\chapter{Global Visibility Sampling Tool}


The proposed visibility preprocessing framework consists of two major
steps.

\begin{itemize}
\item The first step is an aggresive visibility sampling which gives
initial estimate about global visibility in the scene. The sampling
itself involves several strategies which will be described in
section~\ref{sec:sampling}. The imporant property of the aggresive
sampling step is that it provides a fast progressive solution to
global visibility and thus it can be easily integrated into the game
development cycle.

\item The second step is mutual visibility verification. This step
turns the previous aggresive visibility solution into either exact,
conservative or error bound aggresive solution. The choice of the
particular verifier is left on the user in order to select the best
one for a particular scene, application context and time
constrains. For example, in scenes like a forest an error bound
aggresive visibility can be the best compromise between the resulting
size of the PVS (and framerate) and the visual quality. The exact or
conservative algorithm can however be chosen for urban scenes where
ommision of even small objects can be more distructing for the
user. The mutual visibility tool will be described in the next chapter.

\end{itemize}


In traditional visibility preprocessing the view space is
subdivided into view cells and for each view cell the set of visible
objects --- potentially visible set (PVS) is computed. This framewoirk
has been used for conservative, aggresive and exact algorithms.

We propose a different strategy which has several advantages for
sampling based aggresive visibility preprocessing. The stategy is
based on the following fundamental ideas:
\begin{itemize}
\item Compute progressive global visibility instead of sequential from-region visibility
\item Replace the roles of view cells and objects for some parts of the computation
\end{itemize}

Both these points will be addressed in this chapter in more detail.



\section{Related work}
\label{VFR3D_RELATED_WORK}

Below we briefly discuss the related work on visibility preprocessing
in several application areas. In particular we focus on computing
from-region which has been a core of most previous visibility
preprocessing techniques.


\subsection{Aspect graph}

The first algorithms dealing with from-region visibility belong to the
area of computer vision. The {\em aspect
  graph}~\cite{Gigus90,Plantinga:1990:RTH, Sojka:1995:AGT} partitions
the view space into cells that group viewpoints from which the
projection of the scene is qualitatively equivalent. The aspect graph
is a graph describing the view of the scene (aspect) for each cell of
the partitioning. The major drawback of this approach is that for
polygonal scenes with $n$ polygons there can be $\Theta(n^9)$ cells in
the partitioning for unrestricted viewspace. A {\em scale space}
aspect graph~\cite{bb12595,bb12590} improves robustness of the method
by merging similar features according to the given scale.


\subsection{Potentially visible sets}


 In the computer graphics community Airey~\cite{Airey90} introduced
the concept of {\em potentially visible sets} (PVS).  Airey assumes
the existence of a natural subdivision of the environment into
cells. For models of building interiors these cells roughly correspond
to rooms and corridors.  For each cell the PVS is formed by cells
visible from any point of that cell.  Airey uses ray shooting to
approximate visibility between cells of the subdivision and so the
computed PVS is not conservative.

This concept was further elaborated by Teller et
al.~\cite{Teller92phd,Teller:1991:VPI} to establish a conservative
PVS.  The PVS is constructed by testing the existence of a stabbing
line through a sequence of polygonal portals between cells. Teller
proposed an exact solution to this problem using \plucker
coordinates~\cite{Teller:1992:CAA} and a simpler and more robust
conservative solution~\cite{Teller92phd}.  The portal based methods
are well suited to static densely occluded environments with a
particular structure.  For less structured models they can face a
combinatorial explosion of complexity~\cite{Teller92phd}. Yagel and
Ray~\cite{Yagel95a} present an algorithm, that uses a regular spatial
subdivision. Their approach is not sensitive to the structure of the
model in terms of complexity, but its efficiency is altered by the
discrete representation of the scene.

