Observer eciency

The concept of an ideal observer enables the performance of human observers to be compared and their eciency to be calculated. Barlow (1978) re-introduced the concept of the ideal observer into the domain of the visual psychophysics to calculate the performance of the ideal observer and used this to determine and compare the eciency of human observers when detecting various higher density dot patterns sited in lower density dot backgrounds. Barlow (1978) found an upper limit

of 50% eciency for human observers carrying out this task.

Burgess et al. (1981) measured the eciency of human observers when conducting a 2AFC discrimination task where the observers were presented with grating signals embedded in the centres of side-by-side square noise patches. The grating signals were presented as stationary gratings, periodic pulse burst signals or periodic sinusoidal signals and the observers were required to indicate which noise eld contained the grating signal with the greatest amplitude. The eciencies of the human observers were calculated from the ideal observer's performance which was determined using equation 1.47:

d0_Ideal = E

N0 (1.47)

where

d0_Ideal=ideal observer performance E =´_−∞∞ ´ s2(x, y) dxdy =signal energy N0=noise spectral density

and human observer performance from:

d0Human= 2erf i (2P − 1) (1.48)

where

d0_Human=human observer performance erf i =inverse error function

P =proportion of correct responses and eciency from:

Ef f iciency = d 0 Ideal d0_Human !2 × 100% _(1.49)

Burgess et al. (1981) found eciencies ranging from 54% for an aperiodic Gaussian signal to 83% for a 4.6 cycle/degree sine-wave grating.

Legge et al. (1987) proposed that the overall eciency of the human observer could be parti- tioned into two components; one reected by the observer's equivalent noise (reecting the level of internal noise) and a second related to the observer's sampling eciency. Legge et al. (1987) used the equivalent noise technique, described in section 1.1.4, and the established relationship between the threshold signal energy (Et)and noise, described by equation 1.50:

where

Et=signal energy at threshold

k =slope

N = noise spectral density. Noise spectral density refers to the noise power per unit of band- width and is sometimes known as the power spectral density of the noise. Legge et al. (1987) calculated this by multiplying the pixel area with the squared value of the root mean square (rms) contrast of the noise.

Neq=equivalent noise

Note: The parameters k and Neq can be estimated by plotting the performance data as shown

in Figure 1.13 as described on the next page

The observer's sampling eciency is related to the eectiveness by which the observer cross- correlates the signal template with the target signal with mismatch in template size and shape, along with incomplete spatial or temporal summation leading to reductions in sampling eciency (Legge et al., 1987). The study included two separate experiments, one using a 2 cycle/degree sine wave as the target signal, the other using a 13.6 arcmin disc as the target signal, both embedded in a pedestal to facilitate a discrimination task between signal and pedestal, with added external noise that was static in the rst experiment and dynamic in the second (Legge et al., 1987). In both cases the threshold signal energy for discrimination was plotted against the noise spectral density such that the separate contributions of equivalent noise and sampling eciency could be estimated as shown in Figure 1.13, extracted from Legge et al. (1987).

For an ideal observer, with no internal noise and a sampling eciency of 1, the signal detectabil- ity index d0

will be given by:

d0 = r

Et

N (1.51)

And, therefore the signal energy Etrequired by the ideal observer for detection will be:

Et= d

0₂

N (1.52)

For a human observer with internal noise and a sampling eciency of 1, the signal energy Et

required for detection will be:

Et= d

0₂

(N + Neq) (1.53)

Figure 1.13: Illustration of Legge et al. (1987) methodology for partitioning contrast threshold for discrimination into the eect of sampling eciency and the eect of equivalent noise.

reduces, and, therefore, sampling eciency J can be dened as:

J = d

0₂

k (1.54)

and hence, equation 1.50 on page 62 can be rewritten as:

Et=

d02

J (N + Neq) (1.55)

With reference to Figure 1.13, we can determine the contrast threshold for discrimination with no added noise from the intercept on the Y axis and this shows that functions B and C have the same threshold energy, but, from threshold energy alone it would not be possible to identify the contribution of the individual sources of ineciency. However, by adding external noise and plotting Et against noise spectral density N, with the signal detectability index maintained at

d0 = 1, using equation 1.50, we are able to determine the contribution of sampling eciency from the reciprocal of the slope of the function and the (negative) value of the equivalent noise from the intercept on the X axis.

From Figure 1.13, we can see that the ideal observer has a slope of 1 and no equivalent noise, therefore Et= N. An observer with a sampling eciency of 1 but with added equivalent noise of

1 is shown by function A, where Et= N + 1.An observer with a sampling eciency of 1 but with

added equivalent noise of 2 is shown by function B, where Et= N +2.An observer with a sampling

eciency of 0.5 and an equivalent noise of 1 is shown by function C, where Et= 2 (N + 1) .

Using this methodology, Legge et al. (1987) were able to determine that the major contribution to the variation in contrast discrimination, and thus, to the overall eciency of the human visual system for this task, resulted from the variation in equivalent noise, with sampling eciency remaining relatively constant across the range of pedestal contrast.

Pelli (1990) also attempted to consolidate and clarify the components that together determined the overall quantum eciency of vision, breaking down the process of vision from the presentation of the image to the performance of the assigned task, be it detection or discrimination, into three discrete stages. The rst two stages being the production of the photon image at the retina and the production of an eective image within the cortex which, together, Pelli (1990) called transduction. The third stage Pelli (1990) referred to as calculation which represented the observer's use of the eective image in decision making and this is analogous to the sampling eciency of Legge et al. (1987). Calculation eciency is also known as central eciency (Barlow, 1977) and detection eciency (Kersten, 1987).

The rst stage of transduction requires the conversion of photons from the stimulus into an image at the retina, and includes the random nature of the the luminance received at the cornea, the losses experienced within the eye plus the impact of noise added to the display and Pelli (1990) described the signal to noise ratio (SNR) at this stage as:

SN R1=

E

N + Nphoton (1.56)

where

E =signal energy N = added display noise

Nphoton=corneal plus photon noise

The formation of the eective image in the cortex introduces the eect of neural noise and the contrast invariant quotient of this added to the corneal and photon noise can be estimated using the equivalent noise method described in section 1.1.4 such that the SNR at this stage is:

SN R2=

E N + Neq

(1.57)

where

Interestingly, Pelli (1990) has not included the noise resulting from the decision making process in equivalent noise which, by inference, previous researchers have done (Barlow, 1957; Nagaraja, 1964), but includes it in the nal performance measured02 for the task in a similar manner to

Legge et al. (1987). Eciency can be calculated for each stage and the overall quantum eciency is equal to the product of the eciencies at each stage. Hence:

T ransduction ef f iciency (F1) = E N +Neq E N +Nphoton = N + Nphoton N + Neq (1.58) Calculation ef f iciency (F2) = d02 E N +Neq (1.59) and

Overall quantum ef f iciency (F ) = F1F2=

d02 E N +Nphoton

(1.60)

Using these denitions of eciency, Pelli (1990) demonstrated, using data from a number of previous studies that transduction eciency is relatively constant at around 1% and, thus, variation in eciency must result from variation in calculation eciency. Whilst this appears to contradict the conclusion drawn by Legge et al. (1987), this may be explained by the methodology. Legge et al. (1987) showed that sampling eciency within a task remained constant, however acknowledged that sampling eciency between dierent tasks (static noise and dynamic noise) did change, in agreement with the conclusion drawn by Pelli (1990).

1.5 The Eect of Signal Characteristics on Signal Detectabil-