CAPÍTULO IV. RESULTADOS Y DISCUSIÓN
4.1 Análisis descriptivo
4.1.2 Diseño de Procesos
4.1.2.5 Ficha técnica del proceso nivel 0 de la UNAS
Monte-Carlo simulations were run in the Matlab programming environment to investigate the effect of noise in vergence, version and disparity signals on the visual system’s estimation of distance from vergence, depth from scaled disparity and height from scaled angular size. Estimates of each of these variables must be made if the visual system is to make judgements about the 3-D shape of objects, such as when comparing the depth of an object to its width or height (e.g. Bradshaw et al., 2000). Simulations were carried out to model performance in a task where an observer is required to set the depth of a vertically orientated triangle defined by three points to match its height (Figure 3.3). The triangle has a depth
! d and height ! h and is situated at a distance !
D from the observer, who has a half interocular distance
!
I. The geometrically correct half-vergence angle
!
", half-disparity
!
" and half-angular size
!
", are given by equations 3.1, 3.2 and 3.3 respectively5.
! "=arctan
(
I D)
(3.1) ! "=arctan Id D D(
#d)
+I2 $ % & ' ( ) (3.2) ! "=arctan h 2 D # $ % & ' ( (3.3)Simulated noise took the form of
!
n samples of zero-mean Gaussian noise added to the geometrically correct vergence, disparity or angular size signal. This resulted in
the vergence, disparity and angular size signals being represented by Gaussian distributions of signal estimates centred on the geometrically correct signal value with standard deviations corresponding to an estimate of that signals uncertainty (Figure 3.3). The use of zero-mean Gaussian noise introduced no bias into the signals, just variability around their true value. It is an empirical question as to the value of noise to use in the simulations; this will be addressed as the role of noise in each signal is discussed.
Figure 3.3: Diagram showing details of the Monte-Carlo simulations. The upper portion of the diagram shows the task modelled, which was to set the depth,
!
d, of a vertically orientated triangle defined by three points to match its height,
!
h. The observer’s left and right eyes are labelled
!
LE and
!
RE respectively and define the half-interocular distance
!
I. When viewed at a distance D the depth of the triangle projects a half-disparity, ", at the retina and results in the half-vergence angle, ". The half height of the triangle subtends an angle, ". The effect of
noise in vergence, disparity and angular size signals was investigated for the estimation of distance, depth and height. A plan view of this process is shown in the lower portion of the diagram, visual signals are shown in black and perceptual estimates shown in grey. Noisy signals were simulated as Gaussian distributions centred on the geometrically correct signal values. Noise in early visual signals results in a distribution of estimates for world properties such as distance, depth and height, this process was investigated separately for each of the three visual signals. For more information see the accompanying text.
In order to estimate distance from vergence, depth from scaled disparity and height from scaled angular size it is assumed that the visual system has estimates of ocular convergence ! ˆ " , retinal disparity ! ˆ
" , the angle of gaze
!
ˆ
" and the interocular distance
!
ˆ
I . For the simulations reported it is assumed that the visual system has accurate
knowledge of the interocular distance i.e.
!
ˆ
I =I. If the visual system has estimates of these values it could estimate distance
! ˆ D , depth ! ˆ d and height ! ˆ h using the relationships shown in Equations 3.4, 3.5 and 3.6 respectively. It is clear from these equations that vergence noise will affect the estimation of distance and also the estimation of depth and height, whereas noise in the gaze angle will effect only the estimation of height, and noise in disparity only the estimation of depth. This is clearly a simplification as vergence, gaze angle and disparity are unlikely to be completely independent signals. For example, the disparity of an object will depend on the distance of fixation. However, it is worthwhile to understand how noise in each of these signals will effect the estimation of distance and 3-D shape.
! ˆ D =I tan ˆ " (3.4) ! ˆ d =tan ˆ " ˆ D 2+I2
(
)
ˆ D tan ˆ " +I (3.5) ! ˆ h =2 ˆ(
D tan ˆ( )
")
(3.6)Simulations were carried out for noise in each of the signals independently, for instance, when the role of noise in the vergence signal was investigated disparity and the angle of gaze were assumed to be geometrically correct and unaffected by noise. This strategy allows the relative importance of noise in each signal to be determined. Noisy visual signals will result in a distribution of possible values for world properties such as distance, depth and height; these distributions were simulated using Equations 3.4, 3.5 and 3.6. For instance, from a distribution of vergence signal estimates it is possible to calculate the corresponding distribution of distance, depth and height values that this noisy signal would produce when combined with geometrically correct estimates of disparity and gaze angle.
Taking noise in the vergence signal as an example, in the simulations there is a Gaussian distribution of
!
n samples corresponding to the vergence signal. For each of
these
!
n samples it is possible to calculate the corresponding distance value using Equation 3.4. This produces a distribution of distance values, each of these values can then be used to calculate the corresponding depth and height values using Equations 3.5 and 3.6 respectively. This results in a distribution of depth and height values consistent with the noisy vergence signal. For the simulations reported here
!
n=500, 000, unless otherwise stated. The distributions of distance, depth and height are produced by constructing histograms of the
!
n samples of each signal with a bin size equal to 1mm. The frequency of samples in each of these bins is then plotted against the bin value to form the distributions shown.
No restrictions were placed on the possible values of distance, depth and height, to ensure that no bias was introduced into the distributions as a result of imposing limits on the values these variables could take. This means that in some instances these values may take on unrealistic values, this is discussed further below. With noisy visual signals there is a distribution of possible world states consistent with the sensory information, from this distribution the visual system must make a single perceptual estimate regarding a property of the world. This process was modelled as choosing the peak of the distance, depth or height distributions.
Choosing the peak of each distribution has a simple interpretation, consider a distribution of distance estimates from vergence, the peak of the distribution
represents the most likely distance to have produced the current sensory information available from the vergence cue. This is equivalent to maximum likelihood estimation (MLE) where the peak of the likelihood function is chosen as the perceptual estimate (Ernst & Banks, 2002). However, the brain might use alternative decision rules depending on the gains and losses associated with making a given decision on the likelihood or posterior distributions.
For example, the least squares loss function results in a loss equal to the square of the difference between the estimated world state and the actual world state. With this loss function the MLE (or MAP) rule is no longer the best rule to follow in order to maximise expected gains, choosing the mean of the posterior distribution now results in the maximum gain (Mamassian et al., 2002). When a distribution is Gaussian its peak and mean are the same, but when the distribution is skewed, the peak and the mean signal different estimates. Typically estimation and cue integration have been modelled with Gaussian probability distributions (Ernst & Banks, 2002, Hillis et al., 2004), however, this is likely to be a simplification as sensory signals may be non- linearly related to world properties (Hogervorst & Eagle, 1998). This may have important implications for models of cue combination and for explaining biases found in the perception of 3-D shape.