[Paper Review] Reducing the Hausdorff Distance in Medical Image Segmentation with Convolutional Neural Networks

Davood Karimi, Septimiu E. Salcudean|arXiv (Cornell University)|Apr 22, 2019

Advanced Neural Network Applications4 citations

TL;DR

This paper proposes three novel differentiable loss functions for training convolutional neural networks to directly minimize the Hausdorff Distance (HD) in medical image segmentation. By estimating HD through distance transforms, morphological erosion, or multi-scale convolutional kernels, the methods reduce HD by 18–45% without degrading Dice scores, offering a significant improvement in handling large segmentation errors.

ABSTRACT

The Hausdorff Distance (HD) is widely used in evaluating medical image segmentation methods. However, existing segmentation methods do not attempt to reduce HD directly. In this paper, we present novel loss functions for training convolutional neural network (CNN)-based segmentation methods with the goal of reducing HD directly. We propose three methods to estimate HD from the segmentation probability map produced by a CNN. One method makes use of the distance transform of the segmentation boundary. Another method is based on applying morphological erosion on the difference between the true and estimated segmentation maps. The third method works by applying circular/spherical convolution kernels of different radii on the segmentation probability maps. Based on these three methods for estimating HD, we suggest three loss functions that can be used for training to reduce HD. We use these loss functions to train CNNs for segmentation of the prostate, liver, and pancreas in ultrasound, magnetic resonance, and computed tomography images and compare the results with commonly-used loss functions. Our results show that the proposed loss functions can lead to approximately 18-45 % reduction in HD without degrading other segmentation performance criteria such as the Dice similarity coefficient. The proposed loss functions can be used for training medical image segmentation methods in order to reduce the large segmentation errors.

Motivation & Objective

• To address the lack of direct optimization for Hausdorff Distance (HD) in medical image segmentation, despite HD being a key metric for identifying the largest segmentation error.
• To develop differentiable, trainable loss functions that guide CNNs to minimize the maximum boundary error, improving robustness in critical applications.
• To overcome the instability and non-convergence issues of using exact HD as a loss function in deep learning.
• To evaluate the effectiveness of HD-optimized training across diverse organs (prostate, liver, pancreas) and imaging modalities (US, MR, CT).
• To provide practical, computationally feasible alternatives to exact HD minimization that maintain segmentation accuracy.

Proposed method

• Proposes a distance transform-based loss ($f^{ ext{DT}}_{ ext{HD}}$) that computes HD from the segmentation probability map using the Euclidean distance transform of the boundary.
• Introduces a morphological erosion-based loss ($f^{ ext{ER}}_{ ext{HD}}$) that applies erosion to the difference between ground truth and predicted segmentation to estimate HD.
• Develops a convolutional kernel-based loss ($f^{ ext{CV}}_{ ext{HD}}$) using circular/spherical convolution kernels of varying radii to approximate HD estimation.
• Integrates these HD-based losses into the training objective alongside standard losses like Dice, using a weighted combination to balance performance.
• Employs a relaxed, differentiable approximation of HD to avoid the non-differentiability and instability of exact HD minimization.
• Trains U-Net-based CNNs on 2D and 3D datasets from ultrasound, MRI, and CT, comparing performance against standard loss functions.

Experimental results

Research questions

• RQ1 Can differentiable, approximate formulations of Hausdorff Distance be used as effective training objectives in CNN-based medical image segmentation?
• RQ2 Does direct optimization of HD lead to measurable reductions in the largest segmentation error without degrading overall segmentation quality?
• RQ3 How do the computational costs and performance of different HD estimation methods compare in practice?
• RQ4 Why does direct minimization of exact HD fail in deep learning, and can relaxed, differentiable alternatives overcome this?
• RQ5 Can HD-optimized losses improve segmentation performance in challenging scenarios with isolated large errors?

Key findings

• The proposed loss functions reduce Hausdorff Distance by 18–45% across multiple datasets, including 2D prostate ultrasound, 3D liver CT, and pancreas imaging.
• The distance transform-based loss ($f^{ ext{DT}}_{ ext{HD}}$) achieves the best HD reduction but incurs higher computational cost, especially in 3D.
• The convolutional kernel-based loss ($f^{ ext{CV}}_{ ext{HD}}$) offers a good trade-off between performance and efficiency, enabling faster training with limited kernel sizes.
• The morphological erosion-based loss ($f^{ ext{ER}}_{ ext{HD}}$) is computationally efficient but less effective in reducing HD compared to the other two methods.
• Training with exact HD as a loss function fails to converge, confirming the instability of direct HD minimization in deep learning.
• All three proposed loss functions maintain or improve Dice similarity coefficient (DSC), indicating no degradation in overall segmentation accuracy.