Wavelet time-frequency analysis enhances noise-resilient quantification of corneal wave velocity and natural frequency in optical coherence elastography

Abstract

In vivo characterization of corneal biomechanics using dynamic optical coherence elastography (OCE) requires accurate quantification of both the temporal and spectral features for transient, frequency-dependent corneal dynamics. However, conventional time-of-flight (TOF) and fast Fourier transform (FFT) methods are limited in resolving coupled time-frequency behaviors, especially under low signal-to-noise ratio (SNR) conditions. We introduce a wavelet-based method for simultaneous characterization of wave propagation and resonant oscillations in air-pulse OCE. A Morlet wavelet transform was employed, with parameters optimized for wave velocity (F _b = 0.2 and F _c =1.0) and natural frequency (F _b =5, F _c =7). The method was validated against TOF and FFT using agar phantoms, ex vivo porcine corneas (n = 26, IOP: 5–20 mmHg), and an in vivo rabbit model. The wavelet method showed excellent agreement with TOF and FFT under controlled conditions, with mean differences of 0.18 m/s for group velocity (2.3–5.2 m/s) and 0.5 Hz for natural frequency (176–437 Hz) in 1–2% agar phantoms. Critically, it outperformed TOF under challenging conditions, reducing the coefficient of variation by ∼50% (0.27 vs. 0.52) at low-SNR depths and preserving more valid data (86.3% vs. 50.0%) in stressed corneas at 20 mmHg. Its time-frequency localization further enabled the in vivo detection of transient vibrational changes, suggesting the potential to distinguish physiological modulations (e.g., heartbeat, respiration) that are not readily accessible with FFT. This work establishes a robust, noise-tolerant foundation for dynamic corneal elastography, potentially advancing dynamic OCE toward precise, clinically translatable assessment of corneal biomechanics.

1. Introduction

In recent decades, there has been growing interest in measuring and understanding corneal biomechanical properties (e.g., stiffness and elasticity) in ophthalmology [1–3]. These properties are important for diagnosing, managing, and understanding the progression of ocular diseases such as keratoconus [4,5], glaucoma [6–9], and myopia [10], as well as for optimizing outcomes in refractive surgery to avoid the potential risk of iatrogenic ectasia [11–17]. Clinical methods, such as the Ocular Response Analyzer [18] and Corvis ST [19], provide valuable insights into corneal biomechanics. However, their measurements lack the spatio-temporal resolution required to fully characterize the cornea's dynamic, localized mechanical response [20–23]. Brillouin microscopy offers high-resolution, depth-resolved corneal stiffness mapping [24–26], yet its clinical translation is limited by long scanning times, motion artifacts, and the uncertain correlation between its GHz-frequency longitudinal modulus and the low-frequency Young's or shear moduli used in mechanical engineering [27–31].

Dynamic optical coherence elastography (OCE) is a rapidly evolving technique for elastic imaging [32,33]. Air-pulse OCE, in particular, is regarded as a suitable method for imaging delicate tissues like the cornea due to its non-contact, low-force nature [3]. This technique uses a transient air pulse to induce micron-level tissue displacements, which are captured by high-speed optical coherence tomography (OCT) [34,35]. The advancement of phase-sensitive OCT enables this measurement with sub-nanometer displacement sensitivity, a significant enhancement over earlier micrometer-scale, intensity-based methods [36]. The biomechanical properties can be then derived from the stimulation-response relationship. As the propagation velocity of mechanical waves correlates directly with tissue stiffness, the most widely used analytical model is the mechanical wave method, with the time-of-flight (TOF) technique commonly used to determine wave speed by tracking the wavefront envelope in the spatiotemporal domain (speed = distance/time) [37–39]. An alternative method quantifies tissue properties through its natural frequency, defined as the inherent oscillation frequency after disturbance. In a single-degree-of-freedom (SDOF) model, natural frequency exhibits a linear relationship with the square root of Young's modulus, and it is typically derived from the vibrational response using fast Fourier transform (FFT) analysis [40]. More detailed viscoelastic properties can be characterized by analyzing frequency-dependent wave dispersion via 2-D Fourier transform [41–43]. Air-pulse OCE has proven effective in ex vivo studies, quantifying corneal stiffening due to cross-linking [44], high intraocular pressure (IOP) [45], and age [46]. Recent in vivo human studies have successfully demonstrated its capability for tracking wave propagation [47,48] and characterizing natural frequency distributions [49].

Despite these advancements, a fundamental methodological challenge persists: conventional analytical approaches—such as the TOF and FFT-based techniques—operate primarily in either the time or frequency domain and typically assume a stationary, linear tissue response. However, the cornea exhibits spatially heterogeneous and temporally transient mechanical behavior under localized, broadband excitations, resulting in non-stationary vibration patterns that these methods cannot accurately resolve [3]. In the TOF approach, wavefront tracking is highly susceptible to noise-induced waveform distortions, particularly under low signal-to-noise ratio (SNR) conditions [47]. Moreover, this method simplifies complex dispersive wave propagation into a single group velocity, potentially overlooking valuable frequency-dependent viscoelastic information. Conversely, FFT-based analyses (e.g., for natural frequency or dispersion characterization) provide spectral insights but inherently assume signal stationarity within the analysis window, limiting their ability to capture transient frequency shifts essential for assessing time-varying biomechanical behavior—such as those influenced by respiration or cardiac motion during in vivo measurements.

In this study, we present a wavelet-based time-frequency analytical approach for quantifying both mechanical wave velocities (group and phase) and natural frequencies from corneal deformation responses measured via air-pulse OCE. This approach offers three key advantages: (1) It enables independent optimization of temporal and frequency resolution, tailoring the analysis to specific signal features such as wave velocity or natural frequency. (2) It provides precise time-frequency localization, allowing tracking of transient, non-stationary events such as altered wave patterns or resonant frequencies induced by in vivo physiological motions (e.g., respiration or heartbeat). (3) It enhances robustness to noise through time-frequency filtering, thereby improving parameter estimation fidelity—particularly under low SNR conditions. To validate the proposed method, we conducted a comprehensive comparison with TOF and FFT approaches across tissue-mimicking agar phantoms, ex vivo porcine corneas, and an in vivo rabbit cornea. The Morlet wavelet was selected as the basis function for its balanced time-frequency resolution, and its parameters were firstly optimized independently for analyzing wave propagation velocity and natural frequency using 1–2% agar phantoms. The optimization was performed under controlled, high-SNR conditions, where conventional TOF and FFT methods provide stable and well-established reference measurements. The method’s performance was further evaluated under varying SNR conditions (depth range: 0.9–6.8 mm) to assess its noise resilience. A subsequent assessment on 26 ex vivo porcine eyes across an IOP range of 5–20 mmHg was performed to evaluate the capability to resolve stress-dependent anisotropic behavior. Finally, the method’s robustness was tested under in vivo conditions in a rabbit model, demonstrating its performance in the presence of physiological motions such as heartbeat and respiration. Through this study, we aim to establish a reliable, noise-tolerant, and spatially resolved time-frequency analysis framework that enables precise and clinically translatable characterization of corneal biomechanical properties.

