Cédric Jules Étienne

Angelo Arleo

Rémy Allard
email: remy.allard@inserm.fr

Maximizing noise energy for noise-masking studies

Keywords: Masking, Noise, External noise paradigm, Binary noise, Filtered noise

Noise-masking experiments are widely used to investigate visual functions. To be useful, noise generally needs to be strong enough to noticeably impair performance, but under some conditions, noise does not impair performance even when its contrast approaches the maximal displayable limit of 100%. To extend the usefulness of noise-masking paradigms over a wider range of conditions, the present study developed a noise with great masking strength. There are two typical ways of increasing masking strength without exceeding the limited contrast range: use binary noise instead of Gaussian noise or filter out frequencies that are not relevant to the task (i.e., which can be removed without affecting performance). The present study combined these two approaches to further increase masking strength. We show that binarizing the noise after the filtering process substantially increases the energy at frequencies within the pass-band of the filter given equated total contrast ranges. A validation experiment showed that similar performances were obtained using binarized-filtered noise and filtered noise (given equated noise energy at the frequencies within the pass-band) suggesting that the binarization operation, which substantially reduced the contrast range, had no significant impact on performance. We conclude that binarized-filtered noise (and more generally, truncated-filtered noise) can substantially increase the energy of the noise at frequencies within the pass-band. Thus, given a limited contrast range, binarized-filtered noise can display higher energy levels than Gaussian noise and thereby widen the range of conditions over which noise-masking paradigms can be useful.

Introduction

Noise-masking experiments are widely used to investigate visual functions [START_REF] Allard | Editorial: Using visual noise to reveal the computations underlying perception[END_REF][START_REF] Lu | Characterizing Observers Using External Noise and Observer Models: Assessing Internal Representations With External Noise[END_REF][START_REF] Pelli | Why use noise?[END_REF][START_REF] Pelli | The effects of visual noise[END_REF]. 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., [START_REF] Pelli | Why use noise?[END_REF]. 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 [START_REF] Raghavan | Sources of visual noise[END_REF]. 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 [START_REF] Campbell | Optical quality of the human eye[END_REF] 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 [START_REF] Raghavan | Sources of visual noise[END_REF]. 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 [START_REF] Allard | Crowding in a detection task: external noise triggers change in processing strategy[END_REF]Allard & Faubert, 2013, 2014a[START_REF] Allard | To characterize contrast detection, noise should be extended, not localized[END_REF][START_REF] Allard | Contrast sensitivity, healthy aging and noise[END_REF] 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 [START_REF] Allard | Crowding in a detection task: external noise triggers change in processing strategy[END_REF]. 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 [START_REF] Allard | Crowding in a detection task: external noise triggers change in processing strategy[END_REF]Allard & Faubert, 2013, 2014a[START_REF] Allard | To characterize contrast detection, noise should be extended, not localized[END_REF]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 Figure 1) [START_REF] Pelli | The effects of visual noise[END_REF]. 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 Figure 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 [START_REF] Pelli | The effects of visual noise[END_REF]:

𝑒 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑁 • 𝑤 • ℎ • 𝑑 (1)
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 [START_REF] Pelli | The effects of visual noise[END_REF][START_REF] Solomon | The visual filter mediating letter identification[END_REF][START_REF] Stromeyer | Spatial-frequency masking in vision: Critical bands and spread of masking[END_REF]. 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.

Binary noise

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 Figure 1). Given no correlation across samples, binary noise is white (flat energy spectrum) and its energy can also be defined by Equation (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 Figure 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 Figure 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 Figure 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 Figure 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.

Non-white noise

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 [START_REF] Pelli | The effects of visual noise[END_REF]. 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 Figure 1 in which each noise Binary / Gaussian noise check size is set to 4x4 pixels rather than 1x1 pixels). For the same variance, reducing the resolution of the display increases the noise energy (Equation 1) and reduces the upper frequency limit of the noise (black curve in top center graph in Figure 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 Figure 1. Because these apparent edges may have undesirable effects (e.g., [START_REF] Harmon | Masking in visual recognition: effects of two-dimensional filtered noise[END_REF], 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 Figure 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 [START_REF] Pelli | The effects of visual noise[END_REF][START_REF] Stromeyer | Spatial-frequency masking in vision: Critical bands and spread of masking[END_REF]. 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 Figure 1, respectively). As a result, at equal contrast ranges, filtered noise would have higher energy at the frequencies within the pass-band.

Binarized-filtered noise

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 Figure 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 Figure 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 Figure 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 Figure 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 Figure 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 Figure 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 Figure 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.

Experiment: Binarized-filtered noise vs Gaussian noise

Binarized-filtered noise requires less contrast than Gaussian noise to display the same expected energy at the frequencies relevant to the task so both noises are expected to have the same masking strength. The main aim of this experiment was to empirically verify this prediction. If the performance in binarized-filtered noise differs from the one in Gaussian noise (given equated energy levels at frequencies within the pass-band), then the binarizedfiltered noise cannot be considered equivalent to the Gaussian noise. On the other hand, if

Energy gain

Binarized-filtered / filtered noise binarized-filtered noise has the same expected noise energy at the frequencies relevant to the task as Gaussian noise and has the same masking strength, then binarized-filtered noise can be considered as equivalent to Gaussian noise.

Method

Observers

Four naïve observers and one of the authors participated in this study. They had normal or corrected-to-normal vision.

Apparatus

Stimuli were presented on a 22.5-inch LCD monitor designed for psychophysics (VIEWPixx)

with a refresh rate of 120 Hz. At the viewing distance of 1 m, the spatial resolution of the display was 64 pixels/degree of visual angle. The monitor was the only source of light in the room. The output intensity of each color gun was linearized psychophysically using a homemade program.

Stimuli and procedure

The detection task was implemented using a two-interval forced-choice procedure with an interstimulus interval of 500 msec. The noise was continuously displayed, refreshed at every frame and covered the entire screen. The signal was a 4 cycles-per-degree vertical grating presented in only one of the two 500-msec intervals. The spatial window of the signal had a diameter of 1 degrees plus a half-cosine soft edge of 0.25 degrees. A 500-msec sound was audible during each of the two intervals and the task consisted in determining if the target was present at the during the first or second sound by pressing one of two keys.

Two types of noise were used: filtered noise and binarized-filtered noise. For each noise type, the width of the ideal filter was 0.25, 0.5, 1, 2 or ∞ (i.e., unfiltered) octaves above and below the signal spatial frequency of 4 cycles-per-degree, resulting in 10 conditions (2 noise types X 5 filter widths, Figure 4). In all the conditions, the expected energy levels at the unfiltered frequency were equated to 0.22 µ degree 2 second. Contrast thresholds were measured using a 3down1up staircase procedure [START_REF] Levitt | Transformed up-down methods in psychoacoustics[END_REF] with step size of 0.1 log and was interrupted after 12 inversions. Threshold estimation of a staircase was set as the geometric mean of the last 8 inversions. The 10 conditions were blocked and performed in a pseudo-random order. Each staircase was performed 5 times so that each threshold was set as the geometric mean of the 5 staircases.

Results and discussion

The results shown in Figure 5 illustrate the two main outcomes. Most importantly, similar contrast thresholds were obtained for filtered noise and binarized-filtered noise suggesting that binarization had no significant impact on the masking strength (given equated noise energy at frequencies with the pass-band as illustrated in right graph in third row of Figure 1).

This was supported statistically with a two-way ANOVA (2 noise types X 5 filter bandwidths) that showed no simple main effect of noise type (F(1,16)=1.22, p=.332). Thus, binarizing the noise had no significant impact on performance even though it substantially reduced the noise contrast. This suggests that binarizing the noise can be used to reduce the contrast range of the noise without affecting performance. The second important outcome was that the filtering operation had no impact when the filter bandwidth was at least 2 octaves (Figure 5), that is, 1 octave above and below the signal spatial frequency, which is consistent with previous findings [START_REF] Pelli | The effects of visual noise[END_REF][START_REF] Stromeyer | Spatial-frequency masking in vision: Critical bands and spread of masking[END_REF]. Note that statistically, the two-way ANOVA showed a simple main effect of filter bandwidth (F(4,16)=46.4, p<.001), which can be explained by the lower contrast thresholds when many frequencies were filtering out (e.g., <2 octaves bandwidth filters).

