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Probabilistic Deep Discriminant Analysis for Wind Blade Segmentation

Raül Pérez-Gonzalo, Andreas Espersen, Antonio Agudo

TL;DR

The paper tackles the limitation of linear discriminant analysis in non-linear segmentation tasks by introducing Deep Discriminant Analysis (DDA), which directly optimizes a Fisher-like criterion within deep networks. It stabilizes training through sigmoid-bounded outputs and a gradient-friendly reformulation, presenting two DDA losses— $\mathcal{L}_{DDA}^{(\\ln)}$ and $\\mathcal{L}_{DDA}^{(\\Delta)}$—and augmenting them with a probabilistic focal loss to form PDDA. Empirical evaluation on wind blade segmentation shows DDA substantially improves over LDA, while PDDA delivers state-of-the-art-like performance with high accuracy, F1, and mIoU and strong generalization across windfarms. The work demonstrates a data-efficient, robust segmentation approach suitable for wind-energy maintenance in constrained labeling scenarios.

Abstract

Linear discriminant analysis improves class separability but struggles with non-linearly separable data. To overcome this, we introduce Deep Discriminant Analysis (DDA), which directly optimizes the Fisher criterion utilizing deep networks. To ensure stable training and avoid computational instabilities, we incorporate signed between-class variance, bound outputs with a sigmoid function, and convert multiplicative relationships into additive ones. We present two stable DDA loss functions and augment them with a probability loss, resulting in Probabilistic DDA (PDDA). PDDA effectively minimizes class overlap in output distributions, producing highly confident predictions with reduced within-class variance. When applied to wind blade segmentation, PDDA showcases notable advances in performance and consistency, critical for wind energy maintenance. To our knowledge, this is the first application of DDA to image segmentation.

Probabilistic Deep Discriminant Analysis for Wind Blade Segmentation

TL;DR

The paper tackles the limitation of linear discriminant analysis in non-linear segmentation tasks by introducing Deep Discriminant Analysis (DDA), which directly optimizes a Fisher-like criterion within deep networks. It stabilizes training through sigmoid-bounded outputs and a gradient-friendly reformulation, presenting two DDA losses— and —and augmenting them with a probabilistic focal loss to form PDDA. Empirical evaluation on wind blade segmentation shows DDA substantially improves over LDA, while PDDA delivers state-of-the-art-like performance with high accuracy, F1, and mIoU and strong generalization across windfarms. The work demonstrates a data-efficient, robust segmentation approach suitable for wind-energy maintenance in constrained labeling scenarios.

Abstract

Linear discriminant analysis improves class separability but struggles with non-linearly separable data. To overcome this, we introduce Deep Discriminant Analysis (DDA), which directly optimizes the Fisher criterion utilizing deep networks. To ensure stable training and avoid computational instabilities, we incorporate signed between-class variance, bound outputs with a sigmoid function, and convert multiplicative relationships into additive ones. We present two stable DDA loss functions and augment them with a probability loss, resulting in Probabilistic DDA (PDDA). PDDA effectively minimizes class overlap in output distributions, producing highly confident predictions with reduced within-class variance. When applied to wind blade segmentation, PDDA showcases notable advances in performance and consistency, critical for wind energy maintenance. To our knowledge, this is the first application of DDA to image segmentation.
Paper Structure (14 sections, 8 equations, 4 figures, 2 tables)

This paper contains 14 sections, 8 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Deep Discriminant Analysis
  3. Towards Gradient Iterative Optimization
  4. Preventing Exploding Gradients
  5. Direct Deep Discriminant Analysis
  6. Probabilistic Deep Discriminant Analysis
  7. Implementation Details
  8. Experimental Results
  9. Direct Deep Discriminant Analysis
  10. Probabilistic Deep Discriminant Analysis
  11. Quantitative Evaluation
  12. Qualitative Evaluation
  13. Generalization Across Windfarms
  14. Conclusion

Figures (4)

  • Figure 1: Enhancing segmentation via non-linear class separability and probability loss. Probabilistic DDA corresponds specifically to our proposed PDDA framework.
  • Figure 2: Qualitative comparison of LDA and DDA. From left-to-right: input image, ground-truth mask, LDA, DDA$^{(\Delta)}$ and DDA$^{(\ln)}$ estimations with accuracy, F1, and mIoU above.
  • Figure 3: Qualitative comparison of $\mathcal{L}_P$, DDA and PDDA. From left to right columns: input image, $\mathcal{L}_P$, DDA$^{(\Delta)}$, DDA$^{(\ln)}$, PDDA$^{(\Delta)}$ and PDDA$^{(\ln)}$ estimations with accuracy, F1-score and mIoU above.
  • Figure 4: Boxplot results of PDDA$^{(\ln)}$ over the test set for each windfarm to study the robustness of our algorithm.