We examine how the perceived contrast of dynamic noise images depends upon temporal frequency (TF) and mean luminance. A novel stepwise suprathreshold matching paradigm shows that both threshold and suprathreshold contrast sensitivity functions may be described by an inverted-U shape as a function of TF. The shape and the peak TF of the tuning function vary with the conditions under which it is measured. Spatiotemporal vision is weakly band-pass at low luminance levels (0.8 cd/m ^{2}) but becomes more strongly band-pass at high luminances (40–400 cd/m ^{2}). The peak temporal frequencies of the band-pass functions increase with the mean luminance and contrast of the test signals. As a function of increasing image contrast, our results demonstrate that the visual system broadens the spatiotemporal bandwidth of its signal detection mechanisms, especially at high mean luminances. Our results are shown to be consistent with an adaptable signal transmission system in which early luminance-dependent gain control mechanisms, in combination with on-line estimates of contrast via the autocorrelation function lead to an adaptive enhancement of spatiotemporal vision at high temporal frequencies.

*MSE*), defined as the average squared difference between the underlying and extracted signal, is achieved (see Equation A1).

*MSE*between the input and the output signals subject to a constraint placed upon the propagation of information across the communication channel. The propagation constraint can be viewed as analogous to the limited dynamic range attributed to real neurons in the visual system (Laughlin, 1994): a bandwidth constraint. Note that the bandwidth constraint is a critical component of the signal transmission system. In its absence, an optimized signal transmission system would simply swamp the channel noise source with amplified signal at the stage of signal encoding (Diamantaras, Hornik, & Strintzis, 1999; Wainwright, 1999). In many applications, this amplification process would be impractical for a physically realizable system because signal power is often held at a premium.

*G*(

*μ*): The model proposes that the visual signal is first suppressed by early luminance gain control, whose purpose is to maintain a relatively constant mean output (independent of the actual mean luminance of the signal). The neural site of the gain control mechanism, by virtue of its luminance dependency, is probably retinal (Heinrich & Bach, 2001; Laughlin, 1994; Mante et al., 2005; Rushton, 1965; Snippe, Poot, & van Hateren, 2000; Webster, Georgeson, & Webster, 2002; Wilson, 1997). As can be observed from Figure 1, an early gain control mechanism that post-cedes a source of input noise will attenuate both signal and input noise, thus preserving signal-to-noise ratios. The system's response (output) at this stage, although constant by design, will nonetheless deliver a signal-to-noise ratio that increases in proportion with increases in signal by virtue of the attenuation of the input noise variance.

*E*(

*ω*), whose TF tuning is determined by the statistical properties of the transmitted and intrinsic noise signals and the transmission channel constraint. The signal's statistics are represented by the temporal autocorrelation function (TAF), which for this paper is a function of the expected mean, variance, and bandwidth of visual signals ( Equation D1). The TAF may also be thought of as an environmental model (Barlow, 1961; Dakin & Bex, 2003; Foldiak & Barlow, 1989; Langley, 2005; Simoncelli, 2003; van Hateren, 1997). The TAF employed here was first derived by Franks (1969) as a mathematical model for video signals (4). This TAF is given by

*σ*

_{tt}

^{2}and

*μ*represent estimates of the spatiotemporal signal's variance and mean luminance, respectively.

*A*is a rate parameter that controls the bandwidth of the TAF. Smaller values for

*A*imply that visual signals are expected to be more correlated as compared with higher values. In its discrete form (i.e., by taking

*Z*transforms), it is not difficult to show that the parameters depicted in Equation 1 provide a simple characterization of spatiotemporal signals in terms of the expected mean (

*μ*), variance (

*σ*

_{tt}

^{2}), and temporal instantaneous frequency (

*ω*(

*A*)). Numerical values for each parameter may be estimated by a system on-line through separate computations in parallel: the system identification problem (Gelb, 1974; Langley, 2005). Numerical values for the variance (

*σ*

_{tt}

^{2}) linearly increased with stimulus contrast under experimental manipulation, while necessary values for the rate parameter (

*A*) and mean (

*μ*) were estimated from the data.

*ω*” for

*ω*≫

*A,*which is a characteristic of the temporal amplitude spectrum dynamic natural images (Atick, 1992; Bex, Dakin, & Mareschal, 2005; Billock, de Guzman, & Scott Kelso, 2001; Dong & Atick, 1995; van Hateren, 1997).