2. System set-up

2.1. Air-pulse OCE system

The air-pulse OCE system (Fig. 1(a)) has been described extensively in our previous publications [34,35]. Briefly, a transient (3-ms duration), low-pressure (tens to hundreds of Pascals) air pulse was delivered onto the sample surface through a solenoid-controlled cannula; while a synchronized 1300-nm phase-sensitive spectral-domain OCT system was used to record the induced dynamic response. A linear-wavenumber (k) spectrometer employing an optimized grating-prism combination dispersed light optically in the k-domain, improving detection sensitivity in deeper tissue and reducing image reconstruction time. The system operated at a 70-kHz line scan rate with 6.94-mm imaging depth and 1.8-mW sample power.

Fig. 1.

Schematic of the air-pulse OCE setup. (a) The system combines a microliter air-pulse system for mechanical stimulation with a high-resolution, phase-sensitive OCT system for dynamic response detection. (b) Top-view geometry showing the central stimulation point and the radial OCT scanning pattern. OCT M-scans were acquired along 20 radial directions (spanning 18° to 342° in 18° intervals), with 17 measurement points per radial line.

We adopted a sparse radial scanning pattern to match the geometry of the corneal tissue, as illustrated in Fig. 1(b). Air-pulse excitation was applied perpendicular to the sample center every 0.5 s using a 90°-angled curved air cannula [34]. The OCT scan covered 20 radial directions (0–360° in 18° increments), with 17 spatial points per direction (340 measurement points, 170 s total measurement time). Two radial ranges were used in this study: 0.93–2.80 mm for flat samples (e.g., agar phantoms) and 0.37–1.87 mm for curved samples (ex vivo porcine corneas and in vivo rabbit corneas). Measurements were acquired sequentially on a point-by-point basis. At each measurement location, 3000 consecutive A-scans were recorded over a 42.9 ms duration to capture the displacement profile (M-mode) using phase-sensitive detection, defined as: Displacement = (Wavelength × Phase) / (4π). The wave propagation velocity along each radial direction and the vibrational characteristics across the measurement field were subsequently reconstructed from the radial measurement matrix.

2.2. Samples

Three types of samples were utilized across a range of biomechanical conditions: (1) tissue-mimicking agar phantoms (1.0–2.0%) as ideal, homogeneous materials with various stiffness; (2) ex vivo porcine corneas (26 right eyes; IOP: 5–20 mmHg) to represent stress-dependent anisotropic behavior; and (3) an in vivo rabbit cornea to demonstrate performance under physiological conditions with heartbeat and respiration.

We selected 1.0–2.0% agar phantoms as testing samples because their mechanical wave speeds [36] and natural frequency ranges [40] closely approximate those observed in in vivo human corneas [47,49]. The agar samples were freshly prepared prior to measurement following previously described protocols [40]. In brief, agar powder (Biowest agarose 111860, XHLY Co. Ltd., Beijing, China) was mixed with distilled water to the desired concentration. The mixture was sealed, heated, and stirred until complete dissolution. After cooling, the solution was poured into a Petri dish (32 mm inner diameter, 12.5 mm height) and refrigerated for 2 hours to solidify. Prior to OCE measurement, phantoms were equilibrated at room temperature for 30 minutes. All phantoms, which had a consistent weight of ∼11.2 g and densities of 1.113 g/cm³ (1.0%), 1.115 g/cm³ (1.2%), 1.117 g/cm³ (1.5%), and 1.123 g/cm³ (2.0%), were measured directly in their dishes.

Fresh porcine eyes were obtained from a local market. For each specimen, extraocular muscles were carefully dissected, and the globe was secured in a custom 3D-printed holder (Figs. 2(a, b)). IOP was regulated by infusing saline into the vitreous chamber using an automated syringe pump (CTN-W100, Chengyitong Co., Ltd., Beijing, China) and monitored in real-time with a digital pressure gauge (YB-80A, XSENR, Suzhou, China) during OCE measurements [34]. All experiments were completed within 12 hours post-mortem.

Fig. 2.

Experimental setup for porcine corneal measurements. (a) An ex vivo porcine globe was secured in a custom 3D-printed holder. Intraocular pressure (IOP) was maintained and regulated between 5 and 20 mmHg using an automated syringe pump. (b) Positioning schematic of a right porcine cornea. The infusion needle (not shown) was inserted at the nasal side (0°). Anatomical directions are indicated: superior (S), inferior (I), nasal (N), and temporal (T).

This study also involved a healthy adult New Zealand rabbit (female, 3 months, 2.5 kg weight), supplied by the Guangdong Provincial Animal Experiment Center. The experimental procedure was conducted in accordance with the protocols and ethical guidelines approved by the Animal Experiment Ethics Committee of Foshan University. Prior to the experiment, the rabbit underwent systematic ophthalmic examinations to confirm the absence of glaucoma, keratoconus, cataracts, and other ocular diseases that could potentially affect the results. One day before the experiment, the animal received four times of prophylactic antibiotic treatment with levofloxacin eye drops (Cravit, Santen Pharmaceutical Co., Japan). On the day of the experiment, general anesthesia was induced by intramuscular injection of a mixed anesthetic at 0.10 mL/kg, comprising Zoletil 50 (tiletamine hydrochloride, 125 mg; zolazepam hydrochloride, 125 mg; Virbac, Carros Cedex, France) and Sumianxin II (xylazine hydrochloride and ketamine hydrochloride) mixed at a 1:1 volume ratio. Topical corneal anesthesia was additionally achieved using proparacaine hydrochloride eye drops (Alcon, USA) to ensure adequate analgesia and anesthesia throughout the measurement procedure. Following completion of all measurements, the animal was humanely euthanized by intravenous injection of an overdose of sodium pentobarbital (100 mg/kg) in accordance with the American Veterinary Medical Association (AVMA) Guidelines for the Euthanasia of Animals (2020) [50].

3. Quantification methods

3.1. Wave propagation velocity

Wave propagation speed can be used to infer mechanical properties via a specific analytical model (e.g., shear, Rayleigh, or Lamb waves), depending on the coupling medium and boundary conditions. In a basic shear wave model—suitable for incompressible, homogeneous, isotropic bulk tissues, the relation between Young’s modulus (E) to shear modulus (G) is $E\approx 3G$ . In viscoelastic tissues, the shear modulus can be written in a complex form, as

$$G^{\ast }=G^{\prime }+i{G}^{\prime \prime },$$ (1)

where $G^{\prime }$ is the storage shear modulus, associated with the elastic behavior; $G^{\prime \prime }$ is the loss shear modulus, associated with the viscose behavior.

Phase velocity $c_p$ describes the speed of individual wave frequencies and is derived from the dispersion relationship $c_p=\omega /k$ , where k is the wavenumber, and $\omega$ is the angular frequency ( $\omega =2\pi f,f:$ linear frequency). The Phase velocity $c_p$ can be used to relate to shear modulus as

$$c_p=\frac{\omega }{k}=\sqrt{\frac{2{|G^{\ast }|}^2}{\rho (G^{\prime }+|G^{\ast }|)}},$$ (2)

where ρ denotes density.

