End-to-End CNN Approach

• End-to-end CNN approach is a unified design that learns hierarchical features directly from raw data with minimal preprocessing.
• It jointly optimizes convolutional backbones and task-specific heads with a global loss, enabling effective multi-task learning for both classification and regression.
• Empirical results show superior accuracy and regression performance in applications like pulsed eddy current testing, despite sensitivity to domain shifts.

An end-to-end CNN approach denotes a convolutional neural network architecture that is trained to map raw, minimally preprocessed input data directly to one or more high-level task outputs, with all learnable feature extraction and prediction stages optimized jointly according to a task-specific loss. In contrast to multi-stage or feature-engineering pipelines, end-to-end CNNs operate on native signals (e.g., time series, images, audio spectrograms) and dispense with hand-crafted intermediate representations, enabling the network to discover directly useful hierarchical features specific to the end task during stochastic gradient-based training.

1. Architectural Principles and Workflow

The defining characteristic of end-to-end CNNs is their monolithic optimization: all parameters from the initial convolutional layers through (potentially multiple) dense, regression, or segmentation heads are updated via the same global loss. Input preprocessing is restricted to minimal normalization, omitting hand-crafted features or stages. For example, in pulsed eddy current (PEC) nondestructive inspection, the raw 1D time series $x\in\mathbb{R}^{T}$ is fed directly into a stack of 1D convolutional layers, without peak, zero-crossing, or other procedural feature extraction (Fu et al., 2019).

A canonical workflow consists of:

1. Input normalization: per-channel scaling to fixed range (e.g., $[0,1]$), with no extraction of domain-specific features.
2. Shared convolutional backbone: multiple convolutional layers with nonlinearities (e.g., ReLU), batch normalization, and pooling for hierarchical feature extraction.
3. Task-specific output heads: parallel branches for different targets (e.g., softmax classifier, linear regressor), optionally supporting multi-task learning.
4. End-to-end loss computation: all outputs contribute to a global loss, e.g., weighted sum of cross-entropy and regression losses; all parameters are updated via back-propagation.

This structure is succinctly illustrated in the following table, instantiated for the multi-task 1D CNN for PEC signal analysis (Fu et al., 2019):

Input	Feature Backbone	Output Heads (examples)
$x\in\mathbb{R}^{T}$	4 × Conv1D + pooling + GAP → $f\in\mathbb{R}^{64}$	(a) Dense + softmax (classification)<br>(b) Dense + linear (regression)

2. Mathematical Formulation

The distinguishing feature of end-to-end CNNs is the unbroken computational graph from raw input to loss. For 1D or 2D CNN blocks, the convolution is formalized as

$(x*w)_t = \sum_{i=0}^{K-1} w_i x_{t+i},$

with $K$ the kernel width and $x$ the input. Each convolution is followed by a nonlinear activation, usually ReLU:

$a_t = \max(0,z_t),\quad z_t=Wx_{t:t+K-1}+b.$

For a classification task with $M$ classes, the softmax output is:

$$\hat{p}_c = \frac{\exp(z_c)}{\sum_{j=1}^M \exp(z_j)}.$$

For regression, the network head typically comprises linear and/or ReLU layers.

A generic total loss for multi-task classification ($[0,1]$0) and regression ($[0,1]$1) is given by

$[0,1]$2

with standard choices $[0,1]$3 for equal weighting (Fu et al., 2019).

3. Application: Pulsed Eddy Current NDI with End-to-End Multi-Task CNN

In (Fu et al., 2019), a strictly end-to-end 1D CNN is employed for simultaneous flaw classification and depth regression in nondestructive PEC testing:

• Input: $[0,1]$4 sample time series per scanpoint, directly normalized and input; no engineered features.
• Architecture: Four 1D convolutional layers (128, 128, 64, 64 filters respectively, kernel size 3), interleaved with max pooling, culminating in a global average pooling layer ($[0,1]$5 feature vector).
• Outputs:
  • Classification: dense layer with $[0,1]$6 units (flaw type), softmax.
  • Regression: dense layer with $[0,1]$7 or $[0,1]$8 outputs (flaw thicknesses), linear activation.
• Loss: Cross-entropy and mean absolute error summed.
• No explicit regularization or data augmentation; weights updated by Adam optimizer.

Performance, compared to non-end-to-end baselines (Gaussian Process, Decision Tree, SVM for classification; Linear/Bayesian/SVM for regression), demonstrates substantial gains (e.g., 91% classification accuracy vs. Decision Tree at 87.2%, SVM at 61.9%) and lower regression mean squared error (MSE) (e.g., 0.498 vs. SVR at 0.841) (Fu et al., 2019).

Method	Accuracy (%)	Regression MSE
1D CNN (E2E)	91.0	0.204–1.120
GP/SVM/DT	53.6–87.2	0.841–1.133

4. Empirical Benefits and Limitations

Key benefits of end-to-end CNNs in the discussed context (Fu et al., 2019):

• Feature learning: The CNN filters automatically discover relevant temporal features for flaw detection and depth quantification, obviating manual feature engineering.
• Unified multi-task deployment: A single network handles both classification and regression, simplifying system integration and deployment.
• Superior empirical accuracy: State-of-the-art performance versus standard statistical learning baselines.

Limitations include:

• Sensitivity to domain shifts: Models trained on signals with specific acquisition conditions (e.g., “no airgap”) may fail to generalize without appropriate augmentation.
• Absence of explicit regularization: No explicit dropout or weight decay was applied, which could lead to overfitting in smaller or noisier datasets.
• Hyperparameter and validation gaps: No hyperparameter grid search or dedicated validation split is reported, which may limit optimal performance and robustness assessment.

5. Broader Context and Future Directions

While (Fu et al., 2019) focuses on the application to electromagnetic NDE, the end-to-end CNN paradigm has been widely adopted across domains, including medical imaging, speech recognition, and sequential data analysis. In all cases, its efficacy stems from the direct optimization of the relevant task objective, often outperforming manually segmented or multi-stage approaches.

For further robustness, the paper outlines directions such as:

• Domain augmentation: Expanding training data with varied structural conditions (e.g., airgap and lift-off configurations) to address sensitivity to distribution shifts.
• Architectural advances: Incorporating residual connections or attention mechanisms, which have shown practical gains in more general end-to-end CNN research.
• Unsupervised pretraining: Leveraging large unlabeled signal sets for representation learning, followed by supervised fine-tuning, especially in data-scarce regimes.

6. Conclusion

End-to-end CNN approaches represent a unified, data-driven learning methodology where all stages from input to task output are differentiable and jointly optimized. Empirical result tables in (Fu et al., 2019) confirm the practical advantages over conventional decoupled solutions in flaw detection and thickness estimation, reflecting the general utility of end-to-end deep learning models in tasks characterized by complex, high-dimensional signal structure and minimal domain-knowledge preprocessing.