# Adaptive hyperparameter tuning for time-frequency algorithms based on the zeros of the spectrogram

Juan Manuel Miramont, Rémi Bardenet, Pierre Chainais, François Auger

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Juan Manuel Miramont, Rémi Bardenet, Pierre Chainais, François Auger. Adaptive hyperparameter tuning for time-frequency algorithms based on the zeros of the spectrogram. 23rd IEEE Statistical Signal Processing Workshop, IEEE, Jun 2025, Edimbourg, United Kingdom. 10.1109/SSP64130.2025.11073382. hal-05050759

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# Adaptive hyperparameter tuning for time-frequency algorithms based on the zeros of the spectrogram

Juan M. Miramont, Rémi Bardenet, Pierre Chainais
Université de Lille, Centrale Lille, CNRS,
UMR 9189 - CRISTAL, Lille, France

François Auger
Nantes Université
UR 4642 - IREENA, Saint-Nazaire, France

**Abstract**—It is well known that the zeros of the spectrogram of white Gaussian noise are homogeneously distributed in the time-frequency (TF) plane. When a signal is present, large holes are created in this pattern. Recently, several methods have been proposed to detect and use them as masks for signal reconstruction. Yet, no systematic studies for tuning their hyperparameters on real data have been conducted. We investigate one key hyperparameter, the interaction scale $r_0$, intuitively corresponding to the radius of a hole caused by a well-localized signal in the TF plane. We benchmark adaptive tuning protocols for $r_0$ from the spatial statistics literature, which take as input simulated noise realizations and the actual data. We use non-Gaussian noise, audio signals, and the reconstruction performance is assessed with audio quality metrics. Results show that tuning strategies can depend both on the protocol and the data, highlighting the importance of benchmarking such approaches.

**Index Terms**—time-frequency analysis, spectrogram, short-time Fourier transform, spatial statistics.

## I. INTRODUCTION

Time-frequency (TF) representations provide the means to process non-stationary time series, the frequency content of which varies with time, helping to discern patterns caused by signals of interest [1]. In particular, recent times have seen the emergence of signal detection and estimation methods based on the zeros of a spectrogram. [2]–[7]. Theoretical aspects of the zeros of the short-time Fourier transform of white noise have been investigated [3], [6], [8], as well as the zeros of different transforms like the continuous wavelet [9], [10], Kravchuk [4], [5], [10] and Stockwell [11] transforms. More practical research can be seen in for instance [12], [13]. However, many empirical aspects of the use of spectrogram zeros and their applications in signal detection and denoising are yet to be studied.

Fig. 1a illustrates the regular distribution of the zeros of the spectrogram of a white Gaussian noise realization [2]. Indeed, their distribution corresponds to that of the zeros of the planar Gaussian analytic function [3], a well-studied point process exhibiting various degrees of organization [14]. Fig. 1b shows how the distribution of the zeros is locally disrupted by the presence of a signal [2], [3], where larger-than-expected regions without zeros (holes) emerge, roughly corresponding to the TF support $\mathcal{D}$ of the signal or, informally, the region of the TF plane with the essential information on the signal.

This work was supported by grants ERC-2019-STG-851866, ANR20-CHIA-0002, and ASCETE-ANR19-CE48-0001. We thank José Luis Romero for discussions on musical noise in time-frequency methods.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Spectrograms of noise and a mixture of signal and noise. White dots indicate the position of the zeros of the spectrogram. The red dashed line approximates the boundary of the signal domain $\mathcal{D}$.</figcaption></figure>

Hence, aggregating these large holes in the pattern is akin to estimating $\mathcal{D}$, and this is precisely the rationale of the methods proposed in the seminal work of Flandrin [2] and later work [3]. One way to find holes that are statistically too large under the hypothesis of noise only, is to define a proper scale $r_0$ that serves as a threshold that separates large, signal-related holes from smaller ones, which can be attributed to chance. One can then determine the region corresponding to such large holes by either filtering triangles with an edge larger than $2r_0$ from the Delaunay triangulation of the spectrogram zeros (the initial proposition of [2]), or by searching for discs of radius larger than $r_0$ that are devoid from zeros, as done in [3]. Choosing $r_0$ when the noise in the signal is unknown is a tricky task. If $r_0$ is too large, then one risks loosing much of the signal. Choosing $r_0$ too small, however, results in the estimated support including excessive noise. Moreover, the distance between zeros close the TF support of a signal is influenced by external factors such as the signal-to-noise ratio, so that one should fix $r_0$ using the data to be analyzed.