Nevertheless, when measuring contrast detection threshold in noise, the same performance was observed whether the noise was white or had energy only 1 octave above and below the spatial frequency of the target. This suggests that, for a contrast detection task, removing the noise outside this frequency range reduces the contrast range of the noise without affecting performance. Taken together, these two outcomes suggest that binarized-filtered noise at ±1 octave around the signal frequency was equivalent to Gaussian noise. Given that the filtering and binarizing operations substantially reduced the noise contrast range without affecting performance (given equated noise energy at frequencies within the pass-band), increasing the contrast of binarized-filtered would display higher noise energy at the relevant frequencies.

Truncated-filtered noise

The advantage of binarized-filtered noise is that it uses a narrower contrast range than Gaussian noise for a given masking strength, which enables to display higher noise energy (by increasing noise contrast). A potential issue with this noise is that the use of only two luminance intensities introduces sharp edges (e.g., Figure 1, bottom right). Because the position of these edges randomly varies over time, they are less salient than the edges for the low-resolution noises presented above (e.g., see Movie 1 in supplementary material).

Although these edges could potentially have an undesirable effect, the experiment above rather suggests that these edges had a negligible impact for this contrast detection task.

Nevertheless, apparent edges could potentially cause binarized-filtered noise not to be equivalent to Gaussian noise. The present section shows that the visibility of these apparent edges can be substantially attenuated with the small cost of slightly decreasing noise energy (given equated noise contrast). Thus, even though the potential drawback of the binarization operation (i.e., introducing sharp edges) was empirically found to be negligible, the present section nevertheless shows that it can be substantially attenuated.

Above, to compare the energy of binary noise and Gaussian noise (Figure 2), the Gaussian noise was truncated at various truncation thresholds. Figure 2 shows that, for equated contrast ranges, the energy of truncated Gaussian noise approaches the one of binary noise as the truncation threshold approaches 0. This is because at an infinitely small truncation threshold, truncated Gaussian noise is equivalent to binary noise. Thus, by varying the truncation value and equating the noise energy at frequencies within the pass-band, truncated Gaussian noise gradually varies from binary noise (truncation threshold = 0) to Gaussian noise (truncation threshold = ∞).

The same rationale applies to truncated-filtered noise (Figure 3), which gradually varies from binarized-filtered noise (truncation threshold = 0) to filtered noise (truncation threshold = ∞).

Figure 6 shows truncated-filtered noise for different truncation thresholds (±0, 0.5, 1, 2 and ∞ sd). Note that the noise was truncated relative to the standard deviation of the noise after the filtering operation and not relative to the noise standard deviation of the initial Gaussian distribution used to generate the pre-filtered noise. Recall that the experiment above found the same performance whether binarized-filtered noise and filtered noise (i.e., truncated-filtered noises with a truncation threshold set to 0 and ∞, respectively). Given no performance difference between these two extremes, there is no reason to expect a different performance at any intermediate truncation threshold when the expected energy at frequencies within the pass-band is equated. Thus, using truncated-filtered noise could be a good compromise between binarized-filtered noise, which introduces undesired edges, and filtered noise, which extends over a large contrast range.

General discussion

By combining two methods of increasing noise energy given a limited contrast range (namely, filtered noise and binary noise), the present study developed a noise, namely truncated-filtered noise, equivalent to Gaussian noise with respect to a given task, but requiring less contrast to be displayed (see Appendix for detailed algorithm and Matlab code). Truncated-filtered noise was found to have the same masking strength as Gaussian noise when having the same expected energy at frequencies relevant to the task, which required less contrast. Thus, this new noise enables to display higher noise energy and thereby widen the range of conditions under which noise-masking paradigms can be effectively used. For instance, the amount of noise required to affect performance is known to gradually increase as luminance intensity is reduced [START_REF] Pelli | The effects of visual noise[END_REF]. To illustrate the usefulness of this method, we evaluated, for the same stimulus as in the experiment above, the lowest luminance intensity at which the maximal displayable noise could noticeably affect performance (i.e., increase detection threshold by a factor of at least 2). The lowest luminance intensities at which Gaussian, binary, filtered (±1 octave) and truncated-filtered (±1 octave, truncation=1 sd) noises could noticeably affect performance were found to be about 350, 39, 16 and 4 td, respectively. This simple example shows that truncated-filtered noise can be a useful tool to widen the range of conditions under which noise can be effectively be used. Note that maximizing noise energy could also be useful for other types of paradigms requiring strong masking, such as continuous flash suppression [START_REF] Tsuchiya | Continuous flash suppression reduces negative afterimages[END_REF].

Truncated-filtered noise varies, depending on the truncation threshold, along a continuum between binarized-filtered noise (truncation threshold = 0) and filtered noise (truncation threshold = ∞, i.e., no truncation). Empirically, results showed that when equating energy at the frequencies relevant to the task, the same performance was observed for the two extreme truncation thresholds (i.e., binarized-filtered noise and filtered noise). This suggests that truncating the noise at any truncation threshold (while equating noise energy at frequencies within the pass-band) is a useful way of reducing the contrast range of the noise without affecting its masking strength.

The advantage of a low truncation threshold is that it reduces the contrast range required to display a given energy level. An apparent drawback is that it artificially introduces edges as for binarized-filtered noise (i.e., truncation threshold = 0, Figure 1). Truncating the noise at ±1 sd, was found, in the example above (center image of Figure 6), to substantially reduce the appearance of the edges and reduce the energy (given equated total contrast range) by a factor of only 1.4 relative to binarized-filtered noise. We can also note that using a truncation threshold of ±2 sd compared to untruncated-filtered noise (fourth vs last image of Figure 6)

has little impact on the noise appearance and provides the advantage of having a noise that covers a smaller and well-defined contrast range. However, a truncation threshold at ±2 sd reduced energy (given equated total contrast range) by about 2.8 times relative to binarizedfiltered noise. Ultimately, choosing the truncation threshold depends on the experimental paradigm and its constraint (e.g., contrast available and potential issue of apparent edges), but the truncated-filtered noise at ±1 sd appears to be a good compromise, as it requires little additional contrast and substantially reduces the appearance of edges.

For truncated-filtered noise to be equivalent to Gaussian noise, they should have the same masking strength. The current study suggests that truncating (or even binarizing) filtered noise has no impact on performance given equated noise energy at frequencies within the pass-band (see Appendix for equating noise energy). Thus, truncated-filtered noise is equivalent to Gaussian noise, if the filtering operation has no impact on performance. In the current study, the noise was filtered according to the spatial frequency, but it could also be filtered along any other dimension such as temporal frequency or orientation. A priori, however, it is not possible to know what information can be filtered out without causing a observer, which may use additional information only available in truncated-filtered noise (e.g., detect a signal when the luminance of any pixel exceed, even by an infinitely small amount, the contrast range of the truncated noise). Thus, at first sight, using truncated-filtered noise seems to compromise the comparison with the ideal observer even if truncated-filtered noise is equivalent to Gaussian noise for the human observer. However, a simple way to prevent the ideal observer from using additional information only available in truncatedfiltered noise is to compare the human performance in truncated-filtered noise with the ideal performance in Gaussian noise. Indeed, given that the human performance in truncatedfiltered noise is equivalent to the one in Gaussian noise, the human performance in Gaussian noise can be estimated and compared with the ideal performance. Thus, the use of truncatedfiltered noise instead of Gaussian noise does not compromise the comparison with the ideal performance given that both noises are equivalent for human observers. Such a comparison can be performed by quantifying the noise energy of Gaussian and truncated-filtered noise at the unfiltered frequencies (see Appendix for a detailed algorithm and Matlab code).

The current method should not be confused with methods improving the contrast resolution of digital displays, such as the Noisy-bit [START_REF] Allard | The noisy-bit method for digital displays: Converting a 256 luminance resolution into a continuous resolution[END_REF] or bit-stealing [START_REF] Tyler | Colour bit-stealing to enhance the luminance resolution of digital displays on a single pixel basis[END_REF] methods. These methods aim at overcoming practical limitations of the digital display (smallest contrast displayable), whereas the current study rather deals with the theoretical limit of 100% contrast displayable. Thus, these two kinds of method address distinct displayable contrast limitations (smallest and highest displayable contrast) and can be used conjointly if needed.

In sum, the current study developed truncated-filtered noise by combining two methods to increase noise energy at frequencies relevant to the task without increasing noise contrast.

