Performance

In order to measure the relative performances of the software and hardware models, it is necessary to define an error metric. The measurement usually used is the deviation from the correct space-time orientation, where velocity is a 3-D unit direction vector [Barron94]:

(6.2)

A 3D vector is used to try and reduce the over-emphasis of directional errors for small velocities whilst encapsulating speed and direction in a single measure. The angular error between correct velocity and the measured estimate velocity is given by

\j/^

This error measure is calculated for every pixel for which a velocity measure was recovered, and the average error computed.

The hardware implementation ha.s been tested with synthetically generated stimuli in order to compare it to the software version. Three test scenes were used (figure 6.21), a translating sine wave moving in the x-direction with a velocity v = (—1,0), a translating plaid with velocity v = ( - l , —1) and a translating image moving at three different velocities; Vj = (0 .5 ,0 ), V 2=(l,0) and Vg = (1,1). The results of evaluation for the hardware and software versions of the model are shown in Tabled. 1.

Sinel Synthetic translating sine wave Sine2 Synthetic translating plaid

w i t h v = ( - l,0 ) V = (-1 ,-1 )

T r a n s i , 2,3 T h re e fram es from th e realistic tr a n s la tin g im age sequences.

F ig u re 6.21 S tim u li used in ev a lu a tin g th e tw o versions of th e model. T h e T r a n s i ,2 + 3 sequences te s t th e m odel using a realistic scene. All im ages w ere w ere low resolution 64x48 images.

The software version of the model used parameters of <7 = 1.5, ct = 10 and r = 0.25 with an integration zone of 11x11x11. It is observed th at the errors are considerably higher for the hardware version compared to the software version. This can be explained by the quantisation problems at low spatial or temporal frequencies when the

numerical values become small. For higher speeds the difference between the two

Stimulus Software Hardware

Sinel v = ( - l,0 ) 0.6 1.7

Sine2 v = (-1 ,-1 ) 0.35 14.2

Transi v, = (0.5,0) 7.5 10.0

Trans2 = (1,0) 7.0 20.5

TransS Vj = (1,1) 17.5 23.2

T a b le 6.2 A verage erro rs in o p tic al flow ca lc u latio n for softw are and h a rd w a re version of m odel. E rro rs are in degrees an d are c a lc u lated using B a rro n s m e tric (eq u a tio n (6 .3 )).

algorithms is smaller. Figure 6.22 shows the response of the system to various speeds of motion. For the highest speeds the uncertainty becomes larger but overall the response is well behaved.

Measured Velocity of Sine Wave Spatial Frequency 20 Degs/pixel

Correct Measured

■D

Actual Speed

Figure 6.22 T h e velocity m easured from by th e m odel is ro b u st for a wide range of ac tu a l speeds.

M u ltip le M easurem en ts

If the second order temporal derivative is removed from the optic flow calculation then the response to high speeds is greatly diminished (figure 6.23). This is because at high speeds, the values in the first order derivative become very small. If we could make extremely accurate and precise measurements, this would not be a problem, but in a quantised system with a finite dynamic range it is clearly important. Analogous situations can occur when individual spatial filters do not respond strongly to an input. Similar effects would be expected in biological neural systems th a t can only encode a finite dynamic range - multiple measurements are required.

Making multiple measurements using banks of similar sensors reduces noise through averaging and is widely used. Here it is seen th a t it is of equal importance, when considering noisy sensors of limited dynamic range, to make multiple different

measurements. It is always of benefit to have at least one filter tuned for every eventuality (in spatial or temporal frequency, orientation, scale, colour etc.) to give at least one robust measurement.

Figure 6.24 shows the response to multiple orientations of sine wave. It is seens th a t in general the velocity is under-estimated due to the under estimation of derivatives in numerical differentiation.

Measured Velocity of Sine Wave

With and without second order temporal derivatives

7 ^ 4 0 0 1 2 3 4 5 6 7 Correct • Full Model ■No 2nd Order Actual Speed

F ig u re 6.23 T h e second o rd er d eriv a tiv e is im p o rta n t w hen m easu rin g high velocities. W h en it is rem oved, th e m easured velocity is progressively u n d er e s tim a te d above a b o u t 4 p ix els/fram e.

Measured Speed for Sine Wave Translating in Different Directions. Correct Velocity=1.0

1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 ■Vel 0 60 120 180 240 300 360

F ig u re 6.24 M easured speed as a function of o rie n ta tio n . M easured speed is slightly u n d e r­ e s tim a te d due to th e erro rs in num erical differen tiatio n .