WO2025072978A1 - Method and system for denoising sets of evoked potentials - Google Patents

Abstract

Methods and systems for denoising an evoked potential waveform are provided. In a first aspect, a method includes receiving a set of auditory brainstem response (ABR) waveforms. The set of ABR waveforms includes an evoked potential waveform for each ear (s). stimulus level (l), and frequency (ƒ). The method includes generating a set of denoised waveforms by applying a dual-tree complex wavelet transform (DTCWT) to the set of ABR waveforms and minimizing an error between each denoised waveform and its corresponding ABR waveform, while: constraining a total amplitude of complex wavelet coefficients to an amplitude threshold (λ_α); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels (l) to a level threshold (λ_Δι); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies (ƒ) to a frequency threshold (λ_Δƒ)

Description

Attorney Docket No.: 055652.00166 METHOD AND SYSTEM FOR DENOISING SETS OF EVOKED POTENTIALS Statement Regarding Federally Sponsored Research [0001] This invention was made with government support under DC017962 awarded by the National Institutes of Health. The government has certain rights in the invention. Field of the Disclosure [0002] The present disclosure relates to the field of neurophysiology, and more particularly to the analysis of evoked potentials. Background of the Disclosure [0003] Evoked potentials are voltages measured on the scalp using electroencephalogram (EEG) electrodes in response to stimuli. A specific kind of evoked potential is called the auditory brainstem response (ABR). The ABR is frequently used to diagnose hearing loss in infants or other patients who cannot complete behavioral tests of hearing loss. To measure the diagnostic ABR, many evoked potentials are recorded with variations in stimulus level and frequency. Obtaining diagnostic ABR can be difficult due to the time required to capture the necessary evoked potentials. There is a need for techniques for obtaining useful ABR in less time and/or with improved quality. Brief Summary of the Disclosure [0004] In an aspect, the present disclosure provides a computer-implemented method for denoising an evoked potential waveform. The method includes receiving, by one or more processors of a signal processing system, a set of auditory brainstem response (ABR) waveforms, the set of ABR waveforms comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^). The method includes generating, using the one or more processors, a set of denoised waveforms by applying a dual-tree complex wavelet transform (DTCWT) to the set of ABR waveforms and minimizing an error between each denoised waveform and its corresponding ABR waveform, while: constraining a total amplitude of complex wavelet coefficients to an amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a level Attorney Docket No.: 055652.00166 threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a frequency threshold ( ^^_௱^). [0005] In some embodiments, the complex wavelet

are calculated from a set ( ^^_^,^,^) of wavelet coefficients and scaling coefficients resulting from the DTCWT. [0006] In some embodiments, the complex wavelet coefficients are calculated according to ^^_^,^,^ ൌ ^^ ^^_^,^,^, where: é^{1 ^^ 0 0 … 0 0} 0 _{1 ^^ … 0 0} ù ú ú ú , ú [0007] In

denoised waveforms by applying a DTCWT and solving according to: ^_ೞ ^_^ ^_{^ ^} ,

[0008] In and the frequency threshold are predetermined. [0009] In some embodiments, the amplitude threshold, the level threshold, and the frequency threshold are determined using cross-validation. Attorney Docket No.: 055652.00166 [0010] In some embodiments, the method includes repeating the steps of receiving a set of ABR waveforms and generating a corresponding set of denoised waveforms. In some embodiments, each ABR waveform of the set of ABR waveforms is averaged and the set of denoised waveforms is generated using the set of averaged ABR waveforms. [0011] In another aspect, the present disclosure may be embodied as a system for generating a denoised ABR waveform. The system includes an auditory stimulator configured to provide a set of stimuli; one or more electrodes configured to measure auditory brainstem response (ABR); and a processor in electronic communication with the auditory stimulator and the one or more electrodes. The processor is configured to: signal the auditory stimulator to provide a set of auditory stimuli, the set of auditory stimuli varying in stimulus level and frequency; receive a set of auditory brainstem response (ABR) waveforms from the one or more electrodes and resulting from the set of auditory stimuli, the set of ABR waveforms comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^); generate, from the set of ABR waveforms, a corresponding set of denoised waveforms by applying a dual- tree complex wavelet transform (DTCWT) to the set of ABR waveforms and minimizing an error between each denoised waveform and its corresponding ABR waveform, while: constraining a total amplitude of complex wavelet coefficients to an amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a frequency threshold ( ^^_௱^). [0012] In some embodiments, the processor is programmed to calculate the complex wavelet coefficients ( ^^_^,^,^) from a set ( ^^_^,^,^) of wavelet coefficients and scaling coefficients resulting from the DTCWT.

