Mumford–Shah Loss Functional for Image Segmentation With Deep Learning

Abstract

Recent state-of-the-art image segmentation algorithms are mostly based on deep neural networks, thanks to their high performance and fast computation time. However, these methods are usually trained in a supervised manner, which requires large number of high quality ground-truth segmentation masks. On the other hand, classical image segmentation approaches such as level-set methods are formulated in a selfsupervised manner by minimizing energy functions such as Mumford-Shah functional, so they are still useful to help generation of segmentation masks without labels. Unfortunately, these algorithms are usually computationally expensive and often have limitation in semantic segmentation. In this paper, we propose a novel loss function based on Mumford-Shah functional that can be used in deep-learning based image segmentation without or with small labeled data. This loss function is based on the observation that the softmax layer of deep neural networks has striking similarity to the characteristic function in the Mumford-Shah functional. We show that the new loss function enables semisupervised and unsupervised segmentation. In addition, our loss function can be also used as a regularized function to enhance supervised semantic segmentation algorithms. Experimental results on multiple datasets demonstrate the effectiveness of the proposed method.

Imposing boundary‐aware prior into CNNs‐based medical image segmentation

“…We compare the boundary-aware priorinduced loss (L CE + P BA ) with three state-of-the-art segmentation losses that are CE (L CE ) alone, boundary prior-induced loss (L CE + P BD ) [5], and AC prior-induced loss (L CE + P AC ) [2,3], respectively. Table 1 reports the regional accuracy for four losses.…”

“…However, it requires additional computational cost. Recent works [2,3] avoid the computational cost by integrating a smooth prior to the typical losses. Unfortunately, this prior smooths the prediction map everywhere, even across the boundaries resulting in the blank prediction map where the small targets are smoothed out ( Fig.…”

“…We first formally define the boundary-aware prior, then relate it to the existing methods [2][3][4][5] to reveal its distinctive properties. Given a dataset of images and their corresponding label maps, the goal of segmentation is to assign a one-hot label vector y n [ {0, 1} K to an observed pixel x n [ R, where K is the number of labels and n is the location of the pixel.…”

“…CNNs trained with this term will be forced to capture the implicitly geometric properties of boundaries. Relating to active contour (AC) and Mumford-Shah (MS) losses: The smooth term in our prior is closely related to the recently proposed AC [2] and MS [3] losses, which are inspired by the classical variational segmentation algorithms. With some mathematical efforts, we can rewrite these losses as L AC = U + P AC = U + N n K k |∇p nk |, where U is the term of pixel-wise prediction error and ∇p nk is the smooth regulariser.…”

DCSegNet: Deep Learning Framework Based on Divide-and-Conquer Method for Liver Segmentation

“…To this end, it is difficult to train a 3D DCNN with limited training data and hardware resources. In addition, some studies have been carried out to improve the segmentation accuracy by adding loss function constraints, such as the boundary loss [20], the Hausdorff distance [21], the Mumford-Shah loss function [22] and signed distance map [23]. These methods contribute to improving the tissue edge segmentation accuracy of 2D slices, but they can hardly consider the continuity between layers.…”