This novel noise enables to extend the use of noise-masking experiments over a wider range of conditions.

Calculating noise energy

The truncation is a nonlinear operation having a frequency-dependent impact on the noise energy: it uniformly reduces the noise energy of frequencies within the pass-band and introduces energy at frequencies that were filtered out (second to third graph in Figure 7). A simple and efficient way of quantifying the energy reduction at the frequencies within the pass-band is to compare the overall energy of truncated-filtered noise with filtered noise after removing the energy introduced by the truncation operation at the filtered frequencies. This can be obtained by re-filtering the truncated-filtered noise with the same ideal filter M (third to fourth graph in Figure 7) resulting in filtered-truncated-filtered noise (N FTFG ):

. ( 5)

Given the same expected energy distribution across frequencies for N FG and N FTFG besides a scaling factor (second and fourth graphs in Figure 7), the expected energy ratio at the frequencies within the pass-band between these two noises (e FG and e FTFG , respectively) are proportional to their variance ratio: . (8) Thus,

! !"!# ! !" = !"#$"%&' ! !"!# !"#$"%&' ! !" . (6)
! !"# ! ! = !"#$"%&' ! !"!# !"#$"%&' ! !" , (9)
and the energy at the frequencies within the pass-band of the truncated-filtered noise is equal to:

𝑒 !"# = 𝑒 ! • !"#$"%&' ! !"!# !"#$"%&' ! !" . ( 10
)
Given Equation 1 defining the energy of Gaussian noise (e G ), the energy at the frequencies within the pass-band of the truncated-filtered noise is equal to:

𝑒 !"# = 𝜎 ! • 𝑤 • ℎ • 𝑑 • !"#$"%&' ! !"!# !"#$"%&' ! !" , ( 11
)
where σ represents the standard deviation of the Gaussian distribution. Note that since energy is proportional to the noise variance (i.e., mean-squared contrast), it is trivial to equate the energy level of truncated-filtered noise at frequencies within the pass-band with the energy level of Gaussian noise (e G =e TFG ', as was done in the experiment above) by scaling its e FG = e G e FTFG = e TFG

Figure 1 .

 1 Figure 1. Different types of noise (first two top rows) and their corresponding spectral energy distributions (third row) and luminance distributions (bottom row). The first row presents samples of Gaussian noise (left), low-resolution Gaussian noise (center) and filtered noise (right). The second row presents the binarized noises of the first row, namely, binary noise, low-resolution binary noise and binarized-filtered noise, respectively. The energy levels at the relevant frequency were equated across all 6 noises by scaling the noise contrast. See Movie 1 in supplementary material for dynamic noise. The third row represents spectral energy density of the different types of noise. The black solid lines represent Gaussian noise (corresponding images in first row), the grey dashed lines represent the binarized noise (corresponding images in second row) and the arrow represents the relevant (e.g., signal) frequency. The

Figure 2 .

 2 Figure 2. Energy gain of using binary noise instead of truncated Gaussian noise as a function of the truncation threshold given equated total contrast ranges.

Figure 3 .

 3 Figure 3. Energy gain at frequencies within the pass-band when using binarized-filtered noise instead of truncated-filtered noise as a function of the truncation threshold given equated total contrast ranges. The filter was 2-octave wide in the spatial frequency domain.

Figure 4 .

 4 Figure 4. The top left represents Gaussian white noise. The 2 nd to the last column of the top row represents the same noise sample after applying a filter with a width of 4, 2, 1 and 0.5 octaves, respectively, centered on a given spatial frequency (here 16 cycles per image). Each noise at the bottom row represents the same noise sample as its corresponding top row, except that it was binarized and its energy at the frequencies within the pass-band was equated. See Movie 2 in supplementary material to for dynamic noise.

Figure 5 .

 5 Figure 5. Contrast detection threshold as a function of filter bandwidth centered on the spatial frequency of the signal obtained in filtered (circles) and binarized-filtered (squares) noises. Error bars represent standard error of the mean.

Figure 6 .

 6 Figure 6. Truncated-filtered noise with truncation threshold set to ±0, 0.5, 1, 2 and ∞ standard deviations (left to right). The expected noise energies at the relevant frequencies were equated. See Movie 3 in supplementary material for dynamic noise.

Figure 7 .

 7 Figure 7. Evolution of spectral energy distribution of Gaussian noise through various operations: original Gaussian noise (first graph), after applying an ideal filter (second graph), after truncating (third graph) and after re-applying the same ideal filter (right graph).

  Since the filtering operation has no impact on the energy at the frequencies within the passband, the expected energy at the frequencies within the pass-band of filtered noise (e FG , second graph in Figure7) is equal to the expected energy at the same frequencies of Gaussian noise (e G , first graph)energy at the frequencies within the pass-band of filtered-truncated-filtered noise (e FTFG , fourth graph) is equal to the expected energy at the same frequencies of the truncated-filtered noise (e TFG , third graph):

Acknowledgments

Thanks to Daphné Silvestre for helpful comments. This research was supported by ANR -Essilor SilverSight Chair.

noticeable change in performance as it depends on which information is relevant to the visual system for the given task. The processing involved in a typical contrast detection task, for instance, is known to be narrowly tuned in the spatial frequency domain, which explains why a narrow filter (e.g., ±1 octave of the signal spatial frequency) has no effect on performance as observed here and elsewhere [START_REF] Pelli | The effects of visual noise[END_REF][START_REF] Stromeyer | Spatial-frequency masking in vision: Critical bands and spread of masking[END_REF]. But for filtering along other dimensions, the filter that can be used without affecting performance depends on the tuning of the processing for the given task and cannot be known a priori. In sum, the filter that can be used to avoid reducing masking strength cannot be known a priori as it depends on the processing properties relevant to the given task. Obviously, truncated-filtered noise suffers from the same limitations as filtered noise. Nevertheless, once a filter is chosen (and assumed or shown not to have any impact on performance when applied to Gaussian noise), the current study suggests that the noise can be truncated in order to further increase energy and thereby extend the range of conditions over which noise-masking paradigms can be useful. See the Appendix for Matlab code to generate truncated-filtered noise filtered along the spatial frequency, orientation or/and temporal frequency dimensions.

A common use of external noise is to quantify the performance of human observers relative to the performance of an ideal observer [START_REF] Gold | Ideal Observers and Efficiency: Commemorating 50 Years of Tanner and Birdsall: Introduction[END_REF][START_REF] Kersten | Ideal Observer Theory[END_REF]). An ideal observer often has a perfect performance in noiseless condition (e.g., an infinitely small contrast detection threshold), but under noisy conditions, the optimal performance is limited even when using optimally all the available information. If the human performance is close to the ideal performance (e.g., [START_REF] Allard | Different processing strategies underlie voluntary averaging in low and high noise[END_REF]Allard & Faubert, 2013;[START_REF] Baldwin | What Do Contrast Threshold Equivalent Noise Studies Actually Measure? Noise vs. Nonlinearity in Different Masking Paradigms[END_REF], then this means that the observer efficiently integrates all the necessary information to perform the task. Otherwise, some information must be lost, deteriorated or not optimally integrated. Although truncated-filtered noise may be equivalent to Gaussian noise for a human observer, they may not be equivalent for an ideal

Appendix: Generating truncated-filtered noise

To facilitate the use of truncated-filtered noise, this appendix provides a detailed algorithm and Matlab code for generating the noise and calculate its energy level.