[0013] In some embodiments, the processor is programmed to calculate the complex wavelet coefficients ( ^^) according to ^^_^,^,^ ൌ ^^ ^^_^,^,^, where:

Attorney Docket No.: 055652.00166 é^{1 ^^ 0 0 … 0 0} 0 _{0 1 ^} ù ê _{^ … 0 0} ú ú ú , ú [0014] In

the corresponding set of denoised waveforms by applying a DTCWT and solving according to: ^_ೞ ^_^ ^_^ min ^_^imize ^^^ฮ^ ^^ ^ଶ ^_,^,^ െ ^^_^,^,^ฮ , [0015] In

the set of ABR waveforms using parallel acquisition. [0016] In another aspect, the present disclosure may be embodied as a method for denoising an evoked potential waveform. The method includes receiving, by one or more processors of a signal processing system, a set of evoked potential waveforms, the set of evoked potential waveforms comprising an evoked potential waveform for each side and one or more stimulus parameters; and generating, using the one or more processors, a set of denoised waveforms by applying a wavelet transform to the set of evoked potential waveforms and minimizing an error between each denoised waveform and its corresponding evoked potential waveform, while enforcing one or more amplitude constraints on a set of complex wavelet coefficients. [0017] In some embodiments, the wavelet transform is a dual-tree complex wavelet transform (DTCWT). 4 000160.01237 Business 9990964v1 Attorney Docket No.: 055652.00166 [0018] In some embodiments, the one or more amplitude constraints comprise constraining a total amplitude of the set of complex wavelet coefficients to a pre-determined amplitude threshold ( ^^_^). [0019] In some embodiments, the received set of evoked potential waveforms is a set of auditory brainstem response (ABR) waveforms comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^); and the one or more amplitude constraints comprise: constraining a total amplitude of the complex wavelet coefficients to a predetermined amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a predetermined level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a predetermined frequency threshold ( ^^_௱^). [0020] In another aspect, the present disclosure may