Methods which mask the complement of these selected areas are similar to thresholding approaches (like hard thresholding [15], [16]), in the sense that noise is suppressed by subtracting noise-related coefficients from the (discrete) short-time Fourier transform (STFT). It is established that hard and soft spectrogram thresholds can introduce perceptually unpleasant artifacts termed musical noise [17]–[19]. However, the type of artifacts introduced by approaches based on the spectrogram zeros is not well documented. Therefore, aiming for future applications on audio processing, it is natural to investigate the effect of picking a scale $r_0$ on such artifacts.
In this paper, we conduct a statistical comparison of variants of the spatial statistics procedure used in [3], namely varying the functional statistics and the type of test from which we infer the value of $r_0$. We evaluate their impact in signal detection and estimation, resulting in practical recommendations. We focus on realistic noise realizations and, considering potential applications, we evaluate signal estimation performance by means of audio quality and the presence of artifacts. Our benchmarks are run using `mcsm-benchs` [20], a Python-based package that we contribute to run collaborative benchmarks in signal processing, which is of independent interest.

## II. METHODOLOGICAL FRAMEWORK

The STFT of $x \in L^2(\mathbb{R})$ is

$$V_x^g : t, \nu \mapsto \int_{-\infty}^{+\infty} x(u)g(u-t)^*e^{-i2\pi\nu u}du,$$

where $t, \nu \in \mathbb{R}$ and $g \in L^2(\mathbb{R})$ is the so-called analysis window. The spectrogram of $x$ is the squared modulus of $V_x^g$, and can be thought of as the TF distribution of the energy of the signal [1].

Considering the zeros of the spectrogram of a noisy signal as a point process $\mathcal{X}$ in $\mathbb{R}^2$ [14], [21], one can summarize the distribution of $\mathcal{X}$ by functional quantities like the empty space function $F(r) = \mathbb{P}(\mathcal{X} \cap B(0, r) \neq \emptyset), r > 0$, the probability that a Euclidean ball $B(0, r)$ centered at 0 and with radius $r$ contains at least one point of $\mathcal{X}$. When $\mathcal{X}$ is assumed to be translation-invariant, as is the case of the zeros of the spectrogram of white Gaussian noise, the definition of $F$ is independent of the location of the ball. Many such summary functions can be defined, and estimated, to investigate the behavior of spatial point processes that are invariant to rotations and/or translations [21]–[23]. Such estimators of summary functions have been used to test the hypothesis of the data being pure white Gaussian noise, and, in case the test rejects, reconstruct the underlying signal. Henceforth, let $r \mapsto S(r)$ denote a summary function, with $r \mapsto \hat{S}(r)$ an estimator of that function.

### A. Detection tests and summary functions

We consider three possible instances of $S$, the latter two being new in the context of zero-based detection. Our first choice is $S = F$, the empty space function defined above, which has been used in [3]. Yet, the authors there could have equally well used for $S$ the variance-stabilized empty space function $\tilde{F}: r \mapsto \arcsin(\sqrt{F(r)})$ [22], [24]. Another natural summary function is the accumulated persistence function (APF) [25]. The latter is obtained from the persistence diagram of the zeros, a tool from topological data analysis, used to describe typical structures in configurations of points in, e.g., material science. Armed with these summary functions, we compare two Monte Carlo tests for signal detection. Let $\alpha$ be a pre-specified significance level, and $H_0$ the null hypothesis considering that the probed signal is a realization of white Gaussian noise.

