⚡ electrical engineering

Robust Wave Origin Detection from Sensor Array Data via Randomized Hough Transform and Model Fitting

This paper proposes a robust framework for estimating wave origins from sensor array data by combining Butterworth filtering, a 3D randomized Hough transform for candidate identification, and least-squares model fitting to effectively handle noisy and heterogeneous conditions.

Original authors: Sicheng Fan, Jiayi Lu, Xiaodan Fan

Published 2026-09-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sicheng Fan, Jiayi Lu, Xiaodan Fan

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, silent grid of tiny sensors, each one listening intently to the faint electrical whispers of living cells. In the world of biology and medicine, these sensor arrays act like a high-resolution microphone for the heart or the stomach, capturing the moment a wave of electrical activity ripples through tissue. This activity is vital; it tells doctors and scientists how an organ is functioning, whether it is beating in a healthy rhythm or struggling with a disease. However, listening to these signals is rarely a clean experience. The data is often cluttered with static, random electrical noise, and missing pieces, making it incredibly difficult to trace where a wave began, how fast it traveled, and which direction it was heading. For years, researchers have struggled to separate the true signal from the chaos, especially when multiple waves overlap or when the sensors themselves are imperfect.

To solve this, a team of researchers has developed a new way to listen to these electrical storms. They treat the data not just as a collection of isolated points, but as a three-dimensional shape unfolding over time. By combining a specialized noise filter with a mathematical technique originally designed to find lines in images, they can now reconstruct the path of these invisible waves with remarkable clarity. Their method, tested on both computer simulations and real heart cells, successfully identifies the origin and speed of electrical waves even when the data is messy or incomplete. This advancement offers a clearer window into the electrical life of cells, potentially helping researchers understand heart conditions and other biological processes with greater precision.

The core challenge the researchers faced was that electrical waves do not appear as simple dots on a screen. When a wave moves across a grid of sensors, it hits each sensor at a slightly different time. If you plot the location of each sensor and the exact moment it fired, the points form a flat, slanted surface in a three-dimensional space. The researchers realized that if they could find these flat surfaces hidden within the noisy data, they could work backward to find the wave's source. The difficulty lies in the noise. Real-world recordings are filled with random spikes that look like signals but are actually just interference. Traditional methods often get confused by this clutter, mistaking noise for a wave or missing a real wave entirely.

To cut through the confusion, the team first cleaned the data using a digital filter known as a Butterworth filter. Think of this as a sieve that lets the high-pitched, rapid changes of the electrical signal pass through while blocking out the low, rumbling background noise. Once the signal was clean, the researchers looked for sharp peaks, or "spikes," which represent the moment a sensor detected the wave. They then took these spikes and fed them into a process called the Randomized Hough Transform. This is a method that looks for patterns by voting. Imagine throwing a handful of sand onto a table; if the grains happen to land in a straight line, the method recognizes that line even if a few grains are scattered elsewhere. In this case, the "grains" are the signal spikes, and the "lines" are the flat surfaces representing the waves. The algorithm randomly picks a few points to test if they form a plane, and if enough other points agree with that shape, it confirms a wave has been found.

The researchers tested this approach rigorously. First, they created a perfect, computer-generated world where they knew exactly where the waves started and how fast they moved. They introduced artificial noise and missing data to mimic real-world problems. In these simulations, their method proved incredibly robust. It correctly identified the waves almost every time, misclassifying only a tiny fraction of the noise as a signal. Even when up to 80 percent of the signal was missing, the algorithm could still pinpoint the wave's origin and speed with high accuracy. The computer simulations showed that as the noise increased, the method's confidence decreased slightly, but it remained reliable enough to be useful.

Encouraged by the simulations, the team moved to real biological data. They used a grid of 64 sensors to record the electrical activity of cultured heart cells taken from rat embryos. These cells naturally generate waves of electricity that travel across the grid, similar to how a heartbeat travels through the heart muscle. The raw data from these cells was messy, filled with the kind of unpredictable noise found in any living system. After applying their cleaning and pattern-finding steps, the algorithm successfully separated the data into five distinct waves. It determined that all five waves were traveling at speeds between 0.44 and 0.49 detectors per millisecond. More importantly, it traced their paths back to a common direction, revealing that the waves were originating from outside the grid and moving inward, a finding that matched the researchers' biological expectations.

