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This paper presents a method for decoding magnetic texture from magneto-optical images to infer external parameters like magnetic field, temperature, and magnetic history. The approach combines quantitative magneto-optical imaging with deep convolutional neural networks (CNNs) and hand-crafted, physically interpretable features. The authors demonstrate that magnetic textures serve as high-fidelity records of external conditions, enabling accurate multiparametric sensing from a single image.

The paper addresses the challenge of retrieving external conditions (magnetic field, temperature, magnetic history) from magnetic domain patterns. This is difficult due to the nonlinear, multivariate, and ambiguous relationship between domain morphology and these stimuli, which arises from stochastic nucleation and hysteresis. Existing methods often rely on synthetic datasets or binarized images, limiting their quantitative accuracy and applicability to experimental data.

Development of a controlled magneto-optical inference experiment to reconstruct magnetic field, temperature, and magnetic history from single fine-scale, high-contrast polarization maps.
Application of deep convolutional neural networks (CNNs) combined with hand-crafted, physically interpretable descriptors for magnetic texture analysis.
Demonstration that magnetic texture acts as a high-fidelity record of external conditions, enabling accurate single-image multiparametric sensing.
Identification of physically interpretable features that encode information about external conditions, providing insights into the physics of magnetic domain formation.

The method involves acquiring quantitative magneto-optical polarization maps of magnetic domain textures. These maps are then analyzed using two complementary approaches:

Feature-based inference: Hand-crafted, physically interpretable descriptors are extracted from each polarization map. These descriptors form a 20-dimensional feature vector (FV). This FV is then used as input to a regression model, specifically a 9-layer multilayer perceptron, to predict the DC magnetic field magnitude, magnetic history, and temperature. The history state is treated as a regression variable.
Convolutional Neural Network (CNN) inference: A CNN is trained directly on the full spatial information of the magneto-optical rotation maps. This approach aims to exploit fine-scale correlations and higher-order spatial structures that might be missed by global descriptors. The CNN architecture consists of seven convolutional layers with increasing channel depth, followed by pooling, adaptive average pooling, and two fully connected layers. The output predicts the magnetic field, temperature, and magnetic history.

Both approaches involve normalizing inputs and splitting data into training and validation sets. Data augmentation through vertical and horizontal flipping is used for the CNN.

Magnetic Material: Epitaxial Bismuth-substituted yttrium iron garnet (Bi:YIG) film with perpendicular magnetic anisotropy (PMA).
Imaging System: Magneto-optical wide-field microscope with a Stokes-polarization camera (Sony CMOS Pregius Polarsens IMX250MZR Sensor chip).
- Objective: 5x, numerical aperture 0.16.
- Field of view: 916 µm × 1096 µm.
- Exposure time per image: 106.7 ms.
- Pixel size: 3.45 µm.
Parameter Control:
- Magnetic field: Out-of-plane coil in Helmholtz configuration, driven by a four-quadrant voltage-controlled current source. Applied field range: -1 mT to +1 mT.
- Temperature: Water-cooled with active PID control, measured via a thermistor. Temperature range: 21°C to 72°C.
Feature Extraction: 20 hand-crafted features, including first-order statistics (mean, standard deviation of magneto-optical rotation), Fourier analysis descriptors (domain period), and morphological descriptors (number of endpoints).
Feature-based Regression Model: 9-layer multilayer perceptron.
- Optimizer: Adam.
- Loss: Mean-squared error.
- Training time for FV: 19.4 minutes (full dataset, 400 epochs).
CNN Architecture: Seven convolutional layers, group normalization, max-pooling, adaptive average pooling, two fully connected layers.
- Input: Subdivided tiles of magneto-optical maps.
- Data Augmentation: Vertical and horizontal flipping (generating 64 sub-maps).
- Optimizer: Adam.
- Loss: Mean-squared error.
- Training time for CNN: 2,320 minutes (full dataset, 70 epochs).
Dataset: 11,372 quantitative polarization maps acquired across a magnetic field range of -1 mT to +1 mT and a temperature range of 21°C to 72°C.

Datasets: 11,372 quantitative polarization maps.
Metrics:
- FV Approach:
  - Average sensitivity per map for magnetic field (µ₀∆H): 6.5 µT (0.33% FS).
  - Average sensitivity per map for temperature (∆T): 0.17 K (0.33% FS).
  - History state identification: Correctly identified all history states.
- CNN Approach:
  - Average sensitivity per map for magnetic field (µ₀∆H): 3.8 µT (0.19% FS).
  - Average sensitivity per map for temperature (∆T): 0.12 K (0.24% FS).
  - History state identification: No loss of accuracy in inferred history states.
Baselines: The FV approach is compared against the CNN approach.
Numeric Results:
- The CNN achieved enhanced sensitivities compared to the FV approach: µ₀∆HCNN ≈ 3.8 µT vs. µ₀∆HFV ≈ 6.5 µT, and ∆TCNN ≈ 0.12 K vs. ∆TFV ≈ 0.17 K.
- Both approaches showed negligible systematic bias.
- The FV approach reached its performance plateau earlier (approx. 4,700 training maps) but at a higher residual level than the CNN (approx. 5,700 maps).
Runtime/Training Cost:
- FV calculation per image: 16.2 s.
- FV training (full dataset): 19.4 minutes.
- FV inference per image: 90 µs.
- CNN training (full dataset): 2,320 minutes.
- CNN inference per augmented image: 0.138 s.

The paper notes a "weak residual trend with both magnetic field and temperature" in the CNN approach, which they attribute to "slight underfitting of the regression model." However, they state its impact is negligible compared to statistical uncertainty.
The FV approach shows "relatively large fluctuations between different trainings," indicating a "higher sensitivity to initialization and training conditions, reflecting a reduced robustness... compared to the CNN."

Method evidence: "Deep convolutional neural networks are complemented by feature-based neural-network inference using hand-crafted, physically interpretable descriptors of measured magneto-optical image data, linking the decoded information to material-dependent features and exploring their contributions." (Abstract)
Result evidence: "The CNN further improves the inference accuracy by exploiting the full spatial information of the magnetic pattern, resulting in enhanced sensitivities of µ₀∆HCNN ≈ 3.8 µT (0.19% FS) and ∆TCNN ≈ 0.12 K (0.24% FS), without loss of accuracy in the inferred history states." (Page 10)
Limitation evidence: "The observed peak for the FV approach is part of relatively large fluctuations between different trainings. It indicates a higher sensitivity to initialization and training conditions, reflecting a reduced robustness of the FV approach compared to the CNN." (Page 11)

Decoding magnetic texture

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