<figure><img src="image_3.png" /><figcaption>Fig. 2. (a) Upper ($S_{\text{upp}}^{(1)}$) and lower ($S_{\text{low}}^{(1)}$) envelopes, in green, dot-dashed lines, as well as the empirical functional statistic $\hat{S}_0$, in red, dashed line for $S = F$, $S = \tilde{F}$ and $S = \text{APF}$. $r_0$ indicated by a red, dashed line, was hand-tuned. The continuous black line indicates the estimation $\hat{r}_0$ obtained for the given summary function.</figcaption></figure>

1) **Global MAD test:** Fix an integer $k \le m$ such that $\alpha = k/(m+1)$. Next, $m$ spectrograms of independent noise realizations under $H_0$ are generated, and one uses their $m$ sets of zeros to evaluate the corresponding estimators $\hat{S}_1, \dots, \hat{S}_m$. Similarly, denote the functional statistic estimated on the data by $\hat{S}_0$. To derive real numbers from these estimated functions, one natural option is to define $t_j = \|(\hat{S}_j(\cdot) - \hat{S}(\cdot))1_{r \in I}\|_{\infty}, j = 0, \dots, m$, where $I = [r_{\min}, r_{\max}]$ and $p > 0$ are parameters of the test, and $r \mapsto \bar{S}(r)$ is the pointwise average of $\hat{S}_0, \dots, \hat{S}_m$ (which includes the observation being tested) [22], [23].

Let $t_{(0)} \le \dots \le t_{(m)}$ be the sorted values of the $t_j$s. The null hypothesis is rejected if and only if $t_0 > t_{(k)}$, that is, if the maximum absolute deviation (MAD) $t_0$ of $\hat{S}_0$ from $\bar{S}$ in $I$ is one of the largest.

2) **Global rank envelope test:** This more recent test [23] is based on assigning to each estimated function $r \mapsto \hat{S}_j(r)$ a rank $\rho_j = \max\{k: S_{\text{low}}^{(k)}(r) \le \hat{S}_j(r) \le S_{\text{upp}}^{(k)}(r), \forall r \in I\}$. Here, $S_{\text{low}}^{(k)}$ and $S_{\text{upp}}^{(k)}$ are the $k$-th lower and upper envelopes, defined for $r > 0$ as

$$S_{\text{low}}^{(k)}(r) = \min_{j=0,\dots,m} \hat{S}_j(r) \quad \text{and} \quad S_{\text{upp}}^{(k)}(r) = \max_{j=0,\dots,m} \hat{S}_j(r),$$

where $\min(\max)$ is the $k$-th smallest (largest) value. One can consider $\rho_j$ as a measure of how “extreme” $\hat{S}_j$ is when compared with the ensemble $\hat{S}_0, \dots, \hat{S}_m$. If $\hat{S}_0$ is too extreme when compared to the simulated $\hat{S}_j$, one can suspect that it does not come from the null model $H_0$. A range of $p$-values $[p_-, p_+]$ can be computed [23], and the authors respectively define a liberal and a conservative test by rejecting whenever $p_-$, respectively $p_+$, is smaller than $\alpha$. As a rule of thumb, a number $m = 2499$ of simulations is recommended to make these two tests essentially indistinguishable when the significance level is $\alpha = 0.05$ [23].

### B. Estimation of $r_0$ and signal reconstruction

Once a detection test rejects $H_0$, we identify a value $r_0$ among the values of $r$ that caused the rejection (in spatial statistics parlance, the scale of interaction [21]–[23]), and then estimate $\mathcal{D}$ by looking at holes of characteristic scale larger than $r_0$ [3].
<figure><img src="image_4.png" /><figcaption>Fig. 3. Spectrograms of the signals used in Sec. III. (a)-(b) Speech signals synthesized using Google Cloud realistic text-to-speech API [26] ("Peter will keep at the peak" used in [27]). (c) Cello excerpt. (d) Trumpet excerpt. (e) Spectrum of the babble noise used in the numerical experiments.</figcaption></figure>

Formally, we define $\hat{r}_0$ as the value of $r$ corresponding to the maximum difference between $\hat{S}_0(r)$ and the lowest envelope $S_{low}^{(1)}(r)$ for $r \in I$, that is

$$ \hat{r}_0 = \arg\max_{r \in I} |S_{low}^{(1)}(r) - \hat{S}_0(r)|. $$

Every summary function thus yields a different estimator $\hat{r}_0$. Figure 2 shows the upper and lower envelopes, as well as $\hat{S}_0$ for the three proposed summary functions for a known synthetic signal. Once $\hat{r}_0$ is determined, we follow the procedure from [2] and estimate the signal by: (1) Computing a Delaunay triangulation (DT) on the spectrogram zeros; (2) Filtering the triangles by only choosing those whose largest edge is larger than $2r_0$. (3) Taking the union of the surviving triangles to estimate $D$. (4) Computing the signal estimation as $\tilde{x} : t \mapsto \frac{1}{g(0)} \int_D V_x^g(t, v) e^{2i\pi t\nu} dv$. Henceforth, we refer to this procedure as the DT approach.