*E*(

*ω*), that delivers the smallest

*MSE*subject to a transmission constraint ( 1; Equation A5) is given by

*σ*

_{in}and

*σ*

_{ch}are the standard deviations of the input and channel noise signals, respectively (Franks, 1969).

*λ*

_{E}denotes the weight attributed to the bandwidth constraint whose value was assumed to be fixed and hence soft (Diamantaras et al., 1999). From Equation 2, it can be noted that the transfer function of the encoding filter's spatiotemporal gain is inversely proportional to the magnitude of

*λ*

_{E}. Hence, larger values of

*λ*

_{E}will lead to a proportionally smaller signal variance that is transmitted across the communication channel. As illustrated in 2 (Figures 7c and 7d), the transfer function of the encoder is generally low-pass when the assumed signal-to-noise ratio of transmitted signals is low but may be band-pass when high. From Equation 2, it can also be observed that the early gain control mechanism

*G*(

*μ*) will suppress the variance of input noise signals. This suppression necessarily allows signal-to-noise ratios to increase as a function of the visual signal's mean thus justifying a luminance-dependent change in the transfer function of the encoding filter. We also allowed

*σ*

_{tt}, the standard deviation of the TAF, to linearly vary as a function of the root mean square (RMS) contrast of test signals. This assumption is easily justified because one would expect the RMS contrast of test signals to be proportional to

*σ*

_{tt}/

*μ*. The latter manipulation has two effects on the transfer function of the encoding filter: (i) the cut-off TF and (ii) the gain of the encoding filter will both increase (2). Both effects, when incorporated into the design of an adaptive encoder, lead to an adjustment (whitening) in its temporal transfer function that varies as a function of both the mean luminance and contrast of spatiotemporal signals (Mante et al., 2005).

*D*(

*ω*), whose dual purpose is to invert the transformation imposed in the signal at the stage of signal encoding and reduce the contribution to the overall

*MSE*(de-noise) of the system that arises from the sources of uncertainty implied by the channel and input noise assumptions. The TF tuning of the decoder is also determined by the signal and noise statistics but unlike the encoder is generally low-pass thus reflecting the transfer function of the TAF ( 2; Figures 7e and 7f). In line with previous researchers, the transfer function of the decoder was held constant (Atick, Li, & Redlich, 1993; Fairhall, Lewen, Bialek, & de Ruyter van Steveninck, 2001; Langley & Anderson, 2007). The step was taken as a minimal requirement that enabled the signal transmission model to explain the empirical data reported in this paper.

*δE*

^{2}]. We interpret this uncertainty as information about the encoding weights that is not known by the decoder (Langley & Anderson, 2007). The decoding filter that assumes this additional degree of uncertainty (see Figure 1; Equation A8) is given by

*E*

_{u}*(

*ω*) denotes the complex conjugate of an underlying (assumed) encoding filter whose transfer function was constant insofar as the decoder was concerned. From Equation 3, it can be noted that the signal uncertainty from [

*δE*

^{2}] will have the net effect of reducing sensitivity at the lower TFs because of its dependence on the (low-pass) TAF.

- Contrast constancy stems from a broadening in spatiotemporal tuning that is controlled by a
*reduction*in input noise variance levels through an early luminance gain control mechanism and through an*increase*in the variance (contrast) of transmitted signals, or equally a signal determined luminance and contrast-dependent increase in signal-to-noise ratio. - A loss in contrast sensitivity at the lower spatio temporal frequencies owing to an assumed uncertainty [
*δE*^{2}] that exists between the encoder and the decoder.

^{2}or on a Phillips BrightView 19-in. monochrome monitor with a mean luminance of 400 cd/m

^{2}. The luminance gamma functions for each RGB color were separately measured with a Minolota CS100 photometer and were directly corrected in the graphics card's advanced settings control panel to produce linear 8-bit resolution per color. The RGB monitor settings were adjusted so that the luminance of green was twice that of red, which in turn was twice that of blue. This shifted the white point of the monitor to 0.31, 0.28 (

*x,*

*y*) at 40 cd/m

^{2}. A bit-stealing algorithm (Tyler, 1997) was used to obtain 10.8 bits (1785 levels) of luminance resolution on the RGB monitor under the constraint that no RGB value could differ from the others by more than one look up table step. RGB inputs to the monochrome monitor produced a gray scale image that was the linear sum of the RGB inputs and thus gave a 9.6-bit resolution (768 levels). The displays measured 36° horizontally (1152 pixels) and 27° vertically (864 pixels), and were 57 cm from the observer, in a dark room.