Group velocity $c_g$ represents the speed of the wave energy envelope and is described as

$$c_g=\frac{d\omega }{dk}=c_p+k\frac{d{c}_p}{dk}=\frac{c_p^2}{c_p-f\frac{d{c}_p}{df}}.$$ (3)

In practice, the most commonly used method for calculating $c_g$ is the time-of-flight (TOF) method. The group velocity can be represented as

$$c_g=\frac{\mathrm{\Delta }X}{\mathrm{\Delta }T},$$ (4)

Where $\mathrm{\Delta }X$ is the wave traveling distance, and $\mathrm{\Delta }T$ is the traveling time.

For tissues with complex boundary conditions, alternative analytical models can be used to calculate mechanical properties from the measured wave speed. For instance, Lamb wave theory is more appropriate for evaluating dispersive, guided waves that propagate along the surface and through the thickness of thin, plate-like structures such as the skin, cornea, or arterial walls. Detailed descriptions of the analytical models can be found in [37].

3.2. Natural frequency

In viscoelastic materials, natural frequency $(f_n)$ arises from the interaction between elastic restoring forces (stiffness) and inertial resistance (mass), and is modified by viscous damping. For a single-degree-of-freedom (S-DOF) system (e.g., Kelvin-Voigt model):

$$f_n=\frac{\sqrt{K/M}}{2\pi },$$ (5)

where K is the effective spring stiffness (N/m) and M is the mass (kg). The damping ratio (ε) is defined as

$$\epsilon =\frac{C}{4\pi M{f}_n}.$$ (6)

where C is the viscous damping coefficient (N·s/m). The dynamic response to mechanical excitation depends critically on ε: (1) In the under-damped regime $(0\le \epsilon <1)$ , tissue exhibits oscillatory decay (damped sinusoids) at an oscillation frequency $f_d=f_n\sqrt{1-{\epsilon }^2}$ . For soft tissues with low damping $(\epsilon <0.3)$ , $f_d\approx f_n$ since viscous effects are negligible [40]. (2) In the critically-damped regime $(\epsilon =1)$ , tissue returns to equilibrium without oscillation in the minimum possible time. (3) In the over-damped regime $(\epsilon >1)$ , relaxation follows an exponential decay with no oscillatory component.

3.3. Wavelet transform

The wavelet transform decomposes a signal into localized wavelets, providing a time-frequency representation. In this approach, wavelet basis functions ${\psi }_{s,\mathrm{\tau }}(t)$ are generated by scaling and translating a mother wavelet $\psi (t)$ [51]:

$${\psi }_{s,\mathrm{\tau }}(t)=\frac{1}{\sqrt{|s|}}\psi \left(\frac{t-\tau }{s}\right);-\mathrm{\infty }<\tau <+\mathrm{\infty },$$ (7)

Where s is a dimensionless scale parameter and τ controls temporal translation.

The Continuous Wavelet Transform (CTW) of an input signal $x(t)$ (e.g., tissue displacement data measured by OCE) is defined as

$$T(s,\tau )=\frac{1}{\sqrt{|s|}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x(t){\psi }^{\ast }\left(\frac{t-\tau }{s}\right)dt,$$ (8)

where ${\psi }^{\ast }$ denotes the complex conjugate of the mother wavelet. By continuously varying s and τ, the CWT provides multiresolution analysis, capturing both high-frequency transients and low-frequency trends.

In this study, we selected the Morlet wavelet basis for its balanced time-frequency resolution [52]. The Morlet wavelet basis is composed of a Fourier transform kernel and a Gaussian window. The complex Morlet wavelet basis function is defined as:

$$\psi (t)=\frac{1}{\sqrt{\pi F_b}}\underset{\begin{array}{c}\text{Fourier}\\ \text{Kernel}\end{array}}{\underbrace{e^{i2\pi F_{c}t}}}\underset{\begin{array}{c}\text{Gaussian}\\ \text{Window}\end{array}}{\underbrace{e^{-\frac{t^2}{F_b}}}},$$ (9)

where the dimensionless parameters $F_b$ and $F_c$ represent the bandwidth and central frequency, respectively. Substituting Eq. (9) into Eq. (8) yields:

$$T(s,\tau )=\frac{1}{\sqrt{|s|\mathrm{\pi }{F}_b}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x(t)\underset{\begin{array}{c}\text{Fourier}\\ \text{Kernel}\end{array}}{\underbrace{e^{-i2\mathrm{\pi }{F}_c\frac{t-\tau }{s}}}}\underset{\begin{array}{c}\text{Gaussian}\\ \text{Window}\end{array}}{\underbrace{e^{-\frac{{(t-\tau )}^2}{s^2{F}_b}}}}dt.$$ (10)

Notably, the Gaussian window's temporal width $({\sigma }_t=\sqrt{F_b}/2)$ and frequency bandwidth $({\sigma }_f=1/\left(\pi \sqrt{F_b}\right)$ exhibit an inherent trade-off: large $F_b$ values narrow ${\sigma }_f$ to enhance frequency resolution while prolonging ${\sigma }_t$ and degrading temporal localization. Conversely, smaller $F_b$ sharpens time resolution at the expense of frequency bandwidth.

Simultaneously, $F_c$ governs spectral sensitivity through the physical frequency (f) conversion

$$f=\frac{F_c}{s\cdot \mathrm{\Delta }t},$$ (11)

where $\mathrm{\Delta }t$ is the sampling interval. For a given scale s, a higher $F_c$ value indicates that the wavelet transform becomes more sensitive to high-frequency components. Conversely, a decrease in $F_c$ enhances its sensitivity to low-frequency components.

Critically, the parameters $F_b$ and $F_c$ interact synergistically through the total number of oscillation cycles in the wavelet, defined as:

$$N_{osc}=F_c\ast \sqrt{F_b}.$$ (12)

This metric $N_{osc}$ serves as a key indicator for balancing or prioritizing time resolution versus frequency resolution. Smaller $N_{osc}(<1)$ prioritizes temporal resolution, while larger $N_{osc}(\gg 1)$ favors frequency resolution. This interdependence necessitates deliberate optimization of $F_b$ and $F_c$ to meet specific resolution requirements, such as those for wave propagation speed or natural frequency measurements. By doing so, $F_b$ and $F_c$ can be tailored to distinct tasks: improving temporal resolution for wave propagation velocity estimation or enhancing frequency resolution for natural frequency measurement.

3.4. Quantification of mechanical wave speed and natural frequency

A representative air-pulse-induced tissue displacement profile along the wave propagation path is shown in Fig. 3(a), using a 1.2% agar phantom as an example. It consists of two major periods: (1) the primary deformation period (red window), which is directly induced by the air pulse and spans from the onset of displacement to the maximum negative displacement, then recovers to the first positive maximum; and (2) the damped oscillatory motion period (blue window), ranging from the first negative maximum following the primary negative deformation to the end of the measurement window. Using the central stimulation and the radial OCT scanning pattern (Fig. 1(b)), we can measure 360° mechanical wave propagations in tempo-spatial domain, as shown in Fig. 3(b). We can then derive the wave’s group velocity, phase velocity, and natural frequencies from these induced tissue dynamics in Figs. 3(c–p).

Fig. 3.