as a system for generating a denoised evoked potential waveform. The system includes a stimulator configured to provide a set of stimuli to an individual; one or more electrodes configured to measure a response to the set of stimuli and provide a set of evoked potential waveforms; a processor in electronic communication with the stimulator and the one or more electrodes. The processor is configured to: signal the stimulator to provide a set of stimuli; receive a set of evoked potential waveforms from the one or more electrodes and resulting from the set of auditory stimuli, the set of evoked potential waveforms comprising an evoked potential waveform for each side and one or more stimulus parameters; and generate, from the set of evoked potential waveforms, a corresponding set of denoised waveforms by applying a dual-tree complex wavelet transform (DTCWT) to the set of evoked potential waveforms and minimizing an error between each denoised waveform and its corresponding evoked potential waveform, while enforcing one or more amplitude constraints on a set of complex wavelet coefficients. [0021] In some embodiments, the wavelet transform is a dual-tree complex wavelet transform (DTCWT). [0022] In some embodiments, the one or more amplitude constraints comprise constraining a total amplitude of the set of complex wavelet coefficients to a pre-determined amplitude threshold ( ^^_^). Attorney Docket No.: 055652.00166 [0023] In some embodiments, the received set of evoked potential waveforms is a set of auditory brainstem response (ABR) waveforms comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^); and the one or more amplitude constraints comprise: constraining a total amplitude of the complex wavelet coefficients to a predetermined amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a predetermined level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a predetermined frequency threshold ( ^^_௱^). Description of the Drawings [0024] For a fuller understanding of the nature and objects of the disclosure, reference should be made to the following detailed description taken in conjunction with the accompanying drawings. [0025] Figure 1: A chart showing a method according to an embodiment of the present disclosure. [0026] Figure 2: A diagram showing a system according to another embodiment of the present disclosure. [0027] Figure 3: An example wavelet set having 128 wavelet bases corresponding to a signal of length 64. Each row shows ^^_^ in black and ^^_^ in gray for a given time and scale. The bottom two row show real-valued ^^ for the coarsest scale, which is the only scale needed for the basis set to allow reconstruction. The peak amplitude of each base has been normalized for visualization purposes. This is a non-limiting example wavelet set. Others may have, for example, fewer scales or more bases (e.g., in the case of a longer signal). [0028] Figure 4: ABR waveforms after 15 minutes of recording (top set), 56 minutes of recording (middle set), and after 15 minutes of recording with denoising applied (bottom set). Each set is arranged the same: the left and right subsets correspond to measurements made from the left and right ears. Within each subset, stimulus frequencies are arranged by column, increasing from left to right, and stimulus level is arranged by row, decreasing from top to bottom. Denoising allows the responses from 15 minutes of data to be of similar quality to the responses averaged from all 56 minutes of data collected. Attorney Docket No.: 055652.00166 Detailed Description of the Disclosure [0029] The present disclosure allows separately measured evoked potentials, such as the auditory brainstem response to different frequencies and stimulus levels, to be denoised. Denoising is important because it reduces recording time and increases signal to noise ratios. This improves the speed and quality of diagnoses that can be made, such as hearing loss at specific frequencies. [0030] Evoked potentials are voltages measured on the scalp using electroencephalogram (EEG) electrodes in response to stimuli. A specific kind of evoked potential is called the auditory brainstem response (ABR). The ABR is frequently used to diagnose hearing loss in infants or other patients who cannot complete behavioral tests of hearing loss. [0031] To measure the diagnostic ABR, many evoked potentials are recorded with variations in stimulus level and frequency. Because the stimuli used to measure the responses are similar, the responses themselves tend to be similar as well. Specifically, the size and latency of the responses vary smoothly with changing stimulus frequency and level. That the responses to similar stimuli are themselves so similar suggests that there is information in one response about the ones nearby it. This information could lead to drastically improved signal to noise ratios, but it has previously never been used. [0032] The present disclosure improves the signal-to-noise ratio (SNR) of responses by leveraging neighbor responses to similar stimuli. This is accomplished computing a dual-tree complex wavelet transform of the set of responses with regularization that penalizes the size of the coefficients and also the size of the differences between corresponding coefficients of responses at neighboring stimulus frequencies and levels. The presently disclosed techniques are more effective for signal processing of evoked potentials and allow for much improved compute efficiency and performance when compared with previous noise reduction methods. [0033] While this technique was developed using the ABR, it is easily applied to other types of evoked potentials, such as the multi-focal electroretinogram (MFERG), as well as to other sets of signals with similar characteristics, such as head-related impulse responses (which have important application to augmented and virtual reality), etc. In the case of use for MFERG, the neighboring responses would be those responses at adjacent locations on the retina, rather than adjacent frequencies and levels. The present techniques may also be applied to denoise the Attorney Docket No.: 055652.00166 response to a single stimulus across several electrodes at different positions on the scalp, though this application may be less powerful as the noise between electrodes would not be independent. Evoked potentials and signal-to-noise ratio [0034] Evoked potentials are measured brain responses to some stimulus. In humans they are typically recorded using electroencephalography (EEG) electrodes, a safe way of measuring voltages from the scalp which reflect human brain activity. [0035] Because EEG recording electrodes are located outside of the head—i.e., some distance from the neural generators—a recording from a single electrode pair will reflect brain activity from many simultaneous but distinct and unrelated neural processes happening in the brain. [0036] In engineering, noise can be thought of as any signal which is (unavoidably) measured but not desired, or equivalently as uncertainty in the true value of the measured signal. The amplitude of an evoked potential is often less than 1 μV, while the ongoing activity from unrelated processes is substantially larger, often measuring in the tens of microvolts. In the context of measuring an evoked potential, the unrelated brain activity constitutes noise. The amplitude of the evoked potential is small compared to the noise. In other words, there is a low signal-to-noise ratio (SNR). The SNR is the ratio of the energy of the signal ( ^^_ௌ) to the energy of the noise ( ^^_ே), typically expressed in decibels: ^^{^ ^^ ^^ ^^ ൌ 10 log ௌ} ^_{^ ^^ே} ^. ₍₁₎ Thus, if the signal and noise have the same energy, the SNR is 0 dB, if the signal energy is twice that of the noise the SNR is 3 dB, and if the signal energy is half that of the noise the SNR is −3 dB. At high SNRs, a signal is clearly visible, but at low SNRs, the noise dominates and the signal is either uncertain or completely obscured by the noise. [0037] The latter is the case with evoked potentials: a single evoked response is so small as to be invisible in the comparatively huge levels of noise. The remedy for this low SNR is to record many evoked responses to the same stimulus and average them. There is a simple rule for how averaging improves SNR: every time the number or responses averaged is doubled, the SNR improves by ~3 dB. So if the SNR of a single measurement is −21 dB, and the target SNR Attorney Docket No.: 055652.00166 is 0 dB, then a stimulus will need to be presented 128 times. While averaging responses to repeated stimuli offers a way around poor SNR, it also presents an inescapable problem when measuring evoked potentials: lower SNR leads to longer recording time. The diagnostic auditory brainstem response [0038] The auditory brainstem response is a specific kind of evoked potential whose component waves reflect the response of the subcortical auditory system to a short sound such as a click. It comprises several stereotyped components, typically referred to as waves I–VII, which all occur in the first 10–15 seconds (s) following a short toneburst stimulus. For example, tonebursts used to test experimental embodiments of the present disclosure comprised five cycles of a cosine at a given frequency, smoothly ramped on and off. The response to each toneburst tests only a specific part of the frequency range over which humans can hear. [0039] In a diagnostic ABR, an audiologist will measure responses to tonebursts at different frequencies across several different stimulus levels (expressed in dB) so that the lowest level at which the patient responds (i.e., a response waveform is observable) can be recorded. This level is called a “threshold,” and the threshold for each frequency in each ear is the main outcome of the diagnostic ABR on which the determination of hearing loss is based. The parallel ABR [0040] While the diagnostic ABR is typically recorded one waveform at a time, the parallel ABR (pABR) (Polonenko and Maddox, 2019) allows many simultaneous waveforms to be measured all at once, leading to substantial speedups. Specifically, for a given stimulus level, the responses to all stimulus frequencies in both ears are measured. The pABR yields large, complete sets of responses over several continuous stimulus dimensions (viz., stimulus level and frequency). This fact is exploited by the presently-disclosed denoising techniques. [0041] It should be noted that despite its name, the pABR provides responses not only from the brainstem. At longer latencies, the generators of the evoked potentials are cortical. The cortex is anatomically distinct from the brainstem, yet the later cortical waves that can be measured are included here in the term pABR. The wavelet transform Attorney Docket No.: 055652.00166 [0042] It can be advantageous to decompose, or convert, a recorded time-domain signal into a different domain. This process involves a transform which allows the signal to be expressed as a linear combination of many basis functions. The most common is the Fourier transform, which expresses a signal as the sum of many weighted periodic complex exponential functions with varying frequencies. Once transformed, a signal can be manipulated in the frequency domain, and then transformed back to the time domain. For example, the low frequency components of a musical signal could be made larger in the frequency domain, and the modified signal in the time-domain would have more “bass.” [0043] The purpose of a wavelet transform is to decompose a signal into a set of bases which vary in frequency and time, and are as local as possible in each of those dimensions. This differs from a Fourier transform, which expresses a signal as a linear combination of single- frequency components whose energy is distributed over all time (i.e., sinusoids). Each wavelet ψ is defined over scale and time. Each one can be thought of as a finite impulse response (FIR) bandpass filter. [0044] Some embodiments of the presently-disclosed denoising method utilize a version of the wavelet transform called the dual-tree complex wavelet transform (DTCWT). In various embodiments, other transforms may be utilized, and the present disclosure is not necessarily limited to DTCWT. In the DTWCT, there are actually two wavelets at each scale and time, with one being at a phase of ^^/2 relative to the other (i.e., they are approximately orthogonal). These two wavelets, ^^_^ and ^^_^, can be thought of as the real and imaginary parts of a single, approximately analytic wavelet, ^^_^, such that: ^^_^ ൌ ^^_^ ^ ^^ ^^_^ , (2)