*Ω*

_{p}controlled the peak TF and

*b*

_{0.5}controlled the bandwidth, which was fixed at 1 octave. The SF spectrum of the white noise was unchanged. The noise was spatially windowed with a circular aperture whose edges were smoothed with a raised cosine envelope over 0.5°. The onset and offset of the noise were smoothed with a raised cosine over 40 ms (three video frames).

_{TF}, that was under the control of a staircase (Wetherill & Levitt, 1965). The staircase reduced Δ

_{TF}by 1 dB (1/20 log unit) following three correct responses and increased Δ

_{TF}by 1 dB after 1 incorrect response. The staircase was initialized with a Δ

_{TF}of 10% of the standard TF ±3 dB at random and terminated after 10 reversals or 50 trials, whichever occurred first. The observer's task was to fixate a central point then to indicate the location of the target with lower TF by pressing a corresponding button on a number pad. The fixation mark was a 3.75′ square (2 × 2 pixels), whose color was used to provide feedback (green for correct, red for incorrect that was photometrically isoluminant with the background on the RGB monitor; and white for correct, black for incorrect on the monochrome monitor).

^{2}fit (in which the data are weighted by the binomial standard deviation calculated from the observed proportion correct and the number of trials tested at each level). TF discrimination thresholds were estimated from the 75% correct point of the psychometric functions and 95% confidence intervals on these points were calculated with a bootstrap procedure, based on 1000 data sets simulated from the number of experimental trials at each test level (Foster & Bischof, 1991).

^{2}mean luminance, LM depth = 4 cd/m

^{2}), 0.2 (LM = 8 cd/m

^{2}), 0.4 (LM = 16 cd/m

^{2}), or 0.6 (LM = 24 cd/m

^{2}). The RMS contrast of the match noise was under the control of control of a staircase (Wetherill & Levitt, 1965) that reduced the match contrast by 1 dB following two positive responses and increased it by 2 dB after one negative response. The staircase was initialized with a random contrast within ±3 dB of the standard contrast and was terminated after 10 reversals or 50 trials, whichever occurred first. The observer's task was to fixate a 3.75′ central square and to indicate the location of the target with higher apparent contrast by pressing a corresponding button on a number pad. The 15 match TF pairs were randomly interleaved in a single run and the raw data from a minimum of four runs for each condition (at least 180 trials per psychometric function) were combined and were fit with a cumulative normal function by least

*χ*

^{2}fit. Match contrast was estimated from the 50% point of the psychometric function, and 95% confidence intervals on this point were calculated with a bootstrap procedure (Foster & Bischof, 1991).

*λ*

_{ E}whose magnitude is determined by the channel constraint but was fixed in our simulations; the TAF,

*R*

_{ tt}(

*ω*), which is signal dependent and controls the tuning of the encoding filter; and the channel and the input noise variances (

*σ*

_{ch}and

*σ*

_{in}, respectively), the latter of which we have assumed to be controlled by early retinal gain control processes. From our psychophysical experiments, we are unable to make definitive statements concerning the adaptability of each term employed by the signal transmission model. The main difficulty is because the net transfer function of a signal transmission system is principally controlled by the signal-to-noise ratio and the magnitude of channel constraints. Neither term can be constrained by our experiments. Given this difficulty, we allowed two of the model's parameters to vary freely. One was the magnitude of

*G*(

*μ*), which we assumed affected the variance of input noise. The second was the variance (

*σ*

_{ tt}

^{2}) of the TAF for the encoding filter (

*E*(

*ω*)) that, as already mentioned, one could expect to be proportional to the contrast of test signals. All other system parameters were estimated from each subject's empirical data and then held fixed while the mean luminance and the contrast of our test signals were varied. For subject K.L., we collected 169 data points; whereas for subject P.J.B., we collected 142 data points. The signal transmission model ( Equations 1 and 2) was fitted to each subject's empirical data (reported in Experiments 2 and 3 simultaneously) in a single batch. In total, there were 7 parameters (noise variances, Lagrange multipliers) whose values were fixed throughout the simulations, 3 input noise parameters whose values were allowed to vary freely as a function of the mean luminance, and 9 further parameters to model the input signal variances for both the threshold contrast and suprathreshold contrast matching tasks (11 conditions).