Time-frequency analysis of mechanical wave speed and natural frequency characterization in a 1.2% agar phantom. (a) Spatiotemporal displacement profile showing surface wave propagation, with the red window indicating the primary deformation for wave speed analysis and the blue window highlighting the damped oscillation period for natural frequency extraction. (b) 360° wave propagation profile over time. (c) The Morlet wavelet basis function, defined by its bandwidth $\left(F_b\right)$ and central frequency $\left(F_c\right)$ parameters. (d) Wavelet transform results of the full displacement profile $(F_b=1,F_c=1)$ . (e–j) Comparison of wave speed calculation: (e–g) Time-of-Flight (TOF) analysis, showing (e) normalized displacements, (f) extracted travel times, and (g) the resulting group velocity; (h–j) Wavelet-based analysis $(F_b=0.1,F_c=1)$ , showing (h) the time-frequency representation, (i) time-distance relations at selected frequencies, and (j) derived phase and group velocities. The dashed black line in (j) indicates the R² threshold (0.95). (k–n) Comparison of natural frequency quantification: (k, l) Wavelet-based analysis $(F_b=20,F_c=3)$ showing the time-frequency response and the spatiotemporal distribution of the dominant natural frequencies; (m, n) Fast Fourier Transform (FFT) analysis showing the corresponding frequency spectrum and spatial map of the dominant natural frequencies. (o, p) Circumferential (360°) comparison of (o) wave speeds and (p) dominant natural frequencies obtained from the different methods.

Figure 3(c) demonstrates the typical Morlet wavelet basis (as described in Eq. (9)), with the dimensionless parameters $F_b$ and $F_c$ representing the bandwidth and central frequency, respectively. Figure 3(d) shows the Morlet wavelet transform of the entire displacement profile in Fig. 3(a), using initial parameters of $F_b=1$ and $F_c=1$ to provide a balanced trade-off between temporal and frequency resolution (N_osc  = 1, Eq. (12)). The resulting time-frequency map reveals two distinct phases: first, a broadband frequency component (0–8 ms, red arrow) corresponding to the primary deformation used for wave velocity estimation; and second, sustained narrowband oscillations (>8 ms, white arrow) representing the tissue's natural frequency. However, this balanced resolution causes artifacts (orange arrows), as velocity analysis needs sharper time resolution, while natural frequency requires sharper frequency resolution. Therefore, we independently optimized $F_b$ and $F_c$ for each phase. Figures 3(e–p) demonstrate this separate characterization of wave speeds and natural frequencies, while Figs. 4 and 5 detail the parameter optimization for wave velocity and natural frequency, respectively.

Fig. 4.

Comparison of wavelet and time-of-flight (TOF) methods for surface wave velocity quantification in agar phantoms (1-2%). (a–e) Wavelet parameter optimization procedure (demonstrated using 1.5% agar): (a) Representative time-frequency waveforms generated using different $F_b$ and $F_c$ combinations. (b) Corresponding phase velocity dispersion profiles across parameter combinations. (c) Absolute difference between wavelet-derived group velocities and the TOF reference value; the minimum identifies optimal parameters ( $F_b=0.2$ and $F_c=1.0$). (d) Time-frequency representation using optimized parameters. (e) Phase/group velocity dispersion curves with R² values; dashed line indicates reliability threshold (R² ≥ 0.95). (f–j) Group velocity comparison between methods: (f–h) 360° group velocity distributions for 1%, 1.5%, and 2% agar phantoms. (i) Correlation between TOF and wavelet-derived group velocities. (j) Bland-Altman agreement analysis between methods.

Fig. 5.

Comparison of wavelet and Fast Fourier Transform (FFT) methods for natural frequency characterization in agar phantoms (1–2%). (a–e) Wavelet parameter optimization procedure (demonstrated using 1.5% agar): (a) Representative time-frequency waveforms generated using different $F_b$ and $F_c$ combinations. (b) Corresponding dominant natural frequency values extracted across parameter combinations. (c) Absolute difference between wavelet-derived dominant natural frequencies and the FFT reference value; the minimum identifies optimal parameters ( $F_b=5$, $F_c=7$). (d) Time-frequency representation using optimized parameters. (e) Spatiotemporal map of dominant natural frequency distribution. (f–j) Natural frequency comparison between methods: (f–h) Circumferentially resolved natural frequency distributions (360°) for 1%, 1.5%, and 2% agar phantoms, showing directionally averaged values. (i) Correlation between FFT and wavelet-derived natural frequencies. (j) Bland-Altman agreement analysis between methods.

For mechanical wave speed estimation, both the TOF and wavelet methods require linear fitting between wave propagation time and distance—the TOF method performs this fit on the full-spectrum signal, whereas the wavelet method applies it to each frequency sub-spectrum individually. To ensure reliable and consistent velocity estimation, we applied an $R^2$ -based quality control threshold to the linear fitting. As shown in Visualization 1 ^{(2MB, tif)} (Supplementary Material; representative fitting results with $R^2$ ranging from ∼0.4 to 1 for ex vivo porcine corneas at 20 mmHg IOP), fits with smaller $R^2$ (e.g., $R^2\hspace{0.17em}<\hspace{0.17em}0.7$) were typically associated with visibly distorted wavefronts and unreliable velocity estimates, while larger $R^2$ values (e.g., ≥ 0.8) provided better linear fitting results for the propagation velocity. Based on this empirical evaluation, we adopted context-dependent thresholds to balance data quality and retention. A stricter threshold $(R^2\hspace{0.17em}\ge \hspace{0.17em}0.95)$ was applied to homogeneous agar phantoms, where ideal measurement conditions—uniform stiffness, minimal scattering, and absence of physiological motion—enable consistently high-quality fits. This stringent threshold ensures that wavelet parameter optimization is based exclusively on the most reliable data, minimizing the influence of residual artifacts. In contrast, a more lenient threshold $(R^2\ge 0.8)$ was used for ex vivo and in vivo corneas to accommodate increased signal variability arising from tissue heterogeneity and physiological motion, while still excluding clearly unreliable fits. This adaptive strategy balances data quality and spatial coverage across different experimental conditions.

Figures 3(e–g) and 3(h–j) compare wave speed quantification using the TOF and Wavelet transform methods, respectively. The TOF method involved three steps: normalizing the displacement profiles (Fig. 3(e)), calculating wave travel times via cross-correlation (Fig. 3(f)), and performing linear regression to determine the group velocity (3.04 m/s, $R^2=0.99$; Fig. 3(g)). In contrast, the wavelet method (here, $F_b=0.1$ , $F_c=1$ ; Eqs. (10–12)) extracted time-frequency features of the primary deformations at each spatial location (Fig. 3(h)) and tracked time shifts of spectral magnitudes—binned at 10 Hz intervals—across distances. Figure 3(i) shows the time-distance fits for five selected frequencies, and Fig. 3(j) presents the corresponding dispersive phase and group velocities. The scattered phase velocity data were first fitted to a continuous curve, from which the group velocities were then derived using Eq. (3). An R² threshold of 0.95 was applied to define the valid frequency range for wave speed estimation, yielding a final averaged group velocity of 3.08 ± 0.04 m/s across 80–1000 Hz. Conceptually, the wavelet method applies the TOF principle across multiple frequency bands, averaging only the reliable, narrowband velocity estimates to produce a robust final quantitative result.

