Future Work

A possible direction of future work is the development of an analytic angle-formulated scattering model which considers anisotropic light sources, since the intensity is singularly dependent upon only angle. Also, algorithms for volumetric shadows using analytic tech-niques have potential for real-time applications. A discussion for each of these avenues follows.

6.2.1 Analytic Model for Scattering from Anisotropic Light Sources

Although anisotropic light sources reflect many real-world situations, light scattering effects for scenes with anisotropic light sources are difficult to include in any analytic model due to the non-constant nature of source radiance. A possible direction for future work is the development of a model for scattering from a light source such as the commonly used openGL spotlight model. The intensity of the source falls off with angle θ between the source’s illumination axis and a light ray, described by

I(θ) =

where N is the spotlight exponent and θ_c is the critical angle when the intensity drops to 0. The angle-formulated analytic model for airlight(Equation 4.6) is a good starting point

for considering scattering effects for a light source with the intensity distribution of Equa-tion 6.1 due to the anisotropic light source intensity’s dependence on angle. Substituting Equation 6.1 as the source intensity in our angle-formulated analytic model is shown below:

L_a(γ, d_sv, d_vp, κ_t) = κ_te^−κtt

The angle between the illumination axis and the perpendicular bisector to the view ray with respect the source is denoted α, shown in Figure 6.1. The challenge for developing an analytic model appropriate for real-time rendering is movement of the physical parameter α outside of the integral. A substitution using a series of cosine terms cos^Nx =P

iA_icos(ix) was considered but proved impractical due to the large number of terms if N is large.

illumination

Figure 6.1: Diagram of scattering with anisotropic light source

6.2.2 Volumetric Shadows

Rendering volumetric shadows using analytic techniques has potential for real-time appli-cations, as shown by the volumetric shadow algorithm [Biri et al., 2006] of Section 3.2.1.

In addition to the high fill rate for rendering the shadow volume, the algorithm also has problems when rendering shadow polygons of relatively large size. A possible improvement

for the latter problem is to avoid having to sort the shadow polygons, which is required to determine which shadow polygons are themselves in shadow. Instead, a shadow map can be used to resolve the visibility of each shadow polygon. This experimental algorithm is summarized in the following paragraphs, along with the unsolved problems encountered.

The algorithm utilizes our analytic lighting model and requires the construction of the classic shadow maps and shadow volumes. With modification to the fragment shader, it can be easily implemented as an extra rendering pass of shadow planes generated by the shadow volume algorithm.

Airlight with Occluders

To include volumetric shadows, we must consider airlight that never reaches the viewer due to occluders along the view ray. For this purpose, shadow polygons are generated by the shadow volume algorithm to indicate where the view ray potentially enters into and exits out of shadowed regions. To determine whether a point along the view ray intersecting a shadow polygon at a distance x in fact enters into or exits out of shadowed regions can be determined by the binary visibility function V (x). We consider a point visible if it lies at the edge of a shadowed region. Thus, V (x) = 1 only at those points where the view ray enters or exits shadowed regions, and V (x) = 0 otherwise. Refer to Figure 6.2 for an illustration of V (x). Assuming all objects are closed, we may then express the amount of airlight with occluders L⁰_a as

L⁰_a= L_a(d_vp) − Xn

i

V (d_i,f)L_a(d_i,f) − V (d_i,b)L_a(d_i,b) (6.2) where n is the number of shadow polygons intersecting the view ray, d_i,b is the distance to the i^th back-facing shadow polygon intersecting the view ray, and d_i,f is the distance to the i^th front-facing shadow polygon intersecting the view ray.

Implementation Design

Equation 6.2 can be implemented by rendering the shadow polygons of the scene in an extra pass using the fragment shader. The analytic model for airlight, Equation 4.6, is used evaluate the airlight terms L_a in the fragment shader.

Figure 6.2: Diagram for volumetric shadow algorithm.

Shadow maps are used to determine V (d). Lookup of a shadow map in the frag-ment shader at the transformed coordinates for simply distance d along the view ray will not surprisingly generate inaccurate results since a shadow polygon, conceptually the edge of a shadowed region, may or may not be visible according to a shadow map. Instead, V (d) is determined by a lookup of all adjacent texels to the texel corresponding to the transformed coordinates at d. If any one of the adjacent texels is not under shadow according to the shadow map, then V (d) = 1. Otherwise V (d) = 0. Our experiments have shown this is a good indicator for V (d), though there definitely is potential for a more efficient indicator.

The steps for rendering volumetric shadows consist of:

SceneGeometry Pass

1. Render scene geometry. If fragment in shadow, assign contribution to 0.

That is, this step computes L_a(d_vp) of Equation 6.2.

Shadow polygons pass

1. Set depth test write off.

2. Set additive blending.

3. Render front-facing shadow polygons.

That is, this step computesP_n

i V (d_i,f)L_a(d_i,f) of Equation 6.2.

4. Set reverse subtractive blending.

5. Render back-facing polygons.

That is, this step computes -P_n

i V (d_i,b)L_a(d_i,b) of Equation 6.2.

Experimental Results

Volumetric shadows for simple objects can be rendered; however, the shadow polygons for complex objects cause unsolved discontinuous shading problems. One source of this problem is the situation when shadow polygons intersect other shadow polygons. In such a case, fragments at the intersection of two shadow polygons are rendered twice. Additionally, clamping problems occur when the sum of contributions exceed one for the pass when rendering front-facing shadow polygons. Lastly, the fill rate for the shadow polygons greatly affects the performance of the algorithm, such that complex scenes slow the frame rate significantly. Images from some experiments are shown in Figures 6.3, 6.4, 6.5, 6.6, and 6.7.

Figure 6.3: Image of simple torus’ volumetric shadow.

Figure 6.4: Image of simple table’s volumetric shadow.

Figure 6.5: When intersecting shadow polygons exist, fragments at the intersection are incorrectly rendered multiple times with non-zero contribution.

Figure 6.6: Image illustrating clamping problem. RGB values clamp to one incorrectly when the contribution of the scene and front-facing shadow polygons exceeds one. High intensity of light source induces clamping problems, causing general over-darkening of the torus’ hole.

The really dark strips result where more shadow polygons are rendered, since sharp edges in the silhouette may have concave areas(not a fundamental problem in itself, just a result of clamping problem).

Figure 6.7: Image of car’s volumetric shadow. Discontinuous shading problems occur when rendering the shadow of more complex objects. The source of discontinuous shading problems include intersecting shadow polygons, clamping, and perhaps other unforeseen reasons.

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