For illustrative purposes, Fig. 2 also shows $\hat{r}_0$ for each summary function, as well as an “optimal” $r_0$ obtained by searching in a range of $r$ the value that resulted in the best<fnref n="1" /> estimation $\tilde{x}$ of the synthetic signal used in this case.

### III. NUMERICAL EXPERIMENTS AND DISCUSSION

We report here the results obtained for signal detection and estimation<fnref n="2" />. First, the detection power of each test using $S = F$ and $S = \tilde{F}$ was computed for a synthetic linear chirp immersed in white Gaussian noise (as in [3]). Note that $S = \text{APF}$ was used only with the MAD test in this work.

To evaluate signal estimation, we begin by applying a detection test (to be determined later). If a signal is detected, we compute $\hat{r}_0$ using the three proposed summary functions, and compare the quality of the signal estimation $\tilde{x}$ obtained with the DT approach in each case. To contaminate the data, babble noise from the TIMIT database [31] was used with several signal-to-noise ratios<fnref n="3" /> (SNR). We use a cello and a trumpet excerpts, as well as two speech signals. Figure 3 shows their spectrograms and a Welch estimation of the noise spectrum. Performance is measured using two metrics:

<figure><img src="image_5.png" /><figcaption>Fig. 4. Detection power (i.e. the probability of correctly rejecting $H_0$) for several SNRs. Each bar shows the mean for 2000 repetitions of the experiment. The error bars denote the 95% Clopper-Pearson confidence intervals with Bonferroni correction.</figcaption></figure>

perceptual evaluation of speech quality (PESQ) [32], [33] and artifact-related perceptual score (APS) [34], [35]. PESQ was originally conceived to assess the quality of speech over telecommunication networks, although it has lately been used as a more general audio quality metric, even for non-speech signals [36]. Larger values of PESQ indicate a better audio quality. Yet, this metric is not particularly sensitive to the artifacts that result in musical noise [36]. APS, on the other hand, is sensitive to such artifacts: the higher the value of APS, the less musical noise is present in the signal [34].

#### A. Signal detection

Figure 4 summarizes the detection power of the defined tests. For SNR $\ge 10$ dB, the performance of almost all pairs of tests-summary functions is satisfactory, except for the pair APF-global MAD test, which turned out to be the one with the lowest performance. For lower SNRs, the global rank test [23] is the most effective, regardless of the summary function. Although the MAD test increases the detection power for $S = \tilde{F}$, it remains less effective than the rank test. However, the global MAD test needs fewer simulations ($m = 199$ vs. $m = 2499$), making it less computationally expensive. Overall, our results indicate that global tests have a higher power than the point-wise counterparts described in [3], with the advantage of depending on fewer parameters.

Based on these results, the first step of signal estimation comprising the detection of a signal is carried out using the rank envelope test for $F, \tilde{F}$. In the following section, we use the global MAD test for APF because it takes significantly

<footnote n="1">¹In terms of the quality reconstruction factor [28]–[30], only applicable when the noiseless version of the synthetic signal is known.</footnote>

<footnote n="2">²Code available in https://github.com/jmiramont/ssp-2025</footnote>

<footnote n="3">³SNR($x, \xi$) = $10 \log_{10} (\|x\|_2^2 / \|\xi\|_2^2)$ (dB), where $x$ is the signal and $\xi$ the noise.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 5. Audio-quality measured by average PESQ over 100 realizations for several SNRs. Error bars show the 95% confidence interval with Bonferroni correction for multiple comparisons.</figcaption></figure>

less time for relatively long signals. Since the audio excerpts used are easily detectable, this choice has no impact on the reconstruction.