^{2}, blue circles), 0.2 (LM = 8 cd/m

^{2}, green squares), 0.4 (LM = 16 cd/m

^{2}, red triangles), and 0.6 (LM = 24 cd/m

^{2}, pink diamonds) for the two observers. Note also that Figures 3a– 3b and 3c– 3d show the same data but plotted on linear versus logarithmic ordinates. The STAC functions were generated by scaling adjacent TFs by the proportion of contrast gained or lost between each match pair. Any contrast change was then accumulated across the function. Note that one member of every match pair was fixed at the standard contrast/LM so the STAC functions show the overall change in apparent contrast with TF for a given reference RMS contrast. Although this process may accumulate a serial error with distance between any comparison points along the function, the sign of any such error is expected to be approximately equal on average. Furthermore, under contrast constancy, the function is expected to be flat. It was not possible to collect contrast matches at the highest TFs for the 60% contrast condition because the match tended toward impossibly high contrasts that produced look up table overflows.

*match contrast higher*and

*match contrast lower*responses. This process estimated a match TF at which a higher contrast pattern had the same apparent contrast as an 8-Hz standard of lower contrast. This process allowed us to estimate the relative heights of the functions, albeit crudely and suffering from the criticisms of this paradigm we have leveled above. Following this scaling, the STAC of the peak contrast (around 8 Hz) for each contrast matching function became approximately equal to the physical contrast of the noise pattern. This remarkable observation suggests that peak STAC increases approximately linearly with physical contrast, confirming earlier observations (Kulikowski, 1976).

*R*

^{2}values were .991 for subject K.L. and .986 for subject P.J.B. Estimates for the rate parameter

*A*were 3.2 and 2.9 Hz for subjects K.L. and P.J.B., respectively. Figure 3e shows the variance estimate of the TAF for the encoder (

*σ*

_{ tt}

^{2}), which was the only free parameter in the model that was allowed to vary as a function of the contrast of the matching signals. Estimates for

*σ*

_{ tt}

^{2}insofar as model fitting was concerned were constrained to lie on a linear function whose slope and intercept was estimated from the data. In forcing this restriction on the variance parameter for the TAF, we confirmed that the peak in STAC functions linearly increases with physical contrast. Note also that the parameter

*σ*

_{ tt}

^{2}determines the shape of the encoding filter

*E*(

*ω*) and hence the net transfer function (

*H*(

*ω*)) of the signal transmission system. The estimates for

*σ*

_{ tt}

^{2}linearly increased with the contrast of the test signals, which in an adaptive system can lead to a whitening of the transfer function of the STAC matching function. Estimates for the transfer function of the signal transmission system are shown in Figure 3f, which were required to explain the trends in subject K. L.'s data. From the figure, it can be seen that as the contrast of the test signal increases, the filter whitens—that is, the bandwidth increases and the filter becomes increasingly sensitive to high TFs. This observation confirms one of the predictions made by the transmission model; namely, that temporal bandwidth is controlled by underlying signal-to-noise ratios.

^{2}. Contrast thresholds were measured using the same procedures to those given in Experiment 2. The low mean luminance data were collected on the 40 cd/m

^{2}RGB monitor while the subjects wore neutral density filter spectacles (NoIR, Optima Low Vision services) with 2% transmission. To ensure stable levels of light adaptation, subjects were dark or light adapted for a minimum of 10 min before data collection and runs lasted in excess of 30 min. The high mean luminance data were collected on a Phillips Brightview monochrome monitor.

^{2}(green squares, left) and STAC matches at RMS contrasts of 10% (green circles, right) and 40% (green squares, right) are replotted from Figures 3a to 3b for comparison purposes.

^{2}to 8.0 Hz at 400 cd/m

^{2}. At 40% contrast, the peak TF shifted from 5 Hz at 0.8 cd/m

^{2}to 10 Hz at 400 cd/m

^{2}. The peak spatiotemporal sensitivity of the visual system thus increases as a function of both the mean luminance and contrast of the test signals.

^{2}. Both subjects subjectively noted a significant increase in the visibility of high TF stimuli at high mean luminances.