Algorithm

Truncated-filtered noise is the result of applying two successive operations on Gaussian noise: filtering and then truncating. The filtering operation can be represented as:

(2) where the filtered noise N FG results from the application of the ideal band-pass filter M in the Fourier domain to the Gaussian white noise N G (i.e., samples drawn from a Gaussian distribution centered on 0). The filtering operation leaves intact the energy of a range of frequencies and sets to 0 the others (first to second graph in Figure 7). The truncated-filtered noise (N TFG ) results from a truncation implemented independently on each sample i of the filtered noise N FG :

(3) where t represents the truncation threshold in absolute units. Note that the truncation threshold can be converted from standard deviation units (i.e., relative to the standard deviation of the filtered noise N FG , as expressed in the current study) to absolute units by multiplying the relative truncation threshold in standard deviation units (t SD ) by the root-mean-square of the filtered noise:

(4)

( )

contrast by the root-mean-squared contrast ratio between filtered noise and filtered-truncatedfiltered noise:

. ( 12)

Matlab code implementation

To facilitate the creation of truncated-filtered noise, this section provides programing code implementing the algorithm above. This code is also available in a Matlab file in supplementary material. The novelty of the method consists in combining two successive steps: filtering and then truncating. These two simple steps are combined in function computeTruncatedFilteredNoise, which takes two input parameters: an ideal filtering mask to apply in the Fourier domain (Mask) and a truncation threshold (trunc). Since the filtering mask should be the same size as the noise it filters, the size of the mask matrix (including its number of dimensions) determines the size of the noise to generate. After the filtering process, the noise is truncated based on the input parameter trunc, which represents the truncation value in standard deviation of the filtered noise (as in Figure 6). For binarizedfiltered noise, trunc can be set to 0. This function outputs the truncated-filtered noise (NoiseTFG) normalized between -1 and 1 and its energy at unfiltered frequencies (energyTFG in pixels 2 frames). Note that energyTFG is in energy units so scaling the output noise

NoiseTFG by a contrast gain (g) would affect its energy by the square of this gain (g 2 ).

function [NoiseTFG, energyTFG] = computeTruncatedFilteredNoise( Mask, trunc ) % Generates turncated-fitlered noise from an ideal mask % % INPUTS: % Mask: ideal matrix filter (0s and 1s) to apply in the Fourier domain % trunc: truncation threshold in SD % % OUTPUTS: % NoiseTFG: truncated-filtered noise matrix the same size as Mask % normalized between -1 and 1 % energyTFG: energy of truncated-filtered noise in pixels^2 frames %%%%%%%% Gaussian noise (energy=1 pixel^2 x frame) NoiseG = randn(size(Mask)); This function has the advantage of generating truncated-filtered noise for any arbitrary ideal filter (Mask). The specific choice of the ideal filter depends on the experimental paradigm. In the experiment above, the filtering was along the spatial frequency dimension, but it is also common to filter noise along the temporal frequency and orientation dimensions. The following function (computeMask) creates an ideal mask (Mask) filtering information along all or any of these dimensions. This ideal mask of nbPixels X nbPixels X nbFrames elements keeps only the information within pass-band ranges of spatial frequency, orientation and temporal frequency that are specified by the input parameters cutSpatFreq (in cycles per nbPixels), cutOrien (in degrees) and cutTempFreq (in cycles per nbFrames), which are vectors containing two values representing the lower and upper filtering cutoffs. The ideal mask (Mask) generated by the function computeMask can be used as an input parameter of the function computeTruncatedFilteredNoise to generate truncated-filtered noise. The following function (truncatedFilteredNoise) combines these two functions and adds input parameters (pixelsPerSpatUnit and framesPerTempUnit) enabling to convert the units of the cutoff frequency parameters (cutSpatFreq and cutTempFreq). pixelsPerSpatUnit specifies how many noise pixels are to be displayed within a given arbitrary spatial unit (e.g., degrees of visual angle) and cutSpatFreq represents the cutoff spatial frequencies in cycles per spatial units (e.g., cycles per degree of visual angle). Equivalently, framesPerTempUnit specifies how many noise frames are to be displayed within a given arbitrary temporal unit (e.g., seconds) and cutTempFreq represents the cutoff temporal frequencies in cycles per temporal units (e.g., cycles per second). The unit conversion parameters also define the units of the noise energy given by the output parameter energyTFG (e.g., in degrees 2 seconds, when pixelsPerSpatUnit represents the number of pixels per degree of visual angle and framesPerTempUnit represents the number of frames per second). For static noise, framesPerTempUnit can be omitted or set to 1, in which case the units of the energy only represent the two spatial dimensions (e.g., degree 2 ). This truncatedFilteredNoise function provides a simple way of generating truncated-filtered noise. For instance, to generate 240 noise frames of 1080x1080 pixels that are to be refreshed at 120 Hz with 64 pixels per degree of visual angle, calling the following function: would created noise (N) filtered only along the spatial frequency dimension (2 to 8 cycles per degree of visual angle), truncated at 1 sd, normalized between -1 and 1, and of energy e degrees 2 seconds.


Elmer V H Doggen

Bertrand Georgeot

Gabriel Lemarié

Chaos-assisted tunneling in the presence of Anderson localization

Tunneling between two classically disconnected regular regions can be strongly affected by the presence of a chaotic sea in between. This phenomenon, known as chaos-assisted tunneling, gives rise to large fluctuations of the tunneling rate. Here we study chaos-assisted tunneling in the presence of Anderson localization effects in the chaotic sea. Our results show that the standard tunneling rate distribution is strongly modified by localization, going from the known Cauchy distribution in the ergodic regime to a log-normal distribution in the strongly localized case. We develop an analytical single-parameter scaling theory which accurately describes the numerical data, for both a deterministic and a disordered model. Several possible experimental implementations using cold atoms, photonic lattices or microwave billiards are discussed.

Tunneling has been known since early quantum mechanics as a striking example of a purely quantum effect that is classically forbidden. However, the simplest presentation based on tunneling through a one-dimensional barrier does not readily generalize to more generic situations. Indeed, in dimension two or higher or in timedependent systems, the dynamics becomes more complex with various degrees of chaos, and the tunneling effect can become markedly different [1]. An especially spectacular effect of chaos in this context is known as chaos-assisted tunneling [2]: in this case, tunneling is mediated by ergodic states in a chaotic sea, and tunneling amplitudes have reproducible fluctuations by orders of magnitude over small changes of a parameter. This is reminiscent of universal conductance fluctuations which arise in condensed matter disordered systems. There, reproducible fluctuations of the conductance when e.g. a magnetic field is varied are an interferential signature of the disorder configuration [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF].

A typical example of chaos-assisted tunneling arises for systems having a mixed classical dynamics where regular zones with stable trajectories coexist with chaotic regions with ergodic trajectories. In the presence of a discrete symmetry, one can have two symmetric regular zones which are classically disconnected in the sense that classical transport between these two zones is forbidden. In the quantum regime however, transport between these two structures is possible through what is called dynamical tunneling [START_REF] Davis | [END_REF]. Regular eigenstates are regrouped in pairs of symmetric and antisymmetric states whose eigenenergies differ by a splitting δ inversely proportional to the characteristic tunneling time. If a chaotic region is present between the two regular structures, tunneling becomes strongly dependent on the specificities of the energy and phase-space distributions of the chaotic states. It results in strong fluctuations of the tunneling splittings which is known to be well-described by a Cauchy distribution [5]. Chaos-assisted tunneling has been extensively studied in different contexts, both theoretically [6][7][8][9][10][11][START_REF] Keshavamurthy | Dynamical tunneling: theory and experiment[END_REF] and experimentally [START_REF] Dembowski | [END_REF][14][15][16][17][18][19][20][21].

However, states in a chaotic sea are not necessarily ergodic; indeed, it is well-known that in condensed matter, quantum interference effects can stop classical diffusion and induce Anderson localization [22][23][START_REF] Abrahams | years of Anderson Localization[END_REF]. In chaotic systems, a similar effect known as dynamical localization can take place where states are exponentially localized with a characteristic localization length ξ [START_REF] Casati | Stochastic behavior in classical and quantum Hamiltonian systems[END_REF][START_REF] Grempel | [END_REF][27][28].

When the size L of the chaotic sea is much smaller than ξ, chaotic states are effectively ergodic, whereas a new regime arises when ξ ≪ L where strong localization effects should change the standard picture of chaos-assisted tunneling. Indeed, it is well known for disordered systems that in quasi-1D universal Gaussian conductance fluctuations are replaced by much larger fluctuations in the localized regime with a specific characteristic log-normal distribution [29]. Moreover, for chaotic systems, it was shown in [30] that tunneling is drastically suppressed by dynamical localization, and that the Anderson transition between dynamical localization and diffusive transport manifests itself as a sharp enhancement of the average tunneling rate at the transition [31].

