Noise-masking experiments are widely used to investigate visual functions (Allard, Faubert, & Pelli, 2015; Lu & Dosher, 2008; Pelli & Farell, 1999; Pelli, 1981). Masking occurs when the noise noticeably impairs the observer’s performance, but under some conditions, performance remains unaffected even when the noise approaches 100 % contrast. For instance, consider using a noise-masking paradigm to investigate the internal factors limiting contrast sensitivity (e.g., Pelli & Farell, 1999). The noise energy required to noticeably impair the detection threshold is minimal at middle spatial frequencies and gradually increases at low and high spatial frequencies. At low spatial frequencies, the impact of noise is attenuated by sparser and larger receptive fields integrating (~averaging) the noise over large areas and thereby weakening its masking strength (Raghavan, 1995). At high spatial frequencies, the impact of the noise is attenuated by the modulation transfer function of the eye reducing the effective contrast of the stimulus (Campbell & Gubisch, 1966) and therefore requiring high noise energy to noticeably impair performance. As a result, the frequency range over which noise can effectively impair performance is limited by the maximal external noise energy that can be displayed, that is, without exceeding 100 % contrast. Furthermore, the noise energy required to impair detection also increases when reducing luminance intensities, especially at high spatial frequencies (Raghavan, 1995). Displaying more noise energy would widen the frequency and luminance range over which noise-masking paradigms could be implemented. To extend the usefulness of noise-masking paradigms, the present study developed a new noise maximizing the displayable energy.
Many noise-masking paradigms assume, usually implicitly, that the same underlying processing strategy operates in absence and presence of noise. But recent studies by Allard and colleagues (Allard & Cavanagh, 2011; Allard & Faubert, 2013, 2014a, 2014b; Allard, Renaud, Molinatti, & Faubert, 2013) suggested that this noise-invariant processing assumption can be violated when using some types of noise. For instance, contrast detection is known to be immune to crowding, but adding noise that is spatiotemporally localized to the target (i.e., at the potential target locations and turn on and off with target) made a detection task vulnerable to crowding, whereas noise that is spatiotemporally extended (i.e., full-screen, continuously displayed dynamic noise) did not (Allard & Cavanagh, 2011). These results suggest that the processing strategy in localized noise involved processes vulnerable to crowding, whereas the processing strategy in absence of noise and in extended noise did not. The aim of the present study was to increase the noise-masking strength without triggering a change in processing strategy.
To avoid triggering a change in processing strategy, Allard and colleagues (Allard & Cavanagh, 2011; Allard & Faubert, 2013, 2014a, 2014b; Allard et al., 2013) recommended to use full-screen, continuously displayed, dynamic noise. Unfortunately, using dynamic noise resampled at a high temporal rate instead of static noise tends to reduce the masking strength of noise due to temporal integration (i.e., averaging) occurring early in the visual system. The use of dynamic noise can therefore reduce the range of conditions over which noise-masking paradigms can be usefully implemented (i.e., noticeably impair performance). The constraint of using dynamic noise further emphasizes the need to maximize the displayable noise energy.
The standard way of adding visual noise to a stimulus is to introduce luminance variance to each pixel of the display, that is, to add a random luminance value drawn from a Gaussian distribution centered on 0 (left image of the top row of Fig. 1) (Pelli, 1981). Given that the samples are not correlated between each other, such noise is white (flat energy spectrum, black curve in the left graph of the third row of Fig. 1). This standard noise is typically referred to as “Gaussian noise”. For pixel white noise, the expected energy at all frequencies (e) can be defined as (Pelli, 1981):
$$ e= Variance(N)\cdot w\cdot h\cdot d $$
where Variance(N) represents the luminance variance across the samples within the noise matrix N (for samples drawn from a Gaussian distribution with standard deviation σ, the variance is equal to σ
2), and w, h and d represent the width, height and duration of each noise pixel, respectively. Thus, the noise energy is proportional to the variance of the noise (i.e., squared contrast).
A typical way to modify Gaussian noise to increase its effective masking power is to concentrate its energy to frequencies relevant to the processing of the stimulus (Pelli, 1981; Solomon & Pelli, 1994; Stromeyer & Julesz, 1972). Another method simply consists in sampling the noise from a binary distribution instead of Gaussian distribution (e.g., Allard et al., 2013). The present study combined these two approaches to further increase masking power.
The distribution that maximizes the noise energy given a limited contrast range is a binary distribution in which one of two values is randomly selected independently for each sample (left image in the second row of Fig. 1). Given no correlation across samples, binary noise is white (flat energy spectrum) and its energy can also be defined by Eq. (1). Thus, binary noise has the same expected energy level at all frequencies as Gaussian noise given the same variance (left graph in third row of Fig. 1). The variance of binary noise is equal to the variance of Gaussian noise (σ
2) when the two values from which the samples are drawn are ±σ (left graph in bottom row of Fig. 1).