*R*

^{2}values from the model fits were, therefore, the same as those reported in Experiment 2. For variations in mean luminance, however, we allowed the standard deviation of the input noise (

*σ*

_{in}) to vary as a function of mean luminance to simulate the effects of early luminance gain control (

*G*(

*μ*)) on an adaptive transmission system. As in Experiment 2, the variance of the autocorrelation function used to defined the encoding filter (

*E*(

*ω*)) was allowed to vary as a function of the RMS contrast of the test signals.

^{2}mean luminance (blue curves). The reason for the discrepancy is unclear. However, note that STAC could not be measured above 18 Hz for the 10% contrast condition. This was because the standard stimuli were at or near the contrast detection threshold. It is therefore possible that the practical difficulties when conducting matching tasks near to threshold underestimated STAC. As outlined by Brady and Field (1995) and Cannon (1979), near threshold stimuli may include a proportion of 0% contrast matches, lowering STAC overall. This may have affected the predictability of the signal transmission model for the low mean luminance and high TF conditions.

*σ*

_{ tt}

^{2}) for the encoding filter (

*E*(

*ω*)), taken from the model fits for Experiments 2 and 3. The rate of increase in the variance of the encoding filter's TAF itself increased with mean luminance.

^{2}used in this study). Figure 5d shows the resultant TF tuning of the sensitivity of the encoding filter. As luminance increases from 0.8 to 40 cd/m

^{2}, the encoding filter became increasingly band-pass and its peak sensitivity shifted to the higher TFs with a relative loss in sensitivity at lower TFs. Interestingly, the peak TF of the estimated encoding filter's transfer function was found to be around 18 Hz at 40 cd/m

^{2}mean luminance. This TF coincides with the peak in spatiotemporal adaptability of the visual system at threshold contrast levels for this mean luminance (Langley & Bex, in press).

^{2}and temporal frequencies around 37 Hz, were found to be close to 50%. Note that if the above whitening effects are translatable into effects on perception at threshold contrast levels (a fast adaptation mechanism), then one would expect that threshold elevations are temporal low-pass functions, which would be consistent with the predictions made by the adaptation of a sustained spatiotemporal channel. Sustained threshold contrast elevations according to a filter whitening hypothesis could lead to an adaptation-driven facilitation in threshold contrast: but only for the higher spatiotemporal frequency signals; a prediction that the authors' have found some evidence to support (Langley & Bex, in press).

^{2}). Our experiments do not strongly disagree with their results, but we have examined effects of contrast constancy at higher and lower levels of mean luminance than they examined. At high mean luminances, our results show significant whitening (contrast constancy) but only at the highest range of spatiotemporal frequencies examined. Given the contrast range over which we have observed a trend toward constancy (between 10% and 60% contrast), it is difficult to see how a linear model could be capable of explaining our results.

^{2}(Peli, Yang, Goldstein, & Reeves, 1991; Peli, 1995). We did not compare contrast across mean luminance and our stimuli were binocularly viewed on homogenous fields; however, our data (with >10 min adaptation) and model are consistent with a tendency toward contrast constancy, with a relative loss in apparent contrast at high TFs, which favorably compares with the loss in apparent contrast at high SFs for static images (Peli, Arend, & Labianca, 1996).

*σ*

_{tt}

^{2}) of the TAF. In both cases, the underlying rationale for the adaptive whitening of the visual system's TF processes stems from the enhancement in the signal-to-noise ratios of visual signals.

*t*) as a stationary temporal signal we write:

*E*[.] is the expectation operator. An early gain control mechanism that maintained a constant mean square response would thus be observed to adapt to a signal's mean (

*μ*) and variance (

*σ*

_{tt}

^{2}). Thus, an early gain control mechanism whose attenuation is proportional to the mean square of the (early) retinal signals could, if rapid, also lead to an explanation for our empirical results. The rapidity might also be symptomatic of a pseudo nonlinear system (Langley & Anderson, 2007).