In this paper, we study the hitherto unexplored effect of Anderson localization on the statistics of chaosassisted tunneling in quasi-1D systems. We show that the distribution of level splittings changes from the Cauchy distribution characteristic of the ergodic regime to a lognormal distribution in the strongly localized regime. We consider two models to study these effects: a deterministic model having a mixed phase space and whose quantum chaotic states display dynamical localization, and a disordered model based on the famous Anderson model [22]. This allows us to study numerically the two extreme ergodic and localized regimes as well as the full crossover between them. The numerical data are found to follow a one-parameter scaling law with ξ/L. We present a simple analytical theory which accounts for the observed behaviors throughout the full range of parameters studied for both models. This shows that the fluctuations of chaos-assisted tunneling precisely characterize the different physical regimes of transport in these systems.

Models.-The deterministic model that we use is a variant introduced in [30] of the quantum kicked rotor. 6) with Γ and δtyp fitting parameters (see Fig. 2b). In the localized (ergodic) regime the log-normal (Cauchy) distributions overlap almost perfectly with Eq. ( 6). The data correspond to K = 30, pr = 0.4n (n = 2048) and 6400 values of h in a small range around h = 0.85, 0. The Hamiltonian is given by:

Ĥ = T (p) + K cos(x) ∞ n=-∞ δ(t -n), (1)
where ph is momentum, x is a dimensionless position (or phase) with period 2π, t is time and K is the strength of the kicks. The dispersion relation T (p) is chosen such that the phase space is divided into well-separated chaotic and regular (integrable) regions (cf. Ref. [30]):

T (p) = (hp) 2 2 for |p| ≤ p r /2, (2a)
T (p) = ω|p| + ω 0 for |p| > p r /2. ( 2b
)
The value of ω should be irrational, throughout this work we use ω = 2 √ 5 ≈ 4.4721 . . . . ω 0 is chosen such that T (p) is continuous. A discrete symmetry p → -p exists; the region of phase space |p| ≤ p r /2 is strongly chaotic for K ≫ 1. The characteristic size of the chaotic sea is L = p r + K/[h sin(ω/2)] (see Fig. 1b). The two regions |p| > L/2 correspond to two momentum symmetric regular zones. In the following, we will consider an initial state located on a classical torus in the regular part of phase space as obtained using Einstein-Brillouin-Keller quantization (see [30]), with initial mean momentum p (t = 0) = p c + L/2. The distance to the chaotic sea p c is set to p c = 10 such that the coupling to the chaotic sea is constant. In Fig. 1b we portray this initial state superimposed on the classical phase space representation of (2).

The second model we use is a disordered system based on the Anderson model:

ĤA = i =1,L w i a † i a i + i,j a † i a j + H.c. + t c (a † 1 a 2 + a † L a L-1 + H.c.) (3)
where the summation i, j is over indices 1 ≤ |i -j| ≤ 3 (i, j = 1, L), L is the lattice size and a ( †) are annihilation (creation) operators. The on-site energies w i are independent random numbers, uniformly distributed ∈ [-W/2, W/2], with the important constraint that the w i 's are symmetric with respect to the center of the lattice (creating the analog of the discrete symmetry p → -p for model ( 1)). The coefficient t c ≪ 1 represents a weak coupling of edge sites to the disordered chain. Tunneling beyond the nearest neighbor |i -j| > 1 is required to reach the effective quasi-one-dimensional regime, as the Anderson model with only nearest-neighbor hopping permits only localized and ballistic behavior. can identify two extreme regimes in these systems, characterized by the ratio of the localization length ξ to the size of the chaotic/disordered sea L : i) ξ ≫ L: the ergodic regime and ii) ξ ≪ L: the localized regime. We do not consider the ballistic regime where L much smaller than the mean free path.

Splitting statistics for the deterministic model.-

The dynamics of the model ( 1) can be integrated over one period to give the evolution operator Û = e -i 2h T (p) e -i h K cos(x) e -i 2h T (p) , where we have chosen a symmetric version of the map. Tunneling may be studied through the properties of the Floquet eigenstates |Ψ α of Û with associated quasi-energies ε α obeying Û |Ψ α = e iεα |Ψ α . Classically, transport to the chaotic sea or the other regular island is forbidden by the presence of invariant curves. However, in the quantum regime, the initial state, having a certain expectation value of momentum p (t = 0) = p c + L/2, will tunnel trough the chaotic sea to the other side characterized by an oscillation period T osc , so that p (T osc /2) = -p (t = 0). T osc can be identified with 2π/δ, where δ = ε S -ε A is the splitting between the symmetric and anti-symmetric Floquet eigenstates having the largest overlap with the initial state.

In Fig. 1a we show the distributions of ln δ in the different regimes of the chaotic sea: localized ξ ≪ L, ergodic ξ ≫ L and intermediate ξ ≈ L. One can see that the distributions are markedly different: we recover the known Cauchy distribution in the ergodic regime whereas a log-normal distribution is observed in the localized regime. In the crossover regime, an intermediate distribution is obtained with a non-trivial shape to be discussed later. In order to reveal which scales control these behaviors, we considered the splittings δ for many different parameters (see caption of Fig. 1). In Fig. 1c we show that, strikingly, the data follow a single parameter scaling law as a function of ξ/L, where the localization length obeys the known expression [32]

ξ = (K 2 /4h 2 )[1 -2J 2 ( K)(1 -J 2 ( K))] (J 2 denotes the Bessel function) with K = (2K/h) sin h/2.
Splitting statistics for the disordered model. -The disordered model ( 3) can be used to describe chaos-assisted tunneling, with the localization length in position space, dependent on W , analogous to the localization length in momentum space of the deterministic model ( 1), dependent on K and h. The splitting δ is determined by computing the eigenfunctions most strongly overlapping with the first site (due to symmetry, this is equivalent to the last site). The corresponding splitting distributions for this model are shown in Fig. 2a. Strikingly, very similar behavior is observed in this disordered model compared to the deterministic model. In particular, we recover the Cauchy distribution at small values of W where the disordered sea has delocalized ergodic states, whereas for large W (localized states) the distribution has a log-normal shape, with again an intermediate behavior at the crossover. In Fig. 2b we show, for both the deterministic and disordered models, the scaling behaviors of δ typ = exp(ln δ), where ln δ denotes the ensemble average of ln δ, and the fitted parameter Γ related to the width of the distribution (see Eq. ( 6) below). Here we approximate ξ in the disordered model (3) according to ξ ≈ N ξ 1D , where N = 5 is the number of channels (related to the number of neighbors to which particles can hop), and the localization length for a 1D Anderson chain obeys ξ -1 1D = ln(1+W 2 /16)/2+4 arctan(W/4)/W -1 [33]. Analytical theory.-We can derive an expression for the splitting distribution valid in all regimes, including the crossover regime, for both the disordered model ( 3) and the deterministic model (1). Following [5], the splitting between the symmetric and antisymmetric regular states can be obtained from the displacements δ S,A of their quasi-energies due to the coupling to the chaotic/disordered sea. Because this coupling v is classiforbidden, it is exponentially small and a perturbation theory can be used to obtain: δ S,A ≈ v 2 /(E -E S,A ch ), with E S,A ch the energy of the near resonant chaotic state of the same symmetry of the regular state of energy E [34]. The splitting δ is then given by δ = δ S -δ A . In the delocalized ergodic regime, the overlap between chaotic states is large which excludes that symmetric states are resonant concomitantly with antisymmetric states, thus δ S and δ A are uncorrelated. The splitting distribution is then given by the distribution of δ S,A , and follows a Cauchy law [5].