Plantinga proposed a PVS algorithm based on a conservative viewspace
partitioning by evaluating visual
events~\cite{Plantinga:1993:CVP}. The construction of viewspace
partitioning was further studied by Chrysanthou et
al.~\cite{Chrysanthou:1998:VP}, Cohen-Or et al.~\cite{cohen-egc-98}
and Sadagic~\cite{Sadagic}.  Sudarsky and
Gotsman~\cite{Sudarsky:1996:OVA} proposed an output-sensitive
visibility algorithm for dynamic scenes.  Cohen-Or et
al.~\cite{COZ-gi98} developed a conservative algorithm determining
visibility of an $\epsilon$-neighborhood of a given viewpoint that was
used for network based walkthroughs.

Conservative algorithms for computing PVS developed by Durand et
al.~\cite{EVL-2000-60} and Schaufler et al.~\cite{EVL-2000-59}  make
use of several simplifying assumptions to avoid the usage of 4D data
structures.  Wang et al.~\cite{Wang98} proposed an algorithm that
precomputes visibility within beams originating from the restricted
viewpoint region. The approach is very similar to the 5D subdivision
for ray tracing~\cite{Simiakakis:1994:FAS} and so it exhibits similar
problems, namely inadequate memory and preprocessing complexities.
Specialized algorithms for computing PVS in \m25d scenes were proposed
by Wonka et al.~\cite{wonka00}, Koltun et al.~\cite{koltun01}, and
Bittner et al.~\cite{bittner:2001:PG}.

The exact mutual visibility method presented later in the report is
based on method exploting \plucker coordinates of
lines~\cite{bittner02phd,nirenstein:02:egwr,haumont2005}. This
algorithm uses \plucker coordinates to compute visibility in shafts
defined by each polygon in the scene.


\subsection{Rendering of shadows}


The from-region visibility problems include the computation of soft
shadows due to an areal light source. Continuous algorithms for
real-time soft shadow generation were studied by Chin and
Feiner~\cite{Chin:1992:FOP}, Loscos and
Drettakis~\cite{Loscos:1997:IHS}, and
Chrysanthou~\cite{Chrysantho1996a} and Chrysanthou and
Slater~\cite{Chrysanthou:1997:IUS}. Discrete solutions have been
proposed by Nishita~\cite{Nishita85}, Brotman and
Badler~\cite{Brotman:1984:GSS}, and Soler and Sillion~\cite{SS98}. An
exact algorithm computing an antipenumbra of an areal light source was
developed by Teller~\cite{Teller:1992:CAA}.


\subsection{Discontinuity meshing}


Discontinuity meshing is used in the context of the radiosity global
illumination algorithm or computing soft shadows due to areal light
sources.  First approximate discontinuity meshing algorithms were
studied by Campbell~\cite{Campbell:1990:AMG, Campbell91},
Lischinski~\cite{lischinski92a}, and Heckbert~\cite{Heckbert92discon}.
More elaborate methods were developed by
Drettakis~\cite{Drettakis94-SSRII, Drettakis94-FSAAL}, and Stewart and
Ghali~\cite{Stewart93-OSACS, Stewart:1994:FCSb}. These methods are
capable of creating a complete discontinuity mesh that encodes all
visual events involving the light source.

The classical radiosity is based on an evaluation of form factors
between two patches~\cite{Schroder:1993:FFB}. The visibility
computation is a crucial step in the form factor
evaluation~\cite{Teller:1993:GVA,Haines94,Teller:1994:POL,
Nechvile:1996:FFE,Teichmann:WV}. Similar visibility computation takes
place in the scope of hierarchical radiosity
algorithms~\cite{Soler:1996:AEB, Drettakis:1997:IUG, Daubert:1997:HLS}.