*E*(

*ω*) need only smooth temporal signals to reduce input noise for low signal-to-(input) noise ratios. For high signal-to-noise ratios, the same system need only scale (multiplicatively suppress) signals in order that they may be squeezed through a communication channel and recovered. According to the single channel encoding model illustrated in Figure 1, such an encoding scheme predicts that the net temporal transfer function of the visual system would be low-pass, which contradicts the band-pass temporal CSF that is observed empirically. A transient encoding scheme, however, can be justified under the circumstance where there is a constraint on the propagation of signals along the communication channel, in combination with channel noise. In the event of a noisy communication channel, the objective of the encoding filter should maximize the signal-to-noise ratio of the signals transmitted along the communication channel: a leaky-predictive coding strategy. By leaky, we refer to the principal of partial de-correlation (Langley, 2004; Webster, 1996). A global optimal leaky predictive-encoding strategy cannot, however, explain the transient threshold contrast functions reported here and by many other researchers (de Lange, 1958; Kelly, 1961; Roufs, 1972; Snowden et al., 1995). This is because the white assumptions for sources of signal uncertainty, in combination with the low-pass (1/

*ω*) amplitude spectrum attributed to natural scenes (Dong & Atick, 1995), imply that a decoding filter could be introduced into the visual pathways that will invert any encoding transformation thus leading to a low-pass CSF. To explain the loss in visual sensitivity at low TFs, we have assumed, like others (Atick et al., 1993), that there is a loss in information transmitted across a (neural) communication channel. The loss in this paper, arising from the assumption of a fixed decoding filter and source of signal uncertainty that we have posited to exist between the encoding and the decoding transformations.

*λ*

_{ E}in Equation 2) and estimated its value from the empirical data. There were three reasons why this step was taken: (i) contrast discrimination thresholds do not linearly increase with contrast, but rather flatten at high test contrasts (Ross, Speed, & Morgan, 1993), which implies that the communication channel constraint cannot be viewed as hard (see Diamantaras et al., 1999); and (ii) one's perception of contrast is approximately a linear function of increasing contrast in unadapted conditions (Kulikowski, 1976) and Experiment 2; and (iii) the “channel constraint” that best represents information transmission by the visual system is unknown.

^{2}, with the exception of Snowden et al. (1995) who used a 140 cd/m

^{2}monitor, our results imply that the full range of TF and possible SF tuning of the visual system has been underestimated in laboratory conditions. In natural scenes, where mean luminance signals routinely exceed 1000 cd/m

^{2}(Martin, 1983), our results raise the possibility of considerable contrast sensitivity and adaptability in the 50- to 60-Hz range: Note that this corresponds to the frequency of flicker of many artificial illuminating devices. If so, controlled investigation of the spatiotemporal contrast sensitivity of the visual system under natural lighting conditions may prove useful to researchers of normal and abnormal visual function in clinical settings (e.g., Skottun, 2001).

*Y*(

*ω*) =

*D*(

*ω*)[

*E*(

*ω*)

*X*(

*ω*) +

*ɛ*

_{in}] +

*ɛ*

_{ch}refer to observations of the transmitted signal

*X*(

*ω*), and let

*ɛ*

_{in}and

*ɛ*

_{ch}refer to the input and the channel sources of signal uncertainty, respectively. The transfer function of both the encoder

*E*(

*ω*) and the decoder

*D*(

*ω*) that minimizes the

*MSE*between the input and observed signals is to be determined. Given the cascaded placement of

*D*(

*ω*)

*E*(

*ω*), the signal transmission problem is ill-posed, unless additional constraints are introduced. The additional constraint we assume to be penalty incurred by the transmission of high variance signals whose weight is given by the Lagrangian

*λ*

_{ E}. In placing this constraint on the variance of transmitted signals, the computational objective is to design optimal encoding and decoding filters that will deliver a minimum

*MSE*in the respect of the input and output signals subject to a system constraint that imposes a restriction on the variance of signal propagated along the communication channel.

*R*

_{ tt}(

*ω*) denote the TAF, with

*P*

_{a}referring to the maximum permissible variance that may be transmitted across the communication channel. Note that if

*λ*

_{ E}is fixed,

*P*

_{a}can be ignored. Collectively, the

*MSE*for the transmission system plus constraints may be described by the functional

*F*(Franks, 1969):

*F*under the integral with respect to

*D*(

*ω*)

*E*(

*ω*), which gives two stationary conditions:

*D*(

*ω*) and Equation A3 by

*E*(

*ω*). Subtracting the resulting equations gives

*R*

_{ tt}(

*ω*) ≫

*σd*

^{2},

*σ*

_{ch}

^{2}. This gives

*E*(

*ω*) is known in advance but subject to an additional level of uncertainty denoted by