In the localized regime, symmetric and antisymmetric chaotic states have exponentially small overlap, therefore δ S and δ A are strongly correlated. This correlation is at the origin of the departure of the splitting distribution from the Cauchy law (see [2,5] for similar correlations arising due to partial classical barriers in the chaotic sea). Defining the auxiliary quantities E S ch = E loc -δ loc /2 and E A ch = E loc +δ loc /2, with δ loc the splitting of the localized chaotic states, |δ loc | ≪ |E -E loc | in the localized regime, which yields:

δ = |δ S -δ A | ≈ v 2 E -E loc δ loc E -E loc . (4)
In the first factor, v = v ch exp(-2L reg /ξ loc ) where v ch is the usual coupling strength to the chaotic sea in the case where it has ergodic states. exp(-2L reg /ξ loc ) describes the effect of localization of the chaotic state, with localization length ξ loc , located at a distance L reg from the regular border. In the second factor of (4), δ loc = ∆ exp(-2L loc /ξ loc ) where L loc is the distance between the two peaks of the localized (anti)symmetric chaotic state whereas E -E loc ≈ ∆ the mean level spacing. As L = 2L reg + L loc , the splitting can be approximated by the simple expression:

δ = δ ch exp(-2L/ξ loc ) (5)
with δ ch the "standard" chaos-assisted tunneling splitting. The first term of (5) has the known Cauchy distribution P ch (ln δ ch ) = 2e ln δch /[π(1 + e 2 ln δch )], where δ ≡ δ/δ typ . In our case δ ch,typ = exp ln δ ch depends only on t c or p c (see also [11]). Because the inverse localization length 1/ξ loc in quasi-1D has a normal distribution with width ∝ 1/L, the second term δloc ≡ exp(-2L/ξ loc ) of ( 5) has a log-normal distribution, analogous to that of conductance in the strongly localized limit [29,35]:

P loc (ln δloc ) = (1/4Γ √ 2π) exp[-(ln 1/ δloc -2Γ) 2 /8Γ],
where Γ = L/ξ, ξ = exp ln ξ loc denoting the typical localization length. The total distribution P (ln δ) can be obtained by convolution and is given by:

P (ln δ) = 1 4 exp -ln δ + 2Γ 1 + erf -4Γ + ln δ √ 8Γ + exp(2 ln δ)erfc 4Γ + ln δ √ 8Γ . (6)
The expression (6) can be fitted to the distributions obtained numerically (see Fig. 1a and Fig. 2a). A very good agreement is found with the numerical data, both in the two extreme localized and ergodic regimes where (6) describes a log-normal or Cauchy distribution, respectively. The intermediate regime (ξ/L ≈ 1) is characterized by log-normal behavior in the center of the distribution, around δ = δ typ , with Cauchy-type behavior in the tails. The fitted values of δ typ and Γ are represented in Fig. 2b for both models. A good agreement is found with the expected behavior δ typ = δ ch,typ exp(-2L/ξ) and Γ = L/ξ in the localized regime. Experimental implementation.-We have shown that the effects we have presented are observable for both deterministic chaotic or disordered systems, provided they are in the localized regime. It is possible to implement a realistic version of the model (1) using a variant of the well-known cold-atom implementation of the quantum kicked rotor [27,28,36]. The dispersion relation (2) cannot be implemented directly. However, in the atomic kicked rotor, the kicking potential is usually realized through a sequence of short impulses of a stationary wave periodic in time. By truncating the Fourier series of this sequence at some specified harmonic p r , one realizes a mixed system with a classically ergodic chaotic sea between -p r and p r [37] which realizes an experimentally accessible version of (1) (see [38] for experimental details on an analogous system). The disordered model (3) can be implemented readily using photonic lattices [39][40][41]. A spatially symmetric disorder can be implemented in the direction transverse to propagation, and tunneling will result in oscillations in this transverse direction which can be easily measured. Lastly, both chaos-assisted tunneling [START_REF] Dembowski | [END_REF][14][15][16][17] and Anderson localization [42,43] have been observed in microwave chaotic billiards. In this context, both disordered and deterministic models could be implemented and can yield very precise distributions of the tunneling rate.

Conclusion.-We have shown that the presence of localization in the chaotic sea brings a new regime to chaos-assisted tunneling, manifested by a distinctive distribution of tunneling rates. Remarkably, the crossover from the ergodic to the localized regimes is governed by a single-parameter scaling law. The simple analytical theory presented describes accurately the different behaviors for both disordered and deterministic models, both of which could be implemented experimentally.

Our results show that chaos assisted tunneling could be used as a probe of the non-ergodic character of the chaotic sea. It would be interesting to generalize these ideas in higher dimensional systems where localization properties can be markedly different, with e.g. multifractal states at the Anderson transition. knowledge the use of the following software tools and numerical packages: LAPACK [START_REF] Anderson | LAPACK Users' Guide[END_REF], the Fastest Fourier Transform in the West [START_REF] Frigo | Proc. IEEE[END_REF] and Matplotlib [START_REF] Hunter | [END_REF].

FIG. 1 .

 1 FIG. 1. (Color online). (a) Splitting distribution P(ln δ) for the deterministic model (1) in the localized (left), intermediate (middle) and ergodic (right) regimes. Green dashed line: fit with a log-normal distribution. Blue dashed-dotted line: fit by the Cauchy distribution. Red solid line: analytical theory Eq. (6) with Γ and δtyp fitting parameters (see Fig.2b). In the localized (ergodic) regime the log-normal (Cauchy) distributions overlap almost perfectly with Eq. (6). The data correspond to K = 30, pr = 0.4n (n = 2048) and 6400 values of h in a small range around h = 0.85, 0.45, 0.25 respectively. (b) Phase-space (Husimi) distribution of the initial wave function (color shades) superimposed on the classical phase space of the deterministic model (1), with h = 0.25, K = 30, pc = 10 and pr = 1024. (c) Splittings δ as a function of ξ/L showing single-parameter scaling behavior. There are 16 sets (different colors) corresponding to fixed values of K = 20, 30, 40, 50 and pr varies from 0.1n to 0.7n (n = 2048 is the system size) for 1600 values of h ∈ [0.1, 1.5]. Big red dots: typical value of δ averaged over all data sets.

  FIG. 1. (Color online). (a) Splitting distribution P(ln δ) for the deterministic model (1) in the localized (left), intermediate (middle) and ergodic (right) regimes. Green dashed line: fit with a log-normal distribution. Blue dashed-dotted line: fit by the Cauchy distribution. Red solid line: analytical theory Eq. (6) with Γ and δtyp fitting parameters (see Fig.2b). In the localized (ergodic) regime the log-normal (Cauchy) distributions overlap almost perfectly with Eq. (6). The data correspond to K = 30, pr = 0.4n (n = 2048) and 6400 values of h in a small range around h = 0.85, 0.45, 0.25 respectively. (b) Phase-space (Husimi) distribution of the initial wave function (color shades) superimposed on the classical phase space of the deterministic model (1), with h = 0.25, K = 30, pc = 10 and pr = 1024. (c) Splittings δ as a function of ξ/L showing single-parameter scaling behavior. There are 16 sets (different colors) corresponding to fixed values of K = 20, 30, 40, 50 and pr varies from 0.1n to 0.7n (n = 2048 is the system size) for 1600 values of h ∈ [0.1, 1.5]. Big red dots: typical value of δ averaged over all data sets.

FIG. 2 .

 2 FIG. 2. (Color online). (a) Splitting distribution P(ln δ) for the disordered model (3), for W = 3, 1, 1/2 from left to right. Note the strong similarity with the distributions for the deterministic model (1) of Fig. 1a. Colored lines: as in Fig. 1. L = 256, tc = 0.001, N = 5 channels, and 10000 realizations (see text). (b) Scaling behavior of δtyp as a function of ξ/L. Disordered model (3): L = 128 (blue triangles), L = 256 (magenta squares) and L = 512 (green stars), parameters as in (a). Black circles: deterministic model (1), K = 30, 6400 values of h in a small range around 0.25, 0.45, 0.65, 0.85, 1.45, pr = 0.4n (n = 2048). Red solid line: theoretical prediction δtyp = δ ch,typ exp(-2L/ξ), with δ ch,typ fitted to L = 512 data. (c) Scaling behavior of Γ (see Eq. (6)) as a function of ξ/L (symbols defined as in the left panel). Red solid line: theoretical prediction Γ = L/ξ.

We thank CALMIP for providing computational resources.

This work was supported by Programme Investissements d'Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT, by the ANR grant K-BEC No ANR-13-BS04-0001-01 and the ANR program BOLODISS. Furthermore, we ac-


A Kritikakou

T Marty

C Pagetti

C Rochange

M Lauer

M Roy

Multiplexing Adaptive with Classic AUTOSAR? Adaptive Software Control to Increase Resource Utilization in Mixed-Critical Systems

Keywords: Dynamic management, Real-time systems, Run-time monitoring, Multi/Many-core systems, Safety, Resources Utilization

Automotive embedded systems need to cope with antagonist requirements: on the one hand, the users and market pressure push car manufacturers to integrate more and more services that go far beyond the control of the car itself. On the other hand, recent standardization efforts in the safety domain has led to the development of the ISO 26262 norm that defines means and requirements to ensure the safe operation of automotive embedded systems. In particular, it led to the definition of ASIL (Automotive Safety and Integrity Levels), i.e., it formally defines several criticality levels. Handling the increased complexity of new services makes new architectures, such as multi or many-cores, appealing choices for the car industry. Yet, these architectures provide a very low level of timing predictability due to shared resources, which goes in contradiction with timing guarantees required by ISO 26262.