Figures 3(k–n) compare natural frequency quantification using the Wavelet (Figs. 3(k, l)) and FFT methods (Figs. 3(m, n)). Multiple frequency components were observed in the initial analyses (Figs. 3(k, m)); a single-degree-of-freedom (S-DOF) model [40] was applied to identify and map the distributions of the dominant frequency components. For the Wavelet transform, parameters were set to $F_b=20$ and $F_c=3$ to improve frequency resolution (Eqs. (10–12)). This approach enables tracking of natural frequency evolution both over time (during signal decay) and across spatial regions through continuous scale–translation analysis. In contrast, the FFT method provides only the spatial distribution, as it averages frequency content over time and thus loses transient dynamic information.

Figures 3(o, p) further validate these approaches: circumferential mapping over 360° reveals strong agreement between TOF- and wavelet-derived wave speeds (mean and maximum differences: 1.37% and 4.75%, respectively), and between FFT- and wavelet-derived dominant natural frequencies (mean and maximum differences: 0.31% and 0.40%, respectively).

4. Results

4.1. Wavelet time-frequency optimization

To improve measurement reliability, we independently optimized $F_b$ and $F_c$ for wave velocity (Fig. 4) and natural frequency (Fig. 5). This optimization was performed on homogeneous 1–2% agar phantoms. These simple, homogeneous samples provide an ideal benchmark where conventional TOF and FFT methods are known to be accurate under high-SNR conditions, establishing reliable reference values for tuning the wavelet parameters.

Figure 4 demonstrates the wavelet parameter optimization for wave velocity quantification compared to the TOF method. We systematically tested a range of F_b (0.05–0.6) and F_c (0.2–1.0) combinations on a 1.5% agar phantom, finding that different parameters produced distinct time-frequency waveforms (Fig. 4(a)) and phase velocity profiles (Fig. 4(b)). The optimal parameters (F_b  =  0.2 and F_c  =  1.0) were selected by minimizing the difference between wavelet-derived and TOF reference group velocities (0.14 m/s difference from 4.03 m/s; Fig. 4(c)). Using these optimized parameters, we generated time-frequency representations across propagation distances (Fig. 4(d)) and dispersion phase/group velocity curves (Fig. 4(e)), applying an R² ≥ 0.95 threshold to define reliable frequency ranges (104–1000 Hz) for group velocity calculation (4.12 ± 0.17 m/s). Comparative 360° measurements of 1.0–2.0% agar phantoms showed both methods produced closely matched velocities that increased with stiffness (Figs. 4(f–j)): from 2.41 ± 0.05 m/s (TOF) and 2.54 ± 0.04 m/s (wavelet) for 1.0% agar to 4.78 ± 0.32 m/s (TOF) and 4.35 ± 0.30 m/s (wavelet) for 2.0% agar. Bland-Altman analysis confirmed good agreement between methods (mean difference: 0.18 m/s; 95% limits: ±0.55 m/s).

Similarly, Fig. 5 outlines the wavelet parameter optimization for natural frequency, benchmarked against the FFT method. A wide range of F_b (0.01–20) and F_c (0.01–18) values were tested using a 1.5% agar phantom, generating distinctive time-frequency waveforms (Fig. 5(a)) and the resulting dominant natural frequencies (Fig. 5(b)). The choice of parameters was crucial: poor choices resulted in either poorly defined waveforms (Fc ≤ 0.1) or significant measurement errors (e.g., F_b = 0.01–0.5, F_c = 10–18), whereas suitable parameters within a specific range (F_b = 1–20, F_c = 0.2–10) yielded consistent results. The optimal set (F_b = 5, F_c = 7) minimized the difference from the FFT reference (303.41 Hz) to 0.85 Hz (Fig. 5(c)). The resulting time-frequency representation and spatiotemporal distribution of the dominant natural frequencies are shown in Figs. 5(d) and 5(e), respectively. Applying these optimized parameters across 1–2% agar phantoms yielded dominant natural frequencies in close agreement with FFT results (Figs. 5(f–j)): 200.84 ± 0.69 Hz (1.0%), 304.26 ± 0.21 Hz (1.5%), and 406.63 ± 3.71 Hz (2.0%). Bland-Altman analysis confirmed a mean difference of 0.51 Hz with 95% limits of agreement of ±2.24 Hz. Collectively, the results in Figs. 4 and 5 validate the optimized wavelet method as a robust tool for soft material biomechanics characterization, yielding outcomes consistent with established TOF and FFT techniques.

4.2. Time-frequency estimation under different signal-to-noise ratio

After optimizing the wavelet parameters under ideal high-SNR conditions, we evaluated the robustness of the wavelet, TOF, and FFT methods across varying SNR levels. A mirror was used as an ideal sample to characterize detection sensitivity, SNR, and phase stability over a depth range of up to 6.8 mm (Figs. 6(a, b)). Subsequently, OCE measurements were performed on 1–2% agar phantoms at depths ranging from 0.9 mm to 6.8 mm (Figs. 6(c–g)). These depths were selected to emulate varying SNR conditions, as both SNR and the phase detection sensitivity of OCT ( ${\sigma }_{\mathrm{\Delta }\phi }\propto 1/\sqrt{SNR},(SNR\gg 1)$ [53]) degrade systematically with depth due to sensitivity roll-off, thereby providing a controlled gradient of measurement conditions.

Fig. 6.

Effect of signal-to-noise ratio (SNR) on the estimation of group velocities and dominant natural frequencies in agar phantoms (1.0%, 1.5%, 2.0%). (a) The depth-dependent profile of detection sensitivity. SNR was calculated as the ratio of signal magnitude to noise variance. (b) SNRs and phase stability across depth. (c) Representative displacement profiles at different depths (1.5% agar phantom). (d, e) Evaluation of the proposed wavelet method against established techniques: (d) group velocity compared to the Time-of-Flight (TOF) method, and (e) dominant natural frequency compared to the Fast Fourier Transform (FFT) method. (f, g) Coefficient of variation (CV) analysis demonstrating the measurement precision for (f) group velocities and (g) natural frequencies.

Figure 6(a) shows the roll-off of the detection sensitivity ( $20lo{g}_{10}(\mathit{\text{Intensity}}))$ , with the amplitude decreasing from 99.7 dB at 0.35 mm to 70.3 dB at 6.8 mm. The SNR at different depths was derived from this sensitivity profile. Specifically, the signal amplitude at each depth was estimated as the difference between the measured sensitivity and the noise floor (averaged background noise, blue line). The noise component was quantified from the variance of the background noise [54]. Based on these definitions, the SNR was calculated as the ratio of the signal amplitude to the noise variance. This approach enables quantitative estimation of depth-dependent SNR by directly relating the measured sensitivity roll-off to the underlying signal and noise characteristics. Figure 6(b) shows the resulting SNR values, which decrease from approximately 9.4 at a depth of 0.35 mm to 3.4 at 6.8 mm, accompanied by a corresponding decline in phase stability from 3.5 nm (0.9 mm depth) to 6.3 nm (6.8 mm depth). This degradation in displacement detection quality with increasing depth is visually apparent in the noisier signals shown in Fig. 6(c). Figure 6(d) compares the group velocities (mean ± STD) obtained from 360° measurements using the wavelet and TOF methods. The wavelet method demonstrated superior precision, evidenced by smaller error bars than the TOF method, particularly at lower SNRs (greater depths). This enhanced precision is quantified by the coefficient of variation (CV = STD/mean) in Fig. 6(f), where the wavelet method maintained consistently lower CV values for group velocity. For instance, at a depth of 6.8 mm in 2% agar, the CV was 0.27 for the wavelet method compared to 0.52 for the TOF method. In contrast, for dominant natural frequency quantification, the wavelet and FFT methods exhibited comparable variance, as shown by their similar standard deviations and CVs in Figs. 6(e, g). These results validate the wavelet method as a robust technique for assessing tissue biomechanical properties, particularly for wave speed estimation in low-SNR conditions.

