Linear Model Analysis of Real-Time Rendering - Real-Time Rendering: Computer Graphics with Control Engineering - page 27

where Z N = [ y (1) , u (1) , y (2) , u (2) …, y ( N ) , and u ( N )] are the set of recorded inputs
and outputs over a time interval of 1≤ t ≤ N . Since the parameters of the model are
encapsulated in the vector θ, solving Equation (3.10) gives us their numerical values.
Alternatively, a model may be represented in the state-space format whereby the
inputs, outputs, and state variables are expressed as vectors and the differential and
algebraic equations are written in matrix form. This format provides a convenient
and compact way to model and analyse systems with multiple inputs and outputs.
The state-space representation of a discrete time-invariant dynamic system model is
described by the equations below.

( ) = () + () + ()

xk + 1 Ax k uk Ke k (3.11)

() = () + ()

yk Cx kDuk (3.12)

where x ( k ) is the state vector, y ( k ) is the system output, u ( k ) the system input, and e ( k )
the stochastic error. A , B , C , D , and K are the system matrices. The derivation of the
system matrices can be found in common system modelling and control engineering
textbooks such as [1].
Although many model structures are used in the system identification field
we consider primarily the two structures described in this section because of the
advantages they offer in comparison to other model structures. The ARX model
structure offers computational efficiency in polynomial estimation. It is thus pref-
erable in many situations, particularly when model order is high. On the other
hand, state-space equations provide mathematical constructs that leverage linear,
first-order derivative variables that allow convenient computation even for systems
involving multiple-input-multiple-output systems.

3.5 EXPERIMENTS
This section discusses two experiments conducted to derive the system model of
rendering processes and one experiment illustrating the use of a derived model in a
control system.

3.5.1 e xPeRiment 1: s ingle -i nPut -s ingle -o utPut (siso) s ystem
To illustrate how our modelling framework may be applied, we select two applica-
tions that make use of geometry subdivision techniques for rendering 3D objects
with high visual details. These two applications are taken from a popular 3D graph-
ics rendering toolkit (NVIDIA DirectX SDK) designed to help software developers
exploit this subdivision technique in current computer hardware. The choice of the
two applications is based on the current trend [30] for subdivision techniques in
many application domains. The objective was to show that our framework can gener-
ate system models that describe the selected rendering applications accurately.
In Experiment 1, we wanted to establish a SISO rendering system model based
on a sample interactive rendering application. The input and output variables were
chosen as the input geometry (number of vertices) and frame rate, respectively. We
selected the N -patch tessellation application (Figure 3.3) from DirectX SDK that