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Propagating Confidences through CNNs for Sparse Data Regression
Abdelrahman Eldesokey, Michael Felsberg, Fahad Shahbaz Khan
TL;DR
Sparse and irregular sensor inputs challenge CNNs designed for dense data, especially when regression outputs also need reliability estimates. The paper introduces an algebraically constrained convolution layer with continuous confidence propagation and a joint data-error/confidence objective. On KITTI depth completion, it reports superior performance with three times fewer parameters and produces pixel-wise confidence maps for downstream use.
Problem
CNNs are typically designed for dense image data, leaving sparse-input regression without reliable confidence estimates despite applications requiring both predictions and their reliability.
Method
The paper uses algebraically constrained normalized-convolution operators with continuous confidence propagation, non-negative weights, confidence-based scale fusion, and a joint data-error/confidence objective.
Results
The method achieves state-of-the-art results on the KITTI depth benchmark while requiring only 480 parameters, three times fewer than state-of-the-art methods.
Takeaways & Limitations
The approach produces continuous pixel-wise confidence maps that support information fusion, state inference, and decision support.
Takeaways & Limitations
The multi-scale method’s remaining errors are mainly along sharp edges and the horizon, likely because RGB-derived structural information is absent.
Abstract
from arXiv · showhide
In most computer vision applications, convolutional neural networks (CNNs) operate on dense image data generated by ordinary cameras. Designing CNNs for sparse and irregularly spaced input data is still an open problem with numerous applications in autonomous driving, robotics, and surveillance. To tackle this challenging problem, we introduce an algebraically-constrained convolution layer for CNNs with sparse input and demonstrate its capabilities for the scene depth completion task. We propose novel strategies for determining the confidence from the convolution operation and propagating it to consecutive layers. Furthermore, we propose an objective function that simultaneously minimizes the data error while maximizing the output confidence. Comprehensive experiments are performed on the KITTI depth benchmark and the results clearly demonstrate that the proposed approach achieves superior performance while requiring three times fewer parameters than the state-of-the-art methods. Moreover, our approach produces a continuous pixel-wise confidence map enabling information fusion, state inference, and decision support.
1 Introduction
Sparse active sensors create a depth-completion problem requiring dense maps and reliable pixel-wise confidences, while standard CNN regression typically lacks such confidence estimates. The paper introduces an algebraically constrained convolution approach that propagates confidences, jointly optimizes data error and confidence, and achieves state-of-the-art KITTI performance with fewer parameters.
- 1 Introduction: 95% of pixels are missing in the illustrated LiDAR input, creating the high-sparsity conditions under which existing deep learning methods struggle.The figure shows the RGB image, projected LiDAR point cloud, completed depth map, and confidence map; the image is dilated for visibility.
- 1 Introduction: Scene depth completion reconstructs a dense depth map from sparse active-sensor measurements while distinguishing missing values from zero-valued regions.The output also includes pixel-wise confidence to indicate the reliability of predicted depths for applications such as obstacle detection and decision support.
- 1 Introduction: CNN regression commonly lacks confidence estimates even though applications need both the regressed value and its reliability.The paper motivates this distinction using obstacle distance estimation in autonomous vehicles and robotics.
- 1 Introduction: The proposed algebraically constrained convolution determines and propagates confidences through sparse-input networks while enforcing non-negative weights to keep confidences valid.The network receives sparse inputs equipped with confidences and produces dense outputs.
- 1 Introduction: The objective function simultaneously minimizes data error and maximizes output confidence, while the approach achieves state-of-the-art KITTI results using only 480 parameters.The parameter count is three times lower than that of state-of-the-art methods.
2 Related Work
Prior sparse and inpainting methods largely propagate binary masks, whereas this paper treats confidence continuously and derives an algebraically constrained operator from normalized convolution. It also uses confidence-based scale fusion and emphasizes fewer parameters than earlier approaches.
- 2 Related Work: Earlier inpainting and depth-completion methods use binary masks to identify measured or missing pixels and propagate them through filtering, pooling, or thresholding.Examples include Shepard-interpolation convolution, U-Net mask propagation [5], max pooling in the KITTI method [11], and compressed sensing [1].
