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Rendering Equation

Hemispherical Formulation

Let us assume that \(L_{e}(x \rightarrow \theta)\) represents theradiance emitted by the surface at x and in the outgoing direction Θ, and \(L_{r}(x \rightarrow \theta)\) represents the radiance that is reflected by the surface at x in that direction \(\theta\).

By conservation of energy, the total outgoing radiance at a point and in a particular outgoing direction is the sum of the emitted radiance and the radiance reflected at that surface point in that direction.

\[L(x \rightarrow \theta) = L_{e}(x \rightarrow \theta) + L_{r}(x \rightarrow \theta) \]

Putting these equations together,the rendering equation is

\[L(x \rightarrow \theta) = L_{e}(x \rightarrow \theta) + \int_{\Omega} f_{r}(x,\Psi \rightarrow \Theta)L(x \leftarrow \Psi)cos(N_{x},\Psi)dw_{\Psi} \qquad \qquad (2.29) \]

Area Formulation

One popular formulation is arrived at by considering the surfaces of objects in the scene that contribute to the incoming radiance at the point x. This formulation replaces the integration over the hemisphere by integration over all surfaces visible at the point.

Ray-casting operation(\(r(x,\Psi)\)):finds the point on the closest visible object along a ray originating at point x and pointing in the direction \(\Psi\).

\[r(x,\Psi) = \{ y:y = x + t_{intersection}\Psi \}; \]

\[t_{intersection} = min \{t:t > 0,x + t\Psi \in A \} \]

Assuming nonparticipating media, the incoming radiance at x from direction \(\Psi\) is the same as the outgoing radiance from y in the direction \(-\Psi\):

\[L(x \leftarrow \Psi) = L(y \rightarrow -\Psi) \]

Additionally,the solid angle can be recast as follows

\[dw_{\Psi} = dw_{w \leftarrow dA_{y}} = cos(N_{y},-\Psi)\frac{dA_{y}}{r^2_{xy}} \]

Substituting in Equation 2.29, the rendering equation can also be expressed as an integration over all surfaces in the scene as follows:

\[L(x \rightarrow \Theta) = L_{e}(x \rightarrow \Theta) + \int_{A}f_{r}(x,\Psi \rightarrow \Theta)L(y \rightarrow -\Psi) V(x,y)\frac{cos(N_{x},\Psi)cos(N_{y},-\Psi)}{r^2_{xy}} dA_{y} \]

The term G(x,y), called the geometry term, depends on the relative geometry of the surfaces at point x and y:

\[G(x,y) = \frac{cos(N_{x},\Psi)cos(N_{y},-\Psi)}{r^2_{xy}} \]

Direct and Indirect Illumination Formulation

Another formulation of the rendering equation separates out the direct and indirect illumination terms.Direct illumination is the illumination,that arrives at a surface directly from the light sources in a scene; indirect illumination is the light that arrives after bouncing at least once off another surface in the scene.It is often efficient to sample direct illumination using the area formulation of the rendering equation, and the indirect illumination using the hemispherical formulation.