Linear Model Analysis of Real-Time Rendering - Real-Time Rendering: Computer Graphics with Control Engineering - page 44

sort of response (output), the superposition of stimuli yields a superposition of the
respective responses such that:

( ) = () + () +…+

Fx ++…+ x x Fx Fx Fx n () (3.18)

1 2 3 1 2

Using the same principle, we propose that the overall rendering function of an appli-
cation is equivalent to the sum of the individual functions of the batch jobs lined in
the render queue. To further illustrate, consider a 3D scene with n 3D objects, each
with polygon count x n . The total number of polygons X would be:

n

∑ 1

X = x (3.19)

m

m =

Based on the principle of superposition in Equation (3.19), we draw the parallel
analogy that the time taken to render all objects in a scene is equivalent to the sum
of the time taken to render each of the individual 3D objects in the scene as given by
the following equation:

n

∑ 1

() =

FX fx () (3.20)

m

m =

where F is the system model of the parent rendering process and f denotes the system
model of the separate rendering processes, all obtained through black-box modelling.
The assumptions associated with this hypothesis are:

• State changes and context switch overheads between rendering the 3D

objects are negligible.

• All objects render within the linear range of the rendering model of the

application.

• The application's rendering process is largely partitioned by its content as well.

3.8.2 e xPeRiment
To validate our hypothesis, we designed an experiment. A 3D rendering application
able to display multiple and different types of objects in a single scene was selected.
Each type of object was to be rendered in a different way and the number of objects
of a type were to be changed during runtime by a user-specified variable. At any time
during a run, one or more categories of objects could be rendered.
The application was first set to run with display of only a certain type of object.
A data set was defined to consist of a frame rate (output) and total number of
objects/triangle count (input) of a type of object. A series of data sets were collected
over various object counts within a certain range allowed in the application. The
purpose of this step was to collect data so that the rendering process involving one
type of object could be modelled.