For highest criticality level tasks, Worst-Case Execution Time analysis (WCET) is required to guarantee that timing constraints are respected. The WCET analyzers consider the worst-case scenario: whenever a critical task accesses a shared resource in a multi/many-core platform, a WCET analyzer considers that all cores use the same resource concurrently.

To improve the system performance, we proposed in a earlier work an approach where a critical task can be run in parallel with less critical tasks, as long as the real-time constraints are met. When no further interferences can be tolerated, the proposed runtime control suspends the low critical tasks until the termination of the critical task. In an automotive context, the approach can be translated as a highly critical partition, namely a classic AUTOSAR one, that runs on one dedicated core, with several cores running less critical Adaptive AUTOSAR application(s). We briefly describe the design of our proven-correct approach. Our strategy is based on a graph grammar to formally model the critical task as a set of control flow graphs on which a safe partial WCET analysis is applied and used at run-time to control the safe execution of the critical task.

I. INTRODUCTION

The chip market moved to multi/many-core systems due to increased system requirements and power dissipation issues of single-core systems [START_REF] Singh | Mapping on multi/many-core systems: Survey of current and emerging trends[END_REF]. As these systems offer massive computing power, a higher integration of applications is performed in the same platform. The integrated applications have diverse characteristics which create mixed-critical systems [START_REF] Vestal | Preemptive scheduling of multi-criticality systems with varying degrees of execution time assurance[END_REF]. A mixed-critical system consists of applications with different levels of criticality, as described in the ISO 26262 norm, and envisioned in the future Adaptive AUTOSAR platform. The criticality level of an application depends on the potential consequences on the system in case the application fails to meet its timing constraints. The Automotive Safety Integrity Level (ASIL) model [START_REF]ISO26262 Road Vehicles -Functional Safety[END_REF] defines criticality levels depending on the risk and the potential loss due to a failure of the system; the highest level, D, corresponds to the highest criticality level, and 4 other levels have been defined -C, B, A, and QM , where QM corresponds to a level where no particular safety measure should be taken.

Applications with high criticality level require strict guarantees on their correct execution. To ensure these guarantees, real-time task scheduling techniques should use a safe estimation of the Worst-Case Execution Time (WCET) [START_REF] Gatti | Development and certification of avionics platforms on multicore processors[END_REF]. Several WCET estimation techniques exist, but static analysis tools such as AIT or OTAWA, are recommended for high criticality applications. Unfortunately, many/multi-core systems have a dynamic difficult-to-predict behavior. More precisely, the concurrent accesses to the shared resources introduce timing variations, e.g. in the communication network and in the memory hierarchy with variable delays under concurrent requests. Therefore, the effects of possible task interferences have to be upper bounded to guarantee real-time response, usually by assuming full contention under concurrent requests. The result is a safe but pessimistic WCET, in the sense that the cases leading to the worst-case scenarios are unlikely to occur. This leads to over-allocating resources to high criticality applications and may even be the cause of the system unschedulability.

A. Motivation

Let us consider n + 1 independent synchronous partitions T = {τ C , τ 1 , . . . , τ n } where τ C is a classic AUTOSAR partition of high criticality level (ASIL D), period T C and deadline D C ; τ i are partitions of lower criticality level (ASIL levels A -C or QM ) implemented as Adaptive AUTOSAR partitions. Each partition is executed on a different core. Hence, the classic AUTOSAR partition runs on a dedicated core, but may suffer from interference with other partitions.

Two scenarios for the WCET computation of τ C are considered: 1) maximum load (max): all partitions run in parallel, and 2) isolation (iso): the classic AUTOSAR partition τ C runs alone on the system. In max scenario, we assume that due to

τ C Core0 Core1..n τ C D C a) b) τ C c) Core0 Core1..n Core0 Core1..n τ κ τ 2 τ 1 T C Switch τ κ τ 2 τ 1 τ κ τ 2 τ 1 τ κ τ 2 t
Fig. 1. Mixed-critical schedules scenarios the resource sharing, the WCET of τ C is above the deadline, i.e. WCET max > D C , as depicted in Fig. 1(a). In this case, the hard real-time constraints cannot be met. Existing mixedcritical scheduling approaches, such as [START_REF] Anderson | Multicore operating-system support for mixed criticality[END_REF], [START_REF] Mollison | Mixed-criticality real-time scheduling for multicore systems[END_REF], [START_REF] Li | Global mixed-criticality scheduling on multiprocessors[END_REF], assume that the task set is schedulable at least in the highest criticality level, and thus are not directly applicable. Then, a safe solution is to execute τ C in isolation. When the classic AUTOSAR partition terminates and if time slack exists, the Adaptive AUTOSAR partitions are executed. In this case, no conflict occurs with the low criticality tasks. Hence, the WCET is significantly lower and the classic AUTOSAR partition respects its deadline, i.e.

WCET iso ≤ D C , as shown in Fig. 1(b).

Our goal is to increase the task parallelism and to reduce the over-provisioning of resources by combining the benefits and discarding the drawbacks of the previous cases. To achieve that, all AUTOSAR partitions are started and are run in parallel, as long as it is safe. At run-time, if the interferences may lead to a deadline miss of τ C , the lower criticality partitions are suspended until the termination of τ C . If time slack exists, the low criticality partitions are resumed. In this way, the critical classic AUTOSAR is guaranteed to meet its deadline, whereas the low criticality partitions run in parallel improving the resources utilization, as shown in Fig. 1(c).

B. Proposed methodology and contributions

This optimistic mixed-critical schedule can be achieved using an appropriate run-time control mechanism, as proposed by our methodology. We introduce a set of observation points to enable the run-time (online) monitoring of the timing behavior of the critical partition and the control of partition set scenarios. At each observation point, a safety condition is applied to check whether it is still safe to continue the execution of τ C in the maximum load scenario. The safety condition uses the remaining WCET of the critical partition in isolation scenario, which is run-time computed by our lowoverhead algorithm. If the safety condition evaluates that a risk exists of overloading the system and, thus, the critical partition runs too slow, a backup process is applied to guarantee the real-time response of τ C : the low criticality partitions are suspended and τ C runs in isolation. When the critical classic AUTOSAR partition finishes its execution and if time remains until the next release of τ C , the low criticality Adaptive AUTOSAR partitions are resumed. We proved the correctness of the proposed safety condition and the low-overhead runtime algorithm for computing the remaining WCET.

As the computation of the remaining WCET is time consuming, it cannot be performed at run-time. Hence, we proposed a run-time algorithm based on pre-computed data which reflect the program structure (static analysis). Then, the run-time computation involves only basic arithmetic and reflects the actual progress of the critical partition execution. To achieve this goal, during the design-time (offline) analysis, the functions that compose the classic AUTOSAR partition are represented by a set of Extended Control Flow Graphs (ECFGs) with observation points. We propose a graph grammar to formally describe the set of ECFGs under study and to prove the correctness of our run-time computation algorithm. Based on the obtained ECFGs, a safe WCET analysis is applied for the pre-computation of several partial remaining WCETs used by the run-time control. The proposed methodology consists of the design-time analysis to pre-compute the required data and the scenario control applied at run-time, as shown in Fig. 2. A full description of the ideas sketched in this note are detailed in [START_REF] Kritikakou | Monitoring on-line timing information to support mixedcritical workloads[END_REF], [START_REF] Kritikakou | Run-time control to increase task parallelism in mixed-critical systems[END_REF], [START_REF] Kritikakou | Distributed run-time wcet controller for concurrent critical tasks in mixed-critical systems[END_REF].

II. DESIGN-TIME ANALYSIS

A. Maximum load and isolation

This section presents the design-time analysis of our approach by describing the instrumentation of the Classic AUTOSAR partition, the static schedule and the design-time analyzer for the computation of the timing and structure information.

The proposed methodology considers two scenarios for the partitions that are executed on the platform.

Definition 1 (Execution scenarios):

The execution scenarios are 1) Isolated execution (iso), where the classic AUTOSAR partition only is executed on the platform, 2) Maximum load (max), where all partitions are executed concurrently on the platform.