\subsection{Global visibility}

 The aim of {\em global visibility} computations is to capture and
describe visibility in the whole scene~\cite{Durand:1996:VCN}. The
global visibility algorithms are typically based on some form of {\em
line space subdivision} that partitions lines or rays into equivalence
classes according to their visibility classification. Each class
corresponds to a continuous set of rays with a common visibility
classification. The techniques differ mainly in the way how the line
space subdivision is computed and maintained. A practical application
of most of the proposed global visibility structures for 3D scenes is
still an open problem.  Prospectively these techniques provide an
elegant method for ray shooting acceleration --- the ray shooting
problem can be reduced to a point location in the line space
subdivision.


Pocchiola and Vegter introduced the visibility complex~\cite{pv-vc-93}
that describes global visibility in 2D scenes. The visibility complex
has been applied to solve various 2D visibility
problems~\cite{r-tsvcp-95,r-wvcav-97, r-dvpsv-97,Orti96-UVCRC}.  The
approach was generalized to 3D by Durand et
al.~\cite{Durand:1996:VCN}. Nevertheless, no implementation of the 3D
visibility complex is currently known. Durand et
al.~\cite{Durand:1997:VSP} introduced the {\em visibility skeleton}
that is a graph describing a skeleton of the 3D visibility
complex. The visibility skeleton was verified experimentally and  the
results indicate that its $O(n^4\log n)$ worst case complexity is much
better in practice. Pu~\cite{Pu98-DSGIV} developed a similar method to
the one presented in this chapter. He uses a BSP tree in \plucker
coordinates to represent a global visibility map for a given set of
polygons. The computation is performed considering all rays piercing
the scene and so the method exhibits unacceptable memory complexity
even for scenes of moderate size. Recently, Duguet and
Drettakis~\cite{duguet:02:sig} developed a robust variant of the
visibility skeleton algorithm that uses robust epsilon-visibility
predicates.

 Discrete methods aiming to describe visibility in a 4D data structure
were presented by Chrysanthou et al.~\cite{chrysanthou:cgi:98} and
Blais and Poulin~\cite{blais98a}.  These data structures are closely
related to the {\em lumigraph}~\cite{Gortler:1996:L,buehler2001} or
{\em light field}~\cite{Levoy:1996:LFR}. An interesting discrete
hierarchical visibility algorithm for two-dimensional scenes was
developed by Hinkenjann and M\"uller~\cite{EVL-1996-10}. One of the
biggest problems of the discrete solution space data structures is
their memory consumption required to achieve a reasonable
accuracy. Prospectively, the scene complexity
measures~\cite{Cazals:3204:1997} provide a useful estimate on the
required sampling density and the size of the solution space data
structure.


\subsection{Other applications}

 Certain from-point visibility problems determining visibility over a
period of time can be transformed to a static from-region visibility
problem. Such a transformation is particularly useful for antialiasing
purposes~\cite{grant85a}. The from-region visibility can also be used
in the context of simulation of the sound
propagation~\cite{Funkhouser98}. The sound propagation algorithms
typically require lower resolution than the algorithms simulating the
propagation of light, but they need to account for simulation of
attenuation, reflection and time delays.

\section{Algorithm Description}

This section first describes the setup of the global visibility
sampling algorithm. In particular we describe the view cell
representation and the novel concept of from-object based
visibility. The we outline the different visibility sampling
strategies.

\subsection{View Cell Representation}

\begin{figure}[htb]
  \centerline{
    \includegraphics[height=0.35\textwidth,draft=\DRAFTFIGS]{images/vienna_viewcells_01}
     \includegraphics[height=0.35\textwidth,draft=\DRAFTFIGS]{images/vienna_viewcells_07}
    }

  \caption{(left) Vienna viewcells. (right) The view cells are prisms with a triangular base. }
  \label{fig:vienna_viewcells}
\end{figure}


A good partition of the scene into view cells is an essential part of every visibility
system. If they are chosen too large, the PVS (Potentially Visibible Set) 
of a view cells is too large for efficient culling. If they are chosen too small or
without consideration, then neighbouring view cells contain redundant PVS information,
hence boosting the PVS computation and storage costs for the scene. In the left image of figure~\ref{fig:vienna_viewcells} we see view cells of the Vienna model, generated by triangulation of the streets. 
In the closeup (right image) we can see that each triangle is extruded to a given height to form a view cell prism.