4.3. Ex vivo porcine corneal measurement

We further tested the wavelet method on anisotropic, stress-dependent, ex vivo porcine corneas (26 right eyes) under IOP levels from 5 mmHg to 20 mmHg, and compared group velocity quantification with the TOF method (Fig. 7) and dominant natural frequency characterization with the FFT method (Fig. 8).

Fig. 7.

Comparison of wavelet and time-of-flight (TOF) methods for surface wave velocity quantification in ex vivo porcine corneas (right eye, n = 26; IOP: 5–20 mmHg). (a) Representative displacement profiles, with increased noise observed at higher IOP (e.g., 20 mmHg). Colors indicate different measurement distances. (b) TOF group velocity fitting, showing a decrease in as IOP increases. (c) Wavelet-derived phase and group velocities, with average group velocities meeting the R² ≥ 0.8 criterion. (d) Circumferential (360°) comparison of group velocity distributions between methods, with histograms indicating the number of valid measurements per direction that satisfied the R² ≥ 0.8 threshold. (e–g) Quantitative comparison of TOF and wavelet results across IOP levels, including Bland-Altman agreement analysis.

Fig. 8.

Comparison of wavelet and Fast Fourier Transform (FFT) methods for dominant natural frequency characterization in ex vivo porcine corneas (n = 26 right eyes; IOP: 5–20 mmHg). (a) Representative temporal damped oscillation profiles. (b, c) Natural frequency maps from FFT (spatial) and wavelet (spatiotemporal) analyses, respectively. Dominant frequencies were determined using mode-based analysis (≥80% occurrence threshold) applied to discrete frequency values. (d) Circumferential distributions of directionally averaged dominant natural frequencies with histograms showing valid (red; ≥80% mode criterion) and invalid (black) measurements. Invalid data indicate structural heterogeneity rather than measurement error. (e–g) Quantitative comparison across IOP levels using Bland-Altman analysis, showing remarkable method agreement under ex vivo conditions.

For the group velocity comparison (Fig. 7), an $R^2\ge 0.8$ threshold was applied for both methods. Figure 7(a) shows representative displacement profiles with selected, relatively good-quality datasets for each IOP condition. Notably, these displacement signals became noisier at higher IOP levels (e.g., 20 mmHg). This trend is also reflected in the declining fit quality of the TOF method, with $R^2$ decreasing from 0.996 at 5 mmHg to 0.915 at 20 mmHg (Fig. 7(b)). The degradation of displacement signals at higher IOP reflects changes in the measurement conditions. As IOP increases, the stress within the corneal tissue rises, and its layered structure (including air-water boundaries and collagen fiber alignment) complicates wave propagation by inducing reflections, superposition, and mode conversion [34]. This effect is also evident in Fig. 7(c), where the wavelet-derived phase and group velocities show less dispersion in low-IOP situations (5 mm Hg and 10 mmHg), while has much greater dispersive features at high IOP conditions (15 and 20 mmHg). We chose the plateau regions that meet the threshold $(R^2\ge 0.8)$ for group velocity averaging (Fig. 7(c)). Figure 7(d) compares the spatial distribution of the averaged group velocities from both methods that met the $R^2\ge 0.8$ criterion across 360°, with a histogram quantifying the number of valid measurements per direction. Notably, the wavelet method preserved more valid data than the TOF method, with the valid percentages for each IOP as follows: 97.3% (TOF) vs. 98.6% (wavelet) at 5 mmHg, 98.1% (TOF) vs. 99.0% (wavelet) at 10 mmHg, 86.3% (TOF) vs. 93.8% (wavelet) at 15 mmHg, and 50.0% (TOF) vs. 86.3% (wavelet) at 20 mmHg. At the highest IOP (20 mmHg), half of the TOF-derived group velocities were deemed invalid, while 86% of wavelet-derived velocities remained usable, demonstrating the wavelet method’s robustness to noise. Figures 7(e, f) present the group velocity data for both methods across the 26 eyes at different IOPs. The average group velocities for all directions were: 1.95 m/s (TOF) and 2.05 m/s (wavelet) at 5 mmHg, 2.94 m/s (TOF) and 3.03 m/s (wavelet) at 10 mmHg, 3.45 m/s (TOF) and 3.92 m/s (wavelet) at 15 mmHg, and 3.90 m/s (TOF) and 4.52 m/s (wavelet) at 20 mmHg. The Bland-Altman analysis in Fig. 7(g), performed on the overlapping valid dataset, shows a mean difference of 0.25 m/s and 95% limits of agreement of ±1.58 m/s.

Figure 8 compares the performance of wavelet and FFT methods for characterizing dominant natural frequency in ex vivo porcine corneas across different IOP levels. Panel (a) displays representative temporal damping profiles, showing a clear increase in oscillation frequency with elevated IOP. The corresponding spatial distribution of dominant frequencies from FFT analysis is shown in Fig. 8(b), while the spatiotemporal evolution from wavelet analysis is shown in Fig. 8(c), revealing the evolution of frequency components along the wave propagation direction over time.

Since the extracted natural frequencies form discrete datasets rather than continuous distributions, conventional averaging methods are unsuitable for determining representative values. Instead, we employed a mode-based statistical approach, identifying the most frequently occurring dominant natural frequency within each spatial (FFT) or spatiotemporal (wavelet) unit. It is important to clarify the dual meaning of “mode” in this context: in mechanical testing, “modal analysis” refers to the study of vibration patterns and resonant frequencies of a physical system, while in statistics, “mode” specifically denotes the value with the highest occurrence rate in a dataset. Our mode-based statistical approach effectively captures dominant vibrational behavior by selecting values with the highest occurrence rates, avoiding the ambiguity of averaging disparate discrete values. We empirically implemented a threshold (≥80% occurrence rate within a typical frequency range of several tens to over 100 Hz) to ensure consistent frequency selection, then averaged the valid components within this range to determine directional dominant frequencies.

Figure 8(d) shows the 360° distributions of these directionally averaged, mode-selected dominant natural frequencies, with histograms indicating the proportion of valid (red, meeting ≥80% criterion) and invalid (black) measurements. Notably, “invalid” data do not indicate useless measurements but rather reveal higher-order structural details, including anisotropic and heterogeneous tissue properties. While the directional averaged dominant natural frequencies show a homogeneous pattern with increasing values at higher IOP, the histograms demonstrate increasing frequency component diversity. For instance, at 20 mmHg, two Gaussian-shaped mode distributions emerge, contrasting with the single distribution observed at 5-15 mmHg.