*δE*(

*ω*). The observed signal is now given by

*Y*(

*ω*) =

*D*(

*ω*) [(

*E*(

*ω*) +

*δE*(

*ω*))

*X*(

*ω*) +

*ɛ*

_{in}] +

*ɛ*

_{ch}. The optimal decoder

*D*(

*ω*) in which the encoder is assumed known is found by the minimization of the functional

*G*:

*D*(

*ω*) is known and fixed but that the encoder

*E*(

*ω*) can adjust. If the encoder cannot propagate changes in its transfer function to the decoder, but is still subject to a transmission constraint, the optimal encoder is found by differentiating Equation A1 with respect to the encoder

*E*(

*ω*) only. This gives

*R*

_{ tt}(

*ω*) ≫

*σ*

_{in}

^{2},

*σ*

_{ch}

^{2}one obtains

*H*(

*ω*) =

*E*(

*ω*)

*D*(

*ω*) from Equations A5 and A6 is given by

*MSE*may be obtained under low signal-to-input noise ratios by first smoothing the encoded signal prior to transmission. At higher signal-to-input noise ratios, the transfer function of the encoding filter may broaden in spatiotemporal transfer function or whiten. These ideas lead to the following simple principle:

- An optimal signal transmission system should if adaptable adjust its transfer function such that the degree of smoothing depends upon the underlying signal-to-noise ratio: As the signal-to-noise ratio increases, so may the level of smoothing should decrease (Atick, 1992; Langley, 2005; van Hateren, 1992).

- The transfer function of an optimal spatiotemporal encoder should (subject to principle i) approx imately reciprocate the TAF (Dong & Atick, 1995).

*σ*

_{ tt}

^{2}, the variance of the input signal. When the signal-to-noise ratio is low, the encoder's gain is low and its transfer function low-pass. As the signal-to-noise ratio increases, two changes to the transfer function of the encoder take place: (a) its gain increases and (b) its transfer function becomes more band-pass (see principles i and ii). Figure 7d shows the dependency of the encoder's transfer function on the rate parameter

*A*. Because the magnitude of

*A*is determined by the bandwidth of the underlying signal, larger values for

*A*imply a signal whose bandwidth is broader and hence the transfer function of the optimal encoder should also broaden to take into account this additional information.

*D*(

*ω*). The decoder should

*H*(

*ω*) of the signal transmission system ( Equation A10). From the figures, two points can be noted: (i) as the variance of the underlying signal increases, the gain of the system increases in combination with a broadening (whitening) of the system's spatiotemporal response; and (ii) as the rate parameter

*A*increases, a broadening in temporal bandwidth is also observed. From Figures 7g and 7h, it should also be noted that the net transfer function (

*H*(

*ω*)) of the signal transmission system is low-pass.

*d*′ (assumed fixed). In Equation 8,

*t*) and

*t*) refer to two independent samples of the transmission system's output with var[.] denoting a variance operator. At threshold signal levels, we may set

*t*) = 0 and note

*E*[.] and * denote the expectation and convolution operators, respectively;

*h*(

*t*) =

*e*(

*t*) *

*d*(

*t*) is used to refer to the transfer function of the signal transmission system in the signal domain. The definition for the signal threshold (

*t*) follows from the Fourier transform of Equation 9 in which we represent the Fourier transform of a probe signal by an impulse function at its peak tuning frequency (

*ω*

_{p}), which gives

*t*) denote the estimate of an observed signal in one condition and

*t*) denote the estimate of the same signal in a second condition, a perceptual equality is given by

*ω*

_{r}represents the TF of the reference signal. The equation shows that the ratio of the contrast matches approximates the relative changes in the transfer function of the visual system as a function of the TF of the test signals, but only to the extent that the TF transfer function of the visual system does not significantly vary as a function of the variance of the underlying signal. In comparing Equation C3 with Equation C4, it can be seen that both signal thresholds and signal matching function depend upon the underlying sensitivity of the visual system's temporal processors of visual information. In the case of contrast thresholds, the variance of the intrinsic noise sources is largely unknown and must therefore be estimated from the empirical data.

*μ*

^{2}=

^{2}and

*σ*

_{a}

^{2}=

*σ*

_{tt}

^{2}, gives the desired result

*f*spectra in dynamic images and human vision. Physica D. Nonlinear Phenomena, 148, 136–146. [CrossRef]