The study also compared two different ways of modeling the waves. One model assumed the wave started from a specific point nearby, like a stone dropped in a pond creating expanding circles. The other model assumed the wave was coming from very far away, appearing as a straight line as it approached the sensors. In the case of the heart cells, the circular model fit the data slightly better, suggesting the waves were indeed originating from a specific area just outside the sensor grid. Both models, however, agreed on the direction and speed of the waves, reinforcing the reliability of the results. The researchers noted that while their method is highly effective, it is not a magic bullet for every situation. It works best when the waves travel at a consistent speed and in a straight line, and it can struggle if multiple waves from different sources mix together in complex ways.

Ultimately, this work provides a powerful new tool for scientists who study the electrical behavior of living tissues. By turning a chaotic stream of data into a clear picture of moving waves, the method allows researchers to see the invisible mechanics of life with greater detail. The team has made their code and data available to others, inviting further exploration. As they look to the future, they suggest that their approach could be adapted to handle even more complex scenarios, such as irregular sensor grids or non-linear wave patterns, potentially opening new doors in the study of cardiac health and other biological systems. The ability to accurately track these electrical journeys is a significant step forward, turning the static of noise into a clear story of how life moves.

Technical Summary: Robust Wave Origin Detection from Sensor Array Data via Randomized Hough Transform and Model Fitting

1. Problem Statement

The robust characterization of wave-like propagating signals is frequently challenged by the heterogeneity of transmitting media and sensor networks. Detecting individual waves under noisy conditions remains difficult, particularly when utilizing sensor arrays (e.g., microelectrode arrays or focal-plane arrays) that generate spatiotemporal data. While the Hough Transform (HT) is a established tool for geometric model identification, existing applications in wave detection often operate in two-dimensional parameter spaces. However, modern sensor arrays (such as MEAs for bio-electricity or Quantum dot FPAs for infrared) generate data with three dimensions: two spatial coordinates ( $x, y$ ) and time ( $t$ ). Processing this 3D data using traditional deterministic methods is computationally expensive and often inefficient. Furthermore, distinguishing true wave propagation planes from noise spikes in this high-dimensional space requires a robust, efficient algorithm capable of handling missing data and measurement errors.

2. Methodology

The proposed framework consists of a three-stage pipeline designed to estimate wave origins, propagation speeds, and excitation times from sensor array data.

Stage 1: Data Preprocessing and Peak Extraction

Raw time-series signals from each sensor channel are processed to extract significant signal spikes while suppressing noise.

Filtering: A Butterworth high-pass filter is applied to the input time series to remove low-frequency interference.
Peak Detection: A global peak is identified, followed by an iterative search for subsequent peaks. A dynamic threshold, set at the 99.95th percentile of the filtered signal amplitude, is used to mark spikes.
Neighborhood Exclusion: To ensure temporal accuracy, a minimum time gap ( $G$ ) is enforced between detected spikes, excluding the time neighborhood of a detected spike from subsequent searches.
Output: The continuous signal is converted into a discrete set of spike tuples: $\{(x_i, y_i, t_i)\}_{i=1}^n$ , representing the spatial location and occurrence time of each spike.

Stage 2: Randomized Hough Transform (RHT) for Wave Plane Detection

The core innovation involves a Randomized 3D Hough Transform to identify planar structures in the $(x, y, t)$ space. Since a wave propagating at a constant homogeneous speed forms a plane in this 3D space, the algorithm seeks to cluster spikes belonging to the same wavefront.

Optimized Voting: Unlike standard HT, this method employs a randomized approach where points vote for potential planes. An accumulator threshold (8 votes) triggers the formation of a candidate plane.
Cluster Validation: A plane is accepted only if the number of points on it exceeds a specific threshold (set to two-thirds of the total detector units in the array).
Iterative Refinement: Points not initially classified are re-evaluated based on proximity to discovered planes. Points sufficiently close to a valid plane are added to that cluster; unclassified points are treated as noise.
Bio-Electricity Adaptation: For single-source scenarios (e.g., cultured heart cells), an enhanced RHT is proposed. It designates the normal vector of the plane with the most points as the "master" direction. Any other detected plane with a normal vector deviating by more than $5^\circ$ from the master is discarded to filter out aberrant detections.