An advantage of a binary distribution over a Gaussian distribution is that the contrast range is finite and well defined. Theoretically, a Gaussian distribution extends over an infinite range (left graph in bottom row of Fig. 1). Under experimental conditions, however, Gaussian noise is often truncated at 2 or 3 standard deviations (sd). Truncating at ±3 sd has little impact on the noise energy (0.5 % reduction) as few samples fall outside this range. Truncating at ±2 sd still roughly preserves the shape of the Gaussian distribution, but substantially reduces the distribution range while reducing the noise energy by only 8 %. Given that truncated Gaussian noise has a finite and well-defined contrast range, its energy can be compared with the energy of binary noise given the same total contrast range (i.e., the two binary values set to the positive and negative truncation thresholds). As shown in Fig. 2, the energy of binary noise is greater by a factor of 4.3 and 9.0 relative to the energy of Gaussian noise truncated to ±2 and ±3 sd, respectively.
For many experiments, the ideal noise would have a flat energy spectrum over all frequencies. However, to have the same energy level across an infinite range of frequencies, such an ideal noise would require infinitely small samples (e.g., w, h and d infinitely small) and its energy would therefore also be infinitely small for any finite sample variance (Pelli, 1981). In practice, a noise can have a flat energy spectrum over a finite range of frequencies and the maximal displayable noise energy can be increased by concentrating the noise energy over a narrower range of frequencies.
The simplest way of increasing the energy level is to decrease the spatial and/or temporal resolution of the display (e.g., center image in the top row of Fig. 1 in which each noise check size is set to 4×4 pixels rather than 1×1 pixels). For the same variance, reducing the resolution of the display increases the noise energy (Eq. 1) and reduces the upper frequency limit of the noise (black curve in top center graph in Fig. 1) so it can be used when the noise at these frequencies is not relevant to the task. Nevertheless, a drawback of low-resolution noise is that it introduces apparent edges between noise checks forming a grid as it can be seen in the center image in the top row of Fig. 1. Because these apparent edges may have undesirable effects (e.g., Harmon & Julesz, 1973), it is safer to avoid them as their presence could potentially interfere with the processing of the target.
An alternative method that does not introduce artificial edges consists in filtering the noise to remove frequencies that are not relevant to the task (right image in the top row and black curve in the right graph in third row of Fig. 1). Indeed, some frequencies can be removed to reduce the contrast range without affecting performance. For instance, removing the frequencies outside ±1 octave from the spatial frequency of a sine-wave target does not affect detection threshold (Pelli, 1981; Stromeyer & Julesz, 1972). Filtering out information irrelevant to the task (i.e., does not affect performance) is an efficient way of reducing the noise contrast (e.g., Gaussian distribution narrower for filtered noise compared to Gaussian noise, black curves in right and left graphs in bottom row of Fig. 1, respectively). As a result, at equal contrast ranges, filtered noise would have higher energy at the frequencies within the pass-band.
The rationale of the present study was to combine the two approaches described above (i.e., binary noise and non-white noise) to further increase the energy of the noise at the frequencies relevant to the task. For low-resolution noise, the method simply consists in sampling noise elements from a binary distribution instead of a Gaussian distribution (center images in top and second rows of Fig. 1). Low-resolution binary noise is not novel as it has been used before (e.g., Allard et al., 2013). However, the drawback of using low-resolution noise remains: it artificially introduces apparent edges between noise checks. Alternatively, the present study combined binary noise with filtered noise (right image in second row of Fig. 1). Binarizing the noise after the filtering operation substantially increases the energy level for equal contrast range. In other words, the binarized-filtered noise requires much less contrast than (unbinarized-)filtered noise to reach the same energy level at the frequencies within the pass-band (right graphs in third and bottom rows of Fig. 1).
Binarizing filtered noise changes the profile of the spectral density function as it introduces energy at frequencies that were filtered out (e.g., right graph in third row of Fig. 1). However, given that the energy at those frequencies is irrelevant (completely removing them should not affect performance), this small gain in energy should also be negligible for masking experiments. More importantly, the binarizing operation does not affect the expected constant spectral density across the frequencies within the pass-band (right graph in third row of Fig. 1).
Binarized-filtered noise requires less contrast than filtered noise to display the same expected energy at frequencies within the pass-band (right graph in bottom row of Fig. 1). Thus, by equating the total contrast range, the binarized-filtered noise would display more energy at the frequencies within the pass-band than the filtered noise. To illustrate this energy gain, the energy level at frequencies within the pass-band of binarized-filtered noise was compared with the one of filtered noise (filtered 1 octave below and above a given spatial frequency) given equated total contrast range. As for Gaussian noise, the distribution of the samples of filtered noise also follows a Gaussian distribution theoretically extending over a wide contrast range (curves of left and right graphs in bottom row of Fig. 1). Filtered noise was therefore truncated so that it had a finite and well-defined contrast range. Figure 3 compares the energy at the frequencies within the pass-band of binarized-filtered noise with the truncated-filtered noise as a function of the truncation threshold given equated contrast range (i.e., the two binary values set to the positive and negative truncation thresholds). In this particular case, binarizing the noise and equating noise contrast was found to increase the noise energy by a factor of 3.3 and 6.7 when the filtered noise was truncated at ±2 and ±3 sd of its filtered distribution, respectively. This illustrates that binarizing filtered noise substantially increases the energy at the frequencies within the pass-band of filtered noise given equated contrast range.