Quantitative comparison in Figs. 8(e–g) shows remarkable consistency between FFT and wavelet methods across all IOP levels. The directionally averaged dominant natural frequencies for 26 eyes were 138.86 ± 15.23 Hz (FFT) vs 137.61 ± 15.06 Hz (wavelet) at 5 mmHg, 209.89 ± 37.29 Hz vs 209.75 ± 37.10 Hz at 10 mmHg, 355.75 ± 31.45 Hz vs 355.58 ± 31.60 Hz at 15 mmHg, and 688.18 ± 35.68 Hz vs 688.60 ± 35.45 Hz at 20 mmHg. Bland-Altman analysis confirmed excellent agreement with a minimal mean difference of -0.29 Hz and narrow 95% limits of agreement (±4.32 Hz). The absence of time-varying physiological factors (e.g., heartbeat) in this ex vivo setup explains the nearly identical performance of both methods. The following section will demonstrate their performance under in vivo conditions where physiological motions introduce additional complexity.

4.4. In vivo rabbit corneal measurement

To illustrate the time-varying biomechanical behavior of the cornea under physiological conditions such as heartbeat and respiration, we performed 360° in vivo rabbit cornea measurement using the air-pulse OCE system, as shown in Fig. 9. The experimental setup and corneal orientation (superior-S, inferior-I, nasal-N, temporal-T) are shown in Fig. 9(a). Representative spatiotemporal displacement profiles from these four directions (Figs. 9(b–e)) reveal the propagation of air-pulse-induced surface waves, capturing both the initial primary deformation (red boxes) and subsequent damped oscillations (blue boxes).

Fig. 9.

In vivo time-frequency characterization of rabbit cornea using air-pulse OCE. (a) Experimental photograph showing corneal directions: superior (S), inferior (I), nasal (N), temporal (T). (b–e) Spatiotemporal displacement profiles along four representative directions (S, T, I, N). (f–i) Wave speed analysis: (f) TOF-derived group velocities; (g) Wavelet-derived group velocities; (h) R² values for TOF fitting (16/20 directions met R² ≥ 0.8 validity threshold); (i) Circumferential comparison of valid group velocities between methods. (j–n) Dominant natural frequency characterization: (j) FFT-derived spatial frequency distribution; (k) Wavelet-derived spatiotemporal frequency distribution; (l, m) Histograms of valid (red, ≥80% mode criterion) and invalid (black) measurements; (n) Circumferential comparison of dominant natural frequencies between methods.

Wave speed analysis using an R² ≥ 0.8 validity threshold (Figs. 9(f–i)) demonstrates the wavelet method’s superior robustness for in vivo conditions. Group velocities derived from the TOF and wavelet methods for the four representative directions are shown in panels (f) and (g), respectively. Notably, the TOF method failed to produce a valid measurement in the temporal direction (Fig. 9(f)) and achieved only 16/20 valid measurements overall (Fig. 9(h)). In contrast, the wavelet method maintained 100% validity (20/20) by selectively averaging velocities from spectral components that met the reliability threshold. The circumferential comparison of the overlapping valid datasets (Fig. 9(i)) shows strong agreement between methods (mean difference: 8.8%; maximum: 23.2%), confirming the reliability of the wavelet-based approach for in vivo wave velocity estimation.

Natural frequency characterization (Figs. 9(j–n)) demonstrates a potential advantage of the wavelet method for in vivo biomechanical assessment—its ability to resolve temporal dynamics that are not captured by the FFT approach. While both methods yield similar spatial frequency distributions (Figs. 9(j, k)), the FFT result represents a single spectrum averaged over the entire measurement period. In contrast, the wavelet-derived spatiotemporal map captures the temporal evolution of frequency, enabling visualization of transient variations that may be associated with physiological states (e.g., systole vs. diastole, inhalation vs. exhalation) and providing a more nuanced interpretation of in vivo biomechanics. The dominant frequencies were determined using a mode-based statistical approach (≥80% occurrence threshold), with the distributions of valid (red) and invalid (black) measurements shown in Figs. 9(l, m). Both methods achieved approximately 86% valid frequency detection for evaluating fundamental mechanical properties. The circumferential comparison in Fig. 9(n) demonstrates excellent agreement between FFT- and wavelet-derived results, with mean and maximum differences of 0.8% and 1.9%, respectively, for directionally averaged frequencies. This strong agreement in averaged values confirms methodological reliability, while the wavelet approach offers an additional dimension—transient dynamics—that may be essential for comprehensive in vivo characterization.

5. Conclusion and discussion

In this study, we introduced and validated a wavelet-based time-frequency analysis method for quantifying both mechanical wave propagation velocity and natural frequency in air-pulse OCE. The surface wave displacement response was divided into a primary deformation period for wave propagation analysis and a damped oscillation period for natural frequency extraction (Figs. 3–5). The wavelet parameter optimization and performance comparison were conducted under different noise conditions. Optimization was first performed under controlled, high-SNR conditions using well-established reference methods (TOF and FFT, Figs. 4 and 5) to ensure reliable wavelet parameter settings. In contrast, the performance comparison was then conducted under varying SNR conditions (e.g., at different depths, Fig. 6), as well as in ex vivo (Figs. 7 and 8) and in vivo corneal measurements (Fig. 9), where tissue heterogeneity and physiological motion introduced greater signal variability. This two-stage strategy—calibration under ideal conditions followed by evaluation under more challenging conditions—ensures a fair comparison across methods. The Morlet wavelet parameters—bandwidth $\left(F_b\right)$ and central frequency $\left(F_c\right)$ —were independently optimized to balance temporal and spectral resolution, with the optimal combinations being F_b = 0.2, F_c = 1.0 for velocity and F_b = 5, F_c = 7 for natural frequency characterization. Using these optimized parameters, the wavelet method achieved precise and consistent quantification across tissue-mimicking phantoms (Figs. 3–6), ex vivo (Figs. 7, 8), and in vivo corneas (Fig. 9).

The wavelet approach demonstrated superior robustness for wave velocity estimation under challenging conditions, including deeper regions with low SNR (Fig. 6), stressed ex vivo corneas at 20 mmHg with altered collagen fiber tension and consequent complex wave dynamics (Fig. 7), and in vivo environments with time-varying biomechanical behavior (Fig. 9). In low-SNR scenarios, the coefficient of variation for wavelet-derived group velocity was nearly 50% lower than that of TOF at a 6.8 mm depth, highlighting its enhanced reliability. This advantage was further evidenced in ex vivo porcine corneas, where the wavelet method maintained a high proportion of valid measurements (86.3%) at 20 mmHg IOP—a condition under which 50% of the TOF results were invalidated. The method's consistency was also confirmed in vivo, where it yielded valid group velocities in all directions (20/20) compared to the TOF method (16/20, Fig. 9(h)).