where ^^ is the imaginary number, analytically but is instead determined iteratively in order to best match specific design criteria. Wavelets at many scales and lags can be seen in Figure 1, with the real part plotted in black and the imaginary part plotted in gray. [0045] There is also a complement to the wavelet functions called the scaling function, ^^, which can be thought of as a low-pass filter whose cutoff frequency corresponds to the lower cutoff of the ^^_^ at the same scale. Like ^^_^, ^^ is defined over time and scale, but unlike ^^_^, ^^ is purely real. Figure 1 shows the two ^^ functions in the bottom two rows, in blue only because Attorney Docket No.: 055652.00166 they are real. In practice, because the smoothing functions are only needed at the coarsest scale of the DTCWT, they mostly cover a slow trend in the data. Implementation [0046] The standard way of calculating the wavelet transform coefficients is an inner product, and this allows a perfect reconstruction of the signal being transformed. In the present application, a perfect reconstruction may not be desired—instead, it is advantageous to reconstruct the waveforms with the signal present, but the noise minimized. Thus, the present method provides a calculation of a set of waveforms’ wavelet coefficients in a way that penalizes noise (thus prioritizing signal). This may be accomplished through convex optimization, with the specific objective and constraints defined below (see Eq. (6)). While a non-limiting description of a convex optimization is provided, the optimization may also be performed via stochastic gradient descent and/or other techniques, all of which are within the scope of the present disclosure. [0047] Let ^^ _{^^, ^^, ^^} be a vector which represents the ABR waveform in the time-domain, averaged across all recorded stimulus repetitions (i.e., any type of averaging, such as, for example, traditional averaging, inverse-noise-weighted averaging, etc.), for a given ear side ( ^^), stimulus level ( ^^), and stimulus frequency ( ^^). Let ^^ be a matrix whose columns are the wavelet and smoothing function bases, with the wavelets included for all scales and lags and the smoothing function included for all lags of the coarsest scale. The denoised^{^} ^^_^,^,^ can then be computed by finding the coefficients of a linear model that fits ^^ to each waveform in ^^ _{^^, ^^, ^^} while regularizing to reduce noise: ^ ^{^^} _^,^,^ ^{ൌ ^^} _^,^,^ ^{^^் ,} ₍₃₎ where ^^_^,^,^ is the vector of wavelet coefficients that define the response. Without regularization, this would be an alternative (and slower) way of computing the DTCWT (i.e., finding ^^_^,^,^). But by computing the coefficients in this way, the sparsity in the wavelet space can be improved and, more importantly, the differences between corresponding complex wavelet coefficients of responses evoked by adjacent stimuli (e.g., 30 and 40 dB presentation levels or adjacent frequency bands) can be penalized. Attorney Docket No.: 055652.00166 [0048] In order to benefit from the DTCWT’s features, the wavelet coefficients of ^^_^,^,^ can be reorganized to provide a single complex coefficient for each wavelet rather than two separate coefficients. In other words, it is desirable to have coefficients that correspond to ^^_^ instead of ^^_^ and ^^_^ separately. Thus, the coefficients are slightly restructured to yield ^^_^,^,^, where

^^{^} _^,^,^ ^{ൌ ^^ ^^} _{^,^,^ (4)} and é^{1 ^^ 0 0 … 0 0} 0 _{0 1 ^^ … 0 0} ù (5) with all blank