In order to efficiently use view cells with our sampling method, we
require a view cell representation which is

\begin{itemize}
\item optimized for view cell - ray intersection.
\item flexible, i.e., it can represent arbitrary geometry.
\item naturally suited for a hierarchical approach. %(i.e., there is a root view cell containing all others)
\end{itemize}

We meet these requirements by employing spatial subdivisions (i.e., 
KD trees and BSP trees), to store the view cells. The initial view cells are
associated with the leaves. The reason why we chose BSP trees as view cell representation 
is that they are very flexible. View cells forming arbitrary closed meshes can be closely matched.
Therefore we are able to find a view cells with only a few view ray-plane
intersections.  Furthermore, the hierarchical structures can be
exploited as hierarchy of view cells. Interior nodes form larger view cells
containing the children. If neccessary, a leaf can be easily subdivided into smaller view cells.

Currently we consider three different approaches to generate the initial view cell BSP tree.
The third method is not restricted to BSP trees, but BSP trees are prefered 
because of their greater flexibility.


\begin{table}
\centering \footnotesize
\begin{tabular}{|l|c|c|}
  \hline\hline
  View cells & Vienna selection & Vienna full \\\hline\hline
  \#view cells & 105 & 16447 \\\hline\hline
  \#input polygons & 525 & 82235 \\\hline\hline
  BSP tree generation time & 0.016s & 10.328s \\\hline\hline
  view cell insertion time & 0.016s & 7.984s \\\hline\hline
  \#nodes & 1137 & 597933 \\\hline\hline
	\#interior nodes & 568 & 298966\\\hline\hline
	\#leaf nodes  & 569 & 298967\\\hline\hline
	\#splits & 25 & 188936\\\hline\hline
	max tree depth & 13 & 27\\\hline\hline
	avg tree depth & 9.747 & 21.11\\\hline\hlien
	
 \end{tabular}
 \caption{Statistics for view cell BSP tree on the Vienna view cells and a selection of the Vienna view cells.}\label{tab:viewcell_bsp}
\end{table}


\begin{itemize}

\item We use a number of input view cells given in advance. As input view cell 
any closed mesh can be applied. The only requirement is that the
any two view cells do not overlap. 
First the view cell polygons are extracted, and the BSP is build from
these polygons using some global optimizations like tree balancing or
least splits. Then one view cell after the other
is inserted into the tree to find out the leafes where they are contained
in. The polygons of the view cell are filtered down the tree,
guiding the insertion process. Once we reach a leaf and there are no
more polygons left, we terminate the tree subdivision. If we are on
the inside of the last split plane (i.e., the leaf represents the
inside of the view cell), we associate the leaf with the view cell
(i.e., add a pointer to the view cell). One input view cell can
be associated with many leafes, whereas each leafs has only one view cell.
Some statistics about using this method on the vienna view cells set are given in table~\ref{tab:viewcell_bsp}.
However, sometimes a good set of view cells is not available. Or the scene
is changed frequently, and the designer does not want to create new view cells on each change.
In such a case one of the following two methods should rather be chosen, which generate view cells
automatically.


\item We apply a BSP tree subdivision to the scene geometry. Whenever
the subdivision terminates in a leaf, a view cell is associated with the leaf node.
This simple approach is justified because it places the view cell borders 
along some discontinuities in the visibility function.

\item  The view cell generation can be guided by the sampling process. We start with
with a single initial view cell representing the whole space. If a given threshold
is reached during the preprocessing (e.g., the view cell is pierced by too many rays 
resulting in a large PVS), the view cell is subdivided into smaller cells using some criteria.