The proposed methodology initially executes the maximum load scenario and at each observation point of τ C checks whether the low criticality partitions should be suspended. Hence, the following statement should be proved: "The switching from the maximum load scenario to the isolation scenario guarantees that the classic AUTOSAR partition meets the hard real-time deadline D C ."

B. Switching to isolation mode

The switching occurs when the following safety condition holds:

RWCET iso (x) + W max + t RT + ET(x) > D C (1)
where RWCET iso (x) is the remaining WCET of τ C at an observation point x in the isolation scenario, W max is the WCET until the next observation point, t RT is the total time of the proposed run-time control mechanism and ET(x) is the real execution time of τ C until point x. The t RT is the sum of: 1) t Mon (the overhead to monitor the real execution time), 2) t Cnt (the WCET of the run-time control), and 3) t SW (the WCET overhead due to scenario switching).

Theorem 1: If WCET iso ≤ D C , then for any execution with the proposed run-time control, τ C always respects its deadline.

The proof of this theorem can be found in [START_REF] Kritikakou | Run-time control to increase task parallelism in mixed-critical systems[END_REF]. Intuitively, at each observation point, the run-time control computes the worst-case remaining time for the next observation point: if there is a risk that the next observation is reached too late to switch to a safe mode, then the system switches now to isolated mode, guaranteeing a correct timing behavior.

C. Extended Control Flow Graph Representation

A graph grammar is proposed to model the critical partition τ C considered under study. τ C is described by the general expressions of syntax in Table I, which covers a wide range of applications. From the binary code of τ C [START_REF] Cooper | Building a controlflow graph from scheduled assembly code[END_REF], we create a set of control flow graphs (CFGs), where we insert observation points. The CFGs obtained from the abstract syntax of Table I and compiled without optimizations are covered by the proposed grammar. The full description of the graph grammar can be found in [START_REF] Kritikakou | Run-time control to increase task parallelism in mixed-critical systems[END_REF], and a brief schematics of the grammar is shown in Fig. 3 and Fig. 4. A program is a set of functions (Fig. 3), and every function is described by the grammar of Fig. 4, where C is a conditional, N a binary instruction, F i a function, and B a non-terminal.

IN B OU T F i ≡ Fig. 3. Representation of function F i . B (a) ε (b) N (c) B B (d) C B B B (e) B C B (f) Fi (g)

D. Remaining WCET analysis

The remaining WCET of the critical partition heavily depends on the real execution. Hence, at design-time, we process the ECFGs and compute partial WCETs using safe static WCET analysis, but extended and adapted to our methodology. The WCET is computed by writing an Integer Linear Programming (ILP) formulation to express the program execution time as the combination of the individual times of the grammar components weighted by their execution counts. This expression is maximized to find the WCET, with a number of constraints that reflect flow facts, e.g. loop bounds and unfeasible paths.

Our WCET analysis is based on computing the remaining WCET from one observation point x until the end of the program, RWCET y (x), where y ∈ {iso,max}. When the RWCET max (x) is computed, we consider that interferences occur from the parallel partitions, whereas when the RWCET iso (x) is computed no interferences are taken into account. When point x is the entry of the critical partition, i.e. start of F 0 , RWCET y (start) is the total WCET τi,y . When point x is inside the ECFG, we compute the remaining WCET by using constraints to prohibit the execution of all the blocks which do not belong to any path from point x until the end of the ECFG.

Using the remaining WCET analysis of an observation point x, we can compute remaining WCETs between an observation point and its head point.

III. RUN-TIME CONTROL

At each observation point x, the safety condition described above decides whether switching between scenarios is required. As the RWCET iso (x) is modified at each observation point, we propose a low-overhead algorithm to run-time compute this value by efficiently reusing the RWCET iso of the head points. In [START_REF] Kritikakou | Run-time control to increase task parallelism in mixed-critical systems[END_REF], we describe and formally prove our algorithm when the critical partition consists of a finite set of functions, i.e. S = {F 0 , . . . , F n }. We describe below a simplified version of the general algorithm.

The run-time computation of RWCET iso (x) is depicted in Alg. 1. An important point in this algorithm is the notion of level, that corresponds to the nesting level in a loop nestindeed, such loop nests, as well as function calls, allow us to have a very compact representation of remaining worst-case execution times.

Pre-computed data is stored in the memory as constant arrays: level for the nesting level, and d and w for the partial WCETs: d is the WCET from a loop head to an observation point, and w is the WCET between two consecutive instances of a given loop. The algorithm maintains two local values o level (for the previous observed level) and level(x) for the local level. At run-time, the algorithm stores in array last point the last observed point and in array R the computed RWCET iso per level.

o level = 0, last point[0]=start, RWCET iso [0]=WCET iso Output: RWCET iso (x) = R[level[x]] if o level < level[x] then / * condition 1 * / R[level[x]] = R[level[x] -1] -d[x] else if (last point[level[x]] == x) then / * condition 2 * / R[level[x]] = R[level[x]] -w[x] else R[level[x]] = R[level[x] -1] -d[x] end end last point[level[x]]= x o level=level[x]
The general algorithm [START_REF] Kritikakou | Run-time control to increase task parallelism in mixed-critical systems[END_REF] shares many ideas with the above presented one, with the added complexity of function calls, that can be seen, in some sense, as a dynamic version of (static) loop nests.

IV. CONCLUSION

In this work, we presented a methodology to run several partitions of different criticality levels on a single multi or many-core chip, while guaranteeing the real-time response of a critical classic AUTOSAR partition. At design-time analysis, the critical partition is described by a set of ECFGs and partial WCET analysis is applied to compute the required data for the run-time part. At run-time, a low-overhead controller computes the remaining WCET of the critical partition and decides the switching between maximum load scenario and isolation scenario.

We advocate that such a scheme permits a better utilization of system resources, but further work has to carried out to refine this approach. For now, only two possible scheduling scenarios are supported: total isolation for the critical classic AUTOSAR partition, or full parallelism. One cannot assume that all non-classic AUTOSAR applications can be stopped at any time without any consequence on the system. Thus, supported several (statically defined) scheduling strategies, and switching between different strategies depending on run-time evaluation of remaining resources seems a necessary extension to this work in order to be included into real automotive architectures.

More precisely, our method is applicable to several high criticality partitions running on one core. In case the criticality partitions are executed on several cores, the partial RWCETs in the isolation scenario will be computed by considering that more cores are concurrently running etc. This will increase the partial RWCET, but a trade-off exists to explore between the number of active cores and the pessimism of the remaining WCETs. A similar approach could be followed in case in isolation scenario we allow some low criticality partitions to run on other cores during isolation. When the number of criticality levels is increased, we could also explore several cases regarding the parallel execution of partitions during isolation scenario. In addition, we believe that the combination of our approach with time and partitioning methods will lead to further increase in the task parallelization during isolation scenario. In addition, we are developing a methodology to decide the position of the observation points over the ECFGs in order to further reduce the monitoring overhead.

Fig. 2 .

 2 Fig. 2. Overall methodology

  expr ::= <term> | <term> <operator> <term> | <unary-expr> unary-expr ::= <variable> <unary-operator> | <unary-operator> <variable> cond-expr ::= <expr> <conditional-operator> <expr> assignment ::= <unary-expr> <assignment-operator> <expr> instruction ::= <assignment> | <unary-expr> | <>; stat ::= <instruction> | <stat>;<stat> | if (<cond-expr>) then <stat1> else <stat2> | for (expr1; cond-expr; expr2) <stat> | <function-call> ; function-call ::= <return-type> functionName( <parameter-list> ) <stat> return <expr-return>; program ::= <function-call> Definition 2 (Critical partition τ C ): A critical partition τ C is a set of functions S = {F 0 , F 1 , ..., F n }, with F 0 the main function. Each function is represented by an Extended CFG (ECFG). Definition 3 (Observation point): An observation point is a check point in an ECFG where the run-time control is executed. A special observation point named start is defined before the starting of the execution.

Fig. 4 .

 4 Fig. 4. Schematic representation of rewritting grammar rules

ALGORITHM 1 :

 1 Simplified run-time control algorithm, without function calls. Pre-computed data: level, w, d Input: x Data:

TABLE I

 I

		.	APPLICATION MODEL SYNTAX
			Syntax rules
	term	::= <constant> | <variable>