- 2 Related Work: The proposed method replaces binary masks with continuous confidences and derives an algebraically constrained deep convolution operator from normalized convolution.It infers continuous output confidences rather than restricting them to binary values.
- 2 Related Work: Unlike methods using constant averaging filters for confidence propagation [5] [11], the approach enforces positive filters and does not assume uniform confidence across pixels.The paper identifies uniform confidence as generally inappropriate for real-world data.
- 2 Related Work: Confidence-based normalized-convolution scale fusion outperforms standard convolution in multi-scale networks such as U-Net [5] [9].The paper also presents the approach as requiring remarkably fewer parameters than the referenced methods.
Our Approach
The approach extends normalized convolution for sparse inputs by learning non-negative applicability functions, propagating continuous confidences, and training with a loss that balances data accuracy and output confidence. A compact hierarchical multi-scale architecture then fuses features using confidence information.
- Normalized Convolution: The method introduces an algebraically constrained normalized convolution operator that processes sparse inputs while propagating confidence information through consecutive layers.The applicability function is learned and constrained to be non-negative, supporting valid confidence values during network processing.
- Normalized Convolution: Normalized averaging provides the base operation, combining feature values with confidences and normalizing by the accumulated confidence-weighted applicability.With a constant basis, the framework reduces to normalized averaging; with constant confidence, it generalizes standard convolution.
- Confidence Propagation: The network derives output confidences from the convolution operation and propagates them using a measure based on partial-versus-full-confidence Gram matrices.For a constant basis, the confidence propagation can reuse a term already computed during normalized convolution.
- Loss Function: The proposed loss combines a Huber data-error term with a confidence term that maximizes output confidence while preventing unbounded confidence growth.The confidence term is weighted by the reciprocal of the epoch number so it does not dominate after the data error begins converging.
- Multi-scale Architecture: The hierarchical multi-scale architecture shares weights across scales, selects the most confident pixels during downsampling, and fuses scales with normalized convolution.A final 1 × 1 normalized convolution merges channels into a dense output and an output confidence map.
4 Experiments
Experiments on KITTI evaluate the method quantitatively and qualitatively, showing strong benchmark performance, efficient parameter use, and informative confidence maps.
- Experimental Setup: The KITTI inputs contain values at approximately 4% of pixels, and the evaluation uses validation and test sets with incomplete ground truth.The test server images have no accessible ground truth.
- Quantitative Comparisons: NConv-HMS outperforms competing validation methods on every metric except RMSE, where it is slightly inferior to DCCS-3-Layers.The multi-scale architecture uses 480 parameters, fewer than the compared methods.
- Quantitative Comparisons: The method outperforms all published methods on the KITTI depth-completion test benchmark.All test results are taken from the online KITTI depth benchmark.
- Qualitative Analysis: Qualitative examples show that the multi-scale architecture densifies sparse LiDAR inputs while producing confidence maps that indicate output reliability.Confidence is low where neither input points nor ground-truth information is available and tends to be higher in regions with denser point clouds.
- Qualitative Analysis: The multi-scale architecture addresses the single-scale network’s limited receptive field, which otherwise causes failures above the horizon and increased RMSE.For the multi-scale model, errors are mainly concentrated along sharp edges and near the horizon.
5 Conclusion
The paper concludes that algebraically constrained convolutions can process sparse inputs with continuous confidences and achieve strong depth-completion results with fewer parameters. The resulting confidence maps also support reliability-aware downstream uses.
- Conclusion: The method uses continuous rather than binary input masks, propagates convolution-derived confidences, and constrains weights to remain non-negative.Its objective minimizes data error while maximizing output confidence.
- Conclusion: The proposed approach achieves superior performance while requiring significantly fewer parameters.The conclusion attributes this result to the proposed sparse-input convolution framework.
- Conclusion: Continuous pixel-wise confidence maps produce reasonable results for information fusion, state inference, and decision support.These maps extend the depth output with spatially varying reliability information.