and imaginary components corresponding to ^^_^ and ^^_^ and combines them into a single complex- valued coefficient corresponding to ^^_^. The lower-right part of ^^ simply tacks on the real-valued coefficients corresponding to ^^. Because ^^ has fewer rows than columns, the result of Eq. (4) is a vector ^^_^,^,^ that is shorter than ^^_^,^,^, but with no loss of information because ^^_^,^,^ is complex- valued where ^^_^,^,^ is real-valued. It should be noted that ^^_^,^,^ is only used for the constraints of the complex optimization problem described below. [0049] The importance of this reorganization is that ^^_^,^,^ is complex-valued, and thus has a magnitude and phase. This fact is what affords the DTCWT its property of shift-invariance, and what makes it well suited to the current application. [0050] The presently-disclosed denoising technique is based on two assumptions about the ABR, and evoked potentials more generally. The first is that each individual response is spectrotemporally sparse. This means that the large majority of the variance should be explainable by a small number of non-zero values in ^^_^,^,^. This sparsity is apparent when looking at the waveforms: the earlier waves are shorter and only 1 or 2 ms wide, followed by wave V which is a bit wider (8 ms), followed by several later, increasingly long cortical waves. Attorney Docket No.: 055652.00166 Inducing sparsity in the wavelet space (either by thresholding, or regularizing based on the L1 norm of the coefficients, as is done here) is responsible for a small portion of the denoising power of the present technique. [0051] The more powerful assumption about the ABR exploited here is that responses to similar stimuli are similar. This assumption cannot be applied to a single response, which is why the present technique is useful to denoise sets of related responses. The similarity of responses to related stimuli seems obvious but is difficult to exploit due to the way responses differ. For example, consider the ABR to stimuli at two similar levels: 70 and 60 dB. The response to the quieter 60 dB stimulus is smaller, as expected, but it is also later. In terms of the complex coefficients ^^_^,^,^, a change in size and latency can both be succinctly represented by the addition of a complex number, which modifies both the amplitude and phase of the coefficient, corresponding to a change in size and latency of the response (or the response component, more specifically). The difference in responses to neighboring frequencies is similar: the response to a 1 kHz stimulus is later and slightly wider in time than the response to a 2 kHz stimulus. In the time domain (whose basis is the set of lagged unit impulses), changing a single coefficient cannot describe the temporal shift of a response component. [0052] The assumption of similar responses to adjacent stimuli can be exploited by penalizing the magnitude of the difference between corresponding complex wavelet coefficients of adjacent responses. Only these adjacent differences need to be penalized, because if, e.g., the difference between responses 1 and 2 is penalized, as is the difference between 2 and 3, then some level of similarity between 1 and 3 is also enforced, but less so than the adjacent ones, which is appropriate. [0053] With all the above explained, the optimization problem for denoising the ABR is now provided:

Attorney Docket No.: 055652.00166 ^_ೞ ^_^ ^{^^ minimize} _^^^ฮ ^{^ ^^ ଶ} ^_,^,^ ^{െ ^^} _^,^,^ฮଶ (6) [0054]

the total amplitude of the coefficients to ^^_^ (the first constraint) and the total amplitude of the differences between coefficients at adjacent levels to ^^_௱^ (second constraint) and at adjacent frequencies to ^^_௱^ (third constraint). The difference ears is not constrained because

while responses are typically similar, ears do not form a continuous dimension like level and frequency do, and it would make the algorithm less sensitive to asymmetric hearing loss and single-sided deafness. [0055] The limiting parameters ^^_^, ^^_௱^, and ^^_௱^ control the strength of the denoising. In the limit, as they approach infinity, their effect disappears and^ ^^_^,^,^ becomes ^^_^,^,^. As ^^_^ is lowered, the weights of wavelet bases with small amplitudes shrink to 0, preferably with noise components disappearing before signal components. As ^^_௱^ and ^^_௱^ are lowered, responses become more similar across level and frequency,

three parameters can be set too low, such that desired parts of the signal disappear ( ^^_^) or responses to different stimuli become too similar ( ^^_௱^ and / or ^^_௱^ too low), and diagnostic utility is lost. Setting the parameters properly

ways, both quantitative (e.g., cross-validation) and qualitative (determining the amount of denoising allowing the clinician to advantageously see the responses). Pilot results indicate these parameters should be similar across patients, such that normative data can be used to determine them, avoiding computationally costly cross-validation for each person. [0056] The limiting parameters also do not need to be modified as data is collected during a single session. With the parameters sensibly set, the denoising will have a very strong Attorney Docket No.: 055652.00166 effect when very little data has been acquired (and the SNR is very low). As data is acquired and SNR improves, the norms being constrained in Eq. (6) will naturally diminish towards some non-zero minimum, such that limiting them has less of an effect on^{^} ^^_^,^,^. [0057] In the case of denoising evoked potentials resulting from MFERG, the response may be defined for side and location in the visual field. The constraints for level and frequency ( ^^_^^ and ^^_^^) in Eq.6 may be replaced with a single constraint that penalized the difference between adjacent locations (i.e., locations in the visual field). For example, this may be implemented by creating an adjacency matrix where each row had a 1 and −1 in columns corresponding to adjacent locations, such that multiplying the adjacency matrix by the responses would compute an overall difference penalty. Other techniques may be used for these evoked potentials or others. [0058] With reference to Figure 3, in a first aspect, the present disclosure may be embodied as a computer-implemented method 100 for denoising an evoked potential waveform, such as an ABR waveform. The method 100 includes receiving 103, by one or more processors of a signal processing system, a set of evoked potential waveforms (for example, evoked potential signals), such as, for example, a set of ABR waveforms. The received 103 set of evoked potential waveforms may include an evoked potential waveform for each stimulus location (e.g., sides of an individual, physical structures, etc.), stimulus level, and/or each stimulus type. For example, a set of auditory brainstem response (ABR) waveforms may include an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^). [0059] Using the example of ABR waveforms, the method includes generating 106 (e.g., using the one or more processors) a set of denoised waveforms by applying a wavelet transform (e.g., a dual-tree complex wavelet transform (DTCWT)) to the set of evoked ABR waveforms. An error between each denoised waveform and its corresponding ABR waveform is minimized while: (1) constraining a total amplitude of complex wavelet coefficients to an amplitude threshold ( ^^_^); (2) constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a level threshold ( ^^_௱^); and (3) constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a frequency threshold ( ^^_௱^). More broadly stated, the DTCWT is applied to the set of evoked potential

