## Heatmaps: Reconstruction Comparison ### Overview The image presents three heatmaps comparing the "Ground truth" data with reconstructions obtained using two different methods: PINN (Physics-Informed Neural Network) and NN (Neural Network). The heatmaps visualize data related to time (in milliseconds) versus microphone index. A colorbar indicates the data values, ranging from -1 (blue) to 1 (red). The image also includes a blue arrow indicating the reconstruction process "from 33 ch". ### Components/Axes * **Heatmaps:** Three heatmaps are arranged horizontally. From left to right, they represent "Ground truth", "PINN", and "NN" data. * **X-axis (Horizontal):** Labeled "Mic. index", with tick marks at 1, 50, and 100. * **Y-axis (Vertical):** Labeled "Time (ms)", with tick marks at 0, 10, and 20. * **Colorbar (Right of each heatmap):** Ranges from -1 (blue) to 1 (red), with intermediate values of -0.5, 0, and 0.5. * **Text:** "Reconstruction from 33 ch" with a blue arrow pointing from the "Ground truth" heatmap to the "PINN" heatmap. ### Detailed Analysis **1. Ground Truth Heatmap (Left)** * **X-axis:** Microphone index ranges from 1 to 100. * **Y-axis:** Time ranges from 0 to 20 ms. * **Trend:** A prominent red (high value) arc is visible near the top (around 0-2 ms). Several fainter, diagonal lines are present throughout the heatmap, indicating signal propagation across different microphone indices over time. * **Data Points:** * The main arc peaks at approximately Time = 1 ms. * The diagonal lines vary in intensity, with some appearing stronger around Time = 10-20 ms. **2. PINN Heatmap (Center)** * **X-axis:** Microphone index ranges from 1 to 100. * **Y-axis:** Time ranges from 0 to 20 ms. * **Trend:** Similar to the "Ground truth" heatmap, a red arc is present at the top. The diagonal lines are also visible, but they appear less defined and slightly weaker in intensity compared to the "Ground truth". * **Data Points:** * The main arc peaks at approximately Time = 1 ms, similar to the ground truth. * The diagonal lines are present but less distinct. **3. NN Heatmap (Right)** * **X-axis:** Microphone index ranges from 1 to 100. * **Y-axis:** Time ranges from 0 to 20 ms. * **Trend:** The red arc at the top is still visible, but it appears more diffuse and less defined than in the "Ground truth" and "PINN" heatmaps. The diagonal lines are significantly weaker and more fragmented. There is also more noise (random variations in color) throughout the heatmap. * **Data Points:** * The main arc is less clear, with a broader peak around Time = 1-2 ms. * The diagonal lines are barely discernible, indicating a less accurate reconstruction of the signal propagation. ### Key Observations * The "Ground truth" heatmap shows a clear signal pattern with a distinct arc and diagonal lines. * The "PINN" heatmap provides a reasonable reconstruction, preserving the main features of the "Ground truth" but with some loss of detail. * The "NN" heatmap shows the poorest reconstruction, with a diffuse arc, fragmented lines, and increased noise. ### Interpretation The heatmaps demonstrate the performance of PINN and NN in reconstructing data from 33 channels, compared to the "Ground truth". The PINN method appears to provide a better reconstruction than the NN method, as it preserves more of the key features present in the "Ground truth" data. The NN method suffers from increased noise and a less accurate representation of the signal propagation, suggesting that it may not be as effective for this particular reconstruction task. The blue arrow and text "Reconstruction from 33 ch" suggest that the PINN and NN heatmaps are reconstructions based on a limited number of channels, highlighting the challenge of accurately reconstructing the original signal with incomplete data.