For natural frequency quantification, the wavelet method achieved excellent agreement with FFT in controlled settings—from phantoms (Figs. 5 and 6) to ex vivo corneas (Fig. 8)—while offering an additional dimension: the ability to resolve frequency evolution over time. This capability enables the detection of transient variations in corneal vibrational behavior under in vivo conditions (Fig. 9). This distinction is potentially significant because the spatial heterogeneity captured by FFT at a single time point (Fig. 9(j)) could, in principle, arise from temporal variations occurring during specific physiological phases—such as heartbeat, respiration, or ocular pulsation—rather than solely from structural anisotropy. The wavelet’s spatiotemporal representation of frequency distribution allows dynamic visualization of these evolving patterns, which may help differentiate physiological fluctuations from intrinsic biomechanical features in future studies. Such adaptive time-frequency mapping could potentially minimize motion-induced artifacts and provide a more comprehensive depiction of corneal stiffness dynamics. The ability to reveal time-dependent changes in corneal mechanical properties may be advantageous for future clinical translation, particularly as physiological motion and variable SNR often challenge conventional frequency-domain methods. However, we acknowledge that the current study does not establish the specific physiological origins of the observed temporal variations. Future work will focus on investigating these effects using controlled experimental models (e.g., pressure-modulated corneal phantoms that mimic cardiac cycles) and extended in vivo studies. A deeper understanding of these interactions will support the development of dynamic OCE toward physiology-synchronized biomechanical analysis, potentially enabling improved functional assessment and early disease detection in ophthalmic applications.

It should be noted that the wavelet-based approach incurs higher computational complexity and processing time than the TOF and FFT methods. For a typical surface displacement dataset comprising 20 radial directions × 17 spatial points × 3000 time samples (∼1 million data points), TOF analysis requires approximately 0.4 s on a standard desktop computer (Intel i7-10700, 32 GB RAM), while FFT analysis completes in about 1.1 s. In contrast, wavelet-based mechanical wave velocity estimation—implemented using a continuous wavelet transform across 100 scales (binned at 10 Hz intervals)—requires approximately 108 s per dataset, representing a ∼270× increase in computational cost. For natural frequency estimation, the wavelet method does not require scale division and therefore completes in approximately 13 s. Although the increased processing time, particularly for velocity estimation, may limit real-time applications, it could be substantially reduced through parallel computing or GPU acceleration.

In this study, we employed a single-degree-of-freedom (S-DOF) model [40] to focus on dominant frequency components that met a modal-distribution criterion (≥80% occurrence threshold; Figs. 8 and 9). However, biological tissues such as the cornea are inherently multi-degree-of-freedom (M-DOF) systems and exhibit a complex spectrum of vibrational modes. While this simplification prioritized methodological stability and noise resilience by isolating the fundamental resonant response, it inevitably overlooks subtle spatial and temporal variations in stiffness and damping. This may explain why circumferential wave speed measurements exhibit clearer directional anisotropy than natural frequency maps (Figs. 7–9), suggesting that wave propagation is more sensitive to structural anisotropy (e.g., collagen orientation), whereas the dominant natural frequency reflects a more globally averaged stiffness property. Notably, the “invalid” frequency components—those outside the dominant occurrence range—provide valuable information about structural diversity, with their distributions revealing higher-order tissue heterogeneity. Collectively, all frequency components (low- and high-order, valid and invalid) contribute to characterizing the sample's structural complexity. To more comprehensively characterize corneal anisotropy and spatial heterogeneity, future work should treat the cornea as an M-DOF dynamic system and apply modal analysis to the induced deformation responses. Such an approach would enable extraction of multiple modal parameters—including different orders of natural frequencies, damping ratios, and mode shapes—providing deeper insight into localized biomechanical variations and structural complexity. Integrating this framework with wavelet-based time-frequency analysis could further enable multi-mode decomposition and spatially resolved mapping of corneal microstructure and viscoelastic behavior, ultimately extending OCE toward full modal characterization to advance both measurement precision and biomechanical understanding.

For wavelet-based time-frequency analysis, selecting an appropriate basis is critical for accurately characterizing both wave propagation velocity and natural frequency oscillations from air-pulse-induced tissue dynamics. However, no universal standard exists because analytical requirements differ across applications and signal characteristics. In the present study, the Morlet wavelet was selected for its well-balanced time-frequency resolution. This foundational strength is complemented by three practical advantages [51,52]: (1) its Gaussian spectral profile minimizes spurious ripple artifacts that could otherwise be misinterpreted as physical oscillations, enhancing feature discrimination; (2) it preserves the temporal resolution of the original signal features, maintaining timing accuracy during decomposition; and (3) its computational efficiency through FFT-based implementation reduces processing load while ensuring analytical validity. These combined characteristics make the Morlet wavelet particularly suitable for high-resolution analysis of non-stationary signals, where detecting transient features while preserving temporal precision is essential. Notably, the optimized wavelet parameters (F_b = 0.2, F_c = 1.0 for velocity; F_b = 5, F_c = 7 for natural frequency) were determined using 1.0–2.0% agar phantoms with cornea-relevant stiffness. While these values are specific to the present experimental conditions, the underlying optimization workflow is general and can be extended to other biological tissues with different mechanical properties, such as arterial walls (stiffer), skin (softer), or brain tissue (highly viscoelastic with complex dispersion). As an initial exploration, this work establishes a foundation for wavelet analysis in OCE. Future investigations could explore whether other bases offer superior performance. For example, a single adaptive basis with variable bandwidth could simultaneously accommodate both the primary deformation and damped oscillation periods, simplifying the analysis pipeline. Alternatively, a dual-basis approach could potentially enhance feature sensitivity—for example, employing the Mexican Hat (Ricker) or Paul wavelets for their sharp temporal localization to track the wavefront, paired with the Meyer or Shannon wavelets for their high spectral resolution to characterize the resonant oscillations. While the relative performance of these bases remains to be systematically evaluated, the optimization workflow demonstrated in this study provides a useful reference for parameter tuning in future investigations. Continued exploration of alternative wavelet bases and adaptive strategies will help establish optimal basis selection protocols under diverse excitation conditions and tissue types, ultimately advancing wavelet-based OCE toward broader biomechanical and clinical applications.

In summary, the wavelet-based OCE method establishes a robust, flexible, and noise-tolerant analytical foundation for dynamic corneal elastography. It complements conventional TOF and FFT analyses while extending OCE capability toward in vivo assessment of anisotropic and time-varying biomechanical behavior in ocular tissues. By enabling simultaneous temporal and frequency localization, the wavelet approach offers a more physiologically relevant characterization of corneal dynamics. This advancement represents a step toward clinical translation of air-pulse OCE, with potential benefits for early detection of corneal pathologies, quantitative monitoring of therapeutic interventions, and optimization of refractive surgery outcomes. While the method has been validated in phantoms, ex vivo tissues, and in vivo animal models, further studies in controlled and clinical settings are needed. Future work will focus on large-scale in vivo human studies and continued optimization of the wavelet-based OCE method to support its integration into clinical practice.

Funding

National Natural Science Foundation of China 10.13039/501100001809 ( 62575066); Basic and Applied Basic Research Foundation of Guangdong Province 10.13039/501100021171 ( 2024A1515011344); Guangdong Pearl River Talents Program ( 2019ZT08Y105); Guangdong Eye Intelligent Medical Imaging Equipment Engineering Technology Research Center ( 2022E076).

Disclosures

GL, JX and YH: Weiren Meditech Co., Ltd. (C); JQ. and LA: Weiren Meditech Co., Ltd. (E).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.