an error is minimized between each denoised waveform and its corresponding evoked potential waveform while enforcing one or more amplitude constraints on Attorney Docket No.: 055652.00166 a set of complex wavelet coefficients. The one or more amplitude constraints may include, for example, constraining a total amplitude of the set of complex wavelet coefficients to a pre- determined amplitude threshold. The one or more amplitude constraints may include constraining a total amplitude of differences between adjacent complex wavelet coefficients to a predetermined adjacency threshold. [0060] The DTCWT may be applied to the set of evoked potentials (e.g., ABR waveforms) and the denoised evoked potentials may be reconstructed using the set of complex wavelet coefficients. [0061] The complex wavelet coefficients ( ^^_^,^,^) may be calculated from a set ( ^^_^,^,^) of wavelet coefficients and scaling coefficients resulting from the DTCWT. For example, the complex wavelet coefficients are calculated according to ^^_^,^,^ ൌ ^^ ^^_^,^,^, where ^^ is as shown in equation 5 (above). [0062] The corresponding set of denoised waveforms may be generated by applying a DTCWT and solving according to equation 6. [0063] Other techniques may be used to generate the set of denoised waveforms by applying a DTCWT and are within the scope of the present disclosure. For example, a method may include defining an objective function that measures a quality metric of the signal reconstruction and a sparsity of the complex wavelet coefficients. The objective function may be minimized while enforcing one or more amplitude constraints on a set of complex wavelet coefficients. In other examples, other optimizers may be used and are within the scope of the present disclosure. In some embodiments, stochastic gradient descent and/or other techniques may be used. [0064] With reference to Figure 2, the present disclosure may be embodied as a system 10 for generating a denoised evoked potential waveform (for example, an ABR waveform). The system 10 includes a stimulator 20 configured to provide a set of stimuli to an individual. In the example of ABR waveforms, the stimulator may be an auditory stimulator configured to provide a set of auditory stimuli—e.g., to each ear, at a plurality of levels, at a plurality of frequencies, or combinations of these or other stimuli parameters. Attorney Docket No.: 055652.00166 [0065] The system 10 includes one or more electrodes 30 configured to measure evoked potentials resulting from the set of stimuli and produce a set of evoked potential waveforms. A processor 40 is in electronic communication with the stimulator 20 and the one or more electrodes 30. The processor 40 is configured to signal the stimulator to provide a set of stimuli (e.g., provide a set of auditory stimuli varying in stimulus level and frequency). The processor is configured to receive a set of evoked potential waveforms from the one or more electrodes and resulting from the set of stimuli provided by the stimulator. For example, the processor may be configured to receive a set of auditory brainstem response (ABR) waveforms from the one or more electrodes and resulting from the set of auditory stimuli. In the example of ABR waveforms, the received set of ABR waveforms may comprise an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^). [0066] The processor 40 is further configured to generate a corresponding set of denoised waveforms using, for example, any of the methods disclosed herein. For example, the processor may be configured to generate, from the set of ABR waveforms, a corresponding set of denoised waveforms by applying a dual-tree complex wavelet transform (DTCWT) to the set of ABR waveforms and minimizing an error between each denoised waveform and its corresponding ABR waveform, while: constraining a total amplitude of complex wavelet coefficients to an amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a frequency threshold ( ^^_௱^). Modifications to improve performance