Stage 3: Model Fitting and Parameter Estimation

Once spikes are clustered into wavefronts, a least-squares fitting process estimates the physical parameters of the wave. The method adapts to the source distance:

Near-Field (Circular Wavefront Model): Used when the source is close to or inside the array. The model assumes a source at $(x_0, y_0)$ $(x_{0}, y_{0})$ with speed $v$ $v$ . Due to the high coupling of parameters, an alternating minimization strategy (Algorithm 3) is employed:
1. Fix speed $v$ and excitation time $t_0$ to optimize source location $(x_0, y_0)$ using an iterative tabulation method (zooming into a 4x4 lattice).
2. Fix $(x_0, y_0)$ to optimize $v$ and $t_0$ via linear regression (transforming the problem by setting $u = 1/v$ ).
3. Iterate until convergence.
Far-Field (Linear Wavefront Model): Used when the source is distant. The wavefront is approximated as a plane $ax + by + c = 0$. This reduces to a standard linear least-squares regression problem, yielding a closed-form solution for the direction vector $(\tilde{a}, \tilde{b})$ and propagation speed $v$ .

3. Key Contributions

3D Randomized Hough Transform: The paper introduces a modified RHT specifically tailored for wave detection in 3D spatiotemporal sensor array data, addressing the computational inefficiency of deterministic 3D HT.
Hybrid Fitting Framework: It proposes a dual-model approach (Circular vs. Linear) with specialized optimization strategies (alternating minimization for circular, closed-form least squares for linear) to handle varying source distances.
Robustness to Noise and Missing Data: The integration of dynamic thresholding, neighborhood exclusion, and proximity-based re-clustering allows the method to maintain accuracy even with significant noise and signal dropout (missing spikes).
Single-Source Constraint: An enhanced algorithmic step for bio-electricity applications that enforces a dominant propagation direction, effectively filtering noise in single-source biological experiments.

4. Experimental Results

The method was validated through both synthetic simulations and real-world biological data.

Simulation Studies

Linear Wavefront Detection: Using synthetic data mimicking an 8x8 MEA grid with added noise and missing signals, the RHT achieved a mean False Positive Rate (FPR) of 6.5% and a False Negative Rate (FNR) of 3.6%. The algorithm successfully separated multiple waves and identified planes with high accuracy.
Circular Wavefront Fitting: Using a 96x96 grid simulation, the alternating minimization algorithm was tested under varying Signal-to-Noise Ratios (SNR) and signal missing probabilities ( $p$ $p$ ).
- As SNR increased, parameter estimates ( $x_0, y_0, v, t_0$ ) converged closer to ground truth.
- The algorithm remained robust even with high signal missing probabilities (up to $p=0.8$ ), though estimation accuracy for speed $v$ decreased slightly as $p$ increased.
- Measurement error variance ( $\sigma$ ) showed a direct correlation with estimation error; lower noise yielded higher precision.

Real-World Data Study

The framework was applied to extracellular electrical potential signals from cultured cardiomyocytes (Sprague-Dawley rats) recorded on a 64-channel MEA.

Wave Separation: The RHT successfully identified five distinct propagating waves from the raw data, matching the expected number of action potentials.
Parameter Estimation:
- Circular Model: Identified wave sources originating from outside the search space (boundary at $y_0 = -50$ ), indicating a far-field source. Estimated propagation speeds were consistent across waves ($0.44 - 0.49$ detectors/ms).
- Linear Model: Provided direction vectors and speeds that closely matched the circular model results.
Model Comparison: The Circular Wavefront model yielded slightly higher coefficients of determination ( $R^2 \approx 0.94 - 0.96$ ) compared to the Linear model, suggesting a marginally better fit for the specific experimental geometry, though both were highly accurate.

5. Significance and Claims

The authors claim that this work provides a robust and computationally efficient framework for wave origin detection in sensor arrays. By transforming the problem into a 3D plane detection task and utilizing a Randomized Hough Transform, the method overcomes the limitations of traditional 2D processing and deterministic 3D approaches.

Key claims regarding significance include:

Superior Robustness: The method demonstrates high tolerance to noise and missing data, crucial for biological and physical sensor applications where data quality varies.
Closed-Loop Process: The workflow achieves a complete "detection-separation-fitting" cycle, enabling the reconstruction of signal propagation paths, source locations, and velocities.
Practical Applicability: The successful application to real cardiac myocyte data validates the method's utility in pharmacological and biological studies, offering a tool to analyze wave propagation dynamics without requiring prior knowledge of the source location.
Future Scalability: While the current implementation is efficient, the authors note that for very large arrays (e.g., 64x64), GPU parallelism could further accelerate computation. They also suggest that clustering algorithms like DBSCAN could be integrated to handle complex multi-source scenarios more effectively in future iterations.