In order to evaluate the best split plane, we first have to define the characteristics of a
good view cell partition:  The view cells should be quite large, while their PVS stays rather small.
The PVS of each two view cells should be as distinct as possible, otherwise they could be
merged into a larger view cell if the PVSs are too similar. 
E.g., for a building, the perfect view cells are usually the single rooms connected by portals.

Hence we can define some useful criteria for the split: 1) the number of rays should be roughly equal among the new view cells. 2) The split plane should be chosen in a way that the rays are maximal distinct, i.e., the number
of rays contributing to more than one cell should be minimized => the PVSs are also distinct. 3) For BSP trees,
the split plane should be aligned with some scene geometry which is large enough to contribute
a lot of occlusion power. This criterium can be naturally combined with the second one.
As termination criterium we can choose the minimum PVS / piercing ray size or the maximal tree depth.
\end{itemize}

\subsection{From-object based visibility}

 Our framework is based on the idea of sampling visibility by casting
casting rays through the scene and collecting their contributions. A
visibility sample is computed by casting a ray from an object towards
the view cells and computing the nearest intersection with the scene
objects. All view cells pierced by the ray segment can the object and
thus the object can be added to their PVS. If the ray is terminated at
another scene object the PVS of the pierced view cells can also be
extended by this terminating object. Thus a single ray can make a
number of contributions to the progressively computed PVSs. A ray
sample piercing $n$ view cells which is bound by two distinct objects
contributes by at most $2*n$ entries to the current PVSs. Apart from
this performance benefit there is also a benefit in terms of the
sampling density: Assuming that the view cells are usually much larger
than the objects (which is typically the case) starting the sampling
deterministically from the objects increases the probability of small
objects being captured in the PVS.

At this phase of the computation we not only start the samples from
the objects, but we also store the PVS information centered at the
objects. Instead of storing a PVS consting of objects visible from
view cells, every object maintains a PVS consisting of potentially
visible view cells. While these representations contain exactly the
same information as we shall see later the object centered PVS is
better suited for the importance sampling phase as well as the
visibility verification phase.


\subsection{Basic Randomized Sampling}


The first phase of the sampling works as follows: At every pass of the
algorithm visits scene objects sequentially. For every scene object we
randomly choose a point on its surface. Then a ray is cast from the
selected point according to the randomly chosen direction. We use a
uniform distribution of the ray directions with respect to the
halfspace given by the surface normal. Using this strategy the samples
at deterministicaly placed at every object, with a randomization of
the location on the object surface. The uniformly distributed
direction is a simple and fast strategy to gain initial visibility
information.


 The described algorithm accounts for the irregular distribution of the
objects: more samples are placed at locations containing more
objects. Additionally every object is sampled many times depending on
the number of passes in which this sampling strategy is applied. This
increases the chance of even a small object being captured in the PVS
of the view cells from which it is visible.


\subsection{Accounting for View Cell Distribution}

 The first modification to the basic algorithm accounts for irregular
distribution of the view cells. Such a case is common for example in
urban scenes where the view cells are mostly distributed in a
horizontal direction and more view cells are placed at denser parts of
the city. The modification involves replacing the uniformly
distributed ray direction by directions distributed according to the
local view cell directional density. This means placing more samples at
directions where more view cells are located: We select a random
viecell which lies at the halfpace given by the surface normal at the
chosen point. We pick a random point inside the view cell and cast a
ray towards this point.


\subsection{Accounting for Visibility Events}

 Visibility events correspond to appearance and disapearance of
 objects with respect to a moving view point. In polygonal scenes the
events defined by event surfaces defined by three distinct scene
edges. Depending on the edge configuration we distinguish between
vertex-edge events (VE) and tripple edge (EEE) events. The VE surfaces
are planar planes whereas the EEE are in general quadratic surfaces.

 To account for these event we explicitly place samples passing by the
object edges which are directed to edges and/or vertices of other
objects.