[0067] Several modifications can be made in various embodiments to improve performance and are within the scope of the present disclosure. These modifications tend to relate to specific applications. In the case of the ABR, one modification comes from the fact that responses to lower frequency stimuli peak later than those to high frequency stimuli. Thus, performance can be slightly improved by shifting low-frequency responses leftward in time so they are more likely to line up with high frequency responses. Empirical delays on which to base this time shift exist in the literature (Polonenko and Maddox, 2019). A non-limiting example of compensatory delays could be −6.8, −4.3, −2.0, −0.9, 0 ms for 500, 1000, 2000, 4000, 8000 Hz stimuli, respectively. Other delays across other stimulus dimensions could be employed for other Attorney Docket No.: 055652.00166 applications. It should be noted that octave-frequency spacing (as shown in Figure 2) is the most common, but other sets of stimulus frequencies, such as half-octave spacing, could be used. In fact, pilot testing indicated potential advantages to collecting half as many response epochs (each) to twice as many stimulus frequencies at half-octave spacing (500, 707, 1000, 1414 Hz etc.) [0068] A second modification which can be used simultaneously with the presently described technique is SNR-based or Bayesian averaging. The signals that are denoised by this algorithm are averages of the responses collected (so far) in a recording session. However, noise levels when recording EEG vary over time. Instead of weighting every recording epoch equally as in traditional averaging, epochs can be weighted by the inverse of their estimated SNR. This can improve the quality of responses further (Polonenko and Maddox, 2019). [0069] A third modification which can increase the speed of computation is to use an incomplete set of wavelet bases. For example, a simple way to do this is to exclude a wavelet (or smoothing function) scale which is known not to contain much useful information. For example, for an ABR recorded at 10,000 samples/s, the highest frequency scale or scales can be left out without much loss of signal and greatly improved computation times. The decision to leave out coefficients in a time-specific way could also be made, but this runs the risk of being too strong a prior and carries a risk of false positives if not done carefully. [0070] Embodiments of the present disclosure use the information present in a set of ABR waveforms to improve the signal‐to-noise ratio (SNR) of the responses by leveraging neighbor responses to similar stimuli. This can be accomplished mathematically by computing a dual‐tree complex wavelet transform of the set of responses with regularization that penalizes the size of the coefficients and also the size of the differences between corresponding coefficients of responses at neighboring stimulus frequencies and levels. In some embodiments, the process is as follows for the ABR: 1. Record responses to multiple stimulus frequencies (e.g., 500, 1000, 2000, 4000, 8000 Hz) and levels (e.g., 20 through 80 dB in 10 dB steps). This can be done using conventional ABR acquisition techniques, using parallel ABR acquisition, or otherwise. Attorney Docket No.: 055652.00166 2. Compute the set of complex wavelet bases that will be used to reconstruct the responses. 3. Set up a convex optimization problem that describes the set of recorded responses as a linear combination of the complex wavelet bases from the previous step. That problem will minimize the squared prediction error subject to limits on: (a) the sum of the magnitudes of the complex wavelet coefficients; (b) the sum of the magnitudes of the differences of the complex wavelet coefficients between responses to stimuli at neighboring levels and frequencies (the other‐ear response could also be considered but likely should not because doing so could mask asymmetric hearing losses). 4. Solve the convex optimization problem. In embodiments using cross‐validation to find the optimal regularization coefficients, repeat step 3 numerous times in a standard leave‐one‐out scheme. 5. Reconstruct the measured responses from the wavelets and the calculated weights and interpret them as they normally would be. [0071] Furthermore, although the present discussion describes solving the problem as a convex optimization problem, other optimizers may be used and are within the scope of the present disclosure. In some embodiments, stochastic gradient descent and/or other techniques may be used. Such techniques are used in training deep neural networks and can be run on graphics cards. These may provide low-cost alternatives to increase the speed of the computation. [0072] The presently disclosed techniques take advantage of information present in the responses but not used by any existing paradigms. Results [0073] The present denoising technique offers a substantial SNR improvement, especially when very little data has been collected. An example is shown in Figure 2, with responses measured from two ears, five stimulus frequencies, and seven stimulus levels. The top set of responses is based on 15 minutes of pABR data collection, equivalent to 13 seconds per waveform. It is clear that with this limited amount of data, there is substantial noise present. The Attorney Docket No.: 055652.00166 middle set of responses is organized as above, but is based on the entire 56-minute recording and thus has much better SNR with waveforms clearly present. The bottom set of responses uses the same 15 minutes of data as the top set, but has the denoising method applied to it. [0074] It is clear from visual inspection that the denoised responses (Figure 2, bottom set) have much better SNR than the responses from the same data with no denoising (Figure 2, top set). In fact, their SNR approaches, and in some cases exceeds, that of the full data “ground truth” (with ground truth in quotes, because there is always noise). In early pilot experiments, SNR improvements of 4 or 5 dB were seen often. Such SNR improvements reduce the amount of data needed by two thirds, which means denoised pABR recording sessions may be three times faster than standard pABR recording sessions, which are themselves 2–4× faster than traditional ABR methods currently used in the clinic. Thus, combining the present denoising method with previously published methods (e.g., pABR) could allow full sets of diagnostic ABR waveforms to be measured on the order of 10× faster than traditional methods could. [0075] Such a speedup has the potential to be transformative for diagnosis. Current methods involve a clinician choosing which single response to measure, collecting data, and then moving on to the next response, ideally determining the lowest level for each ear and frequency at which a response can be observed. Recent methods, aided by the denoising disclosed here, may allow a clinician to measure every response in every patient, eliminating the threshold search entirely. [0076] Although the present disclosure has been described with respect to one or more particular embodiments, it will be understood that other embodiments of the present disclosure may be made without departing from the spirit and scope of the present disclosure.

Claims

Attorney Docket No.: 055652.00166 What is claimed is: 1. A computer-implemented method for denoising an evoked potential waveform, the method comprising: receiving, by one or more processors of a signal processing system, a set of auditory brainstem response (ABR) waveforms, the set of ABR waveforms comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^); generating, using the one or more processors, a set of denoised waveforms by applying a dual-tree complex wavelet transform (DTCWT) to the set of ABR waveforms and minimizing an error between each denoised waveform and its corresponding ABR waveform, while: constraining a total amplitude of complex wavelet coefficients to an amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a frequency threshold ( ^^_௱^). 2. The computer-implemented method of claim 1, wherein the complex wavelet coefficients ( ^^_^,^,^) are calculated from a set ( ^^_^,^,^) of wavelet coefficients and scaling coefficients resulting from the DTCWT.