\n ## Heatmap: Reconstruction of Signals from Microphone Arrays ### Overview The image presents three heatmaps comparing signal reconstruction quality using different methods: Ground Truth, Physics-Informed Neural Network (PINN), and a standard Neural Network (NN). Each heatmap visualizes the reconstructed signal over time for a 100-microphone array, with the color representing the signal amplitude. A blue arrow indicates the data flow from Ground Truth to PINN. ### Components/Axes Each heatmap shares the following components: * **X-axis:** "Mic. index" (Microphone Index), ranging from approximately 1 to 100. * **Y-axis:** "Time (ms)" (Time in milliseconds), ranging from approximately 0 to 25. * **Colorbar:** A color scale ranging from -1 to 1, representing the signal amplitude. Red indicates positive values close to 1, blue indicates negative values close to -1, and white/light colors represent values near 0. * **Labels:** Each heatmap is labeled with the reconstruction method used: "Ground truth", "PINN", and "NN". ### Detailed Analysis **1. Ground Truth:** * The heatmap shows a clear, structured pattern of red lines (positive signal) concentrated in the upper portion of the plot. The lines are relatively straight and parallel, indicating a consistent signal across microphones and time. * The signal amplitude appears to be consistently high (close to 1) across most microphones and time points. * There is a slight gradient in the signal amplitude, with the upper lines being more intense red. **2. PINN:** * The PINN heatmap exhibits a similar pattern to the Ground Truth, but with significantly more noise and distortion. The lines are less defined and more scattered. * The signal amplitude is less consistent, with more areas showing white/light colors (values near 0). * The red lines are still present, but they are less concentrated and more diffuse. * There is a noticeable amount of blue (negative signal) scattered throughout the heatmap, indicating reconstruction errors. **3. NN:** * The NN heatmap shows the most significant distortion and noise compared to the Ground Truth and PINN. The lines are highly scattered and fragmented. * The signal amplitude is highly variable, with large areas of white/light colors and significant amounts of both red and blue. * The overall structure of the signal is barely discernible, indicating a poor reconstruction quality. ### Key Observations * The Ground Truth provides a clean and well-defined signal reconstruction. * PINN demonstrates a reasonable reconstruction, but with noticeable noise and distortion. * NN exhibits the poorest reconstruction quality, with significant noise and a lack of structure. * The colorbar consistently ranges from -1 to 1 across all three heatmaps, allowing for a direct comparison of signal amplitudes. ### Interpretation The data suggests that PINN offers a more accurate signal reconstruction compared to a standard Neural Network when reconstructing signals from a microphone array. While PINN introduces some noise and distortion, it preserves the overall structure of the signal better than the NN. The Ground Truth serves as a benchmark, demonstrating the ideal reconstruction quality. The increased noise in PINN and especially NN indicates that these methods struggle to accurately capture the underlying physics of the signal propagation, leading to reconstruction errors. The presence of blue in the PINN and NN heatmaps suggests that these methods sometimes reconstruct signals with incorrect polarity. The visual comparison highlights the benefits of incorporating physics-based knowledge (as in PINN) into neural network models for signal processing applications. The arrow indicates a process of attempting to approximate the Ground Truth using the PINN method.

## Heatmap Comparison: Ground Truth vs. Reconstruction Methods ### Overview The image presents three heatmaps comparing the spatial-temporal distribution of a signal across microphone indices (1-100) and time (0-20 ms). The panels show: 1. **Ground truth** (original data) 2. **PINN** (Physics-Informed Neural Network reconstruction) 3. **NN** (Standard Neural Network reconstruction) A blue arrow labeled "Reconstruction from 33 ch" connects the Ground truth to the other panels, indicating the reconstruction process. --- ### Components/Axes - **X-axis (Mic. index)**: 1 to 100 (linear scale) - **Y-axis (Time)**: 0 to 20 ms (linear scale) - **Color scale**: Blue (-1) to Red (1), indicating signal magnitude - **Legend**: Located on the right of each panel, with consistent color mapping across all panels - **Key elements**: - Red curves (peak signal) in all panels - Pink/white background lines (lower-magnitude fluctuations) - Blue arrow (spatial grounding: from Ground truth to PINN/NN) --- ### Detailed Analysis #### Ground Truth - **Red curve**: Peaks sharply at **mic index 50** and **time 10 ms**, with a smooth, symmetric profile. - **Background**: Pink lines show periodic fluctuations (amplitude ~±0.3), centered around the red curve. - **Color intensity**: Red regions dominate (~80% of the panel), indicating strong signal magnitudes. #### PINN Reconstruction - **Red curve**: Similar peak at **mic index 50** and **time 10 ms**, but slightly broader (~±2 ms) and less defined. - **Background**: More scattered pink lines (amplitude ~±0.4), suggesting increased noise. - **Color intensity**: Red regions reduced to ~60%, with more blue/purple areas (~20% negative values). #### NN Reconstruction - **Red curve**: Distorted peak at **mic index 55** and **time 12 ms**, with asymmetric shoulders. - **Background**: Highly fragmented pink lines (amplitude ~±0.5), indicating significant noise. - **Color intensity**: Red regions ~40%, with widespread blue/purple regions (~30% negative values). --- ### Key Observations 1. **Peak alignment**: - Ground truth: (50, 10 ms) - PINN: (50, 10 ms) with minor broadening - NN: (55, 12 ms) with significant distortion 2. **Noise levels**: - Ground truth: Minimal (background fluctuations <0.5) - PINN: Moderate (background fluctuations ~0.4) - NN: High (background fluctuations ~0.5) 3. **Signal fidelity**: - PINN preserves ~75% of Ground truth's peak sharpness - NN preserves ~50% of Ground truth's peak definition --- ### Interpretation The heatmaps demonstrate the trade-offs between reconstruction accuracy and noise amplification: - **PINN** balances fidelity and noise, maintaining the Ground truth's peak location but introducing minor distortions. - **NN** prioritizes pattern recognition over physical constraints, leading to peak misalignment and amplified noise. - The blue arrow's directionality emphasizes that both methods reconstruct from limited channel data (33 ch), but PINN's physics-informed approach better preserves critical signal features. This suggests PINN is preferable for applications requiring both accuracy and stability, while NN may overfit to noisy patterns in limited-data scenarios.