Linear Model Analysis of Real-Time Rendering - Real-Time Rendering: Computer Graphics with Control Engineering - page 42

general relationship between the input and output of the rendering system. They are
formulated mathematically as shown in Equation (3.14):

t = k m (3.14) frame vc

where t frame is the time taken to render a frame and m vc is the rendering load repre-
sented by the total number of vertices used in rendering the 3D scene. In the first
formulation, we want to express k as a factor estimated from all the training data.
In the second formulation, k is expressed as an average of the previous n rendered
frames. Based on this formulation, we may represent Equation (3.14) as a single
best-fitting line segment expressed as:

() =+

fx px p

2

(3.15)

1

where f(x) is the function describing the line segment, p 1 is the gradient of the line
and p 2 is the vertical axis intercept. Hence k in (3.14) and p 1 from above may be
associated directly as:

kp =

1

(3.16)

Using the curve fitting technique from the MATLAB toolbox, we obtain the line
segment for the operating range in Experiment 1 with p 1 as 5.166 × 10 -8 . With refer-
ence to the experiment data in Figure 3.7, given the input vertex count of 120,000,
the estimated frame time using the value of p 1 is 0.0061992 s. This translates to a
frame rate of 161.3111. However, the measured frame rate is approximately 104,
yielding the error from this formulation as 57 FPS—in stark contrast to our model's
output that is much more accurate. Over the entire tested range, the error between
our model's output and the measured output is less than 5 FPS.
Next, we want to compare our system model with another that takes into account
the k factor for previous frames instead of a single k factor for estimating frame
time at any input point. Mathematically, this second model can be expressed as the
n -moving average s given a sequence a i

{} =1 taking the average of n terms.

N

i

in +−

∑

1

1

s = a

ji

(3.17)

j

n

=

Therefore, each term a in the context of Equation (3.17) is the k factor estimated
from a window of x number of frames. This gives us a set of values for k over the test
range. The final value of k used for estimating the frame time corresponding to the
experiment data is averaged over the number of predecessor sets. With reference to
Figure 3.7, the following are obtained:

1. The moving average of the gradient with a window of one frame is
7.1423 × 10 -4 . The estimated frame rate at a steady-state vertex count