3. The computer-implemented method of claim 2, wherein the complex wavelet coefficients are calculated according to ^^_^,^,^ ൌ ^^ ^^_^,^,^, where: é^{1 ^^ 0 0 … 0 0} ^_^ ù ú ú ú . ú

Attorney Docket No.: 055652.00166 4. The computer-implemented method of claim 2, wherein generating the corresponding set of denoised waveforms by applying a DTCWT and solving according to: ^_ೞ ^_^ ^{^^} m_{inimize ^^^ฮ} ^ଶ ^_^ ^{^} _{^^^,^,^ െ ^^^,^,^ฮଶ} , 5. The computer-

threshold, the level threshold, and the frequency threshold are predetermined. 6. The computer-implemented method of claim 1, wherein the amplitude threshold, the level threshold, and the frequency threshold are determined using cross-validation. 7. The computer-implemented method of claim 1, further comprising repeating the steps of receiving a set of ABR waveforms and generating a corresponding set of denoised waveforms. 8. The computer-implemented method of claim 7, wherein each ABR waveform of the set of ABR waveforms is averaged and the set of denoised waveforms is generated using the set of averaged ABR waveforms. 9. A system for generating a denoised ABR waveform, the system comprising: an auditory stimulator configured to provide a set of stimuli; one or more electrodes configured to measure auditory brainstem response (ABR); a processor in electronic communication with the auditory stimulator and the one or more electrodes, the processor configured to: signal the auditory stimulator to provide a set of auditory stimuli, the set of auditory stimuli varying in stimulus level and frequency; receive a set of auditory brainstem response (ABR) waveforms from the one or more electrodes and resulting from the set of auditory stimuli, the set of ABR waveforms Attorney Docket No.: 055652.00166 comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^); generate, from the set of ABR waveforms, a corresponding set of denoised waveforms by applying a dual-tree complex wavelet transform (DTCWT) to the set of ABR waveforms and minimizing an error between each denoised waveform and its corresponding ABR waveform, while: constraining a total amplitude of complex wavelet coefficients to an amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a frequency threshold ( ^^_௱^). 10. The system of claim 9, wherein the processor is programmed to calculate

wavelet coefficients ( ^^_^,^,^) from a set ( ^^_^,^,^) of wavelet coefficients and scaling coefficients resulting from the DTCWT.

11. The system of claim 10, wherein the processor is programmed to calculate the complex wavelet coefficients ( ^^) according to ^^_^,^,^ ൌ ^^ ^^_^,^,^, where: é^{1 ^^ 0 0 … 0 0} 0 _{1 ^^ … 0 0} ù ú ú ú , ú

Attorney Docket No.: 055652.00166 12. The system of claim 10, wherein the processor is programmed to generate the corresponding set of denoised waveforms by applying a DTCWT and solving according to: ^_ೞ ^_^ ^{^^} m_inimize ^ଶ ^_{^ ^^^ฮ} ^{^} _{^^^,^,^ െ ^^^,^,^ฮଶ} , 13. The system of

the set of ABR waveforms using parallel acquisition. 14. A computer-implemented method for denoising an evoked potential waveform, the method comprising: receiving, by one or more processors of a signal processing system, a set of evoked potential waveforms, the set of evoked potential waveforms comprising an evoked potential waveform for each side and one or more stimulus parameters; generating, using the one or more processors, a set of denoised waveforms by applying a wavelet transform to the set of evoked potential waveforms and minimizing an error between each denoised waveform and its corresponding evoked potential waveform, while enforcing one or more amplitude constraints on a set of complex wavelet coefficients. 15. The computer-implemented method of claim 14, wherein the wavelet transform is a dual-tree complex wavelet transform (DTCWT). 16. The computer-implemented method of claim 14, wherein the one or more amplitude constraints comprise constraining a total amplitude of the set of complex wavelet coefficients to a pre-determined amplitude threshold ( ^^_^). 24 000160.01237 Business 9990964v1 Attorney Docket No.: 055652.00166 17. The computer-implemented method of claim 14, wherein the received set of evoked potential waveforms is a set of auditory brainstem response (ABR) waveforms comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^); and the one or more amplitude constraints comprise: constraining a total amplitude of the complex wavelet coefficients to a predetermined amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a predetermined level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a predetermined frequency threshold ( ^^_௱^). 18. A system for generating a denoised evoked potential waveform, the system comprising: A stimulator configured to provide a set of stimuli to an individual; one or more electrodes configured to measure a response to the set of stimuli and provide a set of evoked potential waveforms; a processor in electronic communication with the stimulator and the one or more electrodes, the processor configured to: signal the stimulator to provide a set of stimuli; receive a set of evoked potential waveforms from the one or more electrodes and resulting from the set of auditory stimuli, the set of evoked potential waveforms comprising an evoked potential waveform for each side and one or more stimulus parameters; generate, from the set of evoked potential waveforms, a corresponding set of denoised waveforms by applying a dual-tree complex wavelet transform (DTCWT) to the set of evoked potential waveforms and minimizing an error between each denoised waveform and its corresponding evoked potential waveform, while enforcing one or more amplitude constraints on a set of complex wavelet coefficients. 19. The system of claim 18, wherein the wavelet transform is a dual-tree complex wavelet transform (DTCWT). Attorney Docket No.: 055652.00166 20. The system of claim 18, wherein the one or more amplitude constraints comprise constraining a total amplitude of the set of complex wavelet coefficients to a pre-determined amplitude threshold ( ^^_^). 21. The system of claim 18, wherein the received set of evoked potential waveforms is a set of auditory brainstem response (ABR) waveforms comprising an evoked potential waveform for each ear ( ^^), stimulus level ( ^^), and frequency ( ^^); and the one or more amplitude constraints comprise: constraining a total amplitude of the complex wavelet coefficients to a predetermined amplitude threshold ( ^^_^); constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent stimulus levels ( ^^) to a predetermined level threshold ( ^^_^^); and constraining a total amplitude of the differences between the complex wavelet coefficients at adjacent frequencies ( ^^) to a predetermined frequency threshold ( ^^_௱^).