## B. Signal estimation

Figure 5 shows the performance of signal estimation in terms of PESQ. As baseline, we add the results for the DT approach and a simple Hard Thresholding (HT) of the spectrogram [16] after exploring the performance for a range of their parameters, i.e. a range of $r_0$ values for DT and range of thresholds for HT. This is done for each signal and SNR, so that we are able to choose the best performances of each method to compare with. These baselines are then termed “Best DT” and “Best HT”, correspondingly.

By comparing the results for $F$, $\tilde{F}$ and APF with “Best DT”, we conclude that the estimation of $r_0$ with any of the summary functions is effective. This means that it is possible to estimate a near optimal value of $r_0$ using the proposed methodology without hand-tuning. Rankings between $F$ and $\tilde{F}$ vary significantly with the signal. For instance, $F$ performs better for the cello and the speech-male signal, whereas $\tilde{F}$ is more effective for the trumpet excerpt. In contrast, APF provides a more stable output. This is an interesting result since APF is a novel summary function [25], that relates both the study of point processes from the statistical perspective with tools stemming from algebraic topology, calling for more explorations of this connection in the future. Figure 5 also

<table><caption>TABLE I
PERCEPTUAL QUALITY RESULTS FOR THE SPEECH-FEMALE SIGNAL FROM FIG. 3(A) WITH SNR=10 dB. BEST RESULTS ARE IN BOLD.</caption><thead><tr><th>Metric</th><th>APF</th><th>F</th><th>F&#771;</th><th>Best HT</th></tr></thead><tbody><tr><td>PESQ</td><td><strong>1.90</strong></td><td>1.85</td><td>1.85</td><td>1.84</td></tr><tr><td>95% CI</td><td>(1.87;1.93)</td><td>(1.83;1.87)</td><td>(1.83;1.88)</td><td>(1.81;1.87)</td></tr><tr><td>APS</td><td>40.30</td><td><strong>67.15</strong></td><td>61.76</td><td>64.74</td></tr><tr><td>95% CI</td><td>(35.16;45.23)</td><td>(63.94;70.19)</td><td>(54.83;67.37)</td><td>(59.22;69.66)</td></tr></tbody></table>

shows that, at least through the lens of PESQ, the performance of both DT and HT approaches is comparable (except in the cello excerpt case, which nonetheless has a simpler TF structure than the rest of the studied signals, and for the trumpet recording for which DT seems to perform better).

Table I complements these results by showing the average value of APS for the speech-female signal with SNR = 10 dB. This particular case was chosen because PESQ values are similar for all approaches. Interestingly, determining the threshold using APF results in more audible artifacts (i.e. lower APS than for $F$, $\tilde{F}$), but the overall audio quality is slightly better (higher PESQ). This difference can be attributed to the fact that APF yields a larger $\hat{r}_0$ than $F$ or $\tilde{F}$, hence improving the separation between noise and signal but leaving more small isolated regions in the spectrogram that tend to be responsible for audible artifacts [18], [37].

Considering these results, special attention must be paid when using a zero-based approach for denoising in the case of audio signals: in the absence of post-processing, it might introduce more undesired artifacts than traditional threshold-based methods. It is also worth noticing that links between the DT method and audio artifacts have been the subject of a previous work [37] aiming to quantify the amount of musical noise present in a signal.

When dealing with signals with more intricate TF structure, methods based on the spectrogram zeros could benefit from using a *local* scale, i.e. dependent on the location in the TF plane, rather than one global scale parameter given by $r_0$. Future work will explore this possibility as well as connections between the spectrogram zeros and topological data analysis, inspired by the results obtained for APF.

Further results are shown in the online repository of this paper, as well as online interactive plots. Interested readers can also upload their own methods to be compared with the ones used in this paper thanks to the collaborative benchmarks created with mcsm-benchs.

## IV. CONCLUSION

We investigated the potential of three summary functions used in spatial statistics to estimate the scale parameter of spectrogram zero-based methods, and we studied their implications for signal detection and estimation. For detection, we recommend using the global rank envelope tests with APF, since that summary function yields a more reliable $\hat{r}_0$ for signal estimation than $F$ or $\tilde{F}$. Finally, even though zero-based methods and traditional approaches like HT can produce similar results in terms of general audio quality (similar PESQ), the former may introduce more audio artifacts, requiring postprocessing.
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