casting details. Notice, in particular, the shadows cast by the windows, and the slats in the chair.

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