Anti-aliasing and Continuity
using
Trapezoidal Shadow Maps
Speaker:
Tobias Martin
Co-Author: Tiow-Seng Tan
National University of Singapore
Standard Shadow Maps
Content
 Shadows
in Computer Graphics
 Standard
Shadow Maps (SSM)
 Problems
with SSM


Problem #1: Aliasing/Resolution
Problem #2: Polygon Offset
 Problem
#3: Continuity
 Trapezoidal
 Conclusion
Shadow Maps
Shadows in Computer Graphics

Why?



Spatial relationship between objects is better perceivable
Realism is enhanced
Our Requirements:

Robust shadows in real-time
(Screenshots taken from “shadowcast.exe” by Mark J. Kilgard, NVIDIA)
Standard Shadow Maps (SSM)
1. Shadow Map Generation:
•
Transform scene into post-perspective space
(PPS) of light, i.e.:
vL = PL · CL · W · v
•
Render scene from this space
•
Copy depth buffer into shadow map
2. Shadow Determination:
•
Render scene in eye’s PPS and perform shadow
test per-fragment
•
Fragment is in shadow iff zp > shadowmap[xp, yp],
where xp, yp and zp is fragment in PPS of light
(Screenshots taken from “shadowcast.exe” by Mark J. Kilgard, NVIDIA)
Standard Shadow Maps (SSM)

Problem #1: Resolution Problem
Jagged shadow boundary appears
due to low resolution in shadow map
 Trivial solution:
Increase shadow map size to
increase resolution & improve quality
Standard Shadow Maps (SSM)

Problem #2: Polygon Offset
Due to finite precision,
surfaces can cast wrong
shadows on themselves
Practical Solution:
Adding offset to depth values
to move shadow slightly away
from light
Current Solutions
 Problem

#3: Continuity (1)
Straightforward approximation can result in
shadow discontinuities
e.g. in Perspective Shadow Maps (PSM)
[Stamminger and Drettakis 2002]
e.g. in the Bounding Box Approximation (BB)
[Brabec et al. 2002]
Frame i
Frame i +1
Current Solutions
 Problem

#3: Continuity (2)
What happens if shadow map depends on
dynamic scene geometry?
light
light
scene’s
bounding box
eye
scene’s
bounding box
eye
n frames later
”tight” hull contains shadow occluders
”expanded” hull contains shadow occluders
good shadow map quality
shadow map quality worsened
Trapezoidal Shadow Maps (TSM)
 Solution

Shadow map generation focuses on area
potentially visible to the eye
 Solution

to Problem #2 (Polygon Offset)
Depth values are preserved, so that problem
is not worsened
 Solution

to Problem #1 (Resolution)
to Problem #3 (Continuity)
Frame coherent technique resulting in a
continuous change in shadow map resolution
Trapezoidal Shadow Maps (TSM)
PPS
of light
NT
A
Shadow map resolution
is increased
Shadow aliasing
is reduced
trapezoidal
space
Trapezoidal Shadow Maps (TSM)
 Solution


to Problem #1
SSM algorithm is slightly modified:
Calculate shadow map in the trapezoidal space, and
perform the shadow test there as well, i.e.:
vT = NT · PL · CL · W · v
Except for the calculation of NT the algorithm
can completely be mapped to graphics
hardware!
Trapezoidal Shadow Maps (TSM)

Side effect of the trapezoidal transformation
(a)
Uniform distribution
(b)
Non-uniform distribution
a) Approximation of an eye's frustum with a trapezoid
b) Applying NT may lead to severe polygon offset problems
 Different distribution of z-values require different offsets
Trapezoidal Shadow Maps (TSM)
Solution to (worsened) Problem #2:
 Apply NT only to the x- and y-values
 Maintain z in PPS of light:
 Store (xT , yT , zL) in the shadow map
Simply applying NT
Applying modified NT
Trapezoidal Shadow Maps (TSM)
 1st

pass (Shadow Map Generation)
Vertex stage:
• vT = NT · PL · CL · W · v
• vL = PL · CL · W · v

Fragment stage:
• shadowmap[xT, yT]= zL instead of zT
Use additional texture coordinate vL and
replace zT with zL per fragment
Trapezoidal Shadow Maps (TSM)
 2nd

pass (Shadow Determination)
Vertex stage:
• vE = PE · CE · W · v
• vT = NT · PL · CL · W · v
• vL = PL · CL · W · v

Fragment stage:
• Fragment in shadow iff zL > shadowmap[xT, yT]
 Use additional texture coordinate vL and
use zL for shadow map comparison
Trapezoidal Shadow Maps (TSM)
Solution to Problem #3

Base and Top Lines:
Center Line
2D-Convex Hull
l
lt
CHE

Base & Top Line
lb
Center line governs the choices of top and base line
 top and base line transit smoothly from frame to frame
Trapezoidal Shadow Maps (TSM)

Side Lines: 80% rule governs the choices
0%
Focus region
of the eye
80% line
Trapezoidal
approximation in L
Trapezoidal space
due to 80% rule
Trapezoidal Shadow Maps (TSM)
 Calculate
NT that pL is
mapped to 80% line in T
 Calculate
q using 1D
perspective projection

(Minus signs in the proceedings got eaten up by the printer)
Trapezoidal Shadow Maps (TSM)
Trapezoidal Shadow Maps (TSM)

Indication for Continuity


Plot of the total area covered by
focus region in shadow map
Change in frustum are color encoded
Trapezoidal Shadow Maps (TSM)

General case:



Eye’s frustum E is not
completely within then
light’s frustum L
Algorithm can be
easily adjusted
Singularities are
avoided
light
light
eye
light
eye
eye
Results
 Fantasy

World:
BB vs. PSM vs. TSM
Urban Model:

PSM vs. TSM
Conclusion

TSM addresses
1)
2)
3)
…following the 3 principles





Resolution Problem
Polygon Offset Problem
Continuity Problem
Effectiveness of 80% rule
Simplicity which is difficult to achieve
Continuity rather than “correctness”
Q and A
http://www.comp.nus.edu.sg/~tants/tsm.html
TSM
Literature
BRABEC, S., ANNEN T., AND SEIDEL, H. 2002. Practical Shadow
Mapping. Journal of Graphics Tools 7(4), 9–18.
CROW, F. C. 1977. Shadow Algorithms for Computer Graphics. In
Proceedings of SIGGRAPH 1977, 242–248.
STAMMINGER, M., AND DRETTAKIS, G. 2002. Perspective Shadow
Maps.
WILLIAMS, L. 1978. Casting curved shadows on curved surfaces. In
Proceedings of SIGGRAPH 1978, 270–274.
WOO, A., POULIN, P., AND FOURNIER, A. 1990. A survey of shadow
algorithms. IEEE Computer Graphics and Applications